Doctoral dissertation in mathematics: Coderivatives of normal cone mappings and applications

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY

INSTITUTE OF MATHEMATICS

NGUYEN THANH QUI

CODERIVATIVES OF NORMAL CONE MAPPINGS AND APPLICATIONS

DOCTORAL DISSERTATION IN MATHEMATICS

HANOI - 2014

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY

INSTITUTE OF MATHEMATICS

Nguyen Thanh Qui

CODERIVATIVES OF NORMAL CONE MAPPINGS AND APPLICATIONS

Speciality: Applied Mathematics

Speciality code: 62 46 01 12

DOCTORAL DISSERTATION IN MATHEMATICS

Supervisors:

1. Prof. Dr. Hab. Nguyen Dong Yen

2. Dr. Bui Trong Kien

HANOI - 2014

To my beloved parents and family members

Conﬁrmation

This dissertation was written on the basis of my research works carried at Institute of Mathematics (VAST, Hanoi) under the supervision of Profes- sor Nguyen Dong Yen and Dr. Bui Trong Kien. All the results presented have never been published by others.

Hanoi, January 2014

The author

Nguyen Thanh Qui

Acknowledgments

I would like to express my deep gratitude to Professor Nguyen Dong Yen and Dr. Bui Trong Kien for introducing me to Variational Analysis and Optimiza- tion Theory. I am thankful to them for their careful and eﬀective supervision.

I am grateful to Professor Ha Huy Bang for his advice and kind help. My many thanks are addressed to Professor Hoang Xuan Phu, Professor Ta Duy Phuong, and Dr. Nguyen Huu Tho, for their valuable support.

During my long stays in Hanoi, I have had the pleasure of contacting with the nice people in the research group of Professor Nguyen Dong Yen. In particular, I have got several signiﬁcant comments and suggestions concerning the results of Chapters 2 and 3 from Professor Nguyen Quang Huy. I would like to express my sincere thanks to all the members of the research group.

I owe my thanks to Professor Daniel Frohardt who invited me to work at Department of Mathematics, Wayne State University, for one month (Septem- ber 1–30, 2011). I would like to thank Professor Boris Mordukhovich who gave me many interesting ideas in the ﬁve seminar meetings at the Wayne State University in 2011 and in the Summer School “Variational Analysis and Applications” at Institute of Mathematics (VAST, Hanoi) and Vietnam Institute Advanced Study in Mathematics in 2012.

This dissertation was typeset with LaTeX program. I am grateful to Pro- fessor Donald Knuth who created TeX the program. I am so much thankful to MSc. Le Phuong Quan for his instructions on using LaTeX.

I would like to thank the Board of Directors of Institute of Mathematics (VAST, Hanoi) for providing me pleasant working conditions at the Institute.

I would like to thank the Steering Committee of Cantho University a lot

for constant support and kind help during many years.

Financial supports from the Vietnam National Foundation for Science and Technology Development (NAFOSTED), Cantho University, Institute of

Mathematics (VAST, Hanoi), and the Project “Joint research and training on Variational Analysis and Optimization Theory, with oriented applications in some technological areas” (Vietnam-USA) are gratefully acknowledged.

I am so much indebted to my parents, my sisters and brothers, for their

love and support. I thank my wife for her love and encouragement.

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Table of Notations vi

List of Figures viii

Introduction ix

Chapter 1. Preliminary 1

1 1.1 Basic Deﬁnitions and Conventions . . . . . . . . . . . . . . . .

3 1.2 Normal and Tangent Cones . . . . . . . . . . . . . . . . . . .

6 1.3 Coderivatives and Subdiﬀerential . . . . . . . . . . . . . . . .

9 1.4 Lipschitzian Properties and Metric Regularity . . . . . . . . .

1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Chapter 2. Linear Perturbations of Polyhedral Normal Cone

12 Mappings

12 2.1 The Normal Cone Mapping F(x, b) . . . . . . . . . . . . . . .

16 2.2 The Fr´echet Coderivative of F(x, b) . . . . . . . . . . . . . . .

26 2.3 The Mordukhovich Coderivative of F(x, b) . . . . . . . . . . .

37 2.4 AVIs under Linear Perturbations . . . . . . . . . . . . . . . .

42 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 3. Nonlinear Perturbations of Polyhedral Normal Cone

43 Mappings

43 3.1 The Normal Cone Mapping F(x, A, b) . . . . . . . . . . . . . .

48 3.2 Estimation of the Fr´echet Normal Cone to gphF . . . . . . . .

54 3.3 Estimation of the Limiting Normal Cone to gphF . . . . . . .

3.4 AVIs under Nonlinear Perturbations . . . . . . . . . . . . . . . 59

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Chapter 4. A Class of Linear Generalized Equations 67

4.1 Linear Generalized Equations . . . . . . . . . . . . . . . . . . 67

4.2 Formulas for Coderivatives . . . . . . . . . . . . . . . . . . . . 69

. . . . . . . . . . 70 4.2.1 The Fr´echet Coderivative of N (x, α)

. . . . . . 78 4.2.2 The Mordukhovich Coderivative of N (x, α)

4.3 Necessary and Suﬃcient Conditions for Stability . . . . . . . . 83

4.3.1 Coderivatives of the KKT point set map . . . . . . . . 83

4.3.2 The Lipschitz-like property . . . . . . . . . . . . . . . . 84

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

General Conclusions 92

List of Author’s Related Papers 93

References 94

Table of Notations

IN := {1, 2, . . .} ∅ IR IR++ IR+ IR− IR := IR ∪ {±∞} |x| IRn (cid:107)x(cid:107) IRm×n detA A(cid:62) (cid:107)A(cid:107) X ∗ (cid:104)x∗, x(cid:105) (cid:104)x, y(cid:105) (cid:92)(u, v) B(x, δ) ¯B(x, δ) BX ¯BX posΩ spanΩ dist(x; Ω) {xk} xk → x w∗ → x∗ x∗ k set of positive natural numbers empty set set of real numbers set of x ∈ IR with x > 0 set of x ∈ IR with x ≥ 0 set of x ∈ IR with x ≤ 0 set of generalized real numbers absolute value of x ∈ IR n-dimensional Euclidean vector space norm of a vector x set of m × n-real matrices determinant of a matrix A transposition of a matrix A norm of a matrix A topological dual of a norm space X canonical pairing canonical inner product angle between two vectors u and v open ball with centered at x and radius δ closed ball with centered at x and radius δ open unit ball in a norm space X closed unit ball in a norm space X convex cone generated by Ω linear subspace generated by Ω distance from x to Ω sequence of vectors xk converges to x in norm topology k converges to x∗ in weak* topology x∗

∀x x := y (cid:98)N (x; Ω) N (x; Ω) f : X → Y f (cid:48)(x), ∇f (x) ϕ : X → IR domϕ epiϕ ∂ϕ(x) ∂2ϕ(x, y)

F : X ⇒ Y domF rgeF gphF kerF (cid:98)D∗F (x, y) D∗F (x, y) for all x x is deﬁned by y Fr´echet normal cone to Ω at x limiting normal cone to Ω at x function from X to Y Fr´echet derivative of f at x extended-real-valued function eﬀective domain of ϕ epigraph of ϕ limiting subdiﬀerential of ϕ at x limiting second-order subdiﬀerential of ϕ at x relative to y multifunction from X to Y domain of F range of F graph of F kernel of F Fr´echet coderivative of F at (x, y) Mordukhovich coderivative of F at (x, y)

vii

List of Figures

74 4.1 The sequences {(xk, αk)}k∈IN , {zk}k∈IN , and {uk}k∈IN . . . . . .

viii

Introduction

Motivated by solving optimization problems, the concept of derivative was ﬁrst introduced by Pierre de Fermat. It led to the Fermat stationary princi- ple, which plays a crucial role in the development of diﬀerential calculus and serves as an eﬀective tool in various applications. Nevertheless, many funda- mental objects having no derivatives, no ﬁrst-order approximations (deﬁned by certain derivative mappings) occur naturally and frequently in mathemat- ical models. The objects include nondiﬀerentiable functions, sets with non- smooth boundaries, and set-valued mappings. Since the classical diﬀerential calculus is inadequate for dealing with such functions, sets, and mappings, the appearance of generalized diﬀerentiation theories is an indispensable trend.

In the 1960s, diﬀerential properties of convex sets and convex functions have been studied. The fundamental contributions of J.-J. Moreau and R. T. Rockafellar have been widely recognized. Their results led to the beau- tiful theory of convex analysis [47]. The derivative-like structure for convex functions, called subdiﬀerential, is one of the main concepts in this theory. In contrast to the singleton of derivatives, subdiﬀerential is a collection of subgradients. Convex programming which is based on convex analysis plays a fundamental role in Mathematics and in applied sciences.

In 1973, F. H. Clarke deﬁned basic concepts of a generalized diﬀerentiation theory, which works for locally Lipschitz functions, in his doctoral disserta- tion under the supervision of R. T. Rockafellar. In Clarke’s theory, convexity is a key point; for instance, subdiﬀerential in the sense of Clarke is always a closed convex set. In the later 1970s, the concepts of Clarke have been devel- oped for lower semicontinuous extended-real-valued functions in the works of R. T. Rockafellar, J.-B. Hiriart-Urruty, J.-P. Aubin, and others. Although the theory of Clarke is beautiful due to the convexity used, as well as to the elegant proofs of many fundamental results, the Clarke subdiﬀerential and the Clarke normal cone face with the challenge of being too big, so too

rough, in complicated practical problems where nonconvexity is an inherent property. Despite to this, Clarke’s theory has opened a new chapter in the development of nonlinear analysis and optimization theory (see, e.g., [8], [2]).

In the mid 1970s, to avoid the above-mentioned convexity limitations of the Clarke concepts, B. S. Mordukhovich introduced the notions of limiting normal cone and limiting subdiﬀerential which are based entirely on dual- space constructions. His dual approach led to a modern theory of generalized diﬀerentiation [28] with a variety of applications [29]. Long before the publi- cation of these books, Mordukhovich’s contributions to Variational Analysis had been presented in the well-known monograph of R. T. Rockafellar and R. J.-B. Wets [48].

The limiting subdiﬀerential is generally nonconvex and smaller than the Clarke subdiﬀerential. Similarly, the limiting normal cone to a closed set in a Banach space is nonconvex in general and usually smaller than the Clarke normal cone. Therefore, necessary optimality conditions in nonlinear pro- gramming and optimal control in terms of the limiting subdiﬀerential and limiting normal cone are much tighter than that given by the corresponding Clarke’s concepts. Furthermore, the Mordukhovich criteria for the Lipchitz- like property (that is the pseudo-Lipschitz property in the original terminol- ogy of J.-P. Aubin [1], or the Aubin continuity as suggested by A. L. Dontchev and R. T. Rockafellar [11], [12]) and the metric regularity of multifunctions are remarkable tools to study stability of variational inequalities, generalized equations, and the Karush-Kuhn-Tucker point sets in parametric optimiza- tion problems. Note that if one uses Clarke’s theory then only suﬃcient conditions for stability can be obtained. Meanwhile, Mordukhovich’s theory provides one with both necessary and suﬃcient conditions for stability. An- other advantage of the latter theory is that its system of calculus rules is much more developed than that of Clarke’s theory. So, the wide range of ap- plications and bright prospects of Mordukhovich’s generalized diﬀerentiation theory are understandable.

In the late 1990s, V. Jeyakumar and D. T. Luc introduced the concepts of approximate Jacobian and corresponding generalized subdiﬀerential. It can be seen [18] that using the approximate Jacobian one can establish conditions for stability, metric regularity, and local Lipschitz-like property of the solu- tion maps of parametric inequality systems involving nonsmooth continuous functions and closed convex sets. Calculus rules and various applications of

the approximate Jacobian can be found in the monograph [17]. It is worthy to study relationships between the concepts of coderivative and approximate Jacobian. In [33], the authors show that the Mordukhovich coderivative and the approximate Jacobian have a little in common. These concepts are very diﬀerent, and they require diﬀerent methods of study and lead to results in diﬀerent forms.

As far as we understand, Variational Analysis is a new name of a math- ematical discipline which uniﬁes Nonsmooth Analysis, Set-Valued Analysis with applications to Optimization Theory and equilibrium problems. Many aspects of the theory can be seen in [2], [4], [8], [28], [29], [48].

Let X, W1, W2 are Banach spaces, ϕ : X × W1 → IR is a continuously Fr´echet diﬀerentiable function, Θ : W2 ⇒ X is a multifunction (i.e., a set- valued map) with closed convex values. Consider the minimization problem

(1) min{ϕ(x, w1)| x ∈ Θ(w2)}

depending on the parameters w = (w1, w2), which is given by the data set {ϕ, Θ}. According to the generalized Fermat rule (see, for instance, [20, pp. 85–86]), if ¯x is a local solution of (1) then

0 ∈ f (¯x, w1) + N (¯x; Θ(w2)),

where f (¯x, w1) = ∇xϕ(¯x, w1) denotes the partial derivative of ϕ with respect to ¯x at (¯x, w1) and

N (¯x; Θ(w2)) = {x∗ ∈ X ∗| (cid:104)x∗, x − ¯x(cid:105) ≤ 0, ∀x ∈ Θ(w2)},

with X ∗ being the dual space of X, stands for the normal cone of Θ(w2). This means that ¯x is a solution of the following generalized equation

(2) 0 ∈ f (x, w1) + F(x, w2),

where F(x, w2) := N (x; Θ(w2)) for every x ∈ Θ(w2) and F(x, w2) := ∅ for every x (cid:54)∈ Θ(w2), is the parametric normal cone mapping related to the multifunction Θ(·). Equilibrium problems of the form (2) have been in- vestigated intensively in the literature (see, e.g., [11], [12], [24], [27], [28, Chapter 4], [43]). Necessary and suﬃcient conditions for the Lipschitz-like property of the solution map (w1, w2) (cid:55)→ S(w1, w2) of (2) can be character- ized by using the Mordukhovich criterion. According to the method proposed by A. L. Dontchev and R. T. Rockafellar [11], which has been developed by A. B. Levy and B. S. Mordukhovich [24] and by G. M. Lee and N. D. Yen

[22], one has to compute the Fr´echet and the Mordukhovich coderivatives of F : X × W2 ⇒ X ∗. Such a computation has been done in [11] for the case Θ(w2) is a ﬁxed polyhedral convex set in IRn, and in [54] for the case where Θ(w2) is a ﬁxed smooth-boundary convex set. The problem is rather diﬃcult if Θ(w2) depends on w2.

J.-C. Yao and N. D. Yen [52], [53] ﬁrst studied the case Θ(w2) = Θ(b) := {x ∈ IRn| Ax ≤ b} where A is an m × n matrix, b is a parameter. Some argu- ments from these papers have been used by R. Henrion, B. S. Mordukhovich and N. M. Nam [13] to compute coderivatives of the normal cone mappings to a ﬁxed polyhedral convex set in Banach space. N. M. Nam [32] showed that the results of [52], [53] on normal cone mappings to linearly perturbed polyhedra can be extended to an inﬁnite dimensional setting. N. T. Q. Trang [50] proposed some developments and reﬁnements of the results of [32].

G. M. Lee and N. D. Yen [23] computed the Fr´echet coderivatives of the normal cone mappings to a perturbed Euclidean balls and derived from the results a stability criterion for the Karush-Kuhn-Tucker point set mapping of parametric trust-region subproblems.

As concerning normal cone mappings to nonlinearly perturbed polyhedra, we would like to mention a recent paper [9] where the authors have computed coderivatives of the normal cone to a rotating closed half-space.

The normal cone mapping considered in [23] is a special case of the normal cone mapping to the solution set Θ(w2) = Θ(p) := {x ∈ X| ψ(x, p) ≤ 0} where ψ : X × P → IR is a C2-smooth function deﬁned on the product space of Banach spaces X and P .

More generally, for the solution map

Θ(w2) = Θ(p) := {x ∈ X| Ψ(x, p) ∈ K}

of a parametric generalized equality system with Ψ : X × P → Y being a C2-smooth vector function which maps the product space X × P into a Banach space Y , K ⊂ Y a closed convex cone, the problems of computing the Fr´echet coderivative (respectively, the Mordukhovich coderivative) of the Fr´echet normal cone mapping (x, w2) (cid:55)→ (cid:98)N (x; Θ(w2)) (respectively, of the limiting normal cone mapping (x, w2) (cid:55)→ N (x; Θ(w2))), are interesting, but very diﬃcult. All the above-mentioned normal cone mappings are special cases of the last two normal cone mappings. It will take some time before signiﬁcant advances on these general problems can be done. Some aspects of

xii

this question have been investigated by [14].

It is worthy to stress that coderivatives of normal cone mappings are noth- ing else as the second-order subdiﬀerentials of the indicator functions of the set in question. The concepts of Fr´echet and/or limiting second-order subd- iﬀerentials of extended-real-valued functions are discussed in [28], [37], [30], [5], [6], [7], [31] from diﬀerent points of views.

This dissertation studies some problems related to the generalized diﬀer- entiation theory of Mordukhovich and its applications. Our main eﬀorts concentrate on computing or estimating the Fr´echet coderivative and the Mordukhovich coderivative of the normal cone mappings to

a) linearly perturbed polyhedra in ﬁnite dimensional spaces, as well as in

inﬁnite dimensional reﬂexive Banach spaces,

b) nonlinearly perturbed polyhedra in ﬁnite dimensional spaces,

c) perturbed Euclidean balls.

Applications of the obtained results are used to study the metric regularity property and/or the Lipschitz-like property of the solution maps of some classes of parametric variational inequalities as well as parametric generalized equations.

Our results develop certain aspects of the preceding works [11], [52], [53], [13], [32], and [23]. The four open questions raised in [52] and [23] have been solved in this dissertation. Some of our techniques are new.

The dissertation has four chapters and a list of references.

Chapter 1 collects several basic concepts and facts on generalized diﬀeren- tiation, together with the well-known dual characterizations of the two funda- mental properties of multifunctions: the local Lipschitz-like property deﬁned by J.-P. Aubin and the metric regularity which has origin in Ljusternik’s theorem [16, p. 30].

Chapter 2 studies generalized diﬀerentiability properties of the normal cone mappings associated to perturbed polyhedral convex sets in reﬂexive Banach spaces. The obtained results lead to solution stability criteria for a class of variational inequalities in ﬁnite dimensional spaces under linear perturba- tions. This chapter also answers the two open questions in [52].

Chapter 3 computes the Fr´echet and the Mordukhovich coderivatives of the normal cone mappings studied in the previous chapter with respect to

xiii

total perturbations. As a consequence, solution stability of aﬃne variational inequalities under nonlinear perturbations in ﬁnite dimensional spaces can be addressed by means of the Mordukhovich criterion and the coderivative formula for implicit multifunctions due to A. B. Levy and B. S. Mordukhovich [24, Theorem 2.1].

Based on a recent paper of G. M. Lee and N. D. Yen [23], Chapter 4 presents a comprehensive study of the solution stability of a class of linear generalized equations connected with the parametric trust-region subproblems which are well-known in nonlinear programming. We show that exact formulas for the coderivatives of the normal cone mappings associated to perturbed Euclidean balls can be obtained. Then, combining the formulas with the necessary and the suﬃcient conditions for the local Lipschitz-like property of implicit multifunctions from a paper by G. M. Lee and N. D. Yen [22], we get new results on stability of the Karush-Kuhn-Tucker point set maps of parametric trust-region subproblems. This chapter also solves the two open questions in [23].

The results of Chapter 2 and Chapter 3 were published on the journals Nonlinear Analysis [38], Journal of Mathematics and Applications [39], Acta Mathematica Vietnamica [40], Journal of Optimization Theory and Applica- tions [41]. Chapter 4 is written on the basis of a joint paper by N. T. Qui and N. D. Yen, which has been accepted for publication on SIAM Journal on Optimization [42].

xiv

Chapter 1

Preliminary

In this chapter we review some background material of Variational Analysis; see, e.g., [1], [12], [20], [28], [29], [35], [48] for more details and references. The basic concepts of generalized diﬀerentiation of set-valued mappings and extended-real-valued functions are presented in this chapter are taken from Mordukhovich [28], [29].

1.1 Basic Deﬁnitions and Conventions

Let X be a norm space with the norm usually denoted by (cid:107) · (cid:107). For each x0 ∈ X and δ > 0, we denote by B(x0, δ) the open ball {x ∈ X(cid:12) (cid:12) (cid:107)x−x0(cid:107) < δ}, and let ¯B(x0, δ) stand for the corresponding closed ball. We will write BX and ¯BX respectively for B(0X, 1) and ¯B(0X, 1). Unless otherwise stated, every norm in question in a product norm space is a sum norm. Let Ω be a subset of X. When Ω (cid:54)= ∅, dist(x; Ω) is the distance from x ∈ X to the nonempty set Ω, that is

(cid:107)x − u(cid:107). dist(x; Ω) = inf u∈Ω

If Ω = ∅, we put dist(x; Ω) = +∞ by convention. The negative dual cone of Ω ⊂ X is deﬁned by

Ω∗ := {x∗ ∈ X ∗| (cid:104)x∗, v(cid:105) ≤ 0, ∀v ∈ Ω}

with X ∗ being the dual space of X, and (cid:104)·, ·(cid:105) standing for the canonical pairing between X ∗ and X. For each u∗ ∈ X ∗, we deﬁne

{u∗}⊥ := (cid:8)v ∈ X| (cid:104)u∗, v(cid:105) = 0(cid:9).

When X is a ﬁnite dimensional Euclidean space, the notation (cid:104)·, ·(cid:105) also stands for the canonical inner product in X. In working with X, we keep to the Euclidean norm given by (cid:107)x(cid:107) = (cid:112)(cid:104)x, x(cid:105) for every x ∈ X. In the sequel, x Ω→ ¯x means x → ¯x with x ∈ Ω.

Let F : X ⇒ Y be a set-valued mapping/multifunction between nonempty

sets X and Y . Denote respectively by

domF := (cid:8)x ∈ X| F (x) (cid:54)= ∅(cid:9), rgeF := (cid:8)y ∈ Y | y ∈ F (x) for some x ∈ X(cid:9) the domain and the range of F . The multifunction F : X ⇒ Y is uniquely associated with its graph

gphF := (cid:8)(x, y) ∈ X × Y | y ∈ F (x)(cid:9)

in the product set X ×Y . Note that if X and Y are Banach spaces, then X ×Y is also a Banach space with respect to the sum norm (cid:107)(x, y)(cid:107) = (cid:107)x(cid:107) + (cid:107)y(cid:107) imposed on X × Y unless otherwise stated. In this case, the kernel of F is deﬁned by

kerF := (cid:8)x ∈ X| 0 ∈ F (x)(cid:9).

The image of a set Ω ⊂ X and the inverse image of a set Θ ⊂ Y under F are deﬁned in succession by setting

F (Ω) := (cid:8)y ∈ Y | y ∈ F (x) for some x ∈ X(cid:9)

and

F −1(Θ) := (cid:8)x ∈ X| F (x) ∩ Θ (cid:54)= ∅(cid:9). The inverse mapping to F : X ⇒ Y is the multifunction F −1 : Y ⇒ X with

F −1(y) := {x ∈ X| y ∈ F (x)}.

Observe that domF −1 = rgeF , rgeF −1 = domF , and

(cid:111)

gphF −1 = (cid:8)(y, x) ∈ Y × X| (x, y) ∈ gphF (cid:9).

(cid:12) y ∈ F (x) with (cid:107)x(cid:107) ≤ 1

A multifunction between Banach spaces F : X ⇒ Y is said to be positively homogeneous if 0 ∈ F (0) and F (αx) ⊃ αF (x) for all x ∈ X and α > 0. The latter is equivalent to saying that the graph of F is a cone in X × Y . The norm of a positively homogeneous multifunction F is deﬁned by (cid:110) (cid:107)y(cid:107) (cid:12) (cid:107)F (cid:107) := sup .

1.2 Normal and Tangent Cones

In this section, we recall the concepts of normals and tangents to sets in Banach spaces and discuss their properties and relationships.

Let F : X ⇒ Y be a multifunction between topological spaces X and Y . Following [28] and [48], the sequential Painlev´e-Kuratowski upper/outer limit of F as x → ¯x is deﬁned by

(cid:110) y ∈ Y (cid:12)

(cid:12) exist sequences xk → ¯x and yk → y

(cid:111) .

F (x) = Limsup x→¯x (1.1)

with yk ∈ F (xk) for all k ∈ IN

Note that the limits in expression (1.1) are understood in the sequential sense which contrast to net/topological limits in general topological spaces. When F : X ⇒ X ∗ is a multifunction between a Banach space X and its dual X ∗, we always understand the sequential Painlev´e-Kuratowski upper limit of F as x → ¯x with respect to the norm topology of X and the weak* topology of X ∗. The latter means that

(cid:110) x∗ ∈ X ∗(cid:12)

(cid:12) exist sequences xk → ¯x and x∗ k

F (x) = Limsup x→¯x (1.2)

w∗ → x∗ (cid:111) .

k ∈ F (xk) for all k ∈ IN

with x∗

In what follows, all the reference spaces are real Banach spaces.

Deﬁnition 1.1 (See [28, Deﬁnition 1.1]) Let Ω be a nonempty subset of a Banach space X.

(cid:27) .

(cid:98)Nε(¯x; Ω) :=

Ω →¯x

(1.3) ≤ ε (i) Given ¯x ∈ Ω and ε ≥ 0, we deﬁne the set of ε-normals to Ω at ¯x by (cid:26) x∗ ∈ X ∗(cid:12) (cid:12) (cid:12) limsup (cid:104)x∗, x − ¯x(cid:105) (cid:107)x − ¯x(cid:107)

Ω→ ¯x, and x∗

w∗ → x∗ such that x∗

When ε = 0, elements of (1.3) are called Fr´echet normals and their collection, denoted by (cid:98)N (¯x; Ω), is the Fr´echet normal cone to Ω at ¯x. If ¯x (cid:54)∈ Ω, we put (cid:98)Nε(¯x; Ω) = ∅ for all ε ≥ 0.

(ii) For ¯x ∈ Ω, a vector x∗ ∈ X ∗ is called limiting normal to Ω at ¯x if there k ∈ (cid:98)Nεk(xk; Ω) for

are sequences εk ↓ 0, xk all k ∈ IN . The collection of such normals

(cid:98)Nε(x; Ω)

x→¯x ε↓0

(1.4) N (¯x; Ω) := Limsup

is the limiting normal cone to Ω at ¯x. We put N (¯x; Ω) = ∅ when ¯x (cid:54)∈ Ω.

We see that for each ε ≥ 0 the ε-normal set (cid:98)Nε(¯x; Ω) is convex and closed in the norm topology of X ∗. In contrast to the ε-normal sets, the limiting normal cone may be nonconvex. For instance, given a subset Ω := {(x1, x2) ∈ IR2| x2 ≥ −|x1|} ⊂ IR2, we have (cid:98)N (cid:0)(0, 0); Ω(cid:1) = {0} while

N (cid:0)(0, 0); Ω(cid:1) = {(v, v)| v ≤ 0} ∪ {(v, −v)| v ≥ 0}

is a nonconvex set. Since duality implies convexity, the example shows that the limiting normal cone to a set Ω at a given point ¯x cannot be dual to any tangential approximation of Ω at ¯x in the primal space. Example 1.7 in [28] shows that, in general, the limiting normal cone may not be norm closed in the dual space X ∗ (hence it is not weakly* closed).

(cid:110) x∗ ∈ X ∗(cid:12)

(cid:111) .

(cid:98)N (¯x; Ω) = N (¯x; Ω) =

(cid:12) (cid:104)x∗, x − ¯x(cid:105) ≤ 0, ∀x ∈ Ω

A set Ω ⊂ X is said to be normally regular at ¯x ∈ Ω if N (¯x; Ω) = (cid:98)N (¯x; Ω). From (1.3) and (1.4) it follows that (cid:98)N (¯x; Ω) ⊂ N (¯x; Ω) for any Ω ⊂ X and ¯x ∈ Ω. If Ω is convex, then by Propositions 1.3 and 1.5 in [28] it holds

In this case, both the Fr´echet and the limiting normal cones coincide with the normal cone of Convex Analysis; thus Ω is normally regular at ¯x.

One says that a set Ω ⊂ X is locally closed around ¯x ∈ Ω if there ex- ists δ > 0 for which Ω ∩ ¯B(¯x, δ) is closed. The next theorem establishes a special representation of the limiting normal cone to closed subsets of ﬁnite dimensional spaces.

Theorem 1.1 (See [28, Theorem 1.6]) Let Ω ⊂ IRn be locally closed around ¯x ∈ Ω. Then it holds that

(cid:98)N (x; Ω).

x→¯x

(1.5) N (¯x; Ω) = Limsup

Asplund spaces, which are speciﬁc Banach spaces, have important role [28, Chapter 3] in Variational Analysis. If X is an Asplund space, the expression on the right-hand side of the formula (1.4) can be also simpliﬁed similarly as (1.5).

Deﬁnition 1.2 (See [28, Deﬁnition 2.17] and [35, Deﬁnition 1.22]) A Banach space X is Asplund, or it has the Asplund property, if every convex continu- ous function ϕ : U → IR deﬁned on an open convex subset U of X is Fr´echet diﬀerentiable on a dense subset of U .

An interesting characterization of Asplund spaces is that X is Asplund if and only if every separable closed subspace of X has a separable dual. The next theorem provides us with a formula for computing of the limiting normal cones to closed subsets of Asplund spaces.

Theorem 1.2 (See [28, Theorem 2.35]) Let X be a Banach space. The fol- lowing properties are equivalent:

(i) X is Asplund.

(ii) For every closed set Ω ⊂ X and every ¯x ∈ Ω one has the representation

(cid:98)N (x; Ω).

x→¯x

(1.6) N (¯x; Ω) = Limsup

The Fr´echet normal cone has a tight connection with the concepts of con-

tingent tangent cone and of weak contingent cone.

Deﬁnition 1.3 (See [28, Deﬁnition 1.8]) Let Ω be a subset of a Banach space X and ¯x ∈ Ω.

(i) The set T (¯x; Ω) ⊂ X deﬁned by

t↓0

(1.7) T (¯x; Ω) := Limsup , Ω − ¯x t

where the “ Limsup ” is taken with respect to the norm topology of X, is called the contingent cone to Ω at ¯x.

(ii) If the “ Limsup ” in (i) is taken with respect to the weak topology of X, then the resulting construction, denoted by TW (¯x; Ω), is called the weak contingent cone to Ω at ¯x.

The contingent cone T (¯x; Ω) in Deﬁnition 1.7 was introduced by Bouligand, and it was also introduced independently by Severi. Hence, another, better name for this cone would be the Bouligand-Severi tangent cone. Note that when Ω is convex, the contingent cone T (¯x; Ω) coincides with the notion of tangent cone in the sense of Convex Analysis. This means that T (¯x; Ω) is the topological closure of the cone {λ(x − ¯x)| x ∈ Ω, λ ≥ 0}.

In contrast to the limiting normal cone, the Fr´echet normal cone can be dual of a tangent cone to a set in the primal space. Relationships between the Fr´echet normal cone and the contingent cones are described as follows.

(cid:111) .

(cid:98)N (¯x; Ω) = (cid:0)TW (¯x; Ω)(cid:1)∗

(cid:12) (cid:104)x∗, v(cid:105) ≤ 0, ∀v ∈ TW (¯x; Ω)

= Proposition 1.1 (See [28, Corollary 1.11]) Let X be a reﬂexive space and Ω ⊂ X with ¯x ∈ Ω. Then the Fr´echet normal cone to Ω at ¯x is computed by (cid:110) x∗ ∈ X ∗(cid:12)

Thus, when X is ﬁnite dimensional, one has

(cid:98)N (¯x; Ω) = (cid:0)T (¯x; Ω)(cid:1)∗

1.3 Coderivatives and Subdiﬀerential

The Fr´echet and the Mordukhovich coderivatives of multifunctions [28] are two basic concepts of the generalized diﬀerentiation theory constructed by the dual-space approach. They are deﬁned via the concepts of Fr´echet normal cone and limiting normal cone.

Deﬁnition 1.4 (See [28, Deﬁnition 1.32]) Let F : X ⇒ Y be a multifunction between Banach spaces X and Y .

(i) For any (¯x, ¯y) ∈ X × Y and ε ≥ 0, ε-coderivative of F at (¯x, ¯y) is the

εF (¯x, ¯y) : Y ∗ ⇒ X ∗ deﬁned by

(cid:110)

multifunction (cid:98)D∗

(cid:0)(¯x, ¯y); gphF (cid:1)(cid:111)

(cid:98)D∗

(cid:12) (x∗, −y∗) ∈ (cid:98)Nε

εF (¯x, ¯y)(y∗) =

x∗ ∈ X ∗(cid:12) , ∀y∗ ∈ Y ∗.

(1.8) εF (¯x, ¯y) with ε = 0 is said to be the Fr´echet coderivative

The mapping (cid:98)D∗ of F at (¯x, ¯y) and is denoted by (cid:98)D∗F (¯x, ¯y).

(ii) The Mordukhovich coderivative (or the normal coderivative) of F at

(¯x, ¯y) ∈ gphF is the multifunction D∗F (¯x, ¯y) : Y ∗ ⇒ X ∗ given by

(cid:98)D∗

εF (x, y)(y∗).

(1.9)

D∗F (¯x, ¯y)(¯y∗) = Limsup (x,y)→(¯x,¯y) y∗w∗ → ¯y∗ ε↓0

If (¯x, ¯y) (cid:54)∈ gphF , we put D∗F (¯x, ¯y)(y∗) = ∅ for all y∗ ∈ Y ∗.

It follows from (1.8) that (cid:98)D∗

k ∈ (cid:98)D∗

k) w∗

k, y∗

εkF (xk, yk)(y∗

εF (¯x, ¯y)(y∗) = ∅ for all ε ≥ 0 and y∗ ∈ Y ∗ when (¯x, ¯y) (cid:54)∈ gphF . From (1.9) we see that D∗F (¯x, ¯y)(¯y∗) is the collection of such ¯x∗ ∈ X ∗ for which there are sequences εk ↓ 0, (xk, yk) → (¯x, ¯y), and (x∗ k) for all k ∈ IN . Note that the multifunction D∗F (¯x, ¯y) in (1.9) is uniquely determined

→ (¯x∗, ¯y∗) with (xk, yk) ∈ gphF and x∗

by the limiting normal cone to the graph of F at the point (¯x, ¯y). Namely, the Mordukhovich coderivative of F at the point (¯x, ¯y) is the multifunction D∗F (¯x, ¯y) : Y ∗ ⇒ X ∗, where

(cid:110) x∗ ∈ X ∗(cid:12)

(cid:12) (x∗, −y∗) ∈ N (cid:0)(¯x, ¯y); gphF (cid:1)(cid:111)

D∗F (¯x, ¯y)(y∗) := , ∀y∗ ∈ Y ∗.

(1.10) From (1.6) and (1.10) it is clear that the computation of the Fr´echet normal cone to the graph of a multifunction between Asplund spaces is a crucial step towards a complete diﬀerentiation of that multifunction.

One says that F is graphically regular at a given point (¯x, ¯y) ∈ gphF if

D∗F (¯x, ¯y)(y∗) = (cid:98)D∗F (¯x, ¯y)(y∗), ∀y∗ ∈ Y ∗.

When F ≡ f is a single-valued mapping and ¯y = f (¯x), one writes respectively (cid:98)D∗f (¯x) and D∗f (¯x) for (cid:98)D∗F (¯x, ¯y) and D∗F (¯x, ¯y).

By deﬁnition, f : X → Y is Fr´echet diﬀerentiable at ¯x if there is a contin- uous linear operator ∇f (¯x) : X → Y , called the Fr´echet derivative of f at ¯x, such that

= 0. lim x→¯x

f (x) − f (¯x) − ∇f (¯x)(x − ¯x) (cid:107)x − ¯x(cid:107) Function f : X → Y is said to be strictly diﬀerentiable [28, Deﬁnition 1.13] at ¯x with the strict derivative denoted by ∇f (¯x) if

= 0. f (x) − f (u) − ∇f (¯x)(x − u) (cid:107)x − u(cid:107) lim x→¯x u→¯x

According to [28, Theorem 1.38], if f : X → Y is Fr´echet diﬀerentiable at ¯x, then (cid:98)D∗f (¯x)(y∗) = {∇f (¯x)∗y∗} for all y∗ ∈ Y ∗ with ∇f (¯x)∗ being the adjoint operator of ∇f (¯x). Similarly, if f is strictly diﬀerentiable at ¯x (in particular, if f is continuously Fr´echet diﬀerentiable in a neighborhood of ¯x) with the strict derivative ∇f (¯x), then

D∗f (¯x)(y∗) = (cid:98)D∗f (¯x)(y∗) = {∇f (¯x)∗y∗}, ∀y∗ ∈ Y ∗.

Thus the Fr´echet coderivative (resp., the Mordukhovich coderivative) of mul- tifunctions is a natural extension of the adjoint of the Fr´echet derivative (resp., of the strict derivative) of a single-valued mapping.

Let ϕ : X → IR be an extended-real-valued function deﬁned on a Banach

space X. If ϕ(x) > −∞ for all x ∈ X and its eﬀective domain

domϕ := {x ∈ X| ϕ(x) < ∞}

is nonempty, then ϕ is said to be a proper function. To ϕ we associate the epigraph

epiϕ := {(x, α) ∈ X × IR| α ≥ ϕ(x)}.

Deﬁnition 1.5 (See [28, Deﬁnition 1.77 and 1.118]) Let ϕ : X → IR be ﬁnite at ¯x ∈ X.

(i) The limiting subdiﬀerential of ϕ at ¯x is the set

∂ϕ(¯x) := (cid:8)x∗ ∈ X ∗| (x∗, −1) ∈ N (cid:0)(¯x, ϕ(¯x)); epiϕ(cid:1)(cid:9),

and its elements are called limiting subgradients of ϕ at this point. When ϕ(¯x) = ∞, one puts ∂ϕ(¯x) = ∅.

(ii) For any ¯y ∈ ∂ϕ(¯x), the mapping ∂2ϕ(¯x, ¯y) : X ∗∗ ⇒ X ∗ with the values

∂2ϕ(¯x, ¯y)(u) = (D∗∂ϕ)(¯x, ¯y)(u), ∀u ∈ X ∗∗,

is said to be the limiting second-order subdiﬀerential of ϕ at ¯x relative to ¯y.

In a ﬁnite dimensional setting, as well as in an inﬁnite dimensional setting, the limiting subdiﬀerential theory has been developed successfully; see e.g. [4], [28], [48]. Meanwhile, the limiting second-order subdiﬀerential theory still requires further investigations, although many interesting theoretical results and applications can be found in [5], [6], [30], [31], and the references therein.

 0 

For each subset Ω ⊂ X, an extended-real-valued function δ(· ; Ω) : X → IR,



if x ∈ Ω δ(x; Ω) = ∞ if x (cid:54)∈ Ω,

is called the indicator function of Ω.

Proposition 1.2 (See [28, Proposition 1.79]) Consider a nonempty subset Ω ⊂ X. Then for any ¯x ∈ Ω one has ∂δ(¯x; Ω) = N (¯x; Ω).

The multifunction F : X ⇒ X ∗ with F (x) = N (x; Ω) for all x ∈ X is called the normal cone mapping to Ω. From Proposition 1.2 it follows that if F is the normal cone mapping to Ω and (¯x, ¯x∗) ∈ gphF , then we have

D∗F (¯x, ¯x∗)(u) = (cid:0)D∗∂δ(· ; Ω)(cid:1)(¯x, ¯x∗)(u) = ∂2δ(· ; Ω)(¯x, ¯x∗)(u), ∀u ∈ X ∗∗.

The latter implies that the problem of computing the limiting second-order subdiﬀerential of the indicator function of a set reduces to that of computing coderivatives of the normal cone mapping.

1.4 Lipschitzian Properties and Metric Regularity

Lipschitzian properties of multifunctions play a principal role in many aspects of variational analysis and its applications.

Consider a multifunction F : X ⇒ Y between Banach spaces X and Y . One says that F is Lipschitz continuous on X if there exists a constant (cid:96) > 0 such that

(1.11) F (x) ⊂ F (u) + (cid:96)(cid:107)x − u(cid:107) ¯BY , ∀x, u ∈ X.

If (1.11) holds for all x, u from a neighborhood U of ¯x ∈ X, then F is called locally Lipschitz at ¯x. The multifunction F is said to be locally upper Lipschitz at ¯x ∈ X with the modulus (cid:96) if there exists a neighborhood U of ¯x ∈ X with

F (x) ⊂ F (¯x) + (cid:96)(cid:107)x − ¯x(cid:107) ¯BY , ∀x ∈ U.

The local Lipschitz-like property (called also the pseudo-Lipschitz property [1], or the Aubin property property [12]) of multifunctions plays a fundamen- tal role in Variational Analysis.

Deﬁnition 1.6 (See [28, Deﬁnition 1.40]) Let F : X ⇒ Y with domF (cid:54)= ∅. Given (¯x, ¯y) ∈ gphF , we say that F is locally Lipschitz-like around (¯x, ¯y) with modulus (cid:96) ≥ 0 if there are neighborhoods U of ¯x and V of ¯y such that

(1.12) F (x) ∩ V ⊂ F (u) + (cid:96)(cid:107)x − u(cid:107) ¯BY , ∀x, u ∈ U.

The inﬁmum of all such moduli {(cid:96)} is called the exact Lipschitzian bound of F around (¯x, ¯y) and is denoted by lipF (¯x, ¯y).

The following theorem provides us a criterion for the local Lipschitz-like

property of multifunctions between ﬁnite dimensional spaces.

Theorem 1.3 (See [28, Theorem 4.10]) Let F : X ⇒ Y be a multifunction between ﬁnite dimensional spaces with its graph being locally closed around (¯x, ¯y) ∈ gphF . Then the following are equivalent:

(i) F is locally Lipschitz-like around (¯x, ¯y).

(ii) D∗F (¯x, ¯y)(0) = {0}.

Moreover, in this case it holds lipF (¯x, ¯y) = (cid:107)D∗F (¯x, ¯y)(cid:107) < ∞.

Metric regularity is another important property of multifunctions; see [28], [29] for more details. This property is closely related to Lipschitzian proper- ties of inverse mappings.

Deﬁnition 1.7 (See [28, Deﬁnition 1.47]) Let F : X ⇒ Y with domF (cid:54)= ∅. We say that F is locally metrically regular around a given point (¯x, ¯y) ∈ gphF with modulus µ > 0 if there exist some neighborhoods U of ¯x and V of ¯y, and γ > 0 such that

(1.13) dist(x; F −1(y)) ≤ µ dist(y; F (x))

for all x ∈ U and y ∈ V satisfying dist(y; F (x)) ≤ γ. The inﬁmum of all such moduli {µ}, denoted by regF (¯x, ¯y), is called the exact regularity bound of F around (¯x, ¯y).

A criterion for the metric regularity property of multifunctions between

ﬁnite dimensional spaces is provided in the next theorem.

Theorem 1.4 (See [28, Theorem 4.18]) Let F : X ⇒ Y be a multifunction between ﬁnite dimensional spaces with its graph being locally closed around (¯x, ¯y) ∈ gphF . Then the following are equivalent:

(i) F is locally metrically regular around (¯x, ¯y).

(ii) D∗F −1(¯y, ¯x)(0) = {0}.

Relationships between the local Lipschitz-like property of a multifunction F and the metric regularity property of its inverse F −1 are described as follows.

Theorem 1.5 (See [28, Theorem 1.49]) Let F : X ⇒ Y with domF (cid:54)= ∅. Then F is locally Lipschitz-like around (¯x, ¯y) ∈ gphF if and only if its inverse mapping F −1 : Y ⇒ X is locally metrically regular around (¯y, ¯x) ∈ gphF −1 with the same modulus. Moreover, the latter equivalent to the existence of neighborhoods U of ¯x, V of ¯y and a number (cid:96) > 0 such that

(1.14) F (x) ∩ V ⊂ F (u) + (cid:96)(cid:107)x − u(cid:107) ¯BY , ∀u ∈ U, ∀x ∈ X.

In this case one has the equality lipF (¯x, ¯y) = regF −1(¯y, ¯x).

In contrast to (1.12), there is no restriction on x in (1.14) because of the

localization in both domain and range of F .

1.5 Conclusions

This chapter collects several basic concepts and facts on generalized diﬀer- entiation, together with the well-known dual characterizations of the two fundamental properties of multifunctions: the local Lipschitz-like property and the metric regularity.

Chapter 2

Linear Perturbations of Polyhedral

Normal Cone Mappings

Generalized diﬀerentiability properties of the normal cone mappings allow us to get useful information about solution sensitivity/stability of variational in- equalities with polyhedral convex constraint sets; see e.g. [11], [13], [15], [25], [32], [38], [39], [40], [45], [46], [50], [52], [53]. In this chapter, we diﬀerentiate the normal cone mappings to linearly perturbed polyhedral convex sets and apply the results to solution stability of aﬃne variational inequalities. We will answer two open questions stated by Yao and Yen in [52].

This chapter is written on the basis of the results in [38], [39], and [40].

2.1 The Normal Cone Mapping F(x, b)

(cid:111)

(cid:110)

Let X be a real Banach space with the dual denoted by X ∗. Consider an index i ∈ X ∗| i ∈ T }, and a polyhedral set T = {1, 2, . . . , m}, a vector system {a∗ convex set

(cid:12) (cid:104)a∗

i , x(cid:105) ≤ bi, ∀i ∈ T

(2.1) Θ(b) = x ∈ X(cid:12)

depending on the parameter b = (b1, . . . , bm) ∈ IRm. The numbers b1, . . . , bm are interpreted as the right-hand side perturbations of the linear inequality system

i , x(cid:105) ≤ bi, i ∈ T.

(cid:104)a∗ (2.2)

(cid:9)

For a pair (x, b) ∈ X × IRm, we call

i , x(cid:105) = bi

(2.3) I(x, b) = (cid:8)i ∈ T | (cid:104)a∗

the active index set of Θ(b) at x. For any I ⊂ T , put ¯I = T \I. By bI we denote the vector with the components bi where i ∈ I. We will write bI ≤ 0 (resp., bI ≥ 0, bI = 0) if bi ≤ 0 (resp., bi ≥ 0, bi = 0) for all i ∈ I.

The multifunction F : X × IRm ⇒ X ∗ deﬁned by setting

(2.4) F(x, b) = N (x; Θ(b)), ∀(x, b) ∈ X × IRm,

(cid:8)x∗ ∈ X ∗(cid:12)

is said to be the linearly perturbed polyhedral normal cone mapping to the perturbed polyhedron Θ(b) (or, the normal cone mapping F(·), for short). Here, the set

 

(cid:12) (cid:104)x∗, u − x(cid:105) ≤ 0, ∀u ∈ Θ(b)(cid:9),



if x ∈ Θ(b) N (x; Θ(b)) = ∅, if x (cid:54)∈ Θ(b)

denotes the normal cone to Θ(b) at x in the sense of convex analysis.

It is well-known that the problem of computing the Fr´echet coderivative and the Mordukhovich coderivative [28] of the normal cone mapping of a sys- tem of linear inequalities was solved by Dontchev and Rockafellar [11]. The obtained coderivative formulas were used to establish a complete characteri- zation of the Aubin property (i.e., the local Lipschitz-like property [28]) of the solution map of parametric aﬃne variational inequalities. Recently, extending the results of [11], Yao and Yen [52] have given some upper estimates for the Fr´echet normal cone and the limiting normal cone to the graph of the normal cone mapping of a system of linear inequalities under linear perturbations. The results in [52] are applied to stability analysis of parametric variational inequalities, whose constraint sets are linearly perturbed polyhedra [53].

Further developments of the studies [11], [52], and [53] can be seen in [13], [32], [40], [38], [39]. Note that Henrion, Mordukhovich and Nam [13] have obtained exact formulas for the Fr´echet and the Mordukhovich coderivatives of the normal cone mapping of a system of linear inequalities in reﬂexive Banach spaces. These coderivative formulas help one to study robust stability of variational inequalities in inﬁnite dimensional spaces [13].

Nam [32] extended the results to the case where the linear inequality system in question undergoes linear perturbations. To investigate the Lipschitz sta- bility of parametric variational inequalities in reﬂexive Banach spaces, Nam [32] has also established exact formulas for the Fr´echet and the Mordukhovich coderivatives of F(x, b) under a linear independence assumption which is im- posed on the normal vectors of the active constraints.

Relaxing and then removing that assumption of Nam [32], we will prove an exact formula for the Fr´echet coderivative and several estimates for the Mordukhovich coderivative of F(x, b) in the next sections. In the rest of this section we provide some facts related to the normal cone mapping F(x, b).

(cid:41)

(cid:88)

Let V be a vector space over the real and let J = {1, 2, . . . , r}. For a vector system {vj| j ∈ J} ⊂ V , the convex cone generated by {vj| j ∈ J} is denoted by pos{vj| j ∈ J}. This means that (cid:40)

(cid:12) (cid:12) (cid:12) (cid:12)

j∈J

. pos{vj| j ∈ J} = λjvj λj ≥ 0, ∀j ∈ J

In the sequel, to cover also the case J = ∅, we use the convention pos ∅ = {0}. The following proposition was proved in [32] by using a generalized version of the Farkas lemma [3].

(cid:41)

(cid:40)

Proposition 2.1 (See [32, Lemma 3.1]) Let ¯b ∈ IRm, Θ(¯b) be given by (2.1), and let ¯x ∈ Θ(¯b), I(¯x, ¯b) be deﬁned by (2.3). Then N (¯x; Θ(¯b)) = pos(cid:8)a∗

(cid:88)

i | i ∈ I(¯x, ¯b)(cid:9) (cid:12) (cid:12) (cid:12) (cid:12)

i∈I(¯x,¯b)

(2.5) = λi ≥ 0, ∀i ∈ I(¯x, ¯b) λia∗ i

(cid:110)

and

(cid:12) (cid:104)a∗

(cid:111) i , v(cid:105) ≤ 0, ∀i ∈ I(¯x, ¯b) .

(2.6) T (¯x; Θ(¯b)) = v ∈ X(cid:12)

From Proposition 2.1 it follows that

i | i ∈ I(x, b)(cid:9), ∀(x, b) ∈ X × IRm.

(2.7) F(x, b) = pos(cid:8)a∗

We now show that Lemma 2.1 of [36] can be stated for vector systems in an arbitrary vector space. The proof is similar to that of [36].

Lemma 2.1 Consider a vector system {vj| j ∈ J} ⊂ V . For each nonzero u ∈ pos{vj| j ∈ J}, there exists a subset (cid:101)J ⊂ J such that {vj| j ∈ (cid:101)J} are linearly independent and u ∈ pos{vj| j ∈ (cid:101)J}.

(cid:88)

Proof. There is nothing to prove if {vj| j ∈ J} are linearly independent. Now, we consider the case where {vj| j ∈ J} are linearly dependent. Let

j∈J

u = λjvj, λj ≥ 0 for j ∈ J,

(cid:88)

and u (cid:54)= 0. Without loss of generality we may assume that λj > 0 for all j ∈ J. Since {vj| j ∈ J} are linearly dependent, there exist αj ∈ IR, j ∈ J, not all zero, such that

j∈J

αjvj = 0.

(cid:27)

Choose an index j0 ∈ J satisfying

(cid:12) (cid:12) (cid:12) j ∈ J

(cid:26)|αj| λj

= max > 0. |αj0| λj0

For each j ∈ J, put

αj. µj = λj − λj0 αj0

(cid:19)

(cid:18)

(cid:19)

We see that µj0 = 0 and (cid:18) 1 − · 1 − · µj = λj ≥ λj ≥ 0 for j ∈ J\{j0}. αj λj |αj| λj λj0 αj0 λj0 |αj0|

(cid:88)

Hence

j∈J

j∈J\{j0}

αjvj = λjvj = u. µjvj = µjvj = λjvj − λj0 αj0

This shows that u ∈ pos(cid:8)vj| j ∈ J\{j0}(cid:9). If the vectors (cid:8)vj| j ∈ J\{j0}(cid:9) are linearly dependent, using again the above arguments we can ﬁnd a proper subset J (cid:48) ⊂ J\{j0} such that u ∈ pos{vj| j ∈ J (cid:48)}. Since J is ﬁnite and u (cid:54)= 0, there must exist an index subset (cid:101)J ⊂ J such that {vj| j ∈ (cid:101)J} are linearly (cid:50) independent and u ∈ pos{vj| j ∈ (cid:101)J}.

The following lemma shows that the graph of the normal cone mapping F(·) is closed in the product space X × IRm × X ∗. This property allows us to calculate the limiting normal cone N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) by formula (1.6).

k)}k∈IN ⊂ gphF and (xk, bk, x∗

k → ¯x∗, so x∗

k) → (¯x, ¯b, ¯x∗). Proof. Suppose that {(xk, bk, x∗ We have to show that (¯x, ¯b, ¯x∗) ∈ gphF. If ¯x∗ = 0, then (¯x, ¯b, ¯x∗) ∈ gphF because 0 ∈ N (¯x, Θ(¯b)) = F(¯x, ¯b). We now consider the case that ¯x∗ (cid:54)= 0. We have ¯x∗ (cid:54)= 0 and x∗ k (cid:54)= 0 for all k large enough. Since (xk, bk) → (¯x, ¯b), we have I(xk, bk) ⊂ I(¯x, ¯b) for suﬃciently large indexes k ∈ IN . Thus, without loss of generality we may assume that x∗ k (cid:54)= 0 and

Lemma 2.2 Let F(·) be given by (2.4). Then, the graph of F(·) is closed in the sum norm topology of the product space X × IRm × X ∗.

k ∈ F(xk, bk),

k ∈ pos{a∗ x∗

i | i ∈ I(xk, bk)} = pos{a∗

i | i ∈ (cid:101)I}.

I(xk, bk) = (cid:101)I ⊂ I(¯x, ¯b) for all k ∈ IN . For each k ∈ IN , since x∗ by (2.7) it holds

k → ¯x∗, it follows that

As ﬁnitely generated convex cones are closed and x∗

¯x∗ ∈ pos{a∗

i | i ∈ I(¯x, ¯b)} = F(¯x, ¯b). i | i ∈ (cid:101)I} ⊂ pos{a∗ Hence, (¯x, ¯b, ¯x∗) ∈ gphF. We have shown that gphF is closed.

(cid:50)

2.2 The Fr´echet Coderivative of F(x, b)

(cid:88)

Let (¯x, ¯b, ¯x∗) ∈ gphF with F being deﬁned by (2.4). For simplicity, we will write I for I(¯x, ¯b). Since ¯x∗ ∈ F(¯x, ¯b), from (2.7) it follows that

i∈I(¯x,¯b)

¯x∗ = for some λi ≥ 0, i ∈ I(¯x, ¯b). λia∗ i ,

(cid:27)

(cid:88)

We deﬁne respectively the multiplier set and the index set corresponding to the point (¯x, ¯b, ¯x∗) as follows

i , λi ≥ 0 ∀i ∈ I

(cid:26) (λi)i∈I ∈ IR|I|(cid:12) (cid:12) ¯x∗ = (cid:12)

i∈I

(cid:110)

(2.8) Ξ(¯x, ¯b, ¯x∗) = , λia∗

(cid:111) (cid:12) λi = 0 for some (λj)j∈I ∈ Ξ(¯x, ¯b, ¯x∗) ,

(2.9) i ∈ I(cid:12) I1(¯x, ¯b, ¯x∗) =

(cid:40)

and construct the sets

(cid:12) (cid:12) (x∗, b∗, v) (cid:12) (cid:12)

H(¯x, ¯b, ¯x∗) = x∗ ∈ (cid:0)T (¯x; Θ(¯b)) ∩ {¯x∗}⊥(cid:1)∗ ,

(2.10) v ∈ T (¯x; Θ(¯b)) ∩ {¯x∗}⊥,

(cid:41) ,

i a∗

i , b∗

i∈I b∗

¯I = 0, b∗ I1

(cid:40)

≤ 0 x∗ = − (cid:80)

(cid:12) (cid:12) (x∗, b∗, v) (cid:12) (cid:12)

x∗ ∈ (cid:0)T (¯x; Θ(¯b)) ∩ {¯x∗}⊥(cid:1)∗ , E0(¯x, ¯b, ¯x∗) =

(cid:41)

(2.11) v ∈ T (¯x; Θ(¯b)) ∩ {¯x∗}⊥,

I ≤ 0

i , b∗

i a∗

i∈I b∗

¯I = 0, b∗

, x∗ = − (cid:80)

1, . . . , b∗

m) ∈ IRm and ¯I = T \I. For each λ = (λi)i∈I ∈ Ξ(¯x, ¯b, ¯x∗),

where b∗ = (b∗ we consider

(2.12) I0(λ) = (cid:8)i ∈ I| λi = 0(cid:9),

(cid:40)

and

(cid:12) (cid:12) (x∗, b∗, v) (cid:12) (cid:12)

, x∗ ∈ (cid:0)T (¯x; Θ(¯b)) ∩ {¯x∗}⊥(cid:1)∗ Eλ(¯x, ¯b, ¯x∗) =

(cid:41) .

(2.13) v ∈ T (¯x; Θ(¯b)) ∩ {¯x∗}⊥,

i a∗

i , b∗

i∈I b∗

¯I = 0, b∗

I0(λ) ≤ 0

x∗ = − (cid:80)

(cid:92)

Lemma 2.3 For any (¯x, ¯b, ¯x∗) ∈ gphF, we have

λ∈Ξ(¯x,¯b,¯x∗)

(2.14) H(¯x, ¯b, ¯x∗) = Eλ(¯x, ¯b, ¯x∗).

i ≤ 0 which implies b∗ I1

I0(λ) ≤ 0. Consequently, b∗

≤ 0. Proof. Pick any (x∗, b∗, v) ∈ H(¯x, ¯b, ¯x∗). For each λ ∈ Ξ(¯x, ¯b, ¯x∗), we have I0(λ) ≤ 0. Since (x∗, b∗, v) ∈ H(¯x, ¯b, ¯x∗), (x∗, b∗, v) ∈ Eλ(¯x, ¯b, ¯x∗) whenever b∗ I0(λ) ≤ 0. Thus, H(¯x, ¯b, ¯x∗) ⊂ Eλ(¯x, ¯b, ¯x∗) b∗ ≤ 0. We have I0(λ) ⊂ I1, so b∗ I1 for all λ ∈ Ξ(¯x, ¯b, ¯x∗) which justiﬁes the inclusion “ ⊂ ” in (2.14). Now, ﬁx any (x∗, b∗, v) in the set on the right-hand side of (2.14). Clearly, one has (x∗, b∗, v) ∈ H(¯x, ¯b, ¯x∗) if b∗ ≤ 0. For each i ∈ I1, we have λi = 0 for some I1 λ = (λj)j∈I ∈ Ξ(¯x, ¯b, ¯x∗). Hence, i ∈ I0(λ). Since (x∗, b∗, v) ∈ Eλ(¯x, ¯b, ¯x∗), it (cid:50) holds b∗

The inclusion (2.15) below is proved in Proposition 3.2 of [32] without using any regularity assumption. Under an auxiliary assumption, the inclusion will become an equality.

Theorem 2.1 (See [32, Proposition 3.2]) Let (¯x, ¯b, ¯x∗) ∈ gphF. Then, for every λ ∈ Ξ(¯x, ¯b, ¯x∗), the following inclusion holds

(cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) ⊂ Eλ(¯x, ¯b, ¯x∗),

i | i ∈ I} are

(2.15)

where Eλ(¯x, ¯b, ¯x∗) is given by (2.13). Besides, if the vectors {a∗ linearly independent, then Ξ(¯x, ¯b, ¯x∗) has only one element ¯λ and

(cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) = E¯λ(¯x, ¯b, ¯x∗).

(2.16)

For the case X = IRn, the upper estimate (2.15) was obtained in [52]. Concerning the above mentioned upper estimate for the Fr´echet normal cone

given in [52], an open question was stated in the same paper. In our notation, Question 1 of [52] can be restated as follows.

Question 2.1 For any λ ∈ Ξ(¯x, ¯b, ¯x∗), does the inclusion (2.15) always hold as an equality?

Deﬁnition 2.1 Let {vj}j∈J be a family of ﬁnitely many vectors of a vector space V over the reals. We say that {vj}j∈J is positively linearly independent if from the conditions (cid:80) j∈J λjvj = 0 and λj ≥ 0 for all j ∈ J it follows that λj = 0 for all j ∈ J.

i | i ∈ I(¯x, ¯b)}, the inclusion (2.15) is also not an equality.

The second assertion of Theorem 2.1 solves Question 2.1 in the aﬃrmative i | i ∈ I(¯x, ¯b)} are linearly independent. under the condition that the vectors {a∗ We are going to show that, in general, the inclusion (2.15) does not hold as an equality. Moreover, even under positive linear independence assumption of {a∗

2 = (0, 2) ∈ X ∗ = IR2, ¯b = (0, 0) ∈ IR2,

1 = (0, 1), a∗

Proposition 2.2 The inclusion (2.15) may be strict in some cases.

(cid:110)

(cid:111)

Proof. Let X = IR2, a∗ and ¯x = (0, 0) ∈ X. We have

(cid:12) (cid:104)a∗

i , x(cid:105) ≤ 0, i = 1, 2

(cid:111)

(cid:110)

Θ(¯b) = x ∈ IR2(cid:12) = IR × (−∞, 0],

(cid:12) (cid:12) λi ≥ 0, i = 1, 2

1 + λ2a∗ 2

2} =

i , ¯x(cid:105) = 0(cid:9) = {1, 2}, 1, a∗

(cid:111)

(cid:110)

I(¯x, ¯b) = (cid:8)i | (cid:104)a∗ F(¯x, ¯b) = N (¯x; Θ(¯b)) = pos{a∗ λ1a∗

(cid:12) (cid:104)a∗

i , v(cid:105) ≤ 0, i = 1, 2

= = {0} × [0, +∞), T (¯x; Θ(¯b)) = (cid:0)N (¯x; Θ(¯b))(cid:1)∗ v ∈ IR2(cid:12)

= IR × (−∞, 0].

For any α > 0, put ¯x∗ = (0, α). Since ¯x∗ ∈ {0} × [0, +∞) = F(¯x, ¯b), it holds (¯x, ¯b, ¯x∗) ∈ gphF. We observe that

{¯x∗}⊥ = {(0, α)}⊥ = IR × {0}, T (¯x; Θ(¯b)) ∩ {¯x∗}⊥ = IR × {0}, (cid:0)T (¯x; Θ(¯b)) ∩ {¯x∗}⊥(cid:1)∗ = {0} × IR.

If we use the representation

1 + 0a∗ 2,

¯x∗ = (0, α) = αa∗

2) ∈ IR2, where b∗

1, b∗

2 ≤ 0

1 will be determined later. For any γ ∈ IR, we have

then λ = (α, 0) and I0(λ) = {2}. Choose b∗ = (b∗ and the value b∗

v = (γ, 0) ∈ T (¯x; Θ(¯b)) ∩ {¯x∗}⊥.

Put

1a∗

1 + b∗

2a∗

2) = (0, −b∗

1 − 2b∗

2) ∈ (cid:0)T (¯x; Θ(¯b)) ∩ {¯x∗}⊥(cid:1)∗

. x∗ = −(b∗

k ) and xk = ( 1

2k , 1

k , 1

Hence, (x∗, b∗, v) ∈ Eλ(¯x, ¯b, ¯x∗).

(cid:27)

(cid:18)

(cid:21)

We now show that (x∗, b∗, v) does not belong to the cone (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1). Consider the sequences {bk} and {xk}, where bk = ( 1 2k ), k ∈ IN . We see that bk → ¯b and xk → ¯x as k → ∞. For every k ∈ IN , it holds that

(cid:26) (x1, x2) ∈ IR2(cid:12) (cid:12) (cid:12) x2 ≤

Θ(bk) = = IR × −∞, . , 2x2 ≤ 1 k 1 2k

1 k Then, xk ∈ Θ(bk), I(xk, bk) = {2}, and

2} = {0} × [0, +∞),

k = (0, α + 1

2k ) for k → (0, α) = ¯x∗

k}k∈IN where u∗ k ∈ F(xk, bk), and u∗

F(xk, bk) = N (xk; Θ(bk)) = pos{a∗

for all k ∈ IN . Now, consider the sequence {u∗ all k ∈ IN . For every k ∈ IN , we have u∗ as k → ∞. Note that

(x,b,u∗)

gphF −→ (¯x,¯b,¯x∗)

k − ¯x∗(cid:105)

limsup (cid:104)x∗, x − ¯x(cid:105) + (cid:104)b∗, b − ¯b(cid:105) + (cid:104)v, u∗ − ¯x∗(cid:105) (cid:107)x − ¯x(cid:107) + (cid:107)b − ¯b(cid:107) + (cid:107)u∗ − ¯x∗(cid:107)

k→∞

(cid:10)(0, −b∗

1 − 2b∗

k )(cid:11) + (cid:10)(γ, 0), (0, 1

2k )(cid:11)

≥ limsup

k→∞

2k , 1 2k , 1

2k )(cid:107)

−b∗

1+b∗ 2 k

√

= limsup (cid:104)x∗, xk − ¯x(cid:105) + (cid:104)b∗, bk − ¯b(cid:105) + (cid:104)v, u∗ (cid:107)xk − ¯x(cid:107) + (cid:107)bk − ¯b(cid:107) + (cid:107)u∗ k − ¯x∗(cid:107) 2k )(cid:11) + (cid:10)(b∗ 2), ( 1 2), ( 1 k , 1 1, b∗ k )(cid:107) + (cid:107)(0, 1 k , 1 2k )(cid:107) + (cid:107)( 1 (cid:107)( 1

√ = =: µ. = lim k→∞ b∗ 1 √ 2 + 2 2 + 1

2k + b∗ 1−2b∗ 2 k + 1 2k + 2k 1 = 1, we have

Choosing b∗

√ µ = > 0. 1 √ 2 + 2 2 + 1

This implies that (x∗, b∗, v) (cid:54)∈ (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1), and hence

Eλ(¯x, ¯b, ¯x∗) (cid:54)⊂ (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1).

(cid:50) The proof is complete.

1, a∗

Remark 2.1 We have seen that without the linear independence condition i | i ∈ I(¯x, ¯b)} the equality (2.16) may fail to hold. The vectors {a∗ of {a∗ 2} in the above proof are not linearly independent, but they are positively linearly independent. We have thus shown that the inclusion (2.15) may be strict even i | i ∈ I(¯x, ¯b)} are positively linearly independent. in the case the vectors {a∗

Remark 2.2 As usual, we say that the inequality system (2.2) satisﬁes the Slater condition if there exists x0 ∈ X with (cid:104)a∗ i , x0(cid:105) < bi for all i ∈ T . This condition is a signiﬁcant sign of the stability of the given inequality system. One may hope that the equality (2.16) holds when the Slater condition is satisﬁed. Nevertheless, the proof of Proposition 2.2 overturns the hope. In- i , x0(cid:105) < ¯bi for i = 1, 2 but, as shown in deed, taking x0 = (0, −1) we have (cid:104)a∗ the proof, Eλ(¯x, ¯b, ¯x∗) (cid:54)⊂ (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1).

Using the set E0(¯x, ¯b, ¯x∗) deﬁned by (2.11), we now provide a lower estimate for the Fr´echet normal cone (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1). Our result is an extension of [52, Lemma 4.2] where it was assumed that X = IRn.

Theorem 2.2 If (¯x, ¯b, ¯x∗) ∈ gphF, then the following inclusion holds

(2.17) E0(¯x, ¯b, ¯x∗) ⊂ (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1).

Proof. Let (x∗, b∗, v) ∈ E0(¯x, ¯b, ¯x∗). In order to show that

(x∗, b∗, v) ∈ (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1),

we need to verify the inequality

(x,b,u∗)

gphF −→ (¯x,¯b,¯x∗)

k) gphF−→ (¯x, ¯b, ¯x∗). Then, because Let there be given any sequence (xk, bk, u∗ (xk, bk) → (¯x, ¯b), we must have I(xk, bk) ⊂ I(¯x, ¯b) for all k suﬃciently large. We put I = I(¯x, ¯b) for short. Since

k ∈ pos(cid:8)a∗ u∗

i | i ∈ I(xk, bk)(cid:9) ⊂ pos(cid:8)a∗

i | i ∈ I(cid:9) = N (¯x; Θ(¯b)) = F(¯x, ¯b),

limsup (2.18) ≤ 0. (cid:104)x∗, x − ¯x(cid:105) + (cid:104)b∗, b − ¯b(cid:105) + (cid:104)v, u∗ − ¯x∗(cid:105) (cid:107)x − ¯x(cid:107) + (cid:107)b − ¯b(cid:107) + (cid:107)u∗ − ¯x∗(cid:107)

the condition v ∈ T (¯x; Θ(¯b)) ∩ {¯x∗}⊥ implies that

k(cid:105) ≤ 0.

k − ¯x∗(cid:105) = (cid:104)v, u∗ 20

(2.19) (cid:104)v, u∗

i a∗

i and b∗

¯I = 0 we deduce that

(cid:29)

i∈I b∗ (cid:28)

(cid:88)

From the equalities x∗ = − (cid:80)

i a∗ b∗

i , xk − ¯x

I, bk − ¯b(cid:105)

i∈I (cid:16)

(cid:17)

(cid:88)

− + (cid:104)b∗ (cid:104)x∗, xk − ¯x(cid:105) + (cid:104)b∗, bk − ¯b(cid:105) =

(cid:16) (bk)i − ¯bi

i , ¯x(cid:105) − (cid:104)a∗

(cid:17) i , xk(cid:105)

i∈I

(cid:16)

i∈I (cid:88)

= (cid:104)a∗ + b∗ i b∗ i

i , ¯x(cid:105) − ¯bi + (bk)i − (cid:104)a∗

(cid:17) i , xk(cid:105)

i∈I (cid:88)

= (cid:104)a∗ b∗ i

(cid:16) (bk)i − (cid:104)a∗

(cid:17) i , xk(cid:105)

i∈I

= . b∗ i

I ≤ 0 and (cid:104)a∗

i , xk(cid:105) ≤ (bk)i for all i ∈ I, this implies that

Since b∗

(2.20) (cid:104)x∗, xk − ¯x(cid:105) + (cid:104)b∗, bk − ¯b(cid:105) ≤ 0.

k − ¯x∗(cid:105)

From (2.19) and (2.20) we obtain

k − ¯x∗(cid:107)

k) gphF−→ (¯x, ¯b, ¯x∗) was given (cid:50)

≤ 0, limsup k→∞ (cid:104)x∗, xk − ¯x(cid:105) + (cid:104)b∗, bk − ¯b(cid:105) + (cid:104)v, u∗ (cid:107)xk − ¯x(cid:107) + (cid:107)bk − ¯b(cid:107) + (cid:107)u∗

which yields (2.18) because the sequence (xk, bk, u∗ arbitrarily. The proof is complete.

I ≤ 0

Our next goal is to solve the following question which is the second open question of Yao and Yen in [52], where X is a ﬁnite dimensional Euclidean space.

Question 2.2 Does the inclusion (x∗, b∗, v) ∈ (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) imply b∗ where I = I(¯x, ¯b)? In other words, does (2.17) always hold as an equality?

1 = (1, 0), a∗

2 = (0, 1) ∈ X ∗ = IR2, ¯b = (0, 0) ∈ IR2, and

Proposition 2.3 The inclusion (2.17) may be strict in some cases.

(cid:110)

(cid:111)

Proof. Let X = IR2, a∗ ¯x = (0, 0) ∈ X. We observe that

i , x(cid:105) ≤ 0, i = 1, 2

Θ(¯b) = x ∈ IR2| (cid:104)a∗ = (−∞, 0] × (−∞, 0],

2} = [0, +∞) × [0, +∞).

i , ¯x(cid:105) = 0(cid:9) = {1, 2}, 1, a∗ Given an arbitrary α > 0, we have ¯x∗ = (0, α) ∈ F(¯x, ¯b). This means that (¯x, ¯b, ¯x∗) ∈ gphF. We want to ﬁnd a triplet (x∗, b∗, v) ∈ (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) 21

I(¯x, ¯b) = (cid:8)i | (cid:104)a∗ F(¯x, ¯b) = N (¯x; Θ(¯b)) = pos{a∗

I (cid:54)≤ 0, i.e., there exists i ∈ I = I(¯x, ¯b) such that b∗

i > 0. Note that

with b∗

1 + αa∗

= [0, +∞) × IR. {¯x∗}⊥ = {(0, α)}⊥ = IR × {0}, T (¯x; Θ(¯b)) = (−∞, 0] × (−∞, 0], T (¯x; Θ(¯b)) ∩ {¯x∗}⊥ = (−∞, 0] × {0}, (cid:0)T (¯x; Θ(¯b)) ∩ {¯x∗}⊥(cid:1)∗

1, a∗

We have ¯x∗ = (0, α) = 0a∗ Observe that {a∗

2. Hence, I0(¯λ) = {1}, where ¯λ = (0, α). 2} are linearly independent, thus Ξ(¯x, ¯b, ¯x∗) = {¯λ}. Choose x∗ = (1, −1) ∈ (cid:0)T (¯x; Θ(¯b)) ∩ {¯x∗}⊥(cid:1)∗

1, b∗

2) with b∗

1 = −1 ≤ 0, b∗

2 = 1, and v = (γ, 0) ∈ T (¯x; Θ(¯b)) ∩ {¯x∗}⊥,

2a∗

1 + b∗

1a∗

b∗ = (b∗

where γ ≤ 0. By the choice of (x∗, b∗, v) above, we have the representation 2) and b∗ x∗ = −(b∗ I0(¯λ) ≤ 0. According to Theorem 2.1, we can infer that (x∗, b∗, v) ∈ (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1). Since b∗ 2 = 1 > 0 and 2 ∈ I\I0(¯λ), we have shown that the inclusion (x∗, b∗, v) ∈ (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) does not imply (cid:50) b∗ I ≤ 0. The proof is complete.

The following fact plays an important role in our subsequent investigations.

i | i ∈ I} are positively linearly independent, then

Theorem 2.3 (See [38, Theorem 3.2]) Let (¯x, ¯b, ¯x∗) ∈ gphF and I = I(¯x, ¯b). If the vectors {a∗

(cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) = H(¯x, ¯b, ¯x∗),

(2.21)

where H(¯x, ¯b, ¯x∗) is deﬁned by (2.10).

Let us prove that (2.21) holds without any additional assumption. Thus, i | i ∈ I} can

the positive linear independence assumption on the vectors {a∗ be removed from the formulation of Theorem 2.3

Theorem 2.4 For any (¯x, ¯b, ¯x∗) ∈ gphF, the equality (2.21) is valid.

(cid:92)

Proof. First, we prove the inclusion “ ⊂ ” in (2.21). For every λ ∈ Ξ(¯x, ¯b, ¯x∗), by Theorem 2.1 we deduce that (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) ⊂ Eλ(¯x, ¯b, ¯x∗). Hence,

(cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) ⊂

λ∈Ξ(¯x,¯b,¯x∗)

Eλ(¯x, ¯b, ¯x∗).

From this and Lemma 2.3 we get (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) ⊂ H(¯x, ¯b, ¯x∗).

k − ¯x∗(cid:105)

To justify the opposite inclusion in (2.21), ﬁx any (x∗, b∗, v) ∈ H(¯x, ¯b, ¯x∗). We have to show that (x∗, b∗, v) ∈ (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1). We suppose to the contrary that (x∗, b∗, v) (cid:54)∈ (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1). Then, by the deﬁnition of the Fr´echet normal cone (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1), there exist δ > 0 and a sequence (xk, bk, x∗

k) gphF−→ (¯x, ¯b, ¯x∗) such that (cid:104)x∗, xk − ¯x(cid:105) + (cid:104)b∗, bk − ¯b(cid:105) + (cid:104)v, x∗ (cid:107)xk − ¯x(cid:107) + (cid:107)bk − ¯b(cid:107) + (cid:107)x∗

k − ¯x∗(cid:107) Since I(xk, bk) ⊂ T for all k ∈ IN and (xk, bk) → (¯x, ¯b), we may assume that I(xk, bk) = (cid:101)I ⊂ I for all k ∈ IN . This implies that

k ∈ N (xk, Θ(bk)) ⊂ N (¯x, Θ(¯b)), ∀k ∈ IN. x∗

(2.22) ≥ δ > 0, ∀k ∈ IN.

Hence, by the inclusion (x∗, b∗, v) ∈ H(¯x, ¯b, ¯x∗) and by (2.10),

k − ¯x∗(cid:105) = (cid:104)v, x∗

k(cid:105) ≤ 0.

(2.23) (cid:104)v, x∗

i a∗

i and b∗

i∈I b∗

¯I = 0 imply that

Besides, the relations x∗ = − (cid:80)

(cid:17)

(cid:88)

(cid:16) (bk)i − ¯bi

i , ¯x(cid:105) − (cid:104)a∗

(cid:17) i , xk(cid:105)

i∈I

(cid:104)x∗, xk − ¯x(cid:105) + (cid:104)b∗, bk − ¯b(cid:105) (cid:16) = (cid:104)a∗ + b∗ i b∗ i

i∈I (cid:88)

(cid:16) (bk)i − (cid:104)a∗

(cid:17) i , xk(cid:105)

(cid:17)

(cid:16)

i∈I (cid:88)

(cid:88)

(2.24) = b∗ i

(cid:16) (bk)i − (cid:104)a∗

i , xk(cid:105)

(cid:17) i , xk(cid:105) .

i∈I1

i∈I\I1

= + (bk)i − (cid:104)a∗ b∗ i b∗ i

i ≤ 0 and (cid:104)a∗

i , xk(cid:105) ≤ (bk)i for all i ∈ I1, we have

(cid:88)

Since b∗

(cid:16) (bk)i − (cid:104)a∗

(cid:17) i , xk(cid:105)

i∈I1

i | i ∈ (cid:101)I}. i | i ∈ (cid:101)I}. Suppose now that ¯x∗ (cid:54)= 0. Since x∗

k ∈ pos{a∗

(2.25) ≤ 0. b∗ i

i | i ∈ Jk} are linearly independent and x∗

k ∈ pos{a∗

If ¯x∗ = 0 then it is obvious that k → ¯x∗, there exists k (cid:54)= 0 for all k ≥ k0. For every k ∈ IN , the equality k ∈ F(xk, bk) = N (xk; Θ(bk)) and i | i ∈ (cid:101)I}. By Lemma 2.1, one can ﬁnd Jk ⊂ (cid:101)I i | i ∈ Jk}.

(cid:111) .

(cid:12) a∗

i , i ∈ J, are linearly independent

Γ := Let us show that ¯x∗ ∈ pos{a∗ ¯x∗ ∈ pos{a∗ k0 > 0 such that x∗ I(xk, bk) = (cid:101)I together with inclusion x∗ formula (2.5) yield x∗ such that {a∗ Since (cid:101)I is ﬁnite, the set below is ﬁnite (cid:110) J ⊂ (cid:101)I(cid:12)

i | i ∈ (cid:101)J} for all (cid:96) ∈ IN . This means that

(cid:88)

∈ pos{a∗ Consequently, there must exist (cid:101)J ∈ Γ and a subsequence {k(cid:96)} of {k} such that x∗ k(cid:96)

i ≥ 0, i ∈ (cid:101)J.

i∈ (cid:101)J

= (2.26) for some λk(cid:96) λk(cid:96) i a∗ x∗ k(cid:96)

(cid:16)

i | i ∈ (cid:101)J} we infer that (cid:17)

(cid:88)

Combining (2.26) with the linear independence of {a∗

i =

a∗ i =

i∈ (cid:101)J

λk(cid:96) i a∗ λk(cid:96) i λia∗ i , x∗ k(cid:96) lim (cid:96)→∞ ¯x∗ = lim (cid:96)→∞ = lim (cid:96)→∞

i ≥ 0 for all i ∈ (cid:101)J. Thus,

where λi := lim(cid:96)→∞ λk(cid:96)

i | i ∈ (cid:101)J} ⊂ pos{a∗

i | i ∈ (cid:101)I}.

¯x∗ ∈ pos{a∗

(cid:88)

We have I\I1 = (cid:101)I\I1. Indeed, since (cid:101)I ⊂ I, it holds (cid:101)I\I1 ⊂ I\I1. Conversely, by the inclusion (cid:101)J ⊂ (cid:101)I we have

i =

i∈ (cid:101)J

i∈(cid:101)I

(2.27) ¯x∗ = λia∗ λia∗ i

(cid:16)

(cid:88)

provided that we put λi = 0 for all i ∈ (cid:101)I\ (cid:101)J. By (2.27) and deﬁnition of I1 = I1(¯x, ¯b, ¯x∗) in (2.9), we see that I\ (cid:101)I ⊂ I1. Since (cid:101)I ⊂ I, this implies that I\I1 ⊂ I\(I\ (cid:101)I) = (cid:101)I. Hence I\I1 ⊂ (cid:101)I\I1. The equality I\I1 = (cid:101)I\I1 has been proved. Furthermore, we have (cid:104)a∗ i , xk(cid:105) = (bk)i for any k ∈ IN and i ∈ I(xk, bk) = (cid:101)I. Thus,

(cid:16) (bk)i − (cid:104)a∗

(cid:17) i , xk(cid:105)

i∈I\I1

i∈(cid:101)I\I1

= (2.28) = 0. (bk)i − (cid:104)a∗ b∗ i b∗ i

From (2.24), (2.25) and (2.28), it follows that

k − ¯x∗(cid:105)

(cid:104)x∗, xk − ¯x(cid:105) + (cid:104)b∗, bk − ¯b(cid:105) ≤ 0.

k − ¯x∗(cid:107)

≤ 0, ∀k ∈ IN. Combining the latter with (2.23), we obtain (cid:104)x∗, xk − ¯x(cid:105) + (cid:104)b∗, bk − ¯b(cid:105) + (cid:104)v, x∗ (cid:107)xk − ¯x(cid:107) + (cid:107)bk − ¯b(cid:107) + (cid:107)x∗

This contradicts (2.22). Therefore, we have (x∗, b∗, v) ∈ (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1). (cid:50) The equality (2.21) has been established.

Let us consider an example to see how Theorem 2.4 can be used for prac-

tical computation of the Fr´echet normal cone (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1).

i | i ∈ T } ⊂ X ∗ where T = {1, 2, 3},

1 = (1, 0), a∗ a∗

2 = (0, 1), a∗

3 = (1, 2).

Example 2.1 Let X = IR2 and let {a∗ and

For ¯b = (0, 0, 0) ∈ IR3, ¯x = (0, 0) ∈ X, we have

i , x(cid:105) ≤ 0, i ∈ T (cid:9) = (−∞, 0] × (−∞, 0],

i , ¯x(cid:105) = ¯bi

Θ(¯b) = (cid:8)x ∈ IR2| (cid:104)a∗

(cid:9) = {1, 2, 3}, 2, a∗ 1, a∗

3} = [0, +∞) × [0, +∞).

I(¯x, ¯b) = (cid:8)i | (cid:104)a∗ F(¯x; ¯b) = N (¯x; Θ(¯b)) = pos{a∗

For α > 0, one has ¯x∗ = (0, α) ∈ F(¯x; ¯b), thus (¯x, ¯b, ¯x∗) ∈ gphF. We observe that

{¯x∗}⊥ = {(0, α)}⊥ = IR × {0},

= (−∞, 0] × (−∞, 0],

(cid:0)T (¯x; Θ(¯b)) ∩ {¯x∗}⊥(cid:1)∗

T (¯x; Θ(¯b)) = (cid:0)N (¯x; Θ(¯b))(cid:1)∗ T (¯x; Θ(¯b)) ∩ {¯x∗}⊥ = (−∞, 0] × {0}, = [0, +∞) × IR.

i where λi ≥ 0 for i ∈ I as follows

i∈I λia∗

Since I := I(¯x, ¯b) = {1, 2, 3}, it holds ¯I = T \I = ∅. There is only one way to represent ¯x∗ = (cid:80)

1 + αa∗

2 + 0a∗ 3.

¯x∗ = (0, α) = 0(1, 0) + α(0, 1) + 0(1, 2) = 0a∗

(cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) (cid:26)

Hence, I1 := I1(¯x, ¯b, ¯x∗) = {1, 3}. By (2.21) and (2.10), we obtain

(cid:12) (cid:12) (cid:12) x∗ ∈ [0, +∞) × IR, v ∈ (−∞, 0] × {0}, (x∗, b∗, v)

(cid:27)

1 − b∗

3, −b∗

2 − 2b∗

3), b∗

1 ≤ 0, b∗

3 ≤ 0

(cid:26)(cid:16)

x∗ = (−b∗

(cid:27) .

(cid:17)(cid:12) (cid:12) (cid:12) β1, β3, γ ∈ IR−

= (−β1 − β3, −β2 − 2β3), (β1, β2, β3), (γ, 0)

The next statement is immediate from Theorem 2.4 and the deﬁnition of

the Fr´echet coderivative.

Theorem 2.5 For any (¯x, ¯b, ¯x∗) ∈ gphF, the Fr´echet coderivative of F(·) at

(cid:27)

the point (¯x, ¯b, ¯x∗) is computed by the formula

(cid:98)D∗F(¯x, ¯b, ¯x∗) : X ∗∗ ⇒ X ∗ × IRm, (cid:98)D∗F(¯x, ¯b, ¯x∗)(v) (cid:26) (x∗, b∗) ∈ X ∗ × IRm(cid:12) (cid:12) (x∗, b∗, −v) ∈ (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) (cid:12)

(cid:88)

(2.29)

i a∗ b∗

i , b∗

¯I = 0, b∗ I1

(cid:26) (x∗, b∗) ∈ X ∗ × IRm(cid:12) (cid:12) (cid:12) x∗ = −

i∈I

(cid:27)

= ≤ 0,

× (cid:0)T (¯x; Θ(¯b)) ∩ {¯x∗}⊥(cid:1) (x∗, −v) ∈ (cid:0)T (¯x; Θ(¯b)) ∩ {¯x∗}⊥(cid:1)∗

i | i ∈ T }, (¯x, ¯b, ¯x∗) be the same as in Example 2.1.

for every v ∈ X ∗∗.

(cid:98)D∗F(¯x, ¯b, ¯x∗)(v) (cid:26)

(cid:27)

Example 2.2 Let X, T , {a∗ It follows from (2.29) and the results obtained in Example 2.1 that



Ω, if v = (v1, v2) ∈ IR+ × {0} (x∗, b∗) ∈ IR2 × IR3(cid:12) (cid:12) (x∗, b∗, −v) ∈ (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) (cid:12)   =

∅, if v (cid:54)∈ IR+ × {0},

(cid:27)

(cid:26)(cid:16)

where

(cid:17)(cid:12) (cid:12) (cid:12) β1, β3 ∈ IR−

Ω = . (−β1 − β3, −β2 − 2β3), (β1, β2, β3)

2.3 The Mordukhovich Coderivative of F(x, b)

Following Henrion, Mordukhovich, and Nam [13], for any sets P , Q with P ⊂ Q ⊂ T , we put

i | i ∈ P } + pos{a∗

i | i ∈ Q\P },

(2.30) AQ,P = span{a∗

where the notation spanΩ stands for the linear subspace generated by Ω (when Ω = ∅, we use the convention span∅ = {0}), and

(cid:111) .

(cid:110) x ∈ X(cid:12)

(cid:12) (cid:104)a∗

i , x(cid:105) ≤ 0 ∀i ∈ Q\P

i , x(cid:105) = 0 ∀i ∈ P, (cid:104)a∗

(2.31) BQ,P =

Note that AQ,P ⊂ X ∗ and BQ,P ⊂ X.

(cid:1)∗

Lemma 2.4 (See [13, Lemma 3.3]) If P ⊂ Q ⊂ T , then

(cid:1)∗

(cid:9).

(2.32) = AQ,P

(cid:0)BQ,P := (cid:8)u∗ ∈ X ∗| (cid:104)u∗, x(cid:105) ≤ 0, ∀x ∈ BQ,P

with (cid:0)BQ,P

Lemma 2.5 If X is reﬂexive, then BQ,P which is embedded in X ∗∗ via the canonical embedding X (cid:44)→ X ∗∗ is a weakly* closed set in X ∗∗, and AQ,P is a weakly* closed set in X ∗.

(cid:1)∗

(cid:9).

Proof. It is well-known that any closed convex set in an arbitrary locally convex topological vector space is weakly closed. Since X ∗∗ = X, the weak* topology σ(X ∗∗, X ∗) on X ∗∗ coincides with the weak topology σ(X, X ∗) on X. Note that BQ,P is closed and convex. Hence, BQ,P is weakly closed in X, and thus BQ,P is weakly* closed in X ∗∗. By Lemma 2.4,

= (cid:8)x∗ ∈ X ∗| (cid:104)x∗, x(cid:105) ≤ 0, ∀x ∈ BQ,P

(cid:50) AQ,P = (cid:0)BQ,P It follows that AQ,P is weakly* closed.

The following lemma develops a result in [13, Theorem 3.4], where the

parameter b is ﬁxed.

(2.33) Lemma 2.6 Let (¯x, ¯b, ¯x∗) ∈ gphF, λ = (λi)i∈I ∈ Ξ(¯x, ¯b, ¯x∗) with I = I(¯x, ¯b). For K = {i ∈ I| λi > 0}, it holds (cid:0)T (¯x; Θ(¯b)) ∩ {¯x∗}⊥(cid:1)∗ × (cid:0)T (¯x; Θ(¯b)) ∩ {¯x∗}⊥(cid:1) = AI,K × BI,K.

Proof. Let us prove that

i , x(cid:105) ≤ 0 for i , x(cid:105) = 0 for all i ∈ K and

(2.34) T (¯x; Θ(¯b)) ∩ {¯x∗}⊥ = BI,K.

i , it holds

i∈K λia∗

(cid:88)

Given any x ∈ BI,K, by the construction of BI,K we have (cid:104)a∗ all i ∈ I. Hence, x ∈ T (¯x; Θ(¯b)). Since (cid:104)a∗ ¯x∗ = (cid:80)

i , x(cid:105) = 0.

i∈K

i∈I λia∗

(cid:104)¯x∗, x(cid:105) = λi(cid:104)a∗

i , x(cid:105).

i∈K

Therefore, x ∈ T (¯x; Θ(¯b))∩{¯x∗}⊥. So, BI,K ⊂ T (¯x; Θ(¯b))∩{¯x∗}⊥. Now, ﬁxing any x ∈ T (¯x; Θ(¯b)) ∩ {¯x∗}⊥, we have (cid:104)a∗ i , x(cid:105) ≤ 0 for all i ∈ I, and (cid:104)¯x∗, x(cid:105) = 0. Since (¯x, ¯b, ¯x∗) ∈ gphF and λ = (λi)i∈I ∈ Ξ(¯x, ¯b, ¯x∗) where I = I(¯x, ¯b), by (2.8) we get ¯x∗ = (cid:80) i . As K = {i ∈ I| λi > 0}, it follows from the last equality and the condition (cid:104)¯x∗, x(cid:105) = 0 that (cid:88) 0 = (cid:104)¯x∗, x(cid:105) = λi(cid:104)a∗

i , x(cid:105) ≤ 0 for all i ∈ I, we see that (cid:104)a∗

Since (cid:104)a∗ i , x(cid:105) = 0 for all i ∈ K. Thus x ∈ BI,K. We have shown that BI,K ⊃ T (¯x; Θ(¯b)) ∩ {¯x∗}⊥. Hence the equality (2.34) is valid.

(cid:1)∗

(cid:0)T (¯x; Θ(¯b)) ∩ {¯x∗}⊥(cid:1)∗

By Lemma 2.4 and by (2.34), we obtain

= (cid:0)BI,K = AI,K,

(cid:50) hence establishing (2.33).

Combining the formula for computing (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) in Theorem 2.4 with Lemma 2.6 we obtain the following statement which unlike Corollary 4.1 i | i ∈ I(¯x, ¯b)} are linearly in [38], does not require the assumption that {a∗ independent.

(cid:26)

(cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) =

(cid:12) (cid:12) (x∗, b∗, v) (cid:12) (x∗, v) ∈ AI,K × BI,K,

Lemma 2.7 Let (¯x, ¯b, ¯x∗), I = I(¯x, ¯b), λ = (λi)i∈I, and K be the same as in Lemma 2.6. Then

(cid:88)

(2.35)

(cid:27) ,

i , b∗

¯I = 0, b∗ I1

i∈I

≤ 0 x∗ = − b∗ i a∗

(cid:110)

where ¯I = T \I and I1 = I1(¯x, ¯b, ¯x∗).

(cid:111) ,

(cid:12) P (cid:54)= ∅, x∗ ∈ pos{a∗

i | i ∈ P }

For each (x, b, x∗) ∈ gphF, we put P ⊂ I(x, b)(cid:12) I(x, b, x∗) =

(cid:110) P ∈ I(x, b, x∗)(cid:12)

(cid:111) ,

(cid:12) a∗

i , i ∈ P, are linearly independent

J (x, b, x∗) =

and

 

(cid:98)I(x, b, x∗) =



J (x, b, x∗) if x∗ (cid:54)= 0

J (x, b, x∗) ∪ {∅} if x∗ = 0.

(cid:111) .

(cid:110) x ∈ X(cid:12)

FQ(b) =

(cid:12) (cid:104)a∗

i , x(cid:105) < bi ∀i ∈ T \Q

i , x(cid:105) = bi ∀i ∈ Q, (cid:104)a∗ Now, let (¯x, ¯b, ¯x∗) ∈ gphF, I = I(¯x, ¯b), J = I\I1(¯x, ¯b, ¯x∗), I = I(¯x, ¯b, ¯x∗), and (cid:98)I = (cid:98)I(¯x, ¯b, ¯x∗). Deﬁne

(cid:26)

(cid:91)

For every Q ⊂ T , we deﬁne a pseudo-face of Θ(b) by putting

(cid:12) (cid:12) (cid:12) (x∗, v) ∈ AQ,P × BQ,P , (x∗, b∗, v)

P ⊂Q⊂I, P ∈(cid:98)I

Σ(¯x, ¯b, ¯x∗) =

(2.36)

(cid:27) ,

i a∗ b∗

i , b∗

Q\P ≤ 0

Q = 0, b∗

i∈Q

x∗ = − (cid:80)

(cid:26)

(cid:91)

and

(cid:12) (cid:12) (cid:12) (x∗, v) ∈ AQ,P × BQ,P , (x∗, b∗, v)

P ⊂Q⊂I, P ∈I FQ(¯b)(cid:54)=∅

Σ0(¯x, ¯b, ¯x∗) =

(cid:27) .

(2.37)

i , b∗

i a∗ b∗

Q\J ≤ 0

Q = 0, b∗

i∈Q

x∗ = − (cid:80)

i | i ∈ I(¯x, ¯b)} are positively linearly independent. Theorem 2.6 For any (¯x, ¯b, ¯x∗) ∈ gphF, the estimates

We will show that the sets given by (2.36) and (2.37) are respectively an upper estimate and a lower estimate for the limiting normal cone N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1). Since the Fr´echet normal cone (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) is a subset of the cone N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1), it serves readily as a lower estimate of the latter. In the sequel, we will prove that Σ0(¯x, ¯b, ¯x∗) is not only a lower estimate for N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) but it is also an upper estimate for (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) in the case ¯x∗ (cid:54)= 0. Moreover, by constructing a suitable example, we will show that the inclusion (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) ⊂ Σ0(¯x, ¯b, ¯x∗) may be strict if ¯x∗ (cid:54)= 0. In comparison with Theorem 4.1 in [38], the estimates provided by the following theorem are tighter. Moreover, we do not assume that the vectors {a∗

Σ0(¯x, ¯b, ¯x∗) ⊂ N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) ⊂ Σ(¯x, ¯b, ¯x∗), (2.38) where Σ(¯x, ¯b, ¯x∗) and Σ0(¯x, ¯b, ¯x∗) are given respectively by (2.36) and (2.37), hold. Besides, if ¯x∗ (cid:54)= 0, then

(cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) ⊂ Σ0(¯x, ¯b, ¯x∗) ⊂ N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1).

(2.39)

Proof. Note that gphF is closed by Lemma 2.2 and X is reﬂexive. Hence, according to [28, Theorem 2.35],

(cid:98)N (cid:0)(xk, bk, x∗

k); gphF (cid:1).

gphF −→ (¯x,¯b,¯x∗)

(xk,bk,x∗ k)

k, η∗

k); gphF (cid:1)

k, vk) w∗ k, η∗ (u∗

Limsup N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) =

To obtain the second inclusion in (2.38), we ﬁx an arbitrary element (x∗, b∗, v) of N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1). We have to show that (x∗, b∗, v) ∈ Σ(¯x, ¯b, ¯x∗). k) gphF−→ By the deﬁnition of the limiting normal cone, one can ﬁnd (xk, bk, x∗ (¯x, ¯b, ¯x∗) and (u∗ → (x∗, b∗, v) such that k, vk) ∈ (cid:98)N (cid:0)(xk, bk, x∗ for all k ∈ IN . Since I(xk, bk) ⊂ T and (xk, bk) → (¯x, ¯b) for all k, we may suppose that I(xk, bk) = Q for all k, where Q ⊂ I(¯x, ¯b) is a ﬁxed index set.

k ∈ N (xk; Θ(bk)) = pos{a∗ x∗

i | i ∈ Q} for all k ∈ IN.

Then, by Proposition 2.1,

i | i ∈ (cid:101)P } are linearly independent and

k ∈ pos{a∗ x∗

i | i ∈ (cid:101)P } for all k ∈ IN.

Due to Lemma 2.1 and the Dirichlet principle, by considering a subsequence of {x∗ k} if necessary, we may assume that there exists a subset (cid:101)P ⊂ Q such that the vectors {a∗

(cid:88)

This means that

i a∗ λk

i ≥ 0, i ∈ (cid:101)P .

i∈ (cid:101)P

for some λk x∗ k =

Using again the Dirichlet principle, we can ﬁnd a subsequence {k(cid:96)} of {k} and a subset P ⊂ (cid:101)P such that

(2.40) {i ∈ (cid:101)P | λk(cid:96)

i > 0} = P for all (cid:96) ∈ IN. k(cid:96), vk(cid:96)) ∈ (cid:98)N (cid:0)(xk(cid:96), bk(cid:96), x∗

k(cid:96)

k(cid:96), η∗

); gphF (cid:1), by Lemma 2.7

(cid:88)

For each (cid:96) ∈ IN , since (u∗ we get

k(cid:96), vk(cid:96)) ∈ AQ,P × BQ,P , u∗ k(cid:96)

i , (η∗ k(cid:96)

k(cid:96) 1

i∈Q

= − (u∗ ≤ 0, (2.41) )ia∗ (η∗ k(cid:96) )Q = 0, (η∗ k(cid:96) )I

). If P = ∅, then x∗ k(cid:96)

= 0 for every (cid:96) ∈ IN . (In this )

(cid:88)

.) By the deﬁnition of I1(xk(cid:96), bk(cid:96), x∗ k(cid:96) ). If P (cid:54)= ∅, then x∗ k(cid:96) ) = Q, and thus Q\P = I1(xk(cid:96), bk(cid:96), x∗ k(cid:96) where I k(cid:96) := I1(xk(cid:96), bk(cid:96), x∗ 1 k(cid:96) case, we have ¯x∗ = 0 because ¯x∗ = lim (cid:96)→∞ we get I1(xk(cid:96), bk(cid:96), x∗ k(cid:96)

i∈P

= (2.42) for all (cid:96) ∈ IN. λk(cid:96) i a∗ x∗ k(cid:96)

Hence, from (2.40) and (2.42) we deduce that

) = I k(cid:96) 1 . Q\P = I(xk(cid:96), bk(cid:96))\P ⊂ I1(xk(cid:96), bk(cid:96), x∗ k(cid:96)

k(cid:96) 1

≤ 0 for every (cid:96) ∈ IN , we also have Since (η∗ k(cid:96) )I

)Q\P ≤ 0 for every (cid:96) ∈ IN. (η∗ k(cid:96)

k(cid:96), vk(cid:96)) w∗

Note that the sets AQ,P and BQ,P are weakly* closed by Lemma 2.5. By letting (cid:96) → ∞ and invoking the ﬁrst inclusion in (2.41), from the relation (u∗ → (x∗, v) we get

(2.43) (x∗, v) ∈ AQ,P × BQ,P .

i for

k(cid:96), η∗

k(cid:96), vk(cid:96)) w∗ )i → b∗ Taking into account that (u∗ each i ∈ T . Therefore, by letting (cid:96) → ∞ and by using (2.41) we obtain

(cid:88)

→ (x∗, b∗, v), we have (η∗ k(cid:96)

i a∗ b∗

i , b∗

Q\P ≤ 0.

Q = 0, b∗

i∈Q

(2.44) x∗ = −

By the choice of P and Q, it holds P ⊂ Q ⊂ I and P ∈ (cid:98)I. Hence, by (2.43), (2.44) and (2.36) we can assert that (x∗, b∗, v) ∈ Σ(¯x, ¯b, ¯x∗). This implies that N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) ⊂ Σ(¯x, ¯b, ¯x∗).

i a∗ b∗

i , b∗

Q\J ≤ 0.

Q = 0, b∗

i∈Q

Now, to verify the ﬁrst inclusion in (2.38), we take any (x∗, b∗, v) from Σ0(¯x, ¯b, ¯x∗). Choose P, Q with P ⊂ Q ⊂ I, P ∈ I, and FQ(¯b) (cid:54)= ∅ such that (cid:88) (2.45) (x∗, v) ∈ AQ,P × BQ,P , x∗ = −

Fix any (cid:101)x ∈ FQ(¯b) and consider the sequence

(cid:101)x + (1 − k−1)¯x.

xk = k−1

(cid:88)

It is clear that xk ∈ FQ(¯b) and xk → ¯x as k → ∞. For every k, by putting bk = ¯b we have I(xk, bk) = Q. Since P ∈ I, it follows that P (cid:54)= ∅ and

i∈P

¯x∗ = for some λi ≥ 0, i ∈ P. λia∗ i

(cid:88)

For each k, put

i ∈ N (xk; Θ(bk)) = F(xk, bk).

i∈P

k → ¯x∗ as k → ∞, we have (xk, bk, x∗

k) gphF−→ (¯x, ¯b, ¯x∗). From (2.46) we Since x∗ infer that Q\P ⊂ I1(xk, bk, x∗ k) ⊂ I(xk, bk) = Q for all k ∈ IN . By considering a subsequence of {k}, if necessary, we can assume that I1(xk, bk, x∗ k) = (cid:101)I1 for all k ∈ IN . The inclusion (cid:101)I1 ⊂ Q\J holds. To show this, ﬁx any i ∈ (cid:101)I1. For every k ∈ IN , there exist some µk

(cid:88)

j ≥ 0 for j ∈ Q\{i} satisfying j| j ∈ Q\{i}(cid:9), j ∈ pos(cid:8)a∗

(2.46) (λi + k−1)a∗ x∗ k =

j a∗ µk

j∈Q

x∗ k =

j| j ∈ K} for all (cid:96) ∈ IN . Since {a∗

∈ pos{a∗ := 0. By Lemma 2.1, one can ﬁnd a subset K ⊂ Q\{i} and a j| j ∈ K} are linearly independent j| j ∈ K} are linearly

→ ¯x∗, we obtain where µk i subsequence {k(cid:96)} of {k} such that {a∗ and x∗ k(cid:96) independent and x∗ k(cid:96)

j| j ∈ K}.

(2.47) ¯x∗ ∈ pos{a∗

(cid:88)

Since i (cid:54)∈ K, by the deﬁnition of I1(¯x, ¯b, ¯x∗) and (2.47) we have i ∈ I1(¯x, ¯b, ¯x∗). As J = I\I1(¯x, ¯b, ¯x∗), it follows that i ∈ Q\J. Therefore, the inclusion (cid:101)I1 ⊂ Q\J is valid. Since we have I1(xk, bk, x∗ k) = (cid:101)I1 ⊂ Q\J for each k ∈ IN , from (2.45) it follows that

i a∗ b∗

i , b∗

Q = 0, b∗ (cid:101)I1

i∈Q

≤ 0. (x∗, v) ∈ AQ,P × BQ,P , x∗ = −

Hence, by Lemma 2.7 we deduce that (x∗, b∗, v) ∈ (cid:98)N (cid:0)(xk, bk, x∗ k); gphF (cid:1) for k) gphF−→ (¯x, ¯b, ¯x∗) yields all k ∈ IN . Combining this with the fact that (xk, bk, x∗ (x∗, b∗, v) ∈ N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1). Since (x∗, b∗, v) can be chosen arbitrary in Σ0(¯x, ¯b, ¯x∗), we obtain Σ0(¯x, ¯b, ¯x∗) ⊂ N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1).

(cid:26)

(cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) =

(cid:12) (cid:12) (x∗, b∗, v) (cid:12) (x∗, v) ∈ AQ,P × BQ,P ,

(cid:27)

Finally, let us prove the ﬁrst inclusion in (2.39) under the assumption that ¯x∗ (cid:54)= 0. Fix any λ = (λi)i∈I ∈ Ξ(¯x, ¯b, ¯x∗), where Ξ(¯x, ¯b, ¯x∗) is given by (2.8). Choosing Q = I and P = {i ∈ I| λi > 0}, we get I1 = I\J = Q\J. According to Lemma 2.7,

i a∗

i , b∗

i∈Q b∗

Q = 0, b∗ I1

≤ 0 x∗ = − (cid:80)

(cid:26)

(2.48)

(cid:12) (cid:12) (x∗, b∗, v) (cid:12) (x∗, v) ∈ AQ,P × BQ,P ,

(cid:27) .

i a∗

i , b∗

i∈Q b∗

Q\J ≤ 0

Q = 0, b∗

x∗ = − (cid:80)

i , (cid:98)x(cid:105) = 0 for i ∈ Q and (cid:104)a∗

Since ¯x∗ (cid:54)= 0, we infer that P (cid:54)= ∅. By our choice of P and Q, it holds In addition, we have FQ(¯b) (cid:54)= ∅ because ¯x ∈ FQ(¯b). P ⊂ Q ⊂ I, P ∈ I. Hence, (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) ⊂ Σ0(¯x, ¯b, ¯x∗). The proof is complete. (cid:50)

Remark 2.3 One necessary condition for obtaining precise formulas for the limiting normal cone N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) is that FQ(¯b) (cid:54)= ∅ for Q ⊂ I = I(¯x, ¯b) (see e.g. [13, Theorem 4.2] and [32, Theorem 4.1]). The linear independence i | i ∈ I} ensures that FQ(¯b) (cid:54)= ∅ for every Q ⊂ I. Indeed, assumption of {a∗ since {a∗ i | i ∈ I} are linearly independent, by [13, Theorem 4.2] there exists (cid:98)x ∈ X such that (cid:104)a∗ i , (cid:98)x(cid:105) < 0 for i ∈ T \ Q. This implies that (cid:98)x + ¯x ∈ FQ(¯b). We have seen that there is no regularity assumption for {a∗ i | i ∈ I} in Theorem 4.1. Hence in that theorem only upper and lower estimates can be obtained for N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1).

The next example is designed to show how Theorem 2.6 can be used for getting an upper estimate and a lower estimate of the limiting normal cone N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1). In this example, we will see that the ﬁrst inclusion of (2.39) is strict in general. This means that the lower estimate Σ0(¯x, ¯b, ¯x∗) ⊂ N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) provided by Theorem 2.6 for N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) is better than the natural estimate (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) ⊂ N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1).

Example 2.3 Let X, T , and {a∗ i | i ∈ T } be the same as in Example 2.1. The calculations already done in Example 2.1 assure that (¯x, ¯b, ¯x∗) ∈ gphF, where

¯b = (0, 0, 0) ∈ IR3, ¯x = (0, 0) ∈ X, and ¯x∗ = (0, α) for α > 0.

Besides, I = I(¯x, ¯b) = {1, 2, 3}, ¯I = T \I = ∅, I1 = I1(¯x, ¯b, ¯x∗) = {1, 3}, and J = I\I1 = {2}.

i , λi ≥ 0 for i ∈ I, is

i∈I λia∗

Note that the unique way to represent ¯x∗ = (cid:80)

that

1 + αa∗

2 + 0a∗ 3.

¯x∗ = (0, α) = 0(1, 0) + α(0, 1) + 0(1, 2) = 0a∗

(cid:111)

So, we have

(cid:110) P ⊂ I(cid:12)

(cid:12) P (cid:54)= ∅, ¯x∗ ∈ pos{a∗

(cid:12) 2 ∈ P

i | i ∈ P }

= I = I(¯x, ¯b, ¯x∗) = .

Hence, P and Q satisfy the conditions P ⊂ Q ⊂ I and P ∈ I if and only if one of the following cases occurs:

(a) Q = {1, 2, 3} and P = {1, 2, 3}, (0, 0) ∈ FQ(¯b) (cid:54)= ∅;

(b) Q = {1, 2, 3} and P = {1, 2}, (0, 0) ∈ FQ(¯b) (cid:54)= ∅;

(d) Q = {1, 2, 3} and P = {2}, (0, 0) ∈ FQ(¯b) (cid:54)= ∅;

FQ(¯b) = ∅;

(e) Q = {1, 2} and P = {1, 2},

FQ(¯b) = ∅;

(f ) Q = {1, 2} and P = {2},

FQ(¯b) = ∅;

(g) Q = {2, 3} and P = {2, 3},

FQ(¯b) = ∅;

(h) Q = {2, 3} and P = {2},

(i) Q = {2} and P = {2}, (−1, 0) ∈ FQ(¯b) (cid:54)= ∅.

(cid:88)

(cid:111) ,

(cid:12) (x∗, v) ∈ AQ,P × BQ,P , x∗ = −

i a∗ b∗

i , b∗

Q\P ≤ 0

Q = 0, b∗

i∈Q

(cid:110)

(cid:88)

For each t ∈ Γ := {a, b, c, d, e, f, g, h, i}, deﬁne P, Q as in the case (t) above and put (cid:110) (x∗, b∗, v)(cid:12) Λ(t) =

(cid:111) ,

(cid:98)Λ(t) =

(cid:12) (x∗, v) ∈ AQ,P × BQ,P , x∗ = −

Q\J ≤ 0

i a∗ b∗

i , b∗

Q = 0, b∗

i∈Q

(cid:98)Λ(t)

 

(x∗, b∗, v)(cid:12)

(t) =

Λ0 ∅  if FQ(¯b) (cid:54)= ∅ if FQ(¯b) = ∅.

i | i ∈ P } are linearly independent if and only if P is given as in the case (t) with t ∈ Γ\{a}. Hence

(cid:91)

Since ¯x∗ = (0, α) (cid:54)= 0IR2, we have ∅ (cid:54)∈ (cid:98)I(¯x, ¯b, ¯x∗). We see that {a∗

(t).

t∈Γ

t∈Γ\{a}

(t) can be computed through various realizations of the

Λ0 (2.49) Σ(¯x, ¯b, ¯x∗) = Λ(t) and Σ0(¯x, ¯b, ¯x∗) =

The sets Λ(t) and Λ0 inclusion t ∈ Γ as follows.

1. If t = a, then Q = {1, 2, 3} and P = {1, 2, 3}. We have

i | i ∈ P } = IR2, BQ,P = (AQ,P )∗ = {0IR2}.

i a∗

i and

i∈Q b∗

AQ,P = span{a∗

3) and b∗

3 = (−b∗

2 − 2b∗

2 − b∗

1 − b∗

3, −b∗

1a∗

2a∗

3a∗

1 ≤ 0, b∗

3 ≤ 0.

x∗ = −b∗ Note that FQ(¯b) (cid:54)= ∅. Let x∗ ∈ AQ,P be such that x∗ = − (cid:80) b∗ Q\J ≤ 0. Then, 1 − b∗

(cid:26)(cid:16)

Therefore,

(cid:27) .

(a) =

(cid:17)(cid:12) (cid:12) (cid:12) β1, β3 ∈ IR−

Λ0 (−β1 − β3, −β2 − 2β3), (β1, β2, β3), (0, 0)

2. If t = b, then Q = {1, 2, 3} and P = {1, 2}. We have

1, a∗

2} + pos{a∗

3} = IR2, BQ,P = (AQ,P )∗ = {0IR2}.

AQ,P = span{a∗

Let x∗ ∈ AQ,P be such that x∗ = − (cid:80)

1a∗

1 − b∗

2a∗

2 − b∗

3a∗

3 = (−b∗

i∈Q b∗ 1 − b∗

i a∗ 3, −b∗

i and b∗ 2 − 2b∗

Q\P ≤ 0. Then, 3) and b∗

3 ≤ 0.

x∗ = −b∗

(cid:26)(cid:16)

(cid:27) .

Therefore,

(cid:17)(cid:12) (cid:12) (cid:12) β3 ∈ IR−

Λ(b) = (−β1 − β3, −β2 − 2β3), (β1, β2, β3), (0, 0)

i a∗

i and

i∈Q b∗

Note that FQ(¯b) (cid:54)= ∅. Let x∗ ∈ AQ,P be such that x∗ = − (cid:80) b∗ Q\J ≤ 0. Then,

1 − b∗

3, −b∗

2 − 2b∗

3) and b∗

1 ≤ 0, b∗

3 ≤ 0.

x∗ = (−b∗

(cid:26)(cid:16)

Therefore,

(cid:27) .

(b) =

(cid:17)(cid:12) (cid:12) (cid:12) β1, β3 ∈ IR−

Λ0 (−β1 − β3, −β2 − 2β3), (β1, β2, β3), (0, 0)

Treating the cases t = c, . . . , t = h in a similar manner, we obtain the

following results.

(cid:27)

(cid:26)(cid:16)

3. For t = c,

(cid:17)(cid:12) (cid:12) (cid:12) β1 ∈ IR−

(cid:27)

(cid:26)(cid:16)

, Λ(c) = (−β1 − β3, −β2 − 2β3), (β1, β2, β3), (0, 0)

(cid:17)(cid:12) (cid:12) (cid:12) β1, β3 ∈ IR−

Λ0 . (−β1 − β3, −β2 − 2β3), (β1, β2, β3), (0, 0)

(cid:27)

(cid:26)(cid:16)

4. For t = d,

(cid:17)(cid:12) (cid:12) (cid:12) β1, β3, γ ∈ IR−

, Λ(d) = (−β1 − β3, −β2 − 2β3), (β1, β2, β3), (γ, 0)

(d) = Λ(d).

Λ0

(cid:26)(cid:16)

(cid:27) ,

5. For t = e,

(cid:17)(cid:12) (cid:12) (cid:12) β1, β2 ∈ IR

Λ(e) = (−β1, −β2), (β1, β2, 0), (0, 0)

(e) = ∅.

Λ0

(cid:27)

(cid:26)(cid:16)

6. For t = f ,

(cid:17)(cid:12) (cid:12) (cid:12) β1, γ ∈ IR−

, Λ(f ) = (−β1, −β2), (β1, β2, 0), (γ, 0)

(f ) = ∅.

Λ0

(cid:26)(cid:16)

(cid:27) ,

7. For t = g,

(cid:17)(cid:12) (cid:12) (cid:12) β2, β3 ∈ IR

Λ(g) = (−β3, −β2 − 2β3), (0, β2, β3), (0, 0)

(g) = ∅.

Λ0

(cid:26)(cid:16)

(cid:27) ,

8. For t = h,

(cid:17)(cid:12) (cid:12) (cid:12) β3, γ ∈ IR−

Λ(h) = (−β3, −β2 − 2β3), (0, β2, β3), (γ, 0)

(h) = ∅.

Λ0

The case t = i deserves a special treatment. Detailed arguments are given

below.

9. If t = i, then Q = {2} and P = {2}. We have

2} = {0} × IR, BQ,P = (AQ,P )∗ = IR × {0}.

i , b∗

i a∗

i∈Q b∗

Q\P ≤ 0.

Q = 0, and b∗

AQ,P = span{a∗

Let x∗ ∈ AQ,P be such that x∗ = − (cid:80) Then,

2a∗

2 = (0, −b∗

2),

1 = b∗ b∗

3 = 0.

x∗ = −b∗

(cid:27)

(cid:26)(cid:16)

Hence, b∗ ∈ {0} × IR × {0} and x∗ ∈ {0} × IR. Therefore,

(cid:17)(cid:12) (cid:12) (cid:12) β2, γ ∈ IR

. Λ(i) = (0, −β2), (0, β2, 0), (γ, 0)

(cid:26)(cid:16)

We have FQ(¯b) (cid:54)= ∅ and Q\P = Q\J, thus

(cid:27) .

(i) = Λ(i) =

(cid:17)(cid:12) (cid:12) (cid:12) β2, γ ∈ IR

Λ0 (0, −β2), (0, β2, 0), (γ, 0)

(cid:26)(cid:16)

(i) =

(cid:17)(cid:12) (cid:12) (cid:12) β2, γ ∈ IR

On the basis of the above listed results and (2.49), we obtain the exact formulas for Σ(¯x, ¯b, ¯x∗) and Σ0(¯x, ¯b, ¯x∗). Since ¯x∗ (cid:54)= 0, by (2.39) we have (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) ⊂ Σ0(¯x, ¯b, ¯x∗). According to (2.49), (cid:27) Λ0 ⊂ Σ0(¯x, ¯b, ¯x∗). (0, −β2), (0, β2, 0), (γ, 0)

(cid:27)

(cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) (cid:26)(cid:16)

As shown in Example 2.1, we have

(cid:17)(cid:12) (cid:12) (cid:12) β1, β3, γ ∈ IR−

= . (−β1 − β3, −β2 − 2β3), (β1, β2, β3), (γ, 0)

(i) (cid:54)⊂ (cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1), we deduce that

(cid:98)N (cid:0)(¯x, ¯b, ¯x∗); gphF (cid:1) (cid:36) Σ0(¯x, ¯b, ¯x∗).

Since Λ0

(cid:111)

(cid:12) (u∗, η∗, −v) ∈ Σ(x, b, x∗)

From Theorem 2.6 we obtain easily some upper and lower estimates for values of the Mordukhovich coderivative D∗F(x, b, x∗) : X ∗∗ ⇒ X ∗ × IRm of F at a point (x, b, x∗) ∈ gphF. Namely, putting (cid:110) (u∗, η∗) ∈ X ∗ × IRm(cid:12) Ω(x, b, x∗)(v) =

(cid:110) (u∗, η∗) ∈ X ∗ × IRm(cid:12)

(cid:111) ,

and

(cid:12) (u∗, η∗, −v) ∈ Σ0(x, b, x∗)

Ω0(x, b, x∗)(v) =

we have

for all v ∈ X ∗∗. Ω0(x, b, x∗)(v) ⊂ D∗F(x, b, x∗)(v) ⊂ Ω(x, b, x∗)(v)

The obtained generalized diﬀerentiation properties of the mapping F(x, b) will be applied to establish the conditions for solution stability of parametric aﬃne variational inequalities (AVIs, for short) under linear perturbations in the next section.

2.4 AVIs under Linear Perturbations

In this section we investigate solution stability of variational inequalities with polyhedral convex constraint sets under linear perturbations. We establish necessary and suﬃcient conditions for the local Lipschitz-like and the metric regularity properties of the solution maps of such variational inequalities in ﬁnite dimensional spaces.

(2.50) Consider the implicit multifunction S : IRm × IRn ⇒ IRn deﬁned by S(b, q) = (cid:8)x ∈ IRn| q ∈ M x + F(x, b)(cid:9)

with M ∈ IRn×n, (b, q) ∈ IRm × IRn, and F(x, b) = N (x; Θ(b)) being given by (2.4), where X = IRn. For any pair (b(cid:48), q(cid:48)) ∈ IRm × IRn, it holds

D∗S−1(¯x, ¯b, ¯q)(b(cid:48), q(cid:48))

= (cid:8)x(cid:48) ∈ IRn| (x(cid:48), −b(cid:48), −q(cid:48)) ∈ N ((¯x, ¯b, ¯q); gphS−1)(cid:9) = (cid:8)x(cid:48) ∈ IRn| (−b(cid:48), −q(cid:48), x(cid:48)) ∈ N ((¯b, ¯q, ¯x); gphS)(cid:9) = (cid:8)x(cid:48) ∈ IRn| (−b(cid:48), −q(cid:48)) ∈ D∗S(¯b, ¯q, ¯x)(−x(cid:48))(cid:9).

Hence, we have

(2.51) D∗S−1(¯x, ¯b, ¯q)(0IRm+n) = (cid:8)x(cid:48) ∈ IRn| 0IRm+n ∈ D∗S(¯b, ¯q, ¯x)(−x(cid:48))(cid:9).

Remark 2.4 Setting w = −q and f (x, w) = M x − q, we see that S(b, q) coincides with the solution map of the parametric variational inequality

Find x ∈ Θ(b) subject to (cid:104)f (x, w), u − x(cid:105) ≥ 0, ∀u ∈ Θ(b).

Note that the inclusion q ∈ M x + F(x, b) in (2.50) is a special form of the following generalized equation

0 ∈ f (x, w) + N (x; Θ(b)).

Here, f (x, w) = M x − q is an aﬃne operator.

Remark 2.5 We can represent S(b, q) from (2.50) as the solution set of a standard parametric aﬃne variational inequality as follows

S(b, q) = (cid:8)x ∈ Θ(b)| (cid:104)M x − q, y − x(cid:105) ≥ 0, ∀y ∈ Θ(b)(cid:9).

Lemma 2.8 Let (cid:101)F(x, b, q) = M x − q + F(x, b). Then, the graph of (cid:101)F(·) is closed in the product space IRn × IRm × IRn × IRn.

k)}k∈IN is an arbitrary sequence in gph (cid:101)F k) → (¯x, ¯b, ¯q, ¯x∗). We have to show that (¯x, ¯b, ¯q, ¯x∗) ∈ gph (cid:101)F. k) ∈ gph (cid:101)F, it holds

Proof. Suppose that {(xk, bk, qk, x∗ and (xk, bk, qk, x∗ For every k ∈ IN , since (xk, bk, qk, x∗

k := x∗ z∗

k → ¯z∗, so z∗

k − M xk + qk ∈ F(xk, bk). k → ¯z∗ := ¯x∗ − M ¯x + ¯q. If ¯z∗ = 0, then ¯z∗ ∈ N (¯x, Θ(¯b)) = F(¯x, ¯b). We have z∗ It follows that ¯x∗ ∈ M ¯x− ¯q +F(¯x, ¯b) = (cid:101)F(¯x, ¯b, ¯q). Hence, (¯x, ¯b, ¯q, ¯x∗) ∈ gph (cid:101)F. We now consider the case that ¯z∗ (cid:54)= 0. We have ¯z∗ (cid:54)= 0 and z∗ k (cid:54)= 0 for all k large enough. Since (xk, bk) → (¯x, ¯b), one has I(xk, bk) ⊂ I(¯x, ¯b) for suﬃciently large indexes k ∈ IN . Thus, without loss of generality we may k (cid:54)= 0 and I(xk, bk) = (cid:101)I ⊂ I(¯x, ¯b) for every k ∈ IN . For each assume that z∗ k ∈ IN , from condition (2.52) it follows that

k ∈ pos{a∗ z∗

i | i ∈ I(xk, bk)} = pos{a∗

i | i ∈ (cid:101)I}.

(2.52)

i | i ∈ (cid:101)I} is closed and z∗

k → ¯z∗, it holds

As pos{a∗

i | i ∈ (cid:101)I} ⊂ pos{a∗

i | i ∈ I(¯x, ¯b)} = F(¯x, ¯b).

¯z∗ ∈ pos{a∗

Hence, ¯x∗ ∈ M ¯x − ¯q + F(¯x, ¯b) = (cid:101)F(¯x, ¯b, ¯q). Consequently, (¯x, ¯b, ¯q, ¯x∗) ∈ gph (cid:101)F (cid:50) which completes the proof.

Let any (¯b, ¯q, ¯x) ∈ gphS. Clearly, (¯x, ¯b, ¯x∗) ∈ gphF with ¯x∗ := ¯q − M ¯x.

(cid:26)

(cid:91)

(cid:98)KM,¯q(x(cid:48)) =

For every x(cid:48) ∈ IRn, we deﬁne the following sets

v(cid:48)∈IRn

(cid:27) ,

(b(cid:48), q(cid:48)) ∈ IRm+n(cid:12) (cid:12) (cid:12) (−x(cid:48), b(cid:48), q(cid:48)) ∈ M (cid:62)v(cid:48) × {0IRm+n}

(cid:26)

(cid:91)

−{0IRn+m} × {v(cid:48)} + (cid:98)D∗F(¯x, ¯b, ¯x∗)(v(cid:48)) × {0IRn}

v(cid:48)∈IRn

(cid:27) ,

KM,¯q(x(cid:48)) = (b(cid:48), q(cid:48)) ∈ IRm+n(cid:12) (cid:12) (cid:12) (−x(cid:48), b(cid:48), q(cid:48)) ∈ M (cid:62)v(cid:48) × {0IRm+n}

−{0IRn+m} × {v(cid:48)} + D∗F(¯x, ¯b, ¯x∗)(v(cid:48)) × {0IRn}

(cid:91)

and

(cid:26) (b(cid:48), q(cid:48)) ∈ IRm+n(cid:12) (cid:12) (cid:12) (−x(cid:48), b(cid:48), q(cid:48)) ∈ M (cid:62)v(cid:48) × {0IRm+n}

v(cid:48)∈IRn

(cid:27) .

LM,¯q(x(cid:48)) =

−{0IRn+m} × {v(cid:48)} + Ω(¯x, ¯b, ¯x∗)(v(cid:48)) × {0IRn}

Remark 2.6 We have D∗F(¯x, ¯b, ¯x∗)(v) ⊂ Ω(¯x, ¯b, ¯x∗)(v) for all v ∈ IRn, thus the inclusion KM,¯q(x(cid:48)) ⊂ LM,¯q(x(cid:48)) holds for every x(cid:48) ∈ IRn.

(cid:98)KM,¯q(x(cid:48)) ⊂ (cid:98)D∗S(¯b, ¯q, ¯x)(x(cid:48)) ⊂ D∗S(¯b, ¯q, ¯x)(x(cid:48)) ⊂ KM,¯q(x(cid:48)) ⊂ LM,¯q(x(cid:48))

Theorem 2.7 The following estimates

(cid:98)KM,¯q(x(cid:48)) = (cid:98)D∗S(¯b, ¯q, ¯x)(x(cid:48)) = D∗S(¯b, ¯q, ¯x)(x(cid:48)) = KM,¯q(x(cid:48)) ⊂ LM,¯q(x(cid:48))

hold for all x(cid:48) ∈ IRn. Moreover, if F(·) is graphically regular at (¯x, ¯b, ¯x∗), then

for every x(cid:48) ∈ IRn.

(cid:98)D∗ (cid:101)F(¯z)(v(cid:48)) = M (cid:62)v(cid:48) × {0IRm+n} − {0IRn+m} × {v(cid:48)} + (cid:98)D∗F(¯x, ¯b, ¯x∗)(v(cid:48)) × {0IRn}, D∗ (cid:101)F(¯z)(v(cid:48)) = M (cid:62)v(cid:48) × {0IRm+n} − {0IRn+m} × {v(cid:48)} + D∗F(¯x, ¯b, ¯x∗)(v(cid:48)) × {0IRn}, for every v(cid:48) ∈ IRn. Note that the graph of (cid:101)F(·) is closed by Lemma 2.8. In addition, it is obvious that kerD∗ (cid:101)F(¯z) = {0}. Applying Theorem 3.1 in [22] to the implicit multifunction

Proof. Let (cid:101)F(x, b, q) = M x − q + F(x, b) and let ¯z = (¯x, ¯b, ¯q, 0IRn). Since (¯b, ¯q, ¯x) ∈ gphS, it holds ¯q ∈ M ¯x + F(¯x, ¯b). Therefore, ¯z ∈ gph (cid:101)F. Using the coderivative sum rules [28, Theorem 1.62], we obtain

S(b, q) = (cid:8)x ∈ IRn| 0 ∈ (cid:101)F(x, b, q)(cid:9) 39

(cid:50) and using Remark 2.6, we get all the assertions of the theorem.

Following Deﬁnition 1.7, the map S(·) is locally metrically regular around (¯b, ¯q, ¯x) ∈ gphS if there are µ > 0, γ > 0, and neighborhoods U of ¯x, V of (¯b, ¯q) such that

(2.53) dist(cid:0)(b, q); S−1(x)(cid:1) ≤ µ dist(cid:0)x; S(b, q)(cid:1) for all x ∈ U and (b, q) ∈ V satisfying dist(cid:0)x; S(b, q)(cid:1) ≤ γ. Suppose that S(·) is locally closed around (¯b, ¯q, ¯x). Then, by Theorem 1.4, S(·) is locally metrically regular around (¯b, ¯q, ¯x) if and only if D∗S−1(¯x, ¯b, ¯q)(0) = {0}. By virtue of (2.51), the last equality is equivalent to kerD∗S(¯b, ¯q, ¯x) = {0}.

Theorem 2.8 The following assertions hold

(i) If S(·) is locally metrically regular around (¯b, ¯q, ¯x), then ker (cid:98)KM,¯q = {0}.

(cid:20) (−x(cid:48), 0, 0) ∈ M (cid:62)v(cid:48)× {0IRm+n} − {0IRn+m} × {v(cid:48)}

(cid:21)

(cid:104)

The last equality means that

(cid:105) x(cid:48) = 0

=⇒ . + (cid:98)D∗F(¯x, ¯b, ¯x∗)(v(cid:48)) × {0IRn}

(cid:20) (−x(cid:48), 0, 0) ∈ M (cid:62)v(cid:48)× {0IRm+n} − {0IRn+m} × {v(cid:48)}

(cid:21)

(cid:104)

(ii) If kerLM,¯q = {0}, or equivalently as

(cid:105) x(cid:48) = 0

=⇒ , +D∗F(¯x, ¯b, ¯x∗)(v(cid:48)) × {0IRn}

then S(·) is locally metrically regular around (¯b, ¯q, ¯x).

(iii) If F(·) is graphically regular at (¯x, ¯b, ¯x∗), then S(·) is locally metrically

regular around (¯b, ¯q, ¯x) if and only if ker (cid:98)KM,¯q = {0}.

(cid:12) (cid:107)x − ¯x(cid:107) + (cid:107)b − ¯b(cid:107) + (cid:107)q − ¯q(cid:107) + (cid:107)v(cid:107) ≤ η(cid:9)

(cid:12) (cid:107)x − ¯x(cid:107) + (cid:107)b − ¯b(cid:107) + (cid:107)q − ¯q(cid:107) ≤ η(cid:9) is also

Proof. (i) According to Lemma 2.8, the graph of (cid:101)F(·) is closed. Hence, there exists η > 0 such that

gph (cid:101)F ∩ (cid:8)(x, b, q, v)(cid:12) is closed. Thus, gph (cid:101)F ∩ (cid:8)(x, b, q, 0)(cid:12) closed. This implies that

gphS ∩ (cid:8)(b, q, x)| (cid:107)b − ¯b(cid:107) + (cid:107)q − ¯q(cid:107) + (cid:107)x − ¯x(cid:107) ≤ η(cid:9)

is closed. This meas that S(·) is locally closed around (¯b, ¯q, ¯x). Since S(·) is locally metrically regular around (¯b, ¯q, ¯x), the equality (2.53) holds. By Theorem 2.7, we deduce that ker (cid:98)KM,¯q = {0}.

(ii) From the assumption kerLM,¯q = {0} and the estimates in Theorem 2.7 we obtain kerD∗S(¯b, ¯q, ¯x) = {0}. This shows that S(·) is locally metrically regular around (¯b, ¯q, ¯x).

(iii) Since F(·) is graphically regular at (¯x, ¯b, ¯x∗), by Theorem 2.7 we get

(cid:98)KM,¯q(x(cid:48)) = (cid:98)D∗S(¯b, ¯q, ¯x)(x(cid:48)) = D∗S(¯b, ¯q, ¯x)(x(cid:48)) = KM,¯q(x(cid:48)) for every x(cid:48) ∈ IRn. These equalities implies that ker (cid:98)KM,¯q = kerD∗S(¯b, ¯q, ¯x). Consequently, S(·) is locally metrically regular around (¯b, ¯q, ¯x) if and only if ker (cid:98)KM,¯q = kerD∗S(¯b, ¯q, ¯x) = {0}. (cid:50)

the following equalities

According to Deﬁnition 1.6, the map S(·) is locally Lipschitz-like around (¯b, ¯q, ¯x) ∈ gphS if there exist (cid:96) > 0, and some neighborhoods U of ¯x and V of (¯b, ¯q) such that

(2.54) S(b, q) ∩ U ⊂ S(b(cid:48), q(cid:48)) + (cid:96)(cid:107)(b, q) − (b(cid:48), q(cid:48))(cid:107) ¯BIRn holds for all (b, q), (b(cid:48), q(cid:48)) ∈ V , where ¯BIRn denotes the unit closed ball in IRn. If S(·) is locally closed around (¯b, ¯q, ¯x), then Theorem 1.3 asserts that S(·) is locally Lipschitz-like around (¯b, ¯q, ¯x) if and only if D∗S(¯b, ¯q, ¯x)(0) = {0}.

Theorem 2.9 The following assertions are valid (i) If S(·) is locally Lipschitz-like around (¯b, ¯q, ¯x), then (cid:98)KM,¯q(0) = {0}. (ii) If LM,¯q(0) = {0}, then S(·) is locally Lipschitz-like around (¯b, ¯q, ¯x). (iii) If F(·) is graphically regular at (¯x, ¯b, ¯x∗), then S(·) is locally Lipschitz-like

around (¯b, ¯q, ¯x) if and only if (cid:98)KM,¯q(0) = {0}.

Proof. (i) Applying the argument of the proof of Theorem 2.8, we infer that S(·) is locally closed around (¯b, ¯q, ¯x). Since S(·) is locally Lipschitz-like around (¯b, ¯q, ¯x), we have D∗S(¯b, ¯q, ¯x)(0) = {0}. By Theorem 2.7, it follows that (cid:98)KM,¯q(0) = {0}.

(ii) Using the estimates in Theorem 2.7, from the equality LM,¯q(0) = {0} we deduce that D∗S(¯b, ¯q, ¯x)(0) = {0}. The last equality means that S(·) is locally Lipschitz-like around (¯b, ¯q, ¯x).

(cid:98)KM,¯q(x(cid:48)) = (cid:98)D∗S(¯b, ¯q, ¯x)(x(cid:48)) = D∗S(¯b, ¯q, ¯x)(x(cid:48)) = KM,¯q(x(cid:48))

(iii) If F(·) is graphically regular at (¯x, ¯b, ¯x∗), Theorem 2.7 asserts that

for every x(cid:48) ∈ IRn. Therefore, S(·) is locally Lipschitz-like around (¯b, ¯q, ¯x) if and only if (cid:98)KM,¯q(0) = D∗S(¯b, ¯q, ¯x)(0) = {0}. (cid:50)

2.5 Conclusions

Theorem 2.4 in this chapter gives an exact formula for the Fr´echet normal cone to the graph of the normal cone mappings F(x, b) in reﬂexive Banach spaces, and Theorem 2.6 establishes upper and lower estimates for the lim- iting normal cone to the graph of F(x, b). Based on the results, an exact formula for the Fr´echet coderivative and upper and lower estimates for the values of the Mordukhovich coderivative of F(x, b) are derived. These formu- las yield the necessary conditions and the suﬃcient conditions for the local metric regularity in Theorem 2.8, as well as the necessary conditions and the suﬃcient conditions in Theorem 2.9 for the locally Lipschitz-like property of the solution maps of aﬃne variational inequalities under linear perturba- tions. Proposition 2.2 and Proposition 2.3 respectively answer the ﬁrst and the second open questions raised by Yao and Yen in [52].

Chapter 3

Nonlinear Perturbations of Polyhedral

Normal Cone Mappings

As a continuation of the study of generalized diﬀerentiation of the normal cone mappings presented in the previous chapter, this chapter is devoted to the estimation of the Fr´echet and the limiting normal cones to the graphs of the normal cone mappings to nonlinearly perturbed polyhedral convex sets in ﬁnite dimensional spaces. The obtained estimates are applied to solution stability of aﬃne variational inequalities under nonlinear perturbations.

The presentation given below comes from the results in [41].

3.1 The Normal Cone Mapping F(x, A, b)

Let T = {1, 2, . . . , m} be a given index set. For each pair (A, b) ∈ IRm×n×IRm, we consider the perturbed polyhedral convex set

1 , . . . , A(cid:62)

(3.1)

i , x(cid:105) ≤ bi,

(3.2) Θ(A, b) = (cid:8)x ∈ IRn| Ax ≤ b(cid:9), where A = (aij)m×n ∈ IRm×n and b = (b1, . . . , bm) ∈ IRm are parameters. We interpret A(cid:62) m, where Ai = (ai1 . . . ain) is the i-th row of the matrix A and superscript (cid:62) denotes transposition, as nonlinear perturbations and b1, . . . , bm as the right-hand side perturbations of the system (cid:104)A(cid:62) i ∈ T.

(cid:9)

For every (x, A, b) ∈ IRn × IRm×n × IRm with A = (aij)m×n, b = (b1, . . . , bm), and x ∈ Θ(A, b), let

(3.3)

I(x, A, b) = (cid:8)i ∈ T | Aix = bi 43





denote the active index set corresponding to the triplet (x, A, b). For any subset Γ ⊂ T , we put Γ := T \Γ. The equality AΓ = 0 means that Ai = 0 for all i ∈ Γ. If Γ = {i1, . . . , ir}, then by aΓ,j we denote the column vector

  for any j ∈ {1, . . . , n}.

 

aΓ,j =

ai1j ... airj

The notation aΓ,j ≤ 0 (resp., aΓ,j ≥ 0, aΓ,j = 0) means that aij ≤ 0 (resp., aij ≥ 0, aij = 0) for all i ∈ Γ. The notation bΓ ≤ 0 (resp., bΓ ≥ 0, bΓ = 0) is used whenever bi ≤ 0 (resp., bi ≥ 0, bi = 0) for all i ∈ Γ. As usual, the norm of a matrix A ∈ IRm×n is deﬁned by

(cid:12) x ∈ IRn, (cid:107)x(cid:107) = 1(cid:9),

(cid:107)A(cid:107) = max (cid:8)(cid:107)Ax(cid:107) (cid:12)

m (cid:88)

n (cid:88)

where (cid:107)x(cid:107) and (cid:107)Ax(cid:107) denote, respectively, the Euclidean norms of x ∈ IRn and Ax ∈ IRm. The scalar product of two matrices A = (aij), (cid:101)A = ((cid:101)aij) from IRm×n is given by

i=1

j=1

(cid:104)A, (cid:101)A(cid:105) = aij(cid:101)aij.

The multifunction F : IRn × IRm×n × IRm ⇒ IRn given by

(3.4) F(x, A, b) = N (x; Θ(A, b)), ∀(x, A, b) ∈ IRn × IRm×n × IRm,

(cid:8)ξ∗ ∈ IRn(cid:12)

is said to be the nonlinearly perturbed polyhedral normal cone mapping to the perturbed polyhedron Θ(A, b) (or, the normal cone mapping F(·), for short), where the formula

 

(cid:12) (cid:104)ξ∗, u − x(cid:105) ≤ 0, ∀u ∈ Θ(A, b)(cid:9),



if x ∈ Θ(A, b) N (x; Θ(A, b)) = ∅, if x (cid:54)∈ Θ(A, b)

deﬁnes the normal cone to the set Θ(A, b) at x in the sense of convex analysis.

Specializing Lemma 3.1 from [32], which has been obtained by using a generalized version of the Farkas lemma [3], to the linear inequalities (3.2) we have the following statement.

Proposition 3.1 Let Θ(A, b) be deﬁned by (3.1). For any point x ∈ Θ(A, b), let I = I(x, A, b) with I(x, A, b) being given by (3.3). Then

(cid:41)

(cid:40)

(cid:88)

N (x; Θ(A, b)) = pos(cid:8)A(cid:62)

i | i ∈ I(cid:9) (cid:12) (cid:12) (cid:12) (cid:12)

i∈I

= , λi ≥ 0, ∀i ∈ I λiA(cid:62) i

and

(cid:110) v ∈ IRn(cid:12)

(cid:111) .

(cid:12) (cid:104)A(cid:62)

i , v(cid:105) ≤ 0, ∀i ∈ I

T (x; Θ(A, b)) =

The forthcoming lemma shows that the positive linear independence prop- erty of a ﬁnite system of vectors is stable under small nonlinear perturbations of the vectors.

If the vectors { ¯A(cid:62)

i (Ak)(cid:62)

i∈Γ λk

Lemma 3.1 Let ¯A ∈ IRm×n and Γ ⊂ T . i | i ∈ Γ} are positively linearly independent, then there exists δ > 0 such that for any A ∈ ¯B( ¯A, δ) the vectors {A(cid:62) i | i ∈ Γ} are also positively linearly independent.

(cid:88)

Proof. To obtain a contradiction, suppose that for any δ > 0 there exists some A in ¯B( ¯A, δ) with {A(cid:62) i | i ∈ Γ} being not positively linearly independent. This implies that for every k ∈ IN there exist Ak ∈ ¯B( ¯A, k−1) and λk i ≥ 0 (i ∈ Γ), which are not all zero, such that (cid:80) i = 0. For each k ∈ IN , we put

i :=

i∈Γ

(cid:37)k := and µk ∈ [0, 1] for all i ∈ Γ. λk i > 0, λk i (cid:37)k

i = µi ≥ 0 for all i ∈ Γ. Note that

(cid:88)

Without any loss of generality, we can assume that there exists a subsequence {k(cid:96)} of {k} such that lim(cid:96)→∞ µk(cid:96)

i =

i = 0 for all k ∈ IN.

i∈Γ

(Ak)(cid:62) µk i (Ak)(cid:62) λk i (Ak)(cid:62) λk i (cid:37)k 1 (cid:37)k

(cid:88)

Hence,

i = 0.

i = lim (cid:96)→∞

i∈Γ

i | i ∈ Γ} force

¯A(cid:62) µi µk(cid:96) i (Ak(cid:96))(cid:62)

(cid:88)

This and the assumed positive linear independence of { ¯A(cid:62) µi = 0 for all i ∈ Γ. As

i∈Γ

µk(cid:96) i = 1, µi = lim (cid:96)→∞

(cid:50) we arrive at a contradiction, which completes the proof.

The next proposition shows that the graph of the normal cone mapping F(x, A, b) is locally closed in the product space IRn × IRm×n × IRm × IRn under the positive linear independence condition on the normal vectors of the active constraints. This property allows us to calculate the limiting normal cone N (cid:0)(¯x, ¯A, ¯b, ¯ξ∗); gphF (cid:1) via formula (1.5).

i | i ∈ I(¯x, ¯A, ¯b)} are Proposition 3.2 For any (¯x, ¯A, ¯b, ¯ξ∗) ∈ gphF, if { ¯A(cid:62) positively linearly independent, then gphF is locally closed around (¯x, ¯A, ¯b, ¯ξ∗).

Proof. First, note that there exists δ > 0 such that I(x, A, b) ⊂ I(¯x, ¯A, ¯b) for all (x, A, b) ∈ ¯B(¯x, δ) × ¯B( ¯A, δ) × ¯B(¯b, δ). Indeed, suppose on the contrary that for any δ > 0 there exists (x, A, b) ∈ ¯B(¯x, δ) × ¯B( ¯A, δ) × ¯B(¯b, δ) such that I(x, A, b)\I(¯x, ¯A, ¯b) (cid:54)= ∅. Then, for each k ∈ IN , we can ﬁnd (xk, Ak, bk) in ¯B(¯x, k−1) × ¯B( ¯A, k−1) × ¯B(¯b, k−1) such that I(xk, Ak, bk)\I(¯x, ¯A, ¯b) (cid:54)= ∅. It follows that, for every k ∈ IN , there is an index i ∈ T \I(¯x, ¯A, ¯b) satisfying (cid:104)(Ak)(cid:62) i , xk(cid:105) = (bk)i. By the Dirichlet principle there exist a subsequence {k(cid:96)} of {k} and an index i0 ∈ T \I(¯x, ¯A, ¯b) with (cid:104)(Ak(cid:96))(cid:62) i0, xk(cid:96)(cid:105) = (bk(cid:96))i0 for all (cid:96) ∈ IN . This implies that

i0, ¯x(cid:105) = (cid:104)lim (cid:96)→∞

i0, lim (cid:96)→∞

i0, xk(cid:96)(cid:105) = lim (cid:96)→∞

(cid:104) ¯A(cid:62) (Ak(cid:96))(cid:62) (cid:104)(Ak(cid:96))(cid:62) (bk(cid:96))i0 = ¯bi0. xk(cid:96)(cid:105) = lim (cid:96)→∞

Therefore, we obtain i0 ∈ I(¯x, ¯A, ¯b), a contradiction.

{A(cid:62)

(cid:88)

Due to Lemma 3.1, we can assume without loss of generality that vectors i | i ∈ I(¯x, ¯A, ¯b)} are positively linearly independent for all A ∈ ¯B( ¯A, δ). We now prove that G := gphF ∩ (cid:0) ¯B(¯x, δ) × ¯B( ¯A, δ) × ¯B(¯b, δ) × IRn(cid:1) is a k)} is a sequence in G and (xk, Ak, bk, ξ∗ closed set. Suppose that {(xk, Ak, bk, ξ∗ k) converges to ((cid:98)x, (cid:98)A, (cid:98)b, (cid:98)ξ∗). We have to show that ((cid:98)x, (cid:98)A, (cid:98)b, (cid:98)ξ∗) ∈ G. Note that ((cid:98)x, (cid:98)A, (cid:98)b) ∈ ¯B(¯x, δ) × ¯B( ¯A, δ) × ¯B(¯b, δ) because ¯B(¯x, δ) × ¯B( ¯A, δ) × ¯B(¯b, δ) is closed in IRn×IRm×n×IRm and (xk, Ak, bk) converges to ((cid:98)x, (cid:98)A, (cid:98)b). According to the observation stated at the beginning of this proof, we have I(xk, Ak, bk) ⊂ I(¯x, ¯A, ¯b) for suﬃciently large indexes k ∈ IN . Without loss of generality, we can assume that I(xk, Ak, bk) = (cid:101)I ⊂ I(¯x, ¯A, ¯b) for all k ∈ IN . For each k ∈ IN , it holds

i (Ak)(cid:62) λk

i ≥ 0, i ∈ (cid:101)I.

i∈(cid:101)I

(3.5) for some λk ξ∗ k =

j ≥ λk(cid:96)

i }k∈IN , i ∈ (cid:101)I, are bounded. Indeed, suppose on }k∈IN is unbounded. Then, i0 > (cid:96). There is no loss of generality i0 → +∞ as

We see that the sequences {λk

j∈(cid:101)I λk(cid:96) j converges +∞ as (cid:96) → ∞. Note that

j∈(cid:101)I λk(cid:96)

the contrary that there exists i0 ∈ (cid:101)I such that {λk i0 for every (cid:96) ∈ IN there is k(cid:96) ≥ (cid:96) satisfying λk(cid:96) in assuming that k(cid:96)+1 > k(cid:96) for all (cid:96) ∈ IN . Since (cid:80) (cid:96) → ∞, (cid:37)k(cid:96) := (cid:80)

(cid:80)

:= = ∈ [0, 1] for all i ∈ (cid:101)I. µk(cid:96) i λk(cid:96) i (cid:37)k(cid:96) λk(cid:96) i j∈(cid:101)I λk(cid:96)

(cid:88)

Thus, for each index i ∈ (cid:101)I, any subsequence of {µk(cid:96) i } possesses a convergent subsequence. We can assume that lim(cid:96)→∞ µk(cid:96) = µi ≥ 0 for every i ∈ (cid:101)I. As k → (cid:98)ξ∗, from (3.5) we deduce that ξ∗

i =

i∈(cid:101)I

i | i ∈ (cid:101)I} i = 0 implies i = 1, which

i∈(cid:101)I µi (cid:98)A(cid:62) i∈(cid:101)I µk(cid:96)

µk(cid:96) i (Ak(cid:96))(cid:62) µi (cid:98)A(cid:62) i . ξ∗ k(cid:96) 0 = lim (cid:96)→∞ = lim (cid:96)→∞ 1 (cid:37)k(cid:96)

(cid:88)

On one hand, since (cid:98)A ∈ ¯B( ¯A, δ) and (cid:101)I ⊂ I(¯x, ¯A, ¯b), the vectors { (cid:98)A(cid:62) are positively linearly independent. So, the equality (cid:80) µi = 0 for all i ∈ (cid:101)I. On the other hand, the property (cid:80) holds for all (cid:96) ∈ IN , yields

i∈(cid:101)I

µi = µk(cid:96) i = 1. lim (cid:96)→∞ µk(cid:96) i = lim (cid:96)→∞

We have arrived at a contradiction.

i }k∈IN (i ∈ (cid:101)I), we can assume i = λi ≥ 0 for all

(cid:88)

By the boundedness of all the sequences {λk

i =

i ∈ N ((cid:98)x; Θ( (cid:98)A, (cid:98)b)) = F((cid:98)x, (cid:98)A, (cid:98)b).

(cid:98)ξ∗ = lim (cid:96)→∞

i∈(cid:101)I

that there exists a subsequence {k(cid:96)} of {k} with lim(cid:96)→∞ λk(cid:96) i ∈ (cid:101)I. Combining this with (3.5) yields (cid:88) λi (cid:98)A(cid:62) λk(cid:96) i (Ak(cid:96))(cid:62) ξ∗ k(cid:96) = lim (cid:96)→∞

We have shown that ((cid:98)x, (cid:98)A, (cid:98)b, (cid:98)ξ∗) ∈ gphF. Moreover, we obtain (cid:16) ¯B(¯x, δ) × ¯B( ¯A, δ) × ¯B(¯b, δ) × IRn(cid:17) = G,

(cid:50) ((cid:98)x, (cid:98)A, (cid:98)b, (cid:98)ξ∗) ∈ gphF ∩ which establishes the closedness of G.

On the basis of generalized diﬀerentiability properties of the normal cone mapping F(x, A, b), we discuss solution stability of the following parametric aﬃne variational inequality problem

(3.6) Find x ∈ Θ(A, b) subject to (cid:104)M x − q, u − x(cid:105) ≥ 0, ∀u ∈ Θ(A, b),

where M ∈ IRn×n is a ﬁxed matrix, and A ∈ IRm×n, b ∈ IRm, q ∈ IRn are subject to change. Let S(A, b, q) be the solution set of the problem (3.6) with respect to a parametric triple (A, b, q). As in [43] and [44], the problem (3.6) can be regarded as a linear generalized equation of the form

(3.7) 0 ∈ M x − q + F(x, A, b),

where the multifunction F(x, A, b) is given by (3.4). Hence, S(A, b, q) is also the solution set of the linear generalized equation (3.7). We ﬁrst provide

an evaluation of the Fr´echet normal cone to the graph of the normal cone mapping F(x, A, b). Then an upper estimate for the limiting normal cone to the graph of that normal cone mapping is obtained under a positive linear independence assumption on the normal vectors of the active constraints of (3.2). These results allow us to evaluate the values of the Mordukhovich coderivative of the normal cone mapping under consideration. Finally, using a standard coderivative sum rule, we are able to establish some criteria for solution stability of the problem (3.6) under nonlinear perturbations.

3.2 Estimation of the Fr´echet Normal Cone to gphF

(cid:26)

(cid:88)

We now provide an upper estimate for the Fr´echet normal cone to the graph of the normal cone mapping F(·) given by (3.4). This is a major step to establish an upper estimate for the limiting normal cone to the graph of F(·) which will be discussed in next section. For any (x, A, b, ξ∗) ∈ gphF, we put

(cid:27) ,

i , λi ≥ 0 ∀i ∈ I

(cid:12) (cid:12) ξ∗ = (cid:12)

i∈I

(3.8) Ξ(x, A, b, ξ∗) := (λi)i∈I λiA(cid:62)

(cid:111)

and

(cid:110) i ∈ I(cid:12)

(cid:12) λi = 0 for some (λj)j∈I ∈ Ξ(x, A, b, ξ∗)

(3.9) , I1(x, A, b, ξ∗) :=

where I := I(x, A, b) is deﬁned by (3.3). The dual of a set Ω ⊂ IRn is given by

Ω∗ := (cid:8)u∗ ∈ IRn| (cid:104)u∗, v(cid:105) ≤ 0 ∀v ∈ Ω(cid:9).

For each u ∈ IRn, set

{u}⊥ := (cid:8)v ∈ IRn| (cid:104)u, v(cid:105) = 0(cid:9).

For every λ = (λi)i∈I ∈ Ξ(x, A, b, ξ∗), we deﬁne

(3.10) I0(λ) := {i ∈ I| λi = 0},

and

Eλ(x, A, b, ξ∗) (cid:26) = ,

i , ξ ∈ T (x; Θ(A, b)) ∩ {ξ∗}⊥,

i A(cid:62)

(cid:12) (cid:12) x∗ ∈ (cid:0)T (x; Θ(A, b)) ∩ {ξ∗}⊥(cid:1)∗ (cid:12) (x∗, A∗, b∗, ξ) i∈I b∗

I0(λ),j ≤ 0 if xj < 0, a∗ a∗

I0(λ),j ≥ 0 if xj > 0,

(cid:27) ,

¯I = 0, b∗

I0(λ),j = 0 if xj = 0, A∗ a∗

I0(λ) ≤ 0

I = 0, b∗

ij)m×n ∈ IRm×n, and b∗ = (b∗

1 . . . b∗

x∗ = − (cid:80) (3.11)

where A∗ = (a∗ m)(cid:62) ∈ IRm. Using the abbreviation I1 := I1(x, A, b, ξ∗) with I1(x, A, b, ξ∗) being given by (3.9), we construct the set

(cid:26)

H(x, A, b, ξ∗)

= ,

i , ξ ∈ T (x; Θ(A, b)) ∩ {ξ∗}⊥,

i A(cid:62)

(cid:12) (cid:12) x∗ ∈ (cid:0)T (x; Θ(A, b)) ∩ {ξ∗}⊥(cid:1)∗ (cid:12) (x∗, A∗, b∗, ξ) i∈I b∗

I1,j ≥ 0 if xj > 0,

I1,j ≤ 0 if xj < 0, a∗ a∗

x∗ = − (cid:80) (3.12)

(cid:27) .

¯I = 0, b∗

I1,j = 0 if xj = 0, A∗ a∗

I = 0, b∗ I1

≤ 0

Although the construction of (3.12) is a little bit complicated, it is clear that the set H(x, A, b, ξ∗) can be computed explicitly.

(cid:92)

We ﬁrst clarify a relationship between the family (cid:8)Eλ(x, A, b, ξ∗)(cid:9)

λ∈Ξ(x,A,b,ξ∗) and H(x, A, b, ξ∗). Later, in Theorem 3.1 it will be proved that H(x, A, b, ξ∗) is an upper estimate for the Fr´echet normal cone (cid:98)N (cid:0)(x, A, b, ξ∗); gphF (cid:1). Lemma 3.2 For any (¯x, ¯A, ¯b, ¯ξ∗) ∈ gphF, it holds H(¯x, ¯A, ¯b, ¯ξ∗) =

λ∈Ξ(¯x, ¯A,¯b, ¯ξ∗)

(3.13) Eλ(¯x, ¯A, ¯b, ¯ξ∗).

λ∈Ξ(¯x, ¯A,¯b, ¯ξ∗)

Proof. From (3.9) and (3.10) we infer that (cid:91) I1(¯x, ¯A, ¯b, ¯ξ∗) = I0(λ).

(cid:50) By (3.11) and (3.12), the equality (3.13) follows from this fact.

For any ¯A ∈ IRm×n, we consider the multifunction F ¯A : IRn × IRm ⇒ IRn

deﬁned by setting

(3.14)

F ¯A(x, b) := N (x; Θ( ¯A, b)). 49

(cid:1), then

(cid:88)

The following lemma gives us an upper estimate for the Fr´echet normal cone to the graph of F ¯A at a given point (x, b, ξ∗) ∈ gphF ¯A. Lemma 3.3 (See [52, Lemma 4.1]) If (x∗, b∗, ξ) ∈ (cid:98)N (cid:0)(x, b, ξ∗); gphF ¯A × (cid:0)T (x; Θ( ¯A, b)) ∩ {ξ∗}⊥(cid:1), (x∗, ξ) ∈ (cid:0)T (x; Θ( ¯A, b)) ∩ {ξ∗}⊥(cid:1)∗

i∈I

and x∗ = − b∗ i ¯A(cid:62) i , b∗ ¯I = 0,

i with

i∈I λi

¯A(cid:62)

≤ 0. where I = {i ∈ T | ¯Aix = bi} and ¯I = T \ I. Moreover, if ξ∗ = (cid:80) λi ≥ 0 for all i ∈ I, and I0 = {i ∈ I| λi = 0}, then b∗ I0

The main result can be formulated as follows.

Theorem 3.1 For any (¯x, ¯A, ¯b, ¯ξ∗) ∈ gphF, it holds

(cid:98)N (cid:0)(¯x, ¯A, ¯b, ¯ξ∗); gphF (cid:1) ⊂ H(¯x, ¯A, ¯b, ¯ξ∗).

(3.15)

(cid:92)

Proof. According to Lemma 3.2, (3.15) is equivalent to the inclusion

(cid:98)N (cid:0)(¯x, ¯A, ¯b, ¯ξ∗); gphF (cid:1) ⊂

λ∈Ξ(¯x, ¯A,¯b, ¯ξ∗)

(3.16) Eλ(¯x, ¯A, ¯b, ¯ξ∗).

To obtain (3.16), let us pick any (x∗, A∗, b∗, ξ) ∈ (cid:98)N (cid:0)(¯x, ¯A, ¯b, ¯ξ∗); gphF (cid:1) and λ = (λi)i∈I ∈ Ξ(¯x, ¯A, ¯b, ¯ξ∗) with I = I(¯x, ¯A, ¯b) being given by (3.3). The proof will be completed if we can show that

(3.17)

(x∗, A∗, b∗, ξ) ∈ Eλ(¯x, ¯A, ¯b, ¯ξ∗). Since (x∗, A∗, b∗, ξ) ∈ (cid:98)N (cid:0)(¯x, ¯A, ¯b, ¯ξ∗); gphF (cid:1), we have

(x,A,b,ξ∗)

gphF −→ (¯x, ¯A,¯b, ¯ξ∗)

limsup ≤ 0. (cid:104)x∗, x − ¯x(cid:105) + (cid:104)A∗, A − ¯A(cid:105) + (cid:104)b∗, b − ¯b(cid:105) + (cid:104)ξ, ξ∗ − ¯ξ∗(cid:105) (cid:107)x − ¯x(cid:107) + (cid:107)A − ¯A(cid:107) + (cid:107)b − ¯b(cid:107) + (cid:107)ξ∗ − ¯ξ∗(cid:107)

(x,b,ξ∗)

gphF ¯A−→ (¯x,¯b, ¯ξ∗)

(cid:1). Applying Lemma 3.3, we

limsup ≤ 0. (3.18) Note that (x, ¯A, b, ξ∗) ∈ gphF if and only if ξ∗ ∈ N (x; Θ( ¯A, b)) = F ¯A(x, b), where F ¯A is given by (3.14). Hence, (x, ¯A, b, ξ∗) ∈ gphF if and only if (x, b, ξ∗) ∈ gphF ¯A. Taking A = ¯A, by (3.18) we deduce that (cid:104)x∗, x − ¯x(cid:105) + (cid:104)b∗, b − ¯b(cid:105) + (cid:104)ξ, ξ∗ − ¯ξ∗(cid:105) (cid:107)x − ¯x(cid:107) + (cid:107)b − ¯b(cid:107) + (cid:107)ξ∗ − ¯ξ∗(cid:107)

This means that (x∗, b∗, ξ) ∈ (cid:98)N (cid:0)(¯x, ¯b, ¯ξ∗); gphF ¯A obtain

(x∗, ξ) ∈ (cid:0)T (¯x; Θ( ¯A, ¯b)) ∩ { ¯ξ∗}⊥(cid:1)∗ × (cid:0)T (¯x; Θ( ¯A, ¯b)) ∩ { ¯ξ∗}⊥(cid:1),

(cid:88)

and

i , b∗

¯I = 0, b∗

I0(λ) ≤ 0,

i∈I

x∗ = − ¯A(cid:62) b∗ i

where ¯I = T \I and I0(λ) = {i ∈ I| λi = 0}. By virtue of these relations, based on (3.11) we see that the proof of (3.17) now reduces to establishing the following properties

(3.19)

a∗ I0(λ),j ≤ 0 if xj < 0, a∗ I0(λ),j ≥ 0 if xj > 0, a∗ I0(λ),j = 0 if xj = 0, A∗ ¯I = 0.

¯I = 0. Fix any pair (r, s) ∈ ¯I × {1, . . . , n}. Let b = ¯b and let aij = ¯aij for all i ∈ T and j ∈ {1, . . . , n} with (i, j) (cid:54)= (r, s). Choose ars ∈ (¯ars − ε, ¯ars + ε) and let A = (aij), where ε > 0 is as small as

First, we prove that A∗

and (3.20) ¯Ar ¯x − ε¯xs < br ¯Ar ¯x + ε¯xs < br.

Note that there is t ∈ (0, 1) such that ars = t(¯ars − ε) + (1 − t)(¯ars + ε). Combining this with (3.20), we get

Ar ¯x = t( ¯Ar ¯x − ε¯xs) + (1 − t)( ¯Ar ¯x + ε¯xs) < br.

By the construction of A and b, we obtain

Ai¯x = bi ∀i ∈ I and Ai¯x < bi ∀i ∈ ¯I.

Hence x := ¯x belongs to Θ(A, b), and ξ∗ := ¯ξ∗ satisﬁes the relation

i | i ∈ I} = pos{A(cid:62)

i | i ∈ I} = N (x; Θ(A, b)).

ξ∗ ∈ pos{ ¯A(cid:62)

From (3.18) it follows that

rs = 0.

≤ 0. limsup ars→¯ars a∗ rs(ars − ¯ars) |ars − ¯ars|

¯I = 0.

Since ars ∈ (¯ars −ε, ¯ars +ε) can be chosen arbitrarily, this implies that a∗ Thus, we deduce that A∗

Now, ﬁx any pair (r, s) ∈ I0(λ) × {1, . . . , n}. Choose x = ¯x, b = ¯b, ξ∗ = ¯ξ∗, aij = ¯aij for all i ∈ T and j ∈ {1, . . . , n} satisfying (i, j) (cid:54)= (r, s). Let A = (aij), where ars is chosen as follows:

α) If ¯xs = 0, then ars ∈ (−∞, +∞) is taken arbitrary;

β) If ¯xs > 0, then we choose ars ∈ (−∞, ¯ars);

γ) If ¯xs < 0, then we choose ars ∈ (¯ars, +∞).

Denote the set of all ars ∈ IR satisfying these conditions by Ω. By the choice of (x, A, b, ξ∗) and the condition r ∈ I0(λ), we infer that (x, A, b, ξ∗) ∈ gphF. By (3.18), it holds

(3.21) ≤ 0. a∗ rs(ars − ¯ars) |ars − ¯ars| limsup Ω →¯ars ars

From (3.21) it follows that

rs = 0;

α1) If ¯xs = 0, then a∗

rs ≥ 0;

β1) If ¯xs > 0, then a∗

rs ≤ 0.

γ1) If ¯xs < 0, then a∗

(cid:50) Thus, the properties in (3.19) are valid. The proof is complete.



To have an idea about how the estimation formula (3.15) can be used in practical calculations, consider the following numerical example, which has the origin in [49, Example 2].



(cid:33)

(cid:32) 5 3

Example 3.1 Let T = {1, 2, 3} and F : IR2 × IR3×2 × IR3 ⇒ IR2 be given by F(x, A, b) = N (x; Θ(A, b)) for any (x, A, b) ∈ IR2 × IR3×2 × IR3. Let 

 

  ,

 

  ,

1 3

¯A = ¯b = and ¯x = ∈ Θ( ¯A, ¯b),

1 1 0 −1 0 −1 2 0 0

where Θ(A, b) = {x ∈ IR2| Ax ≤ b}.

We have I = I(¯x, ¯A, ¯b) = {i ∈ T | ¯Ai¯x = bi} = {1}, ¯I = T \ I = {2, 3}.

3, 2

3), we observe that

Choosing ¯ξ∗ = ( 2

1 ∈ pos{ ¯A(cid:62)

1 } = N (¯x; Θ( ¯A, ¯b)).

¯ξ∗ = ¯A(cid:62)

(cid:26)

(cid:27)

2 3 Hence, (¯x, ¯A, ¯b, ¯ξ∗) belongs to gphF. Note that

i , λi ≥ 0 ∀i ∈ I

i∈I λiA(cid:62)

(cid:12) (cid:12) ξ∗ = (cid:80) (cid:12)

(cid:110)

(cid:111)

Ξ(¯x, ¯A, ¯b, ¯ξ∗) = (λi)i∈I

(cid:27) .

(cid:12) (cid:12) ¯ξ∗ = λ1

1 , λ1 ≥ 0

(cid:26)2 3

= = ¯A(cid:62) λ1

(cid:111)

(cid:110) i ∈ I(cid:12)

Thus,

(cid:12) λi = 0 for some (λj)j∈I ∈ Ξ(x, A, b, ξ∗)

= ∅. I1 = I1(¯x, ¯A, ¯b, ¯ξ∗) =

Besides, it holds

3)(cid:9)⊥

3, 2

= (cid:8)(v1, v2)| v1 + v2 = 0(cid:9),

1 , v(cid:105) ≤ 0(cid:9) = (cid:8)(v1, v2)| v1 + v2 ≤ 0(cid:9),

{ ¯ξ∗}⊥ = (cid:8)( 2 T (¯x; Θ( ¯A, ¯b)) = (cid:8)v ∈ IR2| (cid:104) ¯A(cid:62)

(cid:0)T (¯x; Θ( ¯A, ¯b)) ∩ { ¯ξ∗}⊥(cid:1)∗

1 ∈ IR(cid:9).

1, u∗

3) = (0, 0). Hence b∗ = (b∗

2, b∗

¯I = (b∗

= (cid:8)(u∗ T (¯x; Θ( ¯A, ¯b)) ∩ { ¯ξ∗}⊥ = (cid:8)(v1, v2)| v1 + v2 = 0(cid:9) = (cid:8)(v1, −v1)| v1 ∈ IR(cid:9), 1)| u∗



(cid:33)

(cid:32)

Fix an element (x∗, A∗, b∗, ξ) ∈ H(¯x, ¯A, ¯b, ¯ξ∗), where H(¯x, ¯A, ¯b, ¯ξ∗) is deﬁned via (3.12). Since I1 = ∅, one cannot ﬁnd any index i ∈ I1 such that b∗ i ≤ 0. Besides, as ¯I = {2, 3}, b∗ 1, 0, 0), where b∗ 1 ∈ IR. In the accordance with (3.12), taking account of the fact that I1 = ∅ we have

1, a∗

11, a∗

12 ∈ IR.

1 =

 

  with b∗ 

11 a∗ a∗ 12 0 0 0 0

and A∗ = ¯A(cid:62) x∗ = −b∗ 1 −b∗ 1 −b∗ 1

Consequently, from (3.12) we derive the formula for H(¯x, ¯A, ¯b, ¯ξ∗) as follows









(cid:33)

(cid:32)

(cid:33)

 

 

H(¯x, ¯A, ¯b, ¯ξ∗) = 

12, ξ1 ∈ IR

11, a∗

1, a∗ b∗

 

  ,

 

  ,

 

(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)





11 a∗ a∗ 12 0 0 0 0

, . ξ1 −ξ1 −b∗ 1 −b∗ 1 b∗ 1 0 0

(cid:98)N (cid:0)(¯x, ¯A, ¯b, ¯ξ∗); gphF (cid:1) ⊂









(cid:32)

(cid:33)

(cid:32)

(cid:33)

 

 

Combining this with (3.15) we get the following upper estimate for the Fr´echet normal cone (cid:98)N (cid:0)(¯x, ¯A, ¯b, ¯ξ∗); gphF (cid:1):

 

  ,

 

  ,

 

(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)





α, β, γ, µ ∈ IR . , α α µ −µ β γ 0 0 0 0 −α 0 0

The problem of ﬁnding an exact formula for the computation of the Fr´echet

normal cone (cid:98)N (cid:0)(¯x, ¯A, ¯b, ¯ξ∗); gphF (cid:1) remains open.

3.3 Estimation of the Limiting Normal Cone to gphF

The main goal of this section is to establish an upper estimate for the limiting normal cone to the graph of the normal cone mapping F(·) given by (3.4). As a direct consequence, we get an upper estimate for values of the Mordukhovich coderivative of such mapping at a given point on gphF.

Given any matrix A ∈ IRm×n and subsets P, Q of T satisfying P ⊂ Q,

following [13] we put

i | i ∈ P (cid:9) + pos(cid:8)A(cid:62)

i | i ∈ Q \ P (cid:9),

AQ,P (A) = span(cid:8)A(cid:62)

(cid:111)

(cid:110) v ∈ IRn(cid:12)

and

(cid:12) (cid:104)A(cid:62)

i , v(cid:105) = 0 ∀i ∈ P, (cid:104)A(cid:62)

i , v(cid:105) ≤ 0 ∀i ∈ Q \ P

. BQ,P (A) =

(cid:0)BQ,P (A)(cid:1)∗

Lemma 3.4 (See [13, Lemma 3.3]) For any A ∈ IRm×n and P ⊂ Q ⊂ T , we have

= AQ,P (A),

where (cid:0)BQ,P (A)(cid:1)∗ = (cid:8)u ∈ IRn| (cid:104)u, v(cid:105) ≤ 0 ∀v ∈ BQ,P (A)(cid:9).



Lemma 3.5 Let (¯x, ¯A, ¯b, ¯ξ∗) ∈ gphF, I = I(¯x, ¯A, ¯b), (λi)i∈I ∈ Ξ(¯x, ¯A, ¯b, ¯ξ∗), and K = {i ∈ I| λi > 0}. Then, it holds   (3.22)

i = ¯A(cid:62)

i for all i ∈ T and arguing the same as in the proof (cid:50)

AI,K( ¯A) = (cid:0)T (¯x; Θ( ¯A, ¯b)) ∩ { ¯ξ∗}⊥(cid:1)∗ BI,K( ¯A) = T (¯x; Θ( ¯A, ¯b)) ∩ { ¯ξ∗}⊥.

Proof. By setting a∗ of Lemma 2.6, we get the formula (3.22).

Combining (3.12) with (3.22) we obtain the next lemma.

Lemma 3.6 Let (¯x, ¯A, ¯b, ¯ξ∗), I, (λi)i∈I, and K be the same as in Lemma 3.5. Then, we have

H(¯x, ¯A, ¯b, ¯ξ∗) (cid:26) =

i , b∗

(cid:12) (cid:12) (x∗, ξ) ∈ AI,K( ¯A) × BI,K( ¯A), (cid:12) (x∗, A∗, b∗, ξ) I = 0, b∗ i∈I b∗ I1 i

≤ 0, x∗ = − (cid:80) ¯A(cid:62)

¯I = 0, a∗

I1,j = 0 if ¯xj = 0,

(cid:27)

A∗

I1,j ≤ 0 if ¯xj < 0, a∗ a∗

I1,j ≥ 0 if ¯xj > 0

(cid:111) ,

(cid:12) P (cid:54)= ∅, ξ∗ ∈ pos{A(cid:62)

i | i ∈ P }

(cid:110)

(cid:111)

(3.23) For each (x, A, b, ξ∗) ∈ gphF, we put (cid:110) P ⊂ I(x, A, b)(cid:12) I(x, A, b, ξ∗) :=

(cid:12) A(cid:62)

i , i ∈ P, are linearly independent

(3.24) J (x, A, b, ξ∗) := P ∈ I(cid:12)

with I = I(x, A, b, ξ∗), and

 

(cid:98)I(x, A, b, ξ∗) :=



J (x, A, b, ξ∗), if ξ∗ (cid:54)= 0, (3.25) J (x, A, b, ξ∗) ∪ {∅}, if ξ∗ = 0.

(cid:91)

Using the abbreviations I := I(x, A, b) and (cid:98)I := (cid:98)I(x, A, b, ξ∗), we deﬁne

(cid:26) (x∗, A∗, b∗, ξ)(cid:12)

(cid:12) (x∗, ξ) ∈ AQ,P (A) × BQ,P (A),

P ⊂Q⊂I, P ∈(cid:98)I

Σ(x, A, b, ξ∗) :=

i A(cid:62) i∈Q b∗ i , Q\P ≤ 0, A∗

Q = 0, b∗ b∗

Q = 0,

(cid:27) .

Q\P,j ≥ 0 if xj > 0

Q\P,j ≤ 0 if xj < 0, a∗ a∗

x∗ = − (cid:80) (3.26)

Note that the set in (3.26) can be computed explicitly. The forthcoming statement, the result of this section, describes an upper estimate for the limiting normal cone to gphF at a given point.

i | i ∈ I} are positively linearly independent, then

Theorem 3.2 Let any (¯x, ¯A, ¯b, ¯ξ∗) ∈ gphF and I = I(¯x, ¯A, ¯b). If the vectors { ¯A(cid:62)

(3.27) N (cid:0)(¯x, ¯A, ¯b, ¯ξ∗); gphF (cid:1) ⊂ Σ(¯x, ¯A, ¯b, ¯ξ∗).

(cid:98)N (cid:0)(x, A, b, ξ∗); gphF (cid:1).

(x,A,b,ξ∗)

gphF −→ (¯x, ¯A,¯b, ¯ξ∗)

k, η∗

k) gphF−→ (¯x, ¯A, ¯b, ¯ξ∗) and (u∗ k, η∗

k, A∗

Limsup (3.28) Proof. Since the vectors { ¯A(cid:62) i | i ∈ I} are positively linearly independent, gphF is locally closed around (¯x, ¯A, ¯b, ¯ξ∗) by Proposition 3.2. According to [28, Theorem 2.35], it holds N (cid:0)(¯x, ¯A, ¯b, ¯ξ∗); gphF (cid:1) =

Fix any (x∗, A∗, b∗, ξ) ∈ N (cid:0)(¯x, ¯A, ¯b, ¯ξ∗); gphF (cid:1). To justify (3.27), we have to prove that (x∗, A∗, b∗, ξ) ∈ Σ(¯x, ¯A, ¯b, ¯ξ∗). By (3.28), there exist sequences k); gphF (cid:1) k, ξk) ∈ (cid:98)N (cid:0)(xk, Ak, bk, ξ∗ k, A∗ (xk, Ak, bk, ξ∗ for all k ∈ IN such that (u∗ k, ξk) converges to (x∗, A∗, b∗, ξ). Since (xk, Ak, bk) → (¯x, ¯A, ¯b) and I(xk, Ak, bk) ⊂ T for every k ∈ IN , we can assume

k ∈ N (xk; Θ(Ak, bk)) = pos{(Ak)(cid:62) ξ∗

i | i ∈ Q} for all k ∈ IN.

that I(xk, Ak, bk) = Q for all k ∈ IN , where Q is a ﬁxed index set and Q ⊂ I(¯x, ¯A, ¯b). By Proposition 3.1 we have

i | i ∈ (cid:101)P } are linearly independent and

k ∈ pos{(Ak)(cid:62) ξ∗

i | i ∈ (cid:101)P } for all k ∈ IN.

Due to Lemma 2.1 and the Dirichlet principle, by considering a subsequence of {k} if necessary, we can assume that there is a subset (cid:101)P ⊂ Q such that {(Ak)(cid:62)

(cid:88)

Thus, for every k ∈ IN , it holds

i (Ak)(cid:62) λk

i ≥ 0, i ∈ (cid:101)P .

i∈ (cid:101)P

for some λk ξ∗ k =

Using again the Dirichlet principle, we ﬁnd a subsequence {k(cid:96)} of {k} and a subset P ⊂ (cid:101)P such that

{i ∈ (cid:101)P | λk(cid:96)

i > 0} = P for every (cid:96) ∈ IN. k(cid:96), ξk(cid:96)) ∈ (cid:98)N (cid:0)(xk(cid:96), Ak(cid:96), bk(cid:96), ξ∗

k(cid:96), η∗

k(cid:96), A∗

k(cid:96)

); gphF (cid:1), it follows

For each (cid:96) ∈ IN , as (u∗ from Theorem 3.1 that

k(cid:96), A∗

k(cid:96), η∗

k(cid:96), ξk(cid:96)) ∈ H(xk(cid:96), Ak(cid:96), bk(cid:96), ξ∗ k(cid:96)

(u∗ ).

(cid:88)

Therefore, by Lemma 3.6 we have

k(cid:96), ξk(cid:96)) ∈ AQ,P (Ak(cid:96)) × BQ,P (Ak(cid:96)),

i∈Q

= − (u∗ )i(Ak(cid:96))(cid:62) i , u∗ k(cid:96) (η∗ k(cid:96)

k(cid:96) 1

k(cid:96)

(η∗ k(cid:96) )I

k(cid:96)

(A∗ k(cid:96)

1 ,j ≥ 0 if (xk(cid:96))j > 0,

(a∗ k(cid:96) (a∗ k(cid:96) )I

). := I1(xk(cid:96), Ak(cid:96), bk(cid:96), ξ∗ k(cid:96) (η∗ )Q = 0, ≤ 0, k(cid:96) (a∗ )Q = 0, 1 ,j = 0 if (xk(cid:96))j = 0, )I k(cid:96) 1 ,j ≤ 0 if (xk(cid:96))j < 0, )I )ij) and I k(cid:96) = ((a∗ 1 k(cid:96) where A∗ k(cid:96)

If P = ∅, then ξ∗ k(cid:96) By the deﬁnition of I k(cid:96) = 0.) 1 . If P (cid:54)= ∅, then

i∈P

= 0 for all (cid:96) ∈ IN . (In this case, ¯ξ∗ = lim(cid:96)→∞ ξ∗ k(cid:96) 1 , we get I k(cid:96) 1 = Q, and thus Q \ P = I k(cid:96) (cid:88) (3.29) = for every (cid:96) ∈ IN. λk(cid:96) i (Ak(cid:96))(cid:62) ξ∗ k(cid:96)

it holds Q \ P = I(xk(cid:96), Ak(cid:96), bk(cid:96)) \ P ⊂ I k(cid:96)

k(cid:96) 1

≤ 0 for every (cid:96) ∈ IN , we get From (3.29) and the deﬁnition of I k(cid:96) 1 for every (cid:96) ∈ IN . Consequently, since (η∗ k(cid:96) )I

)Q\P ≤ 0 for every (cid:96) ∈ IN. (η∗ k(cid:96)

k, η∗

k, A∗

k, ξk) → (x∗, A∗, b∗, ξ), we have (η∗

k)i → b∗

i as k → ∞, for each )Q\P ≤ 0 for all (cid:96) ∈ IN , by letting (cid:96) → ∞

)Q = 0 and (η∗ k(cid:96)

As (u∗ i ∈ T . Because (η∗ k(cid:96) we obtain

Q\P ≤ 0. ¯A(cid:62)

i and u∗ k(cid:96)

(3.30)

Q = 0 and b∗ b∗ i → − (cid:80) i∈Q b∗ )i(Ak(cid:96))(cid:62) i i | i ∈ P (cid:9) + pos(cid:8) ¯A(cid:62) i ∈ span(cid:8) ¯A(cid:62)

i∈Q(η∗ k(cid:96) i ) ¯A(cid:62)

i∈Q

i , ξk(cid:96)(cid:105) = 0 for i → ¯A(cid:62) i for all i ∈ T and i , ξ(cid:105) ≤ 0 for i ∈ Q \ P .

i , ξk(cid:96)(cid:105) ≤ 0 for i ∈ Q \ P . Since (Ak(cid:96))(cid:62) i , ξ(cid:105) = 0 for i ∈ P and (cid:104) ¯A(cid:62)

= − (cid:80) Since u∗ k(cid:96) (cid:88) x∗ = (−b∗ → x∗, it holds i | i ∈ Q \ P (cid:9) = AQ,P ( ¯A).

i∈Q

k = ((a∗

k, ξk) → (x∗, A∗, b∗, ξ), using the representations A∗ k)ij → a∗

k, η∗ ij) we deduce that (a∗

(3.31) x∗ = − As ξk(cid:96) ∈ BQ,P (Ak(cid:96)), by the deﬁnition of the latter we have (cid:104)(Ak(cid:96))(cid:62) i ∈ P and (cid:104)(Ak(cid:96))(cid:62) ξk(cid:96) → ξ, we infer that (cid:104) ¯A(cid:62) This means that ξ ∈ BQ,P ( ¯A). We have thus shown that (cid:88) (x∗, ξ) ∈ AQ,P ( ¯A) × BQ,P ( ¯A), b∗ i ¯A(cid:62) i .

)Q = 0 for all (cid:96) ∈ IN yields A∗

Q\P,j ≥ 0 if ¯xj > 0. Therefore,

Since (u∗ k, A∗ k)ij) and A∗ = (a∗ ij as k → ∞, for all i ∈ T and j ∈ {1, . . . , n}. Combining this with (A∗ Q = 0. k(cid:96) We now consider an arbitrary index j ∈ {1, . . . , n}. If ¯xj < 0 (resp., ¯xj > 0), then we have (xk(cid:96))j < 0 (resp., (xk(cid:96))j > 0) for all (cid:96) large enough because xk(cid:96) → ¯x as (cid:96) → ∞. Remembering Q \ P ⊆ I k(cid:96) 1 , from these properties we get (a∗ )Q\P,j ≥ 0) for all (cid:96) large enough. By letting (cid:96) → ∞, k(cid:96) we have a∗

Q = 0,

)Q\P,j ≤ 0 (resp., (a∗ k(cid:96) Q\P,j ≤ 0 if ¯xj < 0 and a∗ A∗ a∗ Q\P,j ≤ 0 if ¯xj < 0, a∗ Q\P,j ≥ 0 if ¯xj > 0.

(3.32) From (3.30), (3.31), and (3.32) we conclude that (x∗, A∗, b∗, ξ) ∈ Σ(¯x, ¯A, ¯b, ¯ξ∗). (cid:50) The proof is complete.

(cid:110) (x∗, A∗, b∗)(cid:12)

(cid:111) (cid:12) (x∗, A∗, b∗, −ξ) ∈ Σ(¯x, ¯A, ¯b, ¯ξ∗)

(cid:110)

(3.33) By Theorem 3.2, setting Λ(¯x, ¯A, ¯b, ¯ξ∗)(ξ) =

(cid:12) (x∗, A∗, b∗, −ξ) ∈ N (cid:0)(¯x, ¯A, ¯b, ¯ξ∗); gphF (cid:1)(cid:111) ,

for every ξ ∈ IRn and recalling that (x∗, A∗, b∗)(cid:12) D∗F(¯x, ¯A, ¯b, ¯ξ∗)(ξ) =

we have

(3.34) D∗F(¯x, ¯A, ¯b, ¯ξ∗)(ξ) ⊂ Λ(¯x, ¯A, ¯b, ¯ξ∗)(ξ) for all ξ ∈ IRn.

The inclusion (3.34) provides us with an upper estimate for the values of the Mordukhovich coderivative of F(·) at the given point (¯x, ¯A, ¯b, ¯ξ∗) ∈ gphF.

i | i ∈ (cid:101)I} are linear independent and ¯ξ∗ ∈ pos{ ¯A(cid:62)

Remark 3.1 In connection with (3.27), it is worthy to note that H(¯x, ¯A, ¯b, ¯ξ∗) is a subset of Σ(¯x, ¯A, ¯b, ¯ξ∗) for any (¯x, ¯A, ¯b, ¯ξ∗) ∈ gphF. Indeed, if ¯ξ∗ = 0, we choose P = ∅ and Q = I with I = I(¯x, ¯A, ¯b). We now consider the case that ¯ξ∗ (cid:54)= 0. Since ¯ξ∗ (cid:54)= 0 and ¯ξ∗ ∈ pos{ ¯A(cid:62) i | i ∈ I}, by Lemma 2.1 there exists (cid:101)I ⊂ I such that { ¯A(cid:62) i | i ∈ (cid:101)I}. This implies that there exist multipliers λi ≥ 0 for i ∈ (cid:101)I such that ¯ξ∗ = (cid:80) ¯A(cid:62) i . i∈(cid:101)I λi In this case, we choose P = {i ∈ (cid:101)I| λi > 0} and Q = I. In both two cases of ¯ξ∗, by the deﬁnition of I1 := I1(¯x, ¯A, ¯b, ¯ξ∗) we have Q \ P ⊆ I1. By the choice of P and Q, applying Lemma 3.6 we get (cid:26) H(¯x, ¯A, ¯b, ¯ξ∗) =

i , b∗

(cid:12) (cid:12) (x∗, ξ) ∈ AQ,P ( ¯A) × BQ,P ( ¯A), (cid:12) (x∗, A∗, b∗, ξ) i∈Q b∗ i

Q = 0, b∗ I1

x∗ = − (cid:80) ¯A(cid:62) ≤ 0,

I1,j = 0 if ¯xj = 0,

Q = 0, a∗

(cid:27)

I1,j ≤ 0 if ¯xj < 0, a∗ a∗

I1,j ≥ 0 if ¯xj > 0

A∗

(cid:26)

(3.35)

⊂

(cid:12) (cid:12) (x∗, ξ) ∈ AQ,P ( ¯A) × BQ,P ( ¯A), (cid:12) (x∗, A∗, b∗, ξ) x∗ = − (cid:80) i∈Q b∗ i Q\P ≤ 0, A∗

¯A(cid:62) i ,

Q = 0,

(cid:27) .

Q\P,j ≥ 0 if ¯xj > 0

Q\P,j ≤ 0 if ¯xj < 0, a∗ a∗

b∗ Q = 0, b∗

Note that P ⊂ Q ⊂ I and P ∈ (cid:98)I with (cid:98)I = (cid:98)I(¯x, ¯A, ¯b, ¯ξ∗) given by (3.25). Hence, by the formula (3.26) the last set in (3.35) is a subset of Σ(¯x, ¯A, ¯b, ¯ξ∗). Therefore, from (3.35) we conclude that H(¯x, ¯A, ¯b, ¯ξ∗) ⊂ Σ(¯x, ¯A, ¯b, ¯ξ∗).

The next example is designed to show how the estimate (3.27) can be used

in practical calculations.

1 } = N (¯x; Θ( ¯A, ¯b)).

1 ∈ pos{ ¯A(cid:62)

3, 2

3) = 2

Example 3.2 Let T , F, ¯A, ¯b, ¯x, and ¯ξ∗ be given as in Example 3.1. Recall that I = I(¯x, ¯A, ¯b) = {1}, ¯I = T \ I = {2, 3}, and I1 = I1(¯x, ¯A, ¯b, ¯ξ∗) = ∅, where ¯ξ∗ = ( 2 ¯A(cid:62)

(cid:98)I(¯x, ¯A, ¯b, ¯ξ∗) = J (¯x, ¯A, ¯b, ¯ξ∗) = I(¯x, ¯A, ¯b, ¯ξ∗) = {I}, 58

We now use the formula (3.27) to established an upper estimate for the limiting normal cone N (cid:0)(¯x, ¯A, ¯b, ¯ξ∗); gphF (cid:1). Let I(¯x, ¯A, ¯b, ¯ξ∗), J (¯x, ¯A, ¯b, ¯ξ∗), and (cid:98)I(¯x, ¯A, ¯b, ¯ξ∗) be deﬁned via (3.23), (3.24), and (3.25) respectively. It is easy to verify that

where I = {1}. Thus, the conditions P ∈ (cid:98)I(¯x, ¯A, ¯b, ¯ξ∗) and P ⊂ Q ⊂ I are satisﬁed if and only if P = I and Q = I. Therefore, by (3.26) we obtain









(cid:32)

(cid:33)

(cid:32)

(cid:33)

 

 

Σ(¯x, ¯A, ¯b, ¯ξ∗) = 

1, a∗ b∗

11, a∗

12, ξ1 ∈ IR

 

  ,

 

  ,

 

(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)





11 a∗ a∗ 12 0 0 0 0

i | i ∈ I} = { ¯A(cid:62)

, . ξ1 −ξ1 −b∗ 1 −b∗ 1 b∗ 1 0 0

Comparing this with the result of Example 3.1, we receive the equality Σ(¯x, ¯A, ¯b, ¯ξ∗) = H(¯x, ¯A, ¯b, ¯ξ∗). Since { ¯A(cid:62) 1 } is positively linearly independent, applying (3.27) we get







(cid:32)

(cid:33)

(cid:32)

(cid:33)

 

 

N (cid:0)(¯x, ¯A, ¯b, ¯ξ∗); gphF (cid:1) ⊂ Σ(¯x, ¯A, ¯b, ¯ξ∗) = H(¯x, ¯A, ¯b, ¯ξ∗) 

 

  ,

 

 

  ,

(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)





= , . α, β, γ, µ ∈ IR α α µ −µ β γ 0 0 0 0 −α 0 0

In what follows, we apply the results of this and of the preceding section to study solution stability of parametric aﬃne variational inequalities (AVIs, for short), where both the basic operator and the constraint set undergo nonlinear perturbations.

3.4 AVIs under Nonlinear Perturbations

Let us ﬁrst recall an upper estimate for the values of the Mordukhovich coderivative of the implicit multifunction

S(p) := {x ∈ IRn| 0 ∈ F (x, p)},

(cid:91)

(cid:8)p(cid:48) ∈ IRd| (−x(cid:48), p(cid:48)) ∈ D∗F (¯z)(v(cid:48))(cid:9).

where F : IRn × IRd ⇒ IRm is an arbitrary multifunction. Given any point (¯p, ¯x) ∈ gphS, we set ¯z := (¯x, ¯p, 0) ∈ IRn+d+m. It is clear that ¯z ∈ gphF . For each x(cid:48) ∈ IRn, following [22] we put

v(cid:48)∈IRm

Ω(x(cid:48)) :=

Theorem 3.3 (See [24, Theorem 2.1(a)] and [22, Theorem 3.1]) Let F be locally closed around ¯z ∈ gphF , i.e., there is ρ > 0 such that the intersection of gphF with the closed ball centered at ¯z with radius ρ is a closed set. If kerD∗F (¯z) = {0}, then

D∗S(¯p, ¯x)(x(cid:48)) ⊂ Ω(x(cid:48))

is valid for every x(cid:48) ∈ IRn.

In [22], Lee and Yen have shown that the inclusion D∗S(¯p, ¯x)(x(cid:48)) ⊂ Ω(x(cid:48))

may not hold if the regularly condition kerD∗F (¯z) = {0} is violated.

We now consider the implicit multifunction S : IRm×n × IRm × IRn ⇒ IRn

deﬁned by setting

(3.36) S(A, b, q) = (cid:8)x ∈ IRn| 0 ∈ M x − q + F(x, A, b)(cid:9)

with M ∈ IRn×n being a ﬁxed matrix, (A, b, q) ∈ IRm×n×IRm×IRn a parameter, and F(x, A, b) = N (x; Θ(A, b)) given by (3.4). Note that S(A, b, q) can be represented as the solution set of a parametric aﬃne variational inequality as follows

S(A, b, q) = (cid:8)x ∈ Θ(A, b)| (cid:104)M x − q, u − x(cid:105) ≥ 0, ∀u ∈ Θ(A, b)(cid:9). Fix any ¯x ∈ S( ¯A, ¯b, ¯q) and note that ¯ϑ := (¯x, ¯A, ¯b, ¯q) belongs to gphS−1. For every (A∗, b∗, q∗) ∈ IRm×n × IRm × IRn, by the deﬁnition of the Mordukhovich coderivative we have

D∗S−1( ¯ϑ)(A∗, b∗, q∗)

= (cid:8)x∗ ∈ IRn| (x∗, −A∗, −b∗, −q∗) ∈ N ((¯x, ¯A, ¯b, ¯q); gphS−1)(cid:9) = (cid:8)x∗ ∈ IRn| (−A∗, −b∗, −q∗, x∗) ∈ N (( ¯A, ¯b, ¯q, ¯x); gphS)(cid:9) = (cid:8)x∗ ∈ IRn| (−A∗, −b∗, −q∗) ∈ D∗S( ¯A, ¯b, ¯q, ¯x)(−x∗)(cid:9).

Hence,

(3.37) D∗S−1( ¯ϑ)(0) = (cid:8)x∗ ∈ IRn| 0 ∈ D∗S( ¯A, ¯b, ¯q, ¯x)(−x∗)(cid:9),

where 0 := (0IRm×n, 0IRm, 0IRn).

The following statement is a generalization of Proposition 3.2 where the

(cid:16) ¯B(¯x, δ) × ¯B( ¯A, δ) × ¯B(¯b, δ) × IRn × IRn(cid:17)

case M = 0IRn×n and q = 0IRn was treated.

Lemma 3.7 Let (cid:101)F(x, A, b, q) = M x−q+F(x, A, b) and (¯x, ¯A, ¯b, ¯q, ¯ξ∗) ∈ gph (cid:101)F. i | i ∈ I(¯x, ¯A, ¯b)} are positively linearly independent, then there exists If { ¯A(cid:62) δ > 0 such that (cid:101)G := gph (cid:101)F ∩ is a closed set.

Proof. Arguing as in the ﬁrst part of the proof of Proposition 3.2, we can i | i ∈ I(¯x, ¯A, ¯b)} are positively linearly independent ﬁnd δ > 0 such that {A(cid:62) for all A ∈ ¯B( ¯A, δ), and I(x, A, b) ⊂ I(¯x, ¯A, ¯b) whenever (x, A, b) belongs to ¯B(¯x, δ)× ¯B( ¯A, δ)× ¯B(¯b, δ). We now prove that (cid:101)G is closed. Given any sequence 60

k)} in (cid:101)G with (xk, Ak, bk, qk, ξ∗

k) → ((cid:98)x, (cid:98)A, (cid:98)b, (cid:98)q, (cid:98)ξ∗), we are going {(xk, Ak, bk, qk, ξ∗ to show that ((cid:98)x, (cid:98)A, (cid:98)b, (cid:98)q, (cid:98)ξ∗) ∈ (cid:101)G. Since (xk, Ak, bk) converges to ((cid:98)x, (cid:98)A, (cid:98)b) and ¯B(¯x, δ) × ¯B( ¯A, δ) × ¯B(¯b, δ) is closed, ((cid:98)x, (cid:98)A, (cid:98)b) ∈ ¯B(¯x, δ) × ¯B( ¯A, δ) × ¯B(¯b, δ). Hence, I(xk, Ak, bk) ⊂ I((cid:98)x, (cid:98)A, (cid:98)b) ⊂ I(¯x, ¯A, ¯b) for every k suﬃciently large. We can assume that I(xk, Ak, bk) = (cid:101)I ⊂ I((cid:98)x, (cid:98)A, (cid:98)b) ⊂ I(¯x, ¯A, ¯b) for all k ∈ IN , where (cid:101)I is a ﬁxed index set. As (xk, Ak, bk, qk, ξ∗

k) ∈ gph (cid:101)F, it holds k + qk − M xk ∈ F(xk, Ak, bk) = N (xk; Θ(Ak, bk)) = pos{(Ak)(cid:62) ξ∗

i | i ∈ (cid:101)I}

i ≥ 0 for i ∈ (cid:101)I such

(cid:88)

for every k ∈ IN . Thus, for each k ∈ IN , there exist λk that

i (Ak)(cid:62) λk

i = ξ∗

k + qk − M xk → (cid:98)ξ∗ + (cid:98)q − M (cid:98)x.

i∈(cid:101)I

i | i ∈ I(¯x, ¯A, ¯b)} are positively linearly independent, arguing as in Since { ¯A(cid:62) the second part of the proof of Proposition 3.2 we can ﬁnd a subsequence k(cid:96) of {k} such that

(cid:88)

(3.38)

i =

i ∈ N ((cid:98)x; Θ( (cid:98)A, (cid:98)b)) = F((cid:98)x, (cid:98)A, (cid:98)b).

i∈(cid:101)I

(3.39) λi (cid:98)A(cid:62) λk(cid:96) i (Ak(cid:96))(cid:62) lim (cid:96)→∞

From (3.38) and (3.39) we deduce that (cid:98)ξ∗ + (cid:98)q − M (cid:98)x ∈ F((cid:98)x, (cid:98)A, (cid:98)b). This means (cid:50) that (cid:98)ξ∗ ∈ (cid:101)F((cid:98)x, (cid:98)A, (cid:98)b, (cid:98)q). Therefore, ((cid:98)x, (cid:98)A, (cid:98)b, (cid:98)q, (cid:98)ξ∗) ∈ (cid:101)G.

(cid:26)

(cid:91)

Let ¯w = ( ¯A, ¯b, ¯q, ¯x) ∈ gphS, where the multifunction S(·) is deﬁned by (3.36). Setting ¯ξ∗ = ¯q − M ¯x, we have (¯x, ¯A, ¯b, ¯ξ∗) ∈ gphF. For each x∗ ∈ IRn, we put

(cid:12) (A∗, b∗, q∗) (cid:12) (cid:12) (−x∗, A∗, b∗, q∗) ∈ M (cid:62)ξ × {(0IRm×n, 0IRm, 0IRn)}

ξ∈IRn

K( ¯w)(x∗) =

(cid:27)

+{(0IRn, 0IRm×n, 0IRm)} × {−ξ}

+D∗F(¯x, ¯A, ¯b, ¯ξ∗)(ξ) × {0IRn}

(cid:26)

(cid:91)

and

(cid:12) (A∗, b∗, q∗) (cid:12) (cid:12) (−x∗, A∗, b∗, q∗) ∈ M (cid:62)ξ × {(0IRm×n, 0IRm, 0IRn)}

ξ∈IRn

L( ¯w)(x∗) =

(cid:27) ,

+{(0IRn, 0IRm×n, 0IRm)} × {−ξ}

+Λ(¯x, ¯A, ¯b, ¯ξ∗)(ξ) × {0IRn}

where Λ(¯x, ¯A, ¯b, ¯ξ∗)(ξ) is given by (3.33).

i | i ∈ I(¯x, ¯A, ¯b)} are positively linearly independent.

i | i ∈ I(¯x, ¯A, ¯b)} are positively linearly independent, we

Remark 3.2 According to (3.34), the inclusion K( ¯w)(x∗) ⊂ L( ¯w)(x∗) holds if the vectors { ¯A(cid:62)

Theorem 3.4 If { ¯A(cid:62) obtain the estimates

(3.40) D∗S( ¯w)(x∗) ⊂ K( ¯w)(x∗) ⊂ L( ¯w)(x∗)

for every x∗ ∈ IRn.

Proof. As in Lemma 3.7, we deﬁne (cid:101)F(x, A, b, q) = M x − q + F(x, A, b). Let ¯z = (¯x, ¯A, ¯b, ¯q, 0IRn) ∈ gph (cid:101)F. Then, according to the coderivative sum rules [28, Theorem 1.62], we obtain

D∗ (cid:101)F(¯z)(ξ) = M (cid:62)ξ × {(0IRm×n, 0IRm, 0IRn)}

(3.41)

+ {(0IRn, 0IRm×n, 0IRm)} × {−ξ} + D∗F(¯x, ¯A, ¯b, ¯ξ∗)(ξ) × {0IRn} for every ξ ∈ IRn, where ¯ξ∗ = ¯q −M ¯x. By (3.41) it follows that 0 ∈ D∗ (cid:101)F(¯z)(ξ) if and only if ξ = 0. This means that kerD∗ (cid:101)F(¯z) = {0}. Since the vectors i | i ∈ I(¯x, ¯A, ¯b)} are positively linearly independent, gph (cid:101)F is locally closed { ¯A(cid:62) around ¯z by Lemma 3.7. Applying Theorem 3.3 to the implicit multifunction S(A, b, q) = (cid:8)x ∈ IRn| 0 ∈ (cid:101)F(x, A, b, q)(cid:9),

which is exactly the multifunction S(·) given by (3.36), and recalling (3.41), we get the inclusion D∗S( ¯w)(x∗) ⊂ K( ¯w)(x∗). From Remark 3.2 it holds (cid:50) K( ¯w)(x∗) ⊂ L( ¯w)(x∗). Thus, the inclusions in (3.40) are valid.

As deﬁned in Chapter 1, S(·) is locally metrically regular around the point ( ¯A, ¯b, ¯q, ¯x) ∈ gphS if there are µ > 0, γ > 0, and neighborhoods U of ¯x and V of ( ¯A, ¯b, ¯q) such that

dist(cid:0)(A, b, q); S−1(x)(cid:1) ≤ µ dist(cid:0)x; S(A, b, q)(cid:1) for all x ∈ U and (A, b, q) ∈ V satisfying dist(cid:0)x; S(A, b, q)(cid:1) ≤ γ. If S(·) is locally closed around ( ¯A, ¯b, ¯q, ¯x), then Theorem 1.4 tells us that S(·) is locally metrically regular around ( ¯A, ¯b, ¯q, ¯x) if and only if D∗S−1(¯x, ¯A, ¯b, ¯q)(0) = {0}

with 0 := (0IRm×n, 0IRm, 0IRn). By (3.37), the last equality can be rewritten equivalently as

(3.42) kerD∗S( ¯A, ¯b, ¯q, ¯x) = {0}.

Recall that S(·) is locally Lipschitz-like around ( ¯A, ¯b, ¯q, ¯x) ∈ gphS if there exist a constant (cid:96) > 0 and neighborhoods U of ¯x, V of ( ¯A, ¯b, ¯q) such that

S(A, b, q) ∩ U ⊂ S(A(cid:48), b(cid:48), q(cid:48)) + (cid:96)(cid:107)(A, b, q) − (A(cid:48), b(cid:48), q(cid:48))(cid:107) ¯BIRn for all (A, b, q), (A(cid:48), b(cid:48), q(cid:48)) ∈ V , where ¯BIRn denotes the closed unit ball in IRn. If S(·) is locally closed around ( ¯A, ¯b, ¯q, ¯x), then Theorem 1.3 asserts that S(·) is locally Lipschitz-like around ( ¯A, ¯b, ¯q, ¯x) if and only if

i | i ∈ I(¯x, ¯A, ¯b)}

(3.43) D∗S( ¯A, ¯b, ¯q, ¯x)(0) = {0}.

Theorem 3.5 Let ¯w = ( ¯A, ¯b, ¯q, ¯x) ∈ gphS and the vectors { ¯A(cid:62) be positively linearly independent. The following assertions are valid:

(i) If kerL( ¯w) = {0}, then S(·) is locally metrically regular around ¯w.

i | i ∈ I(¯x, ¯A, ¯b)} are positively linearly independent, Proof. The vectors { ¯A(cid:62) thus the multifunction (cid:101)F(x, A, b, q) = M x − q + F(x, A, b) is locally closed around (¯x, ¯A, ¯b, ¯q, 0IRn) ∈ gph (cid:101)F by Lemma 3.7. Fix ρ > 0 such that the set gph (cid:101)F ∩ (cid:8)(x, A, b, q, ξ∗)(cid:12) (cid:12) (cid:107)x − ¯x(cid:107) + (cid:107)A − ¯A(cid:107) + (cid:107)b − ¯b(cid:107) + (cid:107)q − ¯q(cid:107) + (cid:107)ξ∗(cid:107) ≤ ρ(cid:9)

(ii) If L( ¯w)(0) = {0}, then S(·) is locally Lipschitz-like around ¯w.

is closed. Then, the set

(cid:12) (cid:107)x − ¯x(cid:107) + (cid:107)A − ¯A(cid:107) + (cid:107)b − ¯b(cid:107) + (cid:107)q − ¯q(cid:107) ≤ ρ(cid:9)

gph (cid:101)F ∩ (cid:8)(x, A, b, q, 0IRn)(cid:12)

is also closed. Since (A, b, q, x) ∈ gphS if and only if (x, A, b, q, 0IRn) ∈ gph (cid:101)F, the latter implies the closedness of

(cid:12) (cid:107)A − ¯A(cid:107) + (cid:107)b − ¯b(cid:107) + (cid:107)q − ¯q(cid:107) + (cid:107)x − ¯x(cid:107) ≤ ρ(cid:9).

gphS ∩ (cid:8)(A, b, q, x)(cid:12)

This means that S(·) is locally closed around ¯w. The assumption that i | i ∈ I(¯x, ¯A, ¯b)} are positively linearly independent guaranties (3.40). { ¯A(cid:62) (i) Combining the condition kerL( ¯w) = {0} with (3.40) yields (3.42) which

establishes the local metric regularity of S(·) around ¯w.

i | i ∈ I(¯x, ¯A, ¯b)} Theorem 3.6 Let ¯w = ( ¯A, ¯b, ¯q, ¯x) ∈ gphS. If the vectors { ¯A(cid:62) are positively linearly independent, then there exists δ > 0 such that S(·) is locally metrically regular around any point w ∈ ¯B( ¯w, δ) ∩ gphS.

(ii) Taking account of (3.40) and the assumption L( ¯w)(0) = {0} we obtain (cid:50) (3.43) which ensures that S(·) is locally Lipschitz-like around ¯w.

i | i ∈ I(¯x, ¯A, ¯b)} implies that there is δ > 0 such that {A(cid:62)

Proof. By Lemma 3.1, the positive linear independence of the vectors i | i ∈ I(¯x, ¯A, ¯b)} { ¯A(cid:62) are positively linearly independent for any A ∈ ¯B( ¯A, δ). Choose δ as small as I(x, A, b) ⊂ I(¯x, ¯A, ¯b) for every (A, b, q, x) ∈ ¯B( ¯w, δ) ∩ gphS. For any (cid:98)w = ( (cid:98)A, (cid:98)b, (cid:98)q, (cid:98)x) ∈ ¯B( ¯w, δ) ∩ gphS, since

i | i ∈ I(¯x, ¯A, ¯b)} are positively linearly independent. Because i | i ∈ I((cid:98)x, (cid:98)A, (cid:98)b)} are also positively

(cid:107) (cid:98)A − ¯A(cid:107) ≤ (cid:107) (cid:98)A − ¯A(cid:107) + (cid:107)(cid:98)b − ¯b(cid:107) + (cid:107)(cid:98)q − ¯q(cid:107) + (cid:107)(cid:98)x − ¯x(cid:107) = (cid:107) (cid:98)w − ¯w(cid:107) ≤ δ,

the vectors { (cid:98)A(cid:62) I((cid:98)x, (cid:98)A, (cid:98)b) ⊂ I(¯x, ¯A, ¯b), the vectors { (cid:98)A(cid:62) linearly independent.

(cid:26)

We now consider an arbitrary x∗ ∈ kerL( (cid:98)w). The inclusion x∗ ∈ kerL( (cid:98)w) means that

(cid:12) (cid:12) (−x∗, A∗, b∗, q∗) ∈ M (cid:62)ξ × {(0IRm×n, 0IRm, 0IRn)} (A∗, b∗, q∗) (cid:12)

ξ∈IRn

(cid:27) ,

0 ∈ L( (cid:98)w)(x∗) (cid:91) =

+{(0IRn, 0IRm×n, 0IRm)} × {−ξ} + Λ((cid:98)x, (cid:98)A, (cid:98)b, (cid:98)ξ∗)(ξ) × {0IRn}

i (cid:98)A(cid:62)

i∈Q b∗

1, . . . , b∗

where 0 = (0IRm×n, 0IRm, 0IRn) and (cid:98)ξ∗ = (cid:98)q − M (cid:98)x. The last inclusion implies that 0IRn ∈ {−ξ} and

The positive linear independence of the vectors { (cid:98)A(cid:62)

(−x∗, 0IRm×n, 0IRm) ∈ M (cid:62)ξ × {0IRm×n} × {0IRm} + Λ((cid:98)x, (cid:98)A, (cid:98)b, (cid:98)ξ∗)(ξ). Thus, it holds (−x∗, 0IRm×n, 0IRm) ∈ Λ((cid:98)x, (cid:98)A, (cid:98)b, (cid:98)ξ∗)(0IRn). By (3.33), the last inclusion yields (−x∗, 0IRm×n, 0IRm, 0IRn) ∈ Σ((cid:98)x, (cid:98)A, (cid:98)b, (cid:98)ξ∗). From this and (3.26) we deduce that there exists Q ⊂ I((cid:98)x, (cid:98)A, (cid:98)b) such that x∗ = (cid:80) i , where b∗ = (b∗ m) = 0IRm. Hence, x∗ = 0. We have shown that kerL( (cid:98)w) = {0}. i | i ∈ I((cid:98)x, (cid:98)A, (cid:98)b)} and the property kerL( (cid:98)w) = {0} allow us to apply Theorem 3.5(i) to assert that S(·) is locally metrically regular around (cid:98)w. Since (cid:98)w can be chosen arbitrary in ¯B( ¯w, δ) ∩ gphS, the proof is complete. (cid:50)

Example 3.3 As in [49, Example 2], we consider the parametric nonconvex quadratic programming problem (Pε) of minimizing the function

1 − x2 x2

2 + x1 − εx2

fε(x) = − 1 2

on ∆ = {x ∈ IR2| x ≥ 0, x1 +x2 ≤ 2}, where ε is a parameter. Denote by (cid:98)S(ε) the Karush-Kuhn-Tucker (KKT) point set of (Pε). The KKT point set (cid:98)S(ε)

is the solution set of the following parametric aﬃne variational inequality problem

(3.44) Find x ∈ ∆ subject to (cid:104)∇fε(x), u − x(cid:105) ≥ 0, ∀u ∈ ∆,

(cid:32)

(cid:33)

(cid:32)

(cid:33)

where ∇fε(x) = M x − qε with

M = . and qε = 0 −1 0 −2 −1 ε

We see that ¯x is a solution of (3.44) if and only if ¯x is a solution of the following generalized equation

0 ∈ M ¯x − qε + N (¯x; ∆).

(cid:26)

(cid:27)

(cid:19)

By Example 2 in [49], we have

(cid:98)S(0) =

(cid:18)5 3

(cid:26)

(0, 0), (1, 0), (2, 0), , (0, 2) , ,

(cid:27)

(cid:98)S(ε) =

(cid:18)5 + ε 3

1 3 (cid:19) (2, 0), , , (0, 2) 1 − ε 3

for ε > 0 small enough, and the multifunction ε (cid:55)→ (cid:98)S(ε) is not lower semicon- tinuous at ε = 0. We consider the implicit multifunction deﬁned via (3.36) as follows

S : IR3×2 × IR3 × IR2 ⇒ IR2, S(A, b, q) = (cid:8)x ∈ IR2| 0 ∈ M x − q + F(x, A, b)(cid:9),







(cid:32)

(cid:33)

where Θ(A, b) = {x ∈ IR2| Ax ≤ b} and F(x, A, b) = N (x; Θ(A, b)). Let

 

  ,

 

3, 1

3) ∈ S( ¯A, ¯b, ¯q) and Then, Θ( ¯A, ¯b) = ∆ and S( ¯A, ¯b, ¯q) ≡ (cid:98)S(0). Hence, ¯x = ( 5 ¯w := ( ¯A, ¯b, ¯q, ¯x) ∈ gphS. We use the criterion Theorem 3.5(ii) to verify the local Lipschitz-like property of S around ¯w. For any x∗ ∈ IR2, we have

(cid:91)

(cid:26) (A∗, b∗, q∗)(cid:12)

¯A = ¯b = and ¯q = . −1 0 1 1 −1 0 0 −1 2 0 0

(cid:12) (−x∗, A∗, b∗, q∗) ∈ M (cid:62)ξ × {(0IR3×2, 0IR3, 0IR2)}

ξ∈IR2

L( ¯w)(x∗) =

+{(0IR2, 0IR3×2, 0IR3)} × {−ξ} (cid:27) , +Λ(¯x, ¯A, ¯b, ¯ξ∗)(ξ) × {0IR2}

 

where ¯ξ∗ = ¯q − M ¯x. Hence, the inclusion (A∗, b∗, q∗) ∈ L( ¯w)(0IR2) means that there exists ξ ∈ IR2 such that



(3.45) (−M (cid:62)ξ, A∗, b∗) ∈ Λ(¯x, ¯A, ¯b, ¯ξ∗)(ξ) q∗ = −ξ.

 

By (3.33), we can rewrite (3.45) as follows







(3.46) (−M (cid:62)ξ, A∗, b∗, −ξ) ∈ Σ(¯x, ¯A, ¯b, ¯ξ∗) q∗ = −ξ.



(cid:32)

(cid:33)

(cid:32)

(cid:33)

 

According to Example 3.2, we obtain  

  ,

 

  ,

 

 (cid:12) (cid:12)  (cid:12) (cid:12)  (cid:12)

  . 



, α, β, γ, µ ∈ IR Σ(¯x, ¯A, ¯b, ¯ξ∗) = µ −µ α α −α 0 0 β γ 0 0 0 0



(cid:32)

(cid:33)

(cid:32)

(cid:33)

Therefore, the condition (3.46) is satisﬁed if and only if 

 

  , b∗ = 

 

  , q∗ = −ξ = 

−M (cid:62)ξ = , A∗ = , α α µ −µ β γ 0 0 0 0 −α 0 0

(3.47) where α, β, γ, µ ∈ IR. It is easy to verify that the condition (3.47) forces ξ = 0, α = 0, µ = 0. However, β and γ can be arbitrary chosen. Consequently, b∗ = 0IR3, q∗ = 0IR2, and A∗ may be diﬀerent from 0IR3×2. This implies that L( ¯w)(0IR2) (cid:54)= {(0IR3×2, 0IR3, 0IR2)}. The latter shows that by using the criterion Theorem 3.5(ii) we cannot conclude about the locally Lipschitz-like property of S(·) around ¯w.

3.5 Conclusions

In this chapter, nonlinearly perturbed polyhedral normal cone mappings in ﬁnite dimensional spaces have been studied. Upper estimates for the Fr´echet and the limiting normal cones to the graphs of such normal cone mappings were given respectively in Theorems 3.1 and 3.2. New results on solution stability of parametric aﬃne variational inequalities under nonlinear pertur- bations, shown in Theorems 3.5 and 3.6, are derived from these estimates. The problem of ﬁnding exact formulas for computing the Fr´echet normal cone and the limiting normal cone remains open.

Chapter 4

A Class of Linear Generalized

Equations

Solution stability of a class of linear generalized equations in ﬁnite dimen- sional Euclidean spaces is investigated in this chapter by means of gener- alized diﬀerentiation. We establish exact formulas for the Fr´echet and the Mordukhovich coderivatives of the normal cone mappings to perturbed Eu- clidean balls. Necessary and suﬃcient conditions for the local Lipschitz-like property of the solution maps of such the linear generalized equations are derived from the coderivative formulas for the normal cone mappings. These conditions lead to new results on stability of the parametric trust-region sub- problems. The two open problems stated by Lee and Yen in [23] are solved.

The results presented below are taken from [42].

4.1 Linear Generalized Equations

The concept of generalized equation introduced by Robinson [43] has been recognized as an eﬃcient tool for dealing with various questions in optimiza- tion theory. It is also a uniﬁed framework for studying equilibrium problems. When the basic single-valued operator of the generalized equation is aﬃne and the accompanying set-valued map is the normal cone operator of a ﬁxed closed convex set called the constraint set, one has a linear generalized equa- tion (linear GE for brevity). Robinson [43, Theorem 2] proved that if a linear GE is monotone and the solution set is nonempty and bounded, then the solution map is locally upper Lipschitzian with respect to the parame-

ters describing the aﬃne operator. This important result has found many applications (see, e.g., [55]).

Linear GEs with perturbed constraint sets have been studied in [20] and

[25] (see also the references therein).

In connection with the solution methods [34], [51] and the qualitative study [21] for the trust-region subproblems, we are interested in the linear GEs of the form

(4.1) 0 ∈ Ax + b + N (x; E(α)),

(cid:12) (cid:107)x(cid:107) ≤ α}, and

where symmetric n × n matrix A ∈ IRn×n, vector b ∈ IRn, and real number α > 0 are parameters, E(α) := {x ∈ IRn(cid:12)

 



{v ∈ IRn| (cid:104)v, y − x(cid:105) ≤ 0, ∀y ∈ E(α)}, if x ∈ E(α) (4.2) N (x; E(α)) := ∅, if x (cid:54)∈ E(α)

is the normal cone to E(α) at x. The solution set of (4.1) is denoted by S(A, b, α). Note that (4.1) is a linear GE where the perturbation of the constraint set E(α) is described by parameter α ∈ (0, +∞). Here E(α) is a ball centered at 0 with radius α.

(cid:26)

(cid:27)

If x is a local solution of the optimization problem

(cid:12) x(cid:62)Ax + b(cid:62)x (cid:12) (cid:12) x ∈ E(α)

min (4.3) f (x) = , 1 2

which is called the trust-region subproblem, then (4.1) holds due to the gen- eralized Fermat rule (see, e.g., [20, p. 85]). The trust-region subproblems given by (4.3) are at a frequent use in the development of the trust-region methods [10]. Here and in the sequel, the apex (cid:62) denotes matrix transposi- tion. It is well-known [26] that if x ∈ E(α) is a local minimum of (4.3), then there exists a Lagrange multiplier λ ≥ 0 such that

(4.4) (A + λI)x = −b, λ((cid:107)x(cid:107) − α) = 0,

where I denotes the n × n unit matrix. If x ∈ E(α) and there exists λ ≥ 0 satisfying (4.4), x is said to be a Karush-Kuhn-Tucker point (or a KKT point) of (4.3) and (x, λ) is called a KKT pair. For each KKT point x, the Lagrange multiplier λ is deﬁned uniquely (see, e.g., [21]). Recall [19] that x is a KKT point of (4.3) if and only if

(cid:104)Ax + b, y − x(cid:105) ≥ 0, ∀y ∈ E(α).

Thus, the solution set of (4.1) coincides with the Karush-Kuhn-Tucker point set of (4.3).

The purpose of this chapter is to investigate the stability of (4.1) with re- spect to the perturbations of all the three components of its data set {A, b, α}. Our main tools are the Mordukhovich criterion (see [28, Theorem 4.10] and [48, Theorem 9.40], or Theorem 1.3) for the local Lipschitz-like property of multifunctions between ﬁnite dimensional normed spaces and some lower and upper estimates for coderivatives of implicit multifunctions from [22]. Our results develop furthermore the preceding work of Lee and Yen [23] on the stability of (4.1). To be more precise, we provide a complete solution for the open problems raised in [23, Remarks 3.6 and 3.13] by giving exact for- mulas for the Fr´echet an the Mordukhovich coderivatives of the normal cone mapping (x, α) (cid:55)→ N (x; E(α)). Moreover, we complement the suﬃcient con- ditions for stability of the solution set of (4.1) given in [23, Theorem 5.1] by a more comprehensive necessary and suﬃcient conditions for stability.

This chapter shows how the generalized diﬀerentiation theory [28], [48] can be applied with a success for analyzing a typical polynomial optimization problem of the form (4.3).

Our approach to the analysis of the parametric problem (4.3) is quite dif- ferent from that one adopted by Lee, Tam and Yen [21]. It is worthy to stress that the focus point of [21] is the lower semicontinuity of the solution map of (4.1), while our aim is to characterize the local Lipschitz-like property of that map. The latter is stronger than the inner semicontinuity of the solution map, which is the basis for deﬁning the above-mentioned lower semicontinu- ity. It is still unclear to us whether the inner semicontinuity property [28, p. 42] of a multifunction can be characterized by using coderivatives, or not.

4.2 Formulas for Coderivatives

The normal cone N (x; E(α)) can be computed explicitly. Namely, we have the formula

if (cid:107)x(cid:107) < α

(4.5) N (x; E(α)) = if (cid:107)x(cid:107) = α

 {0},  {µx| µ ≥ 0},  ∅,

if (cid:107)x(cid:107) > α.

 

For every (x, α) ∈ IRn × IR, we put



N (x; E(α)), if α > 0 (4.6) N (x, α) = ∅, if α ≤ 0,

where N (x; E(α)) is given by (4.2). Thus, N : IRn × IR ⇒ IRn is a mul- tifunction with closed convex values. It is called the normal cone mapping of the closed ball E(α). Setting y = −b, w = (A, α), G(x, w) = Ax, and M (x, w) = N (x, α), we can rewrite (4.1) equivalently as

(4.7) y ∈ G(x, w) + M (x, w).

It is clear that

(cid:12) y ∈ G(x, w) + M (x, w)(cid:9).

(4.8) S(A, b, α) = (cid:101)S(w, y) := (cid:8)x ∈ IRn(cid:12)

Hence, the solution map

S : H(n) × IRn × IR ⇒ IRn, (A, b, α) (cid:55)→ S(A, b, α),

of (4.1) can be interpreted as the implicit multifunction

(cid:101)S : W × IRn ⇒ IRn,

(4.9) (w, y) (cid:55)→ (cid:101)S(w, y),

where W := H(n) × IR with H(n) ⊂ IRn×n being the linear subspace of symmetric n × n matrices of IRn×n.

By (cid:92)(u, v) we denote the angle between nonzero vectors u and v in IRn, i.e., (cid:92)(u, v) ∈ [0, π] and (cid:104)u, v(cid:105) = (cid:107)u(cid:107) · (cid:107)v(cid:107) cos(cid:92)(u, v). For each pair u, v ∈ IRn with u = (u1, . . . , un)(cid:62) and v = (v1, . . . , vn)(cid:62), we deﬁne the vector −→uv in IRn by setting −→uv = (v1 − u1, . . . , vn − un)(cid:62). For any x, y, z ∈ IRn, we call (cid:100)xyz the angle between −→yx and −→yz, provided the latter vectors are nonzero.

We are going to obtain exact formulas for the Fr´echet and the Mordukhovich

coderivatives of the normal cone mapping N (x, α) given by (4.6).

Fix any point (x, α, v) ∈ gphN .

4.2.1 The Fr´echet Coderivative of N (x, α)

The following results are due to Lee and Yen [23].

(cid:98)D∗N (x, α, v)(v(cid:48)) = {(0IRn, 0IR)},

Lemma 4.1 (See [23, Lemma 3.1]) If (cid:107)x(cid:107) < α, then v = 0 and

for every v(cid:48) ∈ IRn.

Lemma 4.2 (See [23, Lemma 3.2]) If (cid:107)x(cid:107) = α and v (cid:54)= 0, then v = µx for some µ > 0. If (x(cid:48), α(cid:48)) ∈ (cid:98)D∗N (x, α, v)(v(cid:48)), then

x(cid:48) = − x + µv(cid:48), (cid:104)v(cid:48), x(cid:105) = 0. α(cid:48) α

Lemma 4.1 describes the Fr´echet coderivative (cid:98)D∗N (x, α, v) in the case (cid:107)x(cid:107) < α. Lemma 4.2 gives an upper estimate for the Fr´echet coderivative value (cid:98)D∗(x, α, v)(v(cid:48)) in the case (cid:107)x(cid:107) = α, v (cid:54)= 0. The ﬁrst part of the open problem raised in [23, Remark 3.6] can be reformulated as follows: Is the upper estimate provided by Lemma 4.2 an exact one? The next statement, which answers this question in the aﬃrmative, establishes an exact formula for computing the coderivative (cid:98)D∗N (x, α, v) in the situation (cid:107)x(cid:107) = α and v (cid:54)= 0.

(cid:8)(x(cid:48), α(cid:48)) ∈ IRn × IR(cid:12)

Theorem 4.1 If (cid:107)x(cid:107) = α and v (cid:54)= 0, then v = µx with µ = (cid:107)v(cid:107) · (cid:107)x(cid:107)−1 and, for every v(cid:48) ∈ IRn,

(cid:12) x(cid:48) = − α(cid:48)

 

α x + µv(cid:48)(cid:9),

(cid:98)D∗N (x, α, v)(v(cid:48)) =



if (cid:104)v(cid:48), x(cid:105) = 0

∅,

if (cid:104)v(cid:48), x(cid:105) (cid:54)= 0. (4.10)

Proof. The property v = µx with µ = (cid:107)v(cid:107) · (cid:107)x(cid:107)−1 and the inclusion “ ⊂ ” of (4.10) are immediate from Lemma 4.2.

To prove the opposite inclusion of (4.10), suppose to the contrary that there exists (x(cid:48), α(cid:48)) belonging to the set described by the right-hand side of (4.10) with (x(cid:48), α(cid:48)) (cid:54)∈ (cid:98)D∗N (x, α, v)(v(cid:48)). Then (x(cid:48), α(cid:48), −v(cid:48)) (cid:54)∈ (cid:98)N ((x, α, v); gphN ). So there exist a sequence (xk, αk, vk) gphN−→ (x, α, v) and a constant δ > 0 such that

(4.11) ≥ δ, Pk := (cid:104)x(cid:48), xk − x(cid:105) + α(cid:48)(αk − α) − (cid:104)v(cid:48), vk − v(cid:105) (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk − v(cid:107)

for all k ∈ IN . By the choice of (x(cid:48), α(cid:48)), we have

α x + µv(cid:48), xk − x(cid:105) + α(cid:48)(αk − α) − (cid:104)v(cid:48), vk − v(cid:105)

(cid:104)− α(cid:48) Pk = (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk − v(cid:107)

α x, xk − x(cid:105) + α(cid:48)(αk − α)

(cid:104)− α(cid:48) + = (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk − v(cid:107) (cid:104)µv(cid:48), xk − x(cid:105) − (cid:104)v(cid:48), vk − v(cid:105) (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk − v(cid:107)

α x, xk − x(cid:105) + α(cid:48)(αk − α)

(cid:104)− α(cid:48) = + (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk − v(cid:107) (cid:104)µv(cid:48), xk(cid:105) − (cid:104)v(cid:48), vk(cid:105) (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk − v(cid:107)

= Qk + Rk,

where

α x, xk − x(cid:105) + α(cid:48)(αk − α)

(cid:104)− α(cid:48) , Qk := (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk − v(cid:107)

and

. Rk := (cid:104)µv(cid:48), xk(cid:105) − (cid:104)v(cid:48), vk(cid:105) (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk − v(cid:107)

Since v = µx (cid:54)= 0 and vk → v, we may assume that vk (cid:54)= 0 for all k. Since vk ∈ N (xk, αk), by (4.6) and (4.5) we have vk = µkxk with µk > 0. As µk = (cid:107)vk(cid:107) · (cid:107)xk(cid:107)−1 and xk → x, we must have µk → µ as k → ∞.

If xk = x then αk = α and vk = µkxk = µkx. Combining this with the properties v = µx and (cid:104)v(cid:48), x(cid:105) = 0, we get Pk = 0, contradicting (4.11). We have thus shown that xk (cid:54)= x for all k ∈ IN .

Indeed, otherwise there exist a subse- It holds that limsupk→∞ Rk ≤ 0.

quence {k(cid:96)} of {k} and a constant ρ > 0 such that

(4.12) ≥ ρ, ∀(cid:96) ∈ IN. Rk(cid:96) = (cid:104)µv(cid:48), xk(cid:96)(cid:105) − (cid:104)v(cid:48), vk(cid:96)(cid:105) (cid:107)xk(cid:96) − x(cid:107) + |αk(cid:96) − α| + (cid:107)vk(cid:96) − v(cid:107)

Then we have

Rk(cid:96) ≤

= = . (cid:104)µv(cid:48), xk(cid:96)(cid:105) − (cid:104)v(cid:48), vk(cid:96)(cid:105) (cid:107)xk(cid:96) − x(cid:107) + (cid:107)vk(cid:96) − v(cid:107) (cid:104)µv(cid:48), xk(cid:96)(cid:105) − (cid:104)v(cid:48), µk(cid:96)xk(cid:96)(cid:105) (cid:107)xk(cid:96) − x(cid:107) + (cid:107)vk(cid:96) − v(cid:107) (1 − µk(cid:96)µ−1)(cid:104)µv(cid:48), xk(cid:96)(cid:105) (cid:107)xk(cid:96) − x(cid:107) + (cid:107)vk(cid:96) − v(cid:107)

Since (cid:104)v(cid:48), x(cid:105) = 0, it holds that

Rk(cid:96) ≤

≤ (1 − µk(cid:96)µ−1)(cid:104)µv(cid:48), xk(cid:96) − x(cid:105) (cid:107)xk(cid:96) − x(cid:107) + (cid:107)vk(cid:96) − v(cid:107) (1 − µk(cid:96)µ−1)(cid:104)µv(cid:48), xk(cid:96) − x(cid:105) (cid:107)xk(cid:96) − x(cid:107)

= (1 − µk(cid:96)µ−1)(cid:10)µv(cid:48), (cid:107)xk(cid:96) − x(cid:107)−1(xk(cid:96) − x)(cid:11).

(cid:28)

(cid:29)

There is no loss of generality in assuming that (cid:107)xk(cid:96) − x(cid:107)−1(xk(cid:96) − x) → ξ with (cid:107)ξ(cid:107) = 1. Since µk(cid:96) → µ, we get

→ 0 µv(cid:48), Rk(cid:96) ≤ (1 − µk(cid:96)µ−1) xk(cid:96) − x (cid:107)xk(cid:96) − x(cid:107)

as (cid:96) → ∞. This contradicts (4.12), hence there must exist N0 > 0 such that Rk ≤ δ/2 for all k ≥ N0. Since (4.11) is satisﬁed and Pk = Qk + Rk for all

k ∈ IN , this implies that Qk ≥ δ/2 for all k ≥ N0. For each k ≥ N0, we have

α x, xk − x(cid:105) + α(cid:48)(αk − α)

(cid:104)− α(cid:48) = ≤ Qk = δ 2

α(cid:48) α (cid:104)x, x(cid:105) − α(cid:48) α (cid:104)x, xk(cid:105) + α(cid:48)(αk − α) (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk − v(cid:107) α (cid:104)x, xk(cid:105) + α(cid:48)αk (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk − v(cid:107)

(cid:17)

− α(cid:48) = = (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk − v(cid:107) α α2 − α(cid:48) α(cid:48) α (cid:104)x, xk(cid:105) + α(cid:48)αk − α(cid:48)α (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk − v(cid:107)

α(cid:48)(cid:16) αk − (cid:104)x,xk(cid:105) = . (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk − v(cid:107)

(4.13) (cid:54)= 0. If α(cid:48) < 0,

α < 0. Consequently, we have

Hence, if α(cid:48) = 0 then Qk = 0, which is impossible. Thus α(cid:48) then it follows from (4.13) that αk − (cid:104)x,xk(cid:105)

= ≤ αk < = αk, ααk α (cid:104)x, xk(cid:105) α

(cid:17)

(cid:107)x(cid:107) · (cid:107)xk(cid:107) α a contradiction. It remains to consider the case where α(cid:48) > 0. The subsequent analytical arguments are based on a geometrical construction. Deﬁne the intersection of the ray Oxk with the sphere ∂E(α) := {x ∈ IRn(cid:12) (cid:12) (cid:107)x(cid:107) = α} by zk. Let uk be the orthogonal projection of x on the ray Oxk. (Since xk → x (cid:54)= 0, uk is well deﬁned for k ≥ N0 large enough.) Since xk (cid:54)= 0 for all k, we have

(cid:17)

(cid:16)

(cid:17)

α(cid:48)(cid:16) αk − (cid:104)x,xk(cid:105) ≤ Qk = δ 2

α (cid:107)xk − x(cid:107)

(cid:17)

(cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk − v(cid:107) α(cid:48)(cid:16) αk − (cid:104)x,xk(cid:105) α(cid:48)αk ≤ =

(cid:16) 1 − (cid:104)x,xk(cid:105) (cid:107)x(cid:107)·(cid:107)xk(cid:107)

(cid:17) 1 − cos (cid:92)(x, xk) (cid:107)xk − x(cid:107) (cid:16) (cid:107)zk−x(cid:107)

1 − (cid:104)x,xk(cid:105) ααk (cid:107)xk − x(cid:107) (cid:16) α(cid:48)αk α(cid:48)αk = =

(cid:17) (cid:92)(x, xk)

2 α−1(cid:17)2

(cid:107)xk − x(cid:107) 2α(cid:48)αk sin2 (cid:16) 1 2α(cid:48)αk = = = · . α(cid:48)αk 2α2 (cid:107)xk − x(cid:107) (cid:107)xk − x(cid:107) (cid:107)zk − x(cid:107)2 (cid:107)xk − x(cid:107)

The equality zk = x yields δ/2 ≤ Qk ≤ 0, an absurd. Thus (cid:107)zk − x(cid:107) (cid:54)= 0 for all k ≥ N0 suﬃcient large. From the above it follows that

· · = ≤ Qk ≤ δ 2 α(cid:48)αk 2α2 (cid:107)zk − x(cid:107) (cid:107)uk − x(cid:107) · (cid:107)zk − x(cid:107)−1

· = · < . α(cid:48)αk 2α2 α(cid:48)αk 2α2 α(cid:48) α (cid:107)zk − x(cid:107)2 (cid:107)uk − x(cid:107) (cid:107)zk − x(cid:107) sin (cid:91)Ozkx (cid:107)zk − x(cid:107) sin (cid:91)Ozkx

Thus, for all k large enough,

(4.14) 0 < . δα 2α(cid:48) < (cid:107)zk − x(cid:107) sin (cid:91)Ozkx

Note that since the triangle Ozkx is isosceles and zk → x, the angle (cid:91)Ozkx tends to π/2 as k → ∞. Hence, from (4.14) we deduce that

≤ 0, 0 <

Figure 4.1: The sequences {(xk, αk)}k∈IN , {zk}k∈IN , and {uk}k∈IN

δα 2α(cid:48) an absurd. Thus, the inclusion “ ⊃ ” of (4.10) is valid. The formula (4.10) (cid:50) has been established.

Remark 4.1 For the second part of the proof of Theorem 4.1, let us present another argument dealing with the case where α(cid:48) > 0. In this case we have (cid:17) α(cid:48) (cid:16) ≤ . ≤ Qk = δ 2 αk − (cid:104)x,xk(cid:105) (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk − v(cid:107) α(cid:48)α−1(ααk − (cid:104)x, xk(cid:105)) (cid:107)xk − x(cid:107)

It follows that

≤ = δα α(cid:48) 2(ααk − (cid:104)x, xk(cid:105)) (cid:107)xk − x(cid:107) 2((cid:107)x(cid:107) · (cid:107)xk(cid:107) − (cid:104)x, xk(cid:105)) (cid:107)xk − x(cid:107)

= (4.15)

(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) · (cid:12)(cid:107)xk(cid:107) − (cid:107)x(cid:107)

. = (cid:107)xk − x(cid:107) − (cid:104)xk − x, xk − x(cid:105) + 2(cid:107)x(cid:107) · (cid:107)xk(cid:107) − (cid:107)x(cid:107)2 − (cid:107)xk(cid:107)2 (cid:107)xk − x(cid:107) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)(cid:107)xk(cid:107) − (cid:107)x(cid:107) (cid:12) (cid:107)xk − x(cid:107)

(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ≤ (cid:107)xk − x(cid:107), we have (cid:12)(cid:107)xk(cid:107) − (cid:107)x(cid:107)

(cid:12) (cid:12) (cid:12)

Since

(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) · (cid:12)(cid:107)xk(cid:107) − (cid:107)x(cid:107)

(cid:12) (cid:12) (cid:12)(cid:107)xk(cid:107) − (cid:107)x(cid:107) (cid:107)xk − x(cid:107)

0 ≤ ≤ (cid:107)xk − x(cid:107).

α(cid:48) ≤ 0. We

Therefore, passing k to inﬁnity, from (4.15) we deduce that 0 < δα have arrived at a contradiction.

Remark 4.2 Formula (4.10) shows that if (cid:104)v(cid:48), x(cid:105) = 0, then (cid:98)D∗N (x, α, v)(v(cid:48)) is a straight line in IRn × IR passing through the point (µv(cid:48), 0). To see this, it suﬃces to put ﬁrst α(cid:48) = 0 to get x(cid:48) = µv(cid:48), then let α(cid:48) take an arbitrary real value and compute x(cid:48) = − α(cid:48) α x + µv(cid:48) for each α(cid:48) ∈ IR.

In the case where (cid:107)x(cid:107) = α and v = 0, the following result has been obtained

in [23].

Lemma 4.3 (See [23, Lemma 3.3]) If (cid:107)x(cid:107) = α, v = 0, and (x(cid:48), α(cid:48)) ∈ (cid:98)D∗N (x, α, v)(v(cid:48)), then (cid:104)v(cid:48), x(cid:105) ≥ 0, and there exists γ ∈ IR such that x(cid:48) = γx.

The upper estimate for the Fr´echet coderivative value (cid:98)D∗N (x, α, v)(v(cid:48))

(cid:111)

provided by Lemma 4.3 can be rewritten formally as

(cid:110) (x(cid:48), α(cid:48)) ∈ IRn × IR(cid:12)

(cid:98)D∗N (x, α, v)(v(cid:48)) ⊂

(cid:12) x(cid:48) = γx for some γ ∈ IR

(4.16)

when (cid:104)v(cid:48), x(cid:105) ≥ 0, and (cid:98)D∗N (x, α, v)(v(cid:48)) = ∅ if (cid:104)v(cid:48), x(cid:105) < 0. The estimate (4.16) may be strict.

Example 4.1 Let n = 2. In this case, N is a multifunction between IR2 × IR and IR2. For α = 1, x = (1, 0)(cid:62), and v = (0, 0)(cid:62), we have (x, α, v) ∈ gphN because v ∈ N (x; E(α)). Choosing αk = α = 1, xk = (1 − k−1, 0)(cid:62), and vk = v = (0, 0)(cid:62), we see at once that (xk, αk, vk) gphN−→ (x, α, v). Select v(cid:48) = (1, 0)(cid:62), x(cid:48) = (−1, 0)(cid:62), α(cid:48) ∈ IR, and observe that

(cid:104)v(cid:48), x(cid:105) > 0 and x(cid:48) = γx,

where γ = −1. However, (x(cid:48), α(cid:48)) (cid:54)∈ (cid:98)D∗N (x, α, v)(v(cid:48)). To see this, it suﬃces to note that

gphN −→ (x,α,v)

((cid:101)x,(cid:101)α,(cid:101)v)

limsup

k→∞

≥ limsup = 1 > 0; (cid:104)x(cid:48), (cid:101)x − x(cid:105) + α(cid:48)((cid:101)α − α) − (cid:104)v(cid:48), (cid:101)v − v(cid:105) (cid:107)(cid:101)x − x(cid:107) + |(cid:101)α − α| + (cid:107)(cid:101)v − v(cid:107) (cid:104)x(cid:48), xk − x(cid:105) + α(cid:48)(αk − α) − (cid:104)v(cid:48), vk − v(cid:105) (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk − v(cid:107)

hence (x(cid:48), α(cid:48), −v(cid:48)) (cid:54)∈ (cid:98)N ((x, α, v); gphN ).

Tightening the estimate (4.16) we can get an exact formula for the coderiva-

tive (cid:98)D∗N (x, α, v) in the case (cid:107)x(cid:107) = α and v = 0 as follows.

(cid:12) x(cid:48) = − α(cid:48)

α x, α(cid:48) ≤ 0(cid:9),

(cid:98)D∗N (x, α, v)(v(cid:48)) =



if (cid:104)v(cid:48), x(cid:105) ≥ 0 Theorem 4.2 If (cid:107)x(cid:107) = α and v = 0, then  (cid:8)(x(cid:48), α(cid:48)) ∈ IRn × IR(cid:12) 

∅,

if (cid:104)v(cid:48), x(cid:105) < 0, (4.17)

for every v(cid:48) ∈ IRn.

Proof. Fix any v(cid:48) ∈ IRn. If (cid:104)v(cid:48), x(cid:105) < 0, then (cid:98)D∗N (x, α, v)(v(cid:48)) = ∅ by Lemma 4.3. If (cid:104)v(cid:48), x(cid:105) ≥ 0 and (x(cid:48), α(cid:48)) ∈ (cid:98)D∗N (x, α, v)(v(cid:48)), then by Lemma 4.3 we can select a γ ∈ IR such that

(4.18)

x(cid:48) = γx. α and α(cid:48) ≤ 0. As (x(cid:48), α(cid:48)) ∈ (cid:98)D∗N (x, α, v)(v(cid:48)),

We are going to show that γ = − α(cid:48) we have

gphN −→ (x,α,v)

((cid:101)x,(cid:101)α,(cid:101)v)

limsup (4.19) ≤ 0.

(cid:104)x(cid:48), (cid:101)x − x(cid:105) + α(cid:48)((cid:101)α − α) − (cid:104)v(cid:48), (cid:101)v − v(cid:105) (cid:107)(cid:101)x − x(cid:107) + |(cid:101)α − α| + (cid:107)(cid:101)v − v(cid:107) Choosing αk = α, xk = (1 − k−1)x, and vk = 0 for every k ∈ IN , we can infer that (xk, αk, vk) gphN−→ (x, α, v). Hence, in accordance with (4.19) and (4.18),

0 ≥ lim k→∞ (cid:104)x(cid:48), xk − x(cid:105) + α(cid:48)(αk − α) − (cid:104)v(cid:48), vk − v(cid:105) (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk − v(cid:107)

= = −γα. = lim k→∞ = lim k→∞ (cid:104)γx, (1 − k−1)x − x(cid:105) (cid:107)(1 − k−1)x − x(cid:107) −γ(cid:104)x, x(cid:105) (cid:107)x(cid:107) (cid:104)x(cid:48), xk − x(cid:105) (cid:107)xk − x(cid:107)

Combining this with the condition α > 0, we get γ ≥ 0. Now, for every k ∈ IN , let xk = αkα−1x and vk = v = 0, where αk will be chosen so that αk → α. As (xk, αk, vk) gphN−→ (x, α, v), by (4.19) we have

≤ 0. limsup k→∞ (cid:104)x(cid:48), xk − x(cid:105) + α(cid:48)(αk − α) (cid:107)xk − x(cid:107) + |αk − α|

Thus, for any ε > 0, there exists kε ∈ IN satisfying

(cid:104)x(cid:48), xk − x(cid:105) + α(cid:48)(αk − α) ≤ ε((cid:107)xk − x(cid:107) + |αk − α|), ∀k ≥ kε.

Since x(cid:48) = γx, the latter implies that

γ(cid:104)x, xk − x(cid:105) + α(cid:48)(αk − α) ≤ ε((cid:107)xk − x(cid:107) + |αk − α|), ∀k ≥ kε.

Hence,

α(cid:48)(αk − α) ≤ γ((cid:104)x, x(cid:105) − (cid:104)x, xk(cid:105)) + ε((cid:107)xk − x(cid:107) + |αk − α|)

= γ(α2 − (cid:104)x, αkα−1x(cid:105)) + ε((cid:107)αkα−1x − x(cid:107) + |αk − α|)

= γ(α2 − αkα) + 2ε|αk − α|.

Therefore,

α ≤ γ + 2ε

α ≥ γ − 2ε

(cid:27)

− (4.20) |αk − α|. (α − αk) ≤ γ(α − αk) + α(cid:48) α

(cid:98)D∗N (x, α, v)(v(cid:48)) ⊂

(4.21) x, α(cid:48) ≤ 0 . 2ε α Letting αk ↑ α as k → ∞, from (4.20) we get − α(cid:48) α . Letting αk ↓ α as k → ∞, from (4.20) we obtain − α(cid:48) α . Since ε > 0 can be chosen arbitrary, it follows that γ = −α(cid:48)α−1. As γ ≥ 0 and α > 0, we must have α(cid:48) ≤ 0. Since x(cid:48) = γx = −α(cid:48)α−1x by virtue of (4.18), we have proved that (cid:26) (x(cid:48), α(cid:48)) ∈ IRn+1(cid:12) (cid:12) x(cid:48) = − (cid:12) α(cid:48) α

Let us check the opposite inclusion of (4.21) in the case (cid:104)v(cid:48), x(cid:105) ≥ 0. If one could ﬁnd an element (x(cid:48), α(cid:48)) from the set on the right-hand side of (4.21) with (x(cid:48), α(cid:48)) (cid:54)∈ (cid:98)D∗N (x, α, v)(v(cid:48)), then there would exist a sequence (xk, αk, vk) and a constant δ > 0 such that (xk, αk, vk) gphN−→ (x, α, v) as k → ∞ and

(4.22) ≥ δ, ∀k ∈ IN. Pk := (cid:104)x(cid:48), xk − x(cid:105) + α(cid:48)(αk − α) − (cid:104)v(cid:48), vk − v(cid:105) (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk − v(cid:107)

Since x(cid:48) = −α(cid:48)α−1x with α(cid:48) ≤ 0 and since v = 0, we have

(4.23) Pk = Qk − Rk

where

. Qk := , Rk := (cid:104)− α(cid:48) α x, xk − x(cid:105) + α(cid:48)(αk − α) (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk(cid:107) (cid:104)v(cid:48), vk(cid:105) (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk(cid:107)

We distinguish two cases: (i) (cid:104)v(cid:48), x(cid:105) = 0, (ii) (cid:104)v(cid:48), x(cid:105) > 0.

Case (i): (cid:104)v(cid:48), x(cid:105) = 0. In this case Rk → 0 as k → ∞. Indeed, if vk = 0 for all large k, then Rk = 0 for all k large enough; hence limk→∞ Rk = 0. Otherwise, we may assume that vk (cid:54)= 0 for all k. For every k, since vk ∈ N (xk, αk) \ {0}, by (4.6) and (4.5) there exists µk > 0 such that vk = µkxk. Then, we have (cid:107)vk(cid:107) = µk(cid:107)xk(cid:107) (cid:54)= 0. Consequently,

|Rk| =

We have seen that Rk → 0 as k → ∞.

Case (ii): (cid:104)v(cid:48), x(cid:105) > 0. Since xk → x, this strict inequality yields (cid:104)v(cid:48), xk(cid:105) > 0 for all k large enough. If there is ¯k ∈ IN such that vk = 0 for all k ≥ ¯k, then

Rk = 0 for all k ≥ ¯k. If there exists a subsequence {vk(cid:96)} of {vk} with vk(cid:96) (cid:54)= 0 for all (cid:96) ∈ IN , then vk(cid:96) = µk(cid:96)xk(cid:96), where µk(cid:96) > 0 for all (cid:96). Since (cid:104)v(cid:48), xk(cid:96)(cid:105) > 0 for all (cid:96) suﬃciently large, we have

(cid:104)v(cid:48), vk(cid:96)(cid:105) = (cid:104)v(cid:48), µk(cid:96)xk(cid:96)(cid:105) = µk(cid:96)(cid:104)v(cid:48), xk(cid:96)(cid:105) > 0 ((cid:96) is large enough).

This implies that

0 ≤ Rk(cid:96) =

→ = ≤ (as (cid:96) → ∞). (cid:104)v(cid:48), vk(cid:96)(cid:105) (cid:107)xk(cid:96) − x(cid:107) + |αk(cid:96) − α| + (cid:107)vk(cid:96)(cid:107) (cid:104)v(cid:48), x(cid:105) (cid:104)v(cid:48), xk(cid:96)(cid:105) (cid:104)v(cid:48), vk(cid:96)(cid:105) (cid:107)x(cid:107) (cid:107)xk(cid:96)(cid:107) (cid:107)vk(cid:96)(cid:107)

Since the last property of {Rk} is valid for any subsequence {vk(cid:96)} of {vk} with vk(cid:96) (cid:54)= 0 for all (cid:96), we can assert that 0 ≤ Rk ≤ (cid:107)x(cid:107)−1(cid:104)v(cid:48), x(cid:105) + 1 for all k large enough.

From the above analysis we see that, in both the cases (i) and (ii), there exists an index k0 such that Rk ≥ −δ/2 for all k ≥ k0. Then, by (4.22) and (4.23),

, ∀k ≥ k0. Qk = Pk + Rk ≥ δ + Rk ≥ δ 2

Since α(cid:48) ≤ 0 and (cid:107)xk(cid:107) ≤ αk, for each k ≥ k0 we have

α α2 + α(cid:48)αk − α(cid:48)α

− α(cid:48) = ≤ Qk = δ 2 (cid:104)− α(cid:48) α x, xk − x(cid:105) + α(cid:48)(αk − α) (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk(cid:107)

α (cid:104)x, xk(cid:105)

α (cid:104)x, xk(cid:105) + α(cid:48) (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk(cid:107) α(cid:48)αk − α(cid:48) α (cid:107)x(cid:107) · (cid:107)xk(cid:107)

α(cid:48)αk − α(cid:48) = ≤ (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk(cid:107)

≤ = 0. (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk(cid:107) α(cid:48)αk − α(cid:48)αk (cid:107)xk − x(cid:107) + |αk − α| + (cid:107)vk(cid:107)

This contradiction completes the proof of the opposite inclusion in (4.21), (cid:50) hence establishes (4.17).

Remark 4.3 Theorem 4.2 gives a complete solution for the second part of the open problem raised in [23, Remark 3.6].

4.2.2 The Mordukhovich Coderivative of N (x, α)

Based on the obtained formulas for (cid:98)D∗N (x, α, v)(·), we provide exact for- mulas for the Mordukhovich coderivative D∗N (x, α, v)(·) of the normal cone

In the next two lemmas, we recall some

mapping N (·) in various cases. existing results.

Lemma 4.4 (See [23, Lemma 4.4]) The set gphN is locally closed in the product space IRn × IR × IRn.

Lemma 4.5 (See [23, Lemma 3.7]) If (cid:107)x(cid:107) < α, then v = 0 and

D∗N (x, α, v)(v(cid:48)) = (cid:98)D∗N (x, α, v)(v(cid:48)) = {(0IRn, 0IR)},

for every v(cid:48) ∈ IRn.

By Lemma 4.5, the normal cone mapping N (·) is graphically regular at any point (x, α, v) ∈ gphN with (cid:107)x(cid:107) < α. The forthcoming theorem shows that N (·) is also graphically regular at any point (x, α, v) ∈ gphN with (cid:107)x(cid:107) = α and v (cid:54)= 0.

Theorem 4.3 If (cid:107)x(cid:107) = α and if v (cid:54)= 0, then we have

(cid:8)(x(cid:48), α(cid:48)) ∈ IRn × IR(cid:12)

D∗N (x, α, v)(v(cid:48)) = (cid:98)D∗N (x, α, v)(v(cid:48))

(cid:12) x(cid:48) = − α(cid:48)

 

α x + µv(cid:48)(cid:9),



if (cid:104)v(cid:48), x(cid:105) = 0 =

∅,

if (cid:104)v(cid:48), x(cid:105) (cid:54)= 0 (4.24)

for every v(cid:48) ∈ IRn, where µ := (cid:107)v(cid:107) · (cid:107)x(cid:107)−1.

k) → (x(cid:48), α(cid:48), v(cid:48)) such that

k, v(cid:48)

k, α(cid:48)

Proof. Fix any v(cid:48) ∈ IRn and let (x(cid:48), α(cid:48)) ∈ D∗N (x, α, v)(v(cid:48)) be given arbitrary. By the deﬁnition of the Mordukhovich coderivative, there exist sequences (xk, αk, vk) gphN−→ (x, α, v) and (x(cid:48)

k, α(cid:48)

k) ∈ (cid:98)D∗N (xk, αk, vk)(v(cid:48)

k),

(4.25) (x(cid:48) ∀k ∈ IN.

Since v (cid:54)= 0, we have vk (cid:54)= 0 for all k large enough. For those k, according to Theorem 4.1, (4.25) holds if and only if (cid:107)xk(cid:107) = αk,

k, xk(cid:105) = 0 and x(cid:48)

k = −

(4.26) (cid:104)v(cid:48) xk + v(cid:48) k. α(cid:48) k αk (cid:107)vk(cid:107) (cid:107)xk(cid:107)

k → v(cid:48), we obtain

k → α(cid:48), and v(cid:48)

k → x(cid:48), α(cid:48)

Passing (4.26) to limit as k → ∞ and remembering that xk → x, αk → α, vk → v, x(cid:48)

(cid:104)v(cid:48), x(cid:105) = 0 and x(cid:48) = − v(cid:48). x + α(cid:48) α (cid:107)v(cid:107) (cid:107)x(cid:107)

Thus (x(cid:48), α(cid:48)) ∈ (cid:98)D∗N (x, α, v)(v(cid:48)) by Theorem 4.1. We have shown that D∗N (x, α, v)(v(cid:48)) ⊂ (cid:98)D∗N (x, α, v)(v(cid:48)). Since the reverse inclusion is obvious, (cid:50) combining this with (4.10) we obtain (4.24) for every v(cid:48) ∈ IRn.

The case (x, α, v) ∈ gphN with (cid:107)x(cid:107) = α, v = 0, is treated now. Combining the following theorem with Theorem 4.2, we see that D∗N (x, α, v)(v(cid:48)) (cid:54)= (cid:98)D∗N (x, α, v)(v(cid:48)) for all v(cid:48) from the closed half-space {v(cid:48) ∈ IRn| (cid:104)v(cid:48), x(cid:105) ≤ 0}. So the multifunction N (·) is graphically irregular at any point (x, α, v) ∈ gphN where (cid:107)x(cid:107) = α and v = 0.

Theorem 4.4 Suppose that (cid:107)x(cid:107) = α and v = 0. For every v(cid:48) ∈ IRn, the following hold

(i) If (cid:104)v(cid:48), x(cid:105) (cid:54)= 0, then

(cid:98)D∗N (x, α, v)(v(cid:48)),

 



if (cid:104)v(cid:48), x(cid:105) > 0 D∗N (x, α, v)(v(cid:48)) =

(cid:8)(x(cid:48), α(cid:48)) ∈ IRn+1(cid:12)

if (cid:104)v(cid:48), x(cid:105) < 0 {(0IRn, 0IR)},

(cid:12) x(cid:48) = − α(cid:48)

 

α x, α(cid:48) ≤ 0(cid:9),



if (cid:104)v(cid:48), x(cid:105) > 0 =

{(0IRn, 0IR)},

if (cid:104)v(cid:48), x(cid:105) < 0. (4.27)

(cid:27)

(cid:26)

(ii) If (cid:104)v(cid:48), x(cid:105) = 0, then

(cid:12) (cid:12) x(cid:48) = − (cid:12)

(4.28) x, α(cid:48) ∈ IR . D∗N (x, α, v)(v(cid:48)) = (x(cid:48), α(cid:48)) ∈ IRn × IR α(cid:48) α

k) → (x(cid:48), α(cid:48), v(cid:48)) with

k, α(cid:48)

k, v(cid:48)

Proof. Let (x, α, v) ∈ gphN , (cid:107)x(cid:107) = α, v = 0, and v(cid:48) ∈ IRn. (i) If (cid:104)v(cid:48), x(cid:105) < 0, then D∗N (x, α, v)(v(cid:48)) = {(0IRn, 0IR)}. Indeed, for such v(cid:48), given (x(cid:48), α(cid:48)) ∈ D∗N (x, α, v)(v(cid:48)) one can ﬁnd sequences (xk, αk, vk) gphN−→ (x, α, v) and (x(cid:48)

k, α(cid:48)

k) ∈ (cid:98)D∗N (xk, αk, vk)(v(cid:48)

k),

k) = ∅ by Theorem 4.2 and by the equality (cid:104)v(cid:48)

(4.29) (x(cid:48) ∀k ∈ IN.

k, xk(cid:105) < 0 for large k. Fix for a while k) = ∅ by k, xk(cid:105) < 0. If (cid:107)xk(cid:107) = αk and if vk = 0, then k, xk(cid:105) < 0. k) shown in (4.29) yields k) =

The condition (cid:104)v(cid:48), x(cid:105) < 0 implies that (cid:104)v(cid:48) such an index k. If (cid:107)xk(cid:107) = αk and if vk (cid:54)= 0, then (cid:98)D∗N (xk, αk, vk)(v(cid:48) Theorem 4.1 and by the equality (cid:104)v(cid:48) (cid:98)D∗N (xk, αk, vk)(v(cid:48) Therefore, the nonemptyness of (cid:98)D∗N (xk, αk, vk)(v(cid:48) (cid:107)xk(cid:107) < αk. By Lemma 4.1, the latter implies that (cid:98)D∗N (xk, αk, vk)(v(cid:48)

k, α(cid:48)

k = 0, and v(cid:48)

k = 0, α(cid:48)

k, α(cid:48)

k, v(cid:48)

{(0IRn, 0IR)}. Consequently, (x(cid:48) k) = (0IRn, 0IR) for all k large enough; hence (x(cid:48), α(cid:48)) = limk→∞(x(cid:48) k) = (0IRn, 0IR). This justiﬁes that D∗N (x, α, v)(v(cid:48)) ⊂ {(0IRn, 0IR)}. To get the reverse inclusion, choose xk = (1 − k−1)x, αk = α, k = v(cid:48) for every k ∈ IN . Then, (xk, αk, vk) ∈ vk = 0, x(cid:48) gphN , (xk, αk, vk) → (x, α, v), and (x(cid:48) k) → (0IRn, 0IR, v(cid:48)) as k → ∞. The choice of xk and αk yields (cid:107)xk(cid:107) < (cid:107)x(cid:107) = α = αk. Hence, by Lemma 4.1 we have

k, α(cid:48)

k) ∈ (cid:98)D∗N (xk, αk, vk)(v(cid:48)

k) = {(0IRn, 0IR)},

(x(cid:48) ∀k ∈ IN.

This gives (0IRn, 0IR) ∈ D∗N (x, α, v)(v(cid:48)) and thus establishes the desired equality

(cid:26)

D∗N (x, α, v)(v(cid:48)) = {(0IRn, 0IR)}. Suppose now that (cid:104)v(cid:48), x(cid:105) > 0. Due to (cid:98)D∗N (x, α, v)(v(cid:48)) ⊂ D∗N (x, α, v)(v(cid:48)) and Theorem 4.2, the proof of the equalities

(cid:27) ,

(cid:12) x(cid:48) = −

x, α(cid:48) ≤ 0 (x(cid:48), α(cid:48)) ∈ IRn+1(cid:12) D∗N (x, α, v)(v(cid:48)) = (cid:98)D∗N (x, α, v)(v(cid:48)) = α(cid:48) α

which are stated in (4.27), reduces to checking the fulﬁlment of the inclusion

(4.30)

k) → (x(cid:48), α(cid:48), v(cid:48)) such that (4.29) holds.

k, v(cid:48)

k, α(cid:48)

k, α(cid:48) (a) Consider the situation vk = 0 for all k suﬃciently large. If (cid:107)xk(cid:107) < αk for all large k, then by Lemma 4.5 we get (x(cid:48) k) = (0, 0) for large indexes k. Hence, (x(cid:48), α(cid:48)) = (0, 0) ∈ (cid:98)D∗N (x, α, v)(v(cid:48)) (the last inclusion is ready by Theorem 4.2 and the assumptions (cid:107)x(cid:107) = α, v = 0, and (cid:104)v(cid:48), x(cid:105) > 0). If there exists a subsequence {k(cid:96)} of {k} such that (cid:107)xk(cid:96)(cid:107) = αk(cid:96) for all (cid:96) ∈ IN , then from (4.29) and Theorem 4.2 we can infer that

D∗N (x, α, v)(v(cid:48)) ⊂ (cid:98)D∗N (x, α, v)(v(cid:48)). For any (x(cid:48), α(cid:48)) ∈ D∗N (x, α, v)(v(cid:48)), there exist (xk, αk, vk) gphN−→ (x, α, v) and (x(cid:48)

k(cid:96), xk(cid:96)(cid:105) ≥ 0,

= − (cid:104)v(cid:48) ≤ 0. x(cid:48) k(cid:96) xk(cid:96), α(cid:48) k(cid:96) α(cid:48) k(cid:96) αk(cid:96)

Taking the limits as (cid:96) → ∞, we obtain

(cid:104)v(cid:48), x(cid:105) ≥ 0, x(cid:48) = − x, α(cid:48) ≤ 0. α(cid:48) α

By virtue of (4.17), this yields (x(cid:48), α(cid:48)) ∈ (cid:98)D∗N (x, α, v)(v(cid:48)).

(b) Suppose now that there is a subsequence {k(cid:96)} of {k} such that vk(cid:96) (cid:54)= 0 for all (cid:96) ∈ IN . Then, (cid:107)xk(cid:96)(cid:107) = αk(cid:96) for all (cid:96). From (4.29) and Theorem 4.1 we 81

obtain

k(cid:96), xk(cid:96)(cid:105) = 0 and x(cid:48) k(cid:96)

= − (cid:104)v(cid:48) xk(cid:96) + v(cid:48) k(cid:96) α(cid:48) k(cid:96) αk(cid:96) (cid:107)vk(cid:96)(cid:107) (cid:107)xk(cid:96)(cid:107)

k, α(cid:48)

k, v(cid:48)

for all (cid:96). This obviously leads to (cid:104)v(cid:48), x(cid:105) = 0, a contradiction with the assump- tion that (cid:104)v(cid:48), x(cid:105) > 0. The inclusion (4.30) has been proved.

Thus, if (cid:104)v(cid:48), x(cid:105) (cid:54)= 0, then we get (4.27). (ii) Suppose that (cid:104)v(cid:48), x(cid:105) = 0. For any (x(cid:48), α(cid:48)) ∈ D∗N (x, α, v)(v(cid:48)), there exist sequences (xk, αk, vk) gphN−→ (x, α, v) and (x(cid:48) k) → (x(cid:48), α(cid:48), v(cid:48)) such that (4.29) holds. If vk = 0 for all k large enough then, arguing similarly as in the subcase (a) of the proof of assertion (i), we obtain

(cid:104)v(cid:48), x(cid:105) = 0, x(cid:48) = − x, α(cid:48) ≤ 0. α(cid:48) α

Hence (x(cid:48), α(cid:48)) belongs to the set on the right-hand side of (4.28). If there is a subsequence {k(cid:96)} of {k} such that vk(cid:96) (cid:54)= 0 for all (cid:96) ∈ IN , then by repeating the arguments of subcase (b) of the proof of assertion (i) we have

k(cid:96), xk(cid:96)(cid:105) = 0,

= − (cid:104)v(cid:48) (cid:107)xk(cid:96)(cid:107) = αk(cid:96), xk(cid:96) + and x(cid:48) k(cid:96) v(cid:48) k(cid:96), α(cid:48) k(cid:96) αk(cid:96) (cid:107)vk(cid:96)(cid:107) (cid:107)xk(cid:96)(cid:107)

for all (cid:96). Since v = 0, this implies that

(cid:104)v(cid:48), x(cid:105) = 0, x(cid:48) = − x, and α(cid:48) ∈ IR. α(cid:48) α

Thus, the inclusion “ ⊂ ” in (4.28) is valid.

To verify the inclusion “ ⊃ ” in (4.28), ﬁx any element (x(cid:48), α(cid:48)) in the set on the right-hand side of (4.28). We have to show that (x(cid:48), α(cid:48)) ∈ D∗N (x, α, v)(v(cid:48)). For every k ∈ IN , choose xk = x, αk = α, and vk = µkxk, where µk := k−1. It is clear that (xk, αk, vk) ∈ gphN and (xk, αk, vk) → (x, α, v) as k → ∞. For every k ∈ IN , putting

k = v(cid:48), α(cid:48) v(cid:48)

k = α(cid:48),

k = −

and x(cid:48) x + k−1v(cid:48), α(cid:48) α

we have

k, xk(cid:105) = (cid:104)v(cid:48), x(cid:105) = 0,

(cid:104)v(cid:48) x(cid:48) k = − xk + µkv(cid:48) k.

k, v(cid:48)

k) for k) → (x(cid:48), α(cid:48), v(cid:48)) as k → ∞, we obtain the inclusion (cid:50)

α(cid:48) k αk k) ∈ (cid:98)D∗N (xk, αk, vk)(v(cid:48) k, α(cid:48)

Hence, in accordance with Theorem 4.1, (x(cid:48) k, α(cid:48) all k. Observing (x(cid:48) (x(cid:48), α(cid:48)) ∈ D∗N (x, α, v)(v(cid:48)) which completes the proof of (4.28).

4.3 Necessary and Suﬃcient Conditions for Stability

Conditions for stability of the solution map (A, b, α) (cid:55)→ S(A, b, α) of the linear GE of the form (4.1) are obtained in this section.

4.3.1 Coderivatives of the KKT point set map

As in Section 4.1, we put G(x, w) = Ax and M (x, w) = N (x, α) for every x ∈ IRn and w = (A, α) ∈ W with W = H(n) × IR. Fix a triplet ( ¯A, ¯b, ¯α) ∈ H(n)×IRn×IR. Put ¯w = ( ¯A, ¯α), ¯y = −¯b, and let ¯x ∈ S( ¯A, ¯b, ¯α). Then we have ¯x ∈ (cid:101)S( ¯w, ¯y) with (cid:101)S( ¯w, ¯y) being given by (4.8). Let ¯v = ¯y−G(¯x, ¯w) = −¯b− ¯A¯x.

We will need two more lemmas of [23].

Lemma 4.6 (See [23, Lemma 4.1]) The Mordukhovich coderivative

D∗M (¯x, ¯w, ¯v) : IRn ⇒ IRn × H(n)∗ × IR

of the multifunction M : IRn × W ⇒ IRn, where H(n)∗ is the dual space of H(n), is computed by the formula

(cid:12) (x(cid:48), α(cid:48)) ∈ D∗N (¯x, ¯α, ¯v)(v(cid:48))(cid:9)

D∗M (¯x, ¯w, ¯v)(v(cid:48)) = (cid:8)(x(cid:48), 0H(n)∗, α(cid:48))(cid:12)

for every v(cid:48) ∈ IRn.

Lemma 4.7 (See [23, Lemma 4.3]) For every v(cid:48) ∈ IRn,

∇G(¯x, ¯w)∗(v(cid:48)) = { ¯Av(cid:48)} × {τ (v(cid:48), ¯x)} × {0IR},

i¯xj.

i¯xj) is the n × n matrix whose ij-th element is v(cid:48)

where τ (v(cid:48), ¯x) := (v(cid:48)

(cid:98)D∗M (¯x, ¯w, ¯v) : IRn ⇒ IRn × H(n)∗ × IR

Remark 4.4 Similarly as in Lemma 4.6, the Fr´echet coderivative

(cid:98)D∗M (¯x, ¯w, ¯v)(v(cid:48)) = (cid:8)(x(cid:48), 0H(n)∗, α(cid:48))(cid:12)

(cid:12) (x(cid:48), α(cid:48)) ∈ (cid:98)D∗N (¯x, ¯α, ¯v)(v(cid:48))(cid:9).

(cid:110)

of the multifunction M : IRn × W ⇒ IRn is computed as follows

(cid:98)ΩG,¯y(x(cid:48)) =

(cid:12) (−x(cid:48), w(cid:48), y(cid:48)) ∈ ∇G(¯x, ¯w)∗(v(cid:48)) × {0IRn}

v(cid:48)∈IRn

(cid:111) ,

For each x(cid:48) ∈ IRn, we put (cid:91) (w(cid:48), y(cid:48)) ∈ W × IRn(cid:12)

−{0IRn} × {0W } × {v(cid:48)} + (cid:98)D∗M (¯x, ¯w, ¯v)(v(cid:48)) × {0IRn}

(cid:110)

(cid:91)

and

(cid:12) (−x(cid:48), w(cid:48), y(cid:48)) ∈ ∇G(¯x, ¯w)∗(v(cid:48)) × {0IRn}

v(cid:48)∈IRn

(cid:111) ,

(w(cid:48), y(cid:48)) ∈ W × IRn(cid:12) ΩG,¯y(x(cid:48)) =

−{0IRn} × {0W } × {v(cid:48)} + D∗M (¯x, ¯w, ¯v)(v(cid:48)) × {0IRn}

where ¯v = ¯y − G(¯x, ¯w) = −¯b − ¯A¯x.

Since M : IRn × W ⇒ IRn has a locally closed graph by Lemma 4.4, the

next statement is an immediate corollary of [22, Theorem 4.3].

Theorem 4.5 The inclusions

(cid:98)ΩG,¯y(x(cid:48)) ⊂ (cid:98)D∗ (cid:101)S( ¯w, ¯y, ¯x)(x(cid:48)) ⊂ D∗ (cid:101)S( ¯w, ¯y, ¯x)(x(cid:48)) ⊂ ΩG,¯y(x(cid:48))

(4.31)

hold for all x(cid:48) ∈ IRn. If, in addition, M (·) is graphically regular at (¯x, ¯w, ¯v) ∈ gphM , then

(cid:98)ΩG,¯y(x(cid:48)) = (cid:98)D∗ (cid:101)S( ¯w, ¯y, ¯x)(x(cid:48)) = D∗ (cid:101)S( ¯w, ¯y, ¯x)(x(cid:48)) = ΩG,¯y(x(cid:48))

(4.32)

for every x(cid:48) ∈ IRn.

Combining (4.32) with Lemmas 4.6 and 4.7, Remark 4.4, Lemma 4.5, Theorem 4.3, and the ﬁrst assertion of Theorem 4.4, we obtain exact for- mulas for computing the Fr´echet and the Mordukhovich coderivatives of (cid:101)S(w, y) = S(A, b, α) at the point ( ¯w, ¯y, ¯x) ∈ gph (cid:101)S with the property that M (x, w) = N (x, α) is graphically regular at ¯ω := (¯x, ¯w, ¯v) ∈ gphM . Simi- larly, invoking (4.31), Lemmas 4.6 and 4.7, Remark 4.4, Theorems 4.2 and 4.4, we get explicit estimates for the Fr´echet and the Mordukhovich coderivatives of (cid:101)S(·) at the point ( ¯w, ¯y, ¯x) ∈ gph (cid:101)S where (cid:107)¯x(cid:107) = ¯α, ¯v = −¯b − ¯A¯x = 0.

4.3.2 The Lipschitz-like property

Since gphN is locally closed in the product space IRn×IR×IRn by Lemma 4.4, gphM is also locally closed in IRn × W × IRn. So, both gphS and gph (cid:101)S are respectively locally closed in the product spaces H(n) × IRn × IR × IRn and W × IRn × IRn. Therefore, by the Mordukhovich criterion stated in Theorem 1.3, (cid:101)S(·) is locally Lipschitz-like around ( ¯w, ¯y, ¯x) if and only if

(4.33) D∗ (cid:101)S( ¯w, ¯y, ¯x)(0) = {0}.

(cid:104)

By (4.8) we have

(cid:105) D∗S( ¯A, ¯b, ¯α, ¯x)(0) = {0}

(cid:105) D∗ (cid:101)S( ¯w, ¯y, ¯x)(0) = {0}

⇐⇒ .

(cid:104)

This implies that S(·) is locally Lipschitz-like around ( ¯A, ¯b, ¯α, ¯x) if and only if (cid:101)S(·) is locally Lipschitz-like around ( ¯w, ¯y, ¯x). If M (·) is graphically regular at ¯ω, then by (4.32) we see that (4.33) holds if and only if ΩG,¯y(0) = (cid:98)ΩG,¯y(0) = {0}. In the case where M (·) is graphically irregular at ¯ω, by (4.31) we can infer that

(cid:105) ΩG,¯y(0) = {0}

(cid:105) (cid:98)ΩG,¯y(0) = {0}

(4.34) =⇒ (4.33) =⇒ .

Theorem 4.6 For any ( ¯A, ¯b, ¯α, ¯x) ∈ gphS, the following assertions hold:

(i) If (cid:107)¯x(cid:107) < ¯α, then the map S(·) is locally Lipschitz-like around ( ¯A, ¯b, ¯α, ¯x)

if and only if det ¯A (cid:54)= 0.

(ii) If (cid:107)¯x(cid:107) = ¯α and ¯A¯x + ¯b (cid:54)= 0, then S(·) is locally Lipschitz-like around

( ¯A, ¯b, ¯α, ¯x) if and only if detQ( ¯A, ¯b, ¯α, ¯x) (cid:54)= 0, where (cid:33)

(cid:32) ¯A + µI − 1 ¯α ¯x ¯x(cid:62) 0

(4.35) Q( ¯A, ¯b, ¯α, ¯x) :=

with µ being the unique Lagrange multiplier associated to ¯x.

(cid:0) − ¯Av(cid:48), A(cid:48) − (v(cid:48)

i¯xj), α(cid:48), y(cid:48) + v(cid:48)(cid:1) ∈ D∗M (¯x, ¯w, ¯v)(v(cid:48)) × {0IRn}.

Proof. (i) Suppose that (cid:107)¯x(cid:107) < ¯α. By Lemma 4.5, N (·) is graphically regular at (¯x, ¯α, ¯v). Hence, M (·) is also graphically regular at ¯ω = (¯x, ¯w, ¯v). According to (4.32), S(·) is locally Lipschitz-like around ¯ω if and only if ΩG,¯y(0) = {0}. We see that (w(cid:48), y(cid:48)) = (A(cid:48), α(cid:48), y(cid:48)) belongs to ΩG,¯y(0) if and only if there exists v(cid:48) ∈ IRn such that

(cid:0) − ¯Av(cid:48), α(cid:48), A(cid:48) − (v(cid:48)

i¯xj), y(cid:48) + v(cid:48)(cid:1) ∈ D∗N (¯x, ¯α, ¯v)(v(cid:48)) × {0H(n)∗} × {0IRn}.

According to Lemma 4.6, this is equivalent to

Since D∗N (¯x, ¯α, ¯v)(v(cid:48)) = {(0IRn, 0IR)} by Lemma 4.5, the last inclusion means that

(cid:0) − ¯Av(cid:48), α(cid:48), A(cid:48) − (v(cid:48)

i¯xj), y(cid:48) + v(cid:48)(cid:1) = (0IRn, 0IR, 0H(n)∗, 0IRn).

(4.36)

So, the equality ΩG,¯y(0) = {0} holds if and only if the fulﬁlment of (4.36) for some v(cid:48) ∈ IRn yields A(cid:48) = 0H(n)∗, α(cid:48) = 0IR, and y(cid:48) = 0IRn. The latter means that det ¯A (cid:54)= 0. Indeed, if det ¯A = 0, then there is v(cid:48) (cid:54)= 0 such that − ¯Av(cid:48) = 0. Setting A(cid:48) = τ (v(cid:48), ¯x) = (v(cid:48) i¯xj), α(cid:48) = 0, and y(cid:48) = −v(cid:48), we get (w(cid:48), y(cid:48)) = (A(cid:48), α(cid:48), y(cid:48)) (cid:54)= (0H(n)∗, 0IR, 0IRn) satisfying (4.36). Thus, there exists

v(cid:48) ∈ IRn such that the fulﬁlment of (4.36) does not yield (w(cid:48), y(cid:48)) = (0W , 0IRn). Conversely, if det ¯A (cid:54)= 0, then (4.36) implies that − ¯Av(cid:48) = 0; hence v(cid:48) = 0. Substituting v(cid:48) = 0 into (4.36) yields A(cid:48) = 0, α(cid:48) = 0, and y(cid:48) = 0.

(cid:0) − ¯Av(cid:48), α(cid:48), A(cid:48) − (v(cid:48)

i¯xj), y(cid:48) + v(cid:48)(cid:1) ∈ D∗N (¯x, ¯α, ¯v)(v(cid:48)) × {0H(n)∗} × {0IRn}.

(ii) Suppose that (cid:107)¯x(cid:107) = ¯α and ¯A¯x + ¯b (cid:54)= 0. As in the case (i), S(·) is locally Lipschitz-like around ¯ω if and only if ΩG,¯y(0) = {0}. Moreover, (w(cid:48), y(cid:48)) ∈ ΩG,¯y(0) if and only if there exists v(cid:48) ∈ IRn such that



¯α ¯x + µv(cid:48)

Since ¯v = −¯b − ¯A¯x (cid:54)= 0, Theorem 4.3 tells us that the last inclusion can be rewritten equivalently as

i¯xj)

(4.37)

− ¯Av(cid:48) = − α(cid:48)  (cid:104)v(cid:48), ¯x(cid:105) = 0 α(cid:48) ∈ IR, A(cid:48) = (v(cid:48)  y(cid:48) + v(cid:48) = 0IRn

¯α ¯x = 0

 

with µ := (cid:107)¯v(cid:107) · (cid:107)¯x(cid:107)−1. If λ is the Lagrange multiplier corresponding to ¯x ∈ S( ¯A, ¯b, ¯α), then ( ¯A¯x + λI)¯x = −¯b. So, λ¯x = −¯b − ¯A¯x = ¯v. It follows that λ = (cid:107)¯v(cid:107) · (cid:107)¯x(cid:107)−1. Thus, µ is the Lagrange multiplier corresponding to ¯x. Clearly, ΩG,¯y(0) = {0} if and only if from (4.37), with v(cid:48) ∈ IRn being chosen arbitrarily, it follows that A(cid:48) = 0H(n)∗, α(cid:48) = 0IR, and y(cid:48) = 0IRn. The latter is equivalent to saying that



 ( ¯A + µI)v(cid:48) − α(cid:48)  ¯x(cid:62)v(cid:48) = 0  v(cid:48) ∈ IRn, α(cid:48) ∈ IR

(cid:33) (cid:32)

(cid:33)

 

(4.38) =⇒ v(cid:48) = 0 α(cid:48) = 0.

(cid:33) (cid:35) (cid:32) 0 0



= =⇒ Since (4.38) can be rewritten equivalently as (cid:34) (cid:32) ¯A + µI − 1 ¯α ¯x ¯x(cid:62) 0 v(cid:48) α(cid:48) v(cid:48) = 0 α(cid:48) = 0,

(cid:50) the condition ΩG,¯y(0) = {0} means that detQ( ¯A, ¯b, ¯α, ¯x) is nonzero, where Q( ¯A, ¯b, ¯α, ¯x) has been deﬁned by (4.35). The proof of the theorem is complete.

Theorem 4.7 Let ( ¯A, ¯b, ¯α, ¯x) ∈ gphS be such that (cid:107)¯x(cid:107) = ¯α and ¯A¯x + ¯b = 0. Then, the following hold

(i) If S(·) is locally Lipschitz-like around ( ¯A, ¯b, ¯α, ¯x), then the constraint qual-

¯α ¯x = 0

 

iﬁcation below is satisﬁed



 ¯Av(cid:48) − α(cid:48)  (cid:104)v(cid:48), ¯x(cid:105) ≥ 0  v(cid:48) ∈ IRn, α(cid:48) ≤ 0

=⇒ (4.39) v(cid:48) = 0 α(cid:48) = 0.

(cid:33)

(ii) If det ¯A (cid:54)= 0, detQ1( ¯A, ¯b, ¯α, ¯x) (cid:54)= 0, where

(cid:32) ¯A − 1 ¯α ¯x ¯x(cid:62) 0

(4.40) , Q1( ¯A, ¯b, ¯α, ¯x) :=

and (4.39) is satisﬁed, then S(·) is locally Lipschitz-like around ( ¯A, ¯b, ¯α, ¯x).

(cid:0) − ¯Av(cid:48), A(cid:48) − (v(cid:48)

i¯xj), α(cid:48), y(cid:48) + v(cid:48)(cid:1) ∈ (cid:98)D∗M (¯x, ¯w, ¯v)(v(cid:48)) × {0IRn}.

(i) Suppose that S(·) is locally Lipschitz-like around ( ¯A, ¯b, ¯α, ¯x). Proof. Then we have D∗S( ¯A, ¯b, ¯α, ¯x)(0) = {0}. Thus, D∗ (cid:101)S( ¯w, ¯y, ¯x)(0) = {0}. By (4.34), the latter implies that (cid:98)ΩG,¯y(0) = {0}. Observe that (w(cid:48), y(cid:48)) ∈ (cid:98)ΩG,¯y(0) if and only if there exists v(cid:48) ∈ IRn such that

(cid:0) − ¯Av(cid:48), α(cid:48), A(cid:48) − (v(cid:48)

i¯xj), y(cid:48) + v(cid:48)(cid:1) ∈ (cid:98)D∗N (¯x, ¯α, ¯v)(v(cid:48)) × {0H(n)∗} × {0IRn}.

Due to Remark 4.4, the last inclusion means that



¯α ¯x

By virtue of Theorem 4.2, this means that

i¯xj)

(4.41)

− ¯Av(cid:48) = − α(cid:48)  (cid:104)v(cid:48), ¯x(cid:105) ≥ 0 α(cid:48) ≤ 0, A(cid:48) = (v(cid:48)  y(cid:48) = −v(cid:48).

Therefore, the condition (cid:98)ΩG,¯y(0) = {0} is equivalent to saying that (4.39) holds.

(cid:0) − ¯Av(cid:48), A(cid:48) − (v(cid:48)

i¯xj), α(cid:48), y(cid:48) + v(cid:48)(cid:1) ∈ D∗M (¯x, ¯w, ¯v)(v(cid:48)) × {0IRn}

(ii) Suppose that det ¯A (cid:54)= 0, detQ1( ¯A, ¯b, ¯α, ¯x) (cid:54)= 0 with Q1( ¯A, ¯b, ¯α, ¯x) given by (4.40), and (4.39) is satisﬁed. As we have seen in the proof of Theo- rem 4.6(i), (w(cid:48), y(cid:48)) ∈ ΩG,¯y(0) if and only if there exists v(cid:48) ∈ IRn such that

i¯xj), y(cid:48) + v(cid:48)(cid:1) ∈ D∗N (¯x, ¯α, ¯v)(v(cid:48)) × {0H(n)∗} × {0IRn}. (4.42)

or, equivalently, (cid:0) − ¯Av(cid:48), α(cid:48), A(cid:48) − (v(cid:48)

(cid:54)= 0. Hence − ¯Av(cid:48)



i¯xj)

If (cid:104)v(cid:48), ¯x(cid:105) < 0, then (− ¯Av(cid:48), α(cid:48)) (cid:54)∈ D∗N (¯x, ¯α, ¯v)(v(cid:48)). Indeed, the inequal- (cid:54)= 0 because det ¯A (cid:54)= 0. By ity (cid:104)v(cid:48), ¯x(cid:105) < 0 yields v(cid:48) Theorem 4.4(i) and by the condition (cid:104)v(cid:48), ¯x(cid:105) < 0, we have (− ¯Av(cid:48), α(cid:48)) (cid:54)∈ D∗N (¯x, ¯α, ¯v)(v(cid:48)). Therefore, (4.42) is equivalent to

(cid:104)v(cid:48), ¯x(cid:105) ≥ 0  (− ¯Av(cid:48), α(cid:48)) ∈ D∗N (¯x, ¯α, ¯v)(v(cid:48)) A(cid:48) = (v(cid:48)  y(cid:48) = −v(cid:48).



So, the equality ΩG,¯y(0) = {0} holds if and only if

i¯xj)

 A(cid:48) = 0  α(cid:48) = 0  y(cid:48) = 0

 

=⇒ (4.43)

(cid:104)v(cid:48), ¯x(cid:105) > 0 (− ¯Av(cid:48), α(cid:48)) ∈ D∗N (¯x, ¯α, ¯v)(v(cid:48)) A(cid:48) = (v(cid:48) y(cid:48) = −v(cid:48)



and

i¯xj)

 A(cid:48) = 0  α(cid:48) = 0  y(cid:48) = 0.

=⇒ (4.44)

¯α ¯x = 0

 

(cid:104)v(cid:48), ¯x(cid:105) = 0  (− ¯Av(cid:48), α(cid:48)) ∈ D∗N (¯x, ¯α, ¯v)(v(cid:48)) A(cid:48) = (v(cid:48)  y(cid:48) = −v(cid:48) By Theorem 4.4(i), (4.43) means that



 ¯Av(cid:48) − α(cid:48)  (cid:104)v(cid:48), ¯x(cid:105) > 0  v(cid:48) ∈ IRn, α(cid:48) ≤ 0

¯α ¯x = 0

 

=⇒ (4.45) v(cid:48) = 0 α(cid:48) = 0.



=⇒ (4.46) v(cid:48) = 0 α(cid:48) = 0. Since (4.39) is satisﬁed by our assumptions, (4.45) is valid. By virtue of Theorem 4.4(ii), (4.44) is equivalent to  ¯Av(cid:48) − α(cid:48)  ¯x(cid:62)v(cid:48) = 0  v(cid:48) ∈ IRn, α(cid:48) ∈ IR

(cid:33) (cid:32)

(cid:33)

(cid:33) (cid:35)

 

or, equivalently,

(cid:32) 0 0

(cid:34) (cid:32) ¯A − 1 ¯α ¯x ¯x(cid:62) 0



= =⇒ v(cid:48) α(cid:48) v(cid:48) = 0 α(cid:48) = 0.

The latter holds because detQ1( ¯A, ¯b, ¯α, ¯x) (cid:54)= 0 where Q1( ¯A, ¯b, ¯α, ¯x) is given by (4.40). We have shown that under the assumptions made, the equality ΩG,¯y(0) = {0} holds. This implies that S(·) is locally Lipschitz-like around ( ¯A, ¯b, ¯α, ¯x), and completes the proof. (cid:50)

(cid:110)

We now analyze Theorems 4.6 and 4.7 by four examples. The ﬁrst two show how Theorems 4.6 can recognize stability/instability of S(·) in the situation where (cid:107)¯x(cid:107) = ¯α and ¯A¯x + ¯b (cid:54)= 0. The third example illustrates a good association of the necessary stability condition and the suﬃcient stability condition provided by Theorem 4.7 for the case where (cid:107)¯x(cid:107) = ¯α and ¯A¯x + ¯b = 0. The last example shows that the necessary stability condition given in Theorem 4.7(i) can recognize instability of many linear GEs.

(cid:12) (cid:12) x = (x1, x2)(cid:62) ∈ IR2, x2

2 + x1

1 + x2

2 ≤ 1

min (4.47) f (x) = −4x2 Example 4.2 Following [51] and [23], we consider the problem (cid:111) .

(cid:33)

Here we have

(cid:32) 0 0 0 −8

(cid:32) 1 0

¯A = , ¯b = , ¯α = 1.

(cid:110)

√ Using the necessary optimality condition (4.4) we ﬁnd that √ (−1, 0)(cid:62), (−1/8, S( ¯A, ¯b, ¯α) = 63/8)(cid:62), (−1/8, − 63/8)(cid:62)(cid:111) .

√ 63/8)(cid:62)





1 8 √

The Lagrange multiplier corresponding to the KKT point ¯x := (−1/8, is λ = 8. Hence,

 

  =

63 8 0

detQ( ¯A, ¯b, ¯α, ¯x) = det , 63 8 8 0 − 1 8 0 0 − √ 63 8

√



and we see that the stability criterion (4.35) is satisﬁed. Observe that ¯A¯x + ¯b (cid:54)= 0. Thanks to Theorem 4.6(ii), we can infer that S(·) is locally Lipschitz-like around ( ¯A, ¯b, ¯α, ¯x) ∈ gphS. By symmetry, we see at once that √ 63/8)(cid:62) is also valid for the KKT point the assertions made for ¯x = (−1/8, 63/8)(cid:62). For the KKT point (cid:98)x = (−1, 0)(cid:62) and the associated La- (−1/8, − grange multiplier λ = 1, we ﬁnd that ¯A(cid:98)x + ¯b (cid:54)= 0 and

 

 1 0 1   = −7. 0 −7 0 0 0 −1

detQ( ¯A, ¯b, ¯α, (cid:98)x) = det

So, by Theorem 4.6(ii), S(·) is also locally Lipschitz-like around ( ¯A, ¯b, ¯α, (cid:98)x) ∈ gphS.

(cid:110)

(cid:111)

Example 4.3 As in [51] and [23], we consider the problem

(cid:12) (cid:12) x = (x1, x2, x3)(cid:62) ∈ IR3, x2

2+x2

3)+x1

1+x2

2+x2

3 ≤ 1

min (4.48) f (x) = −4(x2





with the data tube ( ¯A, ¯b, ¯α) given by

  ,

 

 1  0  , 0

 0 0  0 −8  0

¯b = ¯α = 1. ¯A =

(cid:111)

0 0 0 −8

(cid:110) (−1/8, x2, x3)(cid:62)(cid:12)

(cid:12) x2

2 + x2

3 = 63/64

∪ Using (4.4) we obtain (−1, 0, 0)(cid:62)(cid:111) (cid:110) S( ¯A, ¯b, ¯α) =



For the KKT point (cid:98)x := (−1, 0, 0)(cid:62) with the associated Lagrange multiplier λ = 1, a computation similar to that given in Example 4.2 shows that S(·) is locally Lipschitz-like around the point ( ¯A, ¯b, ¯α, (cid:98)x) ∈ gphS. To complete the stability analysis, ﬁx any t ∈ [0, 2π) and consider the KKT point √ √ 63/8) sin t, ( 63/8) cos t(cid:1)(cid:62) xt := (cid:0) − 1/8, (

√

1 8 63

√

with the associated Lagrange multiplier λ = 8. Note that 

√

    

    

8 sin t 8 cos t 0

= 0. detQ( ¯A, ¯b, ¯α, xt) = det − −

8 0 0 − 1 8 0 0 0 8 cos t

0 0 0 8 sin t Since ¯A¯xt + ¯b (cid:54)= 0 for any t ∈ [0, 2π), applying Theorem 4.6(ii) we deduce that the map S(·) is not locally Lipschitz-like around any point ( ¯A, ¯b, ¯α, xt) for t ∈ [0, 2π).

√ √ 2, √ √ 2,







  = 0

  −



 



Example 4.4 Consider (4.1) with n = 2, ¯A = I, ¯b = −( 2)(cid:62), ¯α = 2, 2)(cid:62). We have (cid:107)¯x(cid:107) = 2 = ¯α and ¯v := −¯b − ¯A¯x = 0. The and ¯x = ( necessary condition for stability of S(·) provided by Theorem 4.7(i) is as follows: √ 2 √ α(cid:48) 2 2 =⇒ v(cid:48) = 0 α(cid:48) = 0.

 

2)(cid:62) ∈ IR2

1, v(cid:48)

v(cid:48) 1 v(cid:48) 2 v(cid:48) 1 + v(cid:48) 2 ≥ 0 α(cid:48) ≤ 0, v(cid:48) = (v(cid:48)





It is not diﬃcult to see that this condition is satisﬁed. As det ¯A (cid:54)= 0, the suﬃ- cient stability condition from Theorem 4.7(ii) reduces to detQ1( ¯A, ¯b, ¯α, ¯x) (cid:54)= 0, where the matrix Q1( ¯A, ¯b, ¯α, ¯x) is given by (4.40). We have

 

  = 2 (cid:54)= 0.

√ √ detQ1( ¯A, ¯b, ¯α, ¯x) =

1 0 √ 2 0 − 1 − √ 2 2/2 2/2 0

Thus S(·) is locally Lipschitz-like around ( ¯A, ¯b, ¯α, ¯x) ∈ gphS.

Example 4.5 (A class of unstable problems) Let ¯A ∈ H(n) be not positive deﬁnite, i.e., among the eigenvalue set {¯λ1, ¯λ2, . . . , ¯λn} of ¯A there is an element ¯λi0 ≤ 0. Since det( ¯A − ¯λi0I) = 0, there exists ¯x ∈ IRn with (cid:107)¯x(cid:107) = 1 such that ( ¯A − ¯λi0I)¯x = 0. Let ¯b = − ¯A¯x and let ¯α = 1. We claim that S(·) is not locally Lipschitz-like around ( ¯A, ¯b, ¯α, ¯x) ∈ gphS. To see this, it suﬃces to check that (4.39) is violated. For v(cid:48) := ¯x, we have v(cid:48) (cid:54)= 0 and (cid:104)v(cid:48), ¯x(cid:105) > 0. Choose α(cid:48) = ¯λi0 ≤ 0. The condition ¯Av(cid:48) − α(cid:48) ¯α ¯x = 0 in (4.39) is equivalent to ( ¯A − ¯λi0I)¯x = 0. Since the latter is guaranteed by the choice of ¯x, we conclude that (4.39) fails to holds. Our claim has been proved.

4.4 Conclusions

Solution stability of a class of linear generalized equations in ﬁnite dimen- sional Euclidean spaces is studied by means of generalized diﬀerentiation. Exact formulas for the Fr´echet and Mordukhovich coderivatives of the normal cone mappings of perturbed Euclidean balls are obtained in Theorems 4.1, 4.2, 4.3, and 4.4. These exact formulas solve the open problems raised by Lee and Yen in [23]. In Theorems 4.6 and 4.7, necessary and suﬃcient conditions for the local Lipschitz-like property of the solution maps of such linear gener- alized equations are derived from the obtained coderivative formulas. Since the trust-region subproblems in nonlinear programming can be regarded as linear generalized equations, these conditions lead to new results on stability of the parametric trust-region subproblems. A series of useful examples have been provided to illustrate the solution stability criteria for this type of linear generalized equations.

General Conclusions

The main results of this dissertation include:

1. An exact formula for the Fr´echet coderivative and some upper and lower estimates for the Mordukhovich coderivative of the normal cone mappings to linearly perturbed polyhedral convex sets in reﬂexive Banach spaces.

2. Upper estimates for the Fr´echet and the limiting normal cone to the graphs of the normal cone mappings to nonlinearly perturbed polyhedral convex sets in ﬁnite dimensional spaces.

3. Exact formulas for the Fr´echet and the Mordukhovich coderivatives of

the normal cone mappings to perturbed Euclidean balls.

4. Conditions for the local Lipschitz-like property and local metric regularity of the solution maps of parametric aﬃne variational inequalities under linear/nonlinear perturbations, and conditions for the local Lipschitz-like property of the solution maps of a class of linear generalized equations in ﬁnite dimensional spaces.

The problem of ﬁnding coderivative estimates for nonlinearly perturbed polyhedral normal cone mappings requires further investigations, although it has been studied by several authors.

The general problem of computing the Fr´echet coderivative (resp., the Mordukhovich coderivative) of the mapping (x, p) (cid:55)→ (cid:98)N (x; Θ(p)) (resp., of the mapping (x, p) (cid:55)→ N (x; Θ(p))), where

Θ(p) := {x ∈ X| Ψ(x, p) ∈ K}

with Ψ : X × P → Y being a C2-smooth vector function which maps the product X × P of two Banach spaces into another Banach space Y , and K ⊂ Y is a closed convex cone, is our next target. In this direction, we have obtained some preliminary results on computing coderivatives of the normal cone mappings to parametric sets with smooth boundaries.

List of Author’s Related Papers

1. N. T. Qui, Linearly perturbed polyhedral normal cone mappings and

applications, Nonlinear Anal., 74 (2011), pp. 1676–1689.

2. N. T. Qui, New results on linearly perturbed polyhedral normal cone

mappings, J. Math. Anal. Appl., 381 (2011), pp. 352–364.

3. N. T. Qui, Upper and lower estimates for a Fr´echet normal cone, Acta

Math. Vietnam., 36 (2011), pp. 601–610.

4. N. T. Qui, Nonlinear perturbations of polyhedral normal cone mappings and aﬃne variational inequalities, J. Optim. Theory Appl., 153 (2012), pp. 98–122.

5. N. T. Qui and N. D. Yen, A class of linear generalized equations,

SIAM J. Optim., (2014). [Accepted for publication]

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Doctoral dissertation in mathematics: Coderivatives of normal cone mappings and applications

INSTITUTE OF MATHEMATICS

CODERIVATIVES OF NORMAL CONE MAPPINGS AND APPLICATIONS

INSTITUTE OF MATHEMATICS

Nguyen Thanh Qui

CODERIVATIVES OF NORMAL CONE MAPPINGS AND APPLICATIONS

Conﬁrmation

Acknowledgments

Contents

Table of Notations

List of Figures

Introduction

Chapter 1

Preliminary

1.1 Basic Deﬁnitions and Conventions

1.2 Normal and Tangent Cones

1.3 Coderivatives and Subdiﬀerential

1.4 Lipschitzian Properties and Metric Regularity

1.5 Conclusions

Chapter 2

Linear Perturbations of Polyhedral

Normal Cone Mappings

2.1 The Normal Cone Mapping F(x, b)

2.2 The Fr´echet Coderivative of F(x, b)

2.3 The Mordukhovich Coderivative of F(x, b)

2.4 AVIs under Linear Perturbations

2.5 Conclusions

Chapter 3

Nonlinear Perturbations of Polyhedral

Normal Cone Mappings

3.1 The Normal Cone Mapping F(x, A, b)

3.2 Estimation of the Fr´echet Normal Cone to gphF

3.3 Estimation of the Limiting Normal Cone to gphF

3.4 AVIs under Nonlinear Perturbations

3.5 Conclusions

Chapter 4

A Class of Linear Generalized

Equations

4.1 Linear Generalized Equations

4.2 Formulas for Coderivatives

4.3 Necessary and Suﬃcient Conditions for Stability

4.4 Conclusions

General Conclusions

List of Author’s Related Papers

References

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