Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 857520, 13 pages doi:10.1155/2011/857520
Research Article Second-Order Contingent Derivative of the Perturbation Map in Multiobjective Optimization
Q. L. Wang1 and S. J. Li2
1 College of Sciences, Chongqing Jiaotong University, Chongqing 400074, China 2 College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China
Correspondence should be addressed to Q. L. Wang, wangql97@126.com
Received 14 October 2010; Accepted 24 January 2011
Academic Editor: Jerzy Jezierski
Copyright q 2011 Q. L. Wang and S. J. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Some relationships between the second-order contingent derivative of a set-valued map and its profile map are obtained. By virtue of the second-order contingent derivatives of set-valued maps, some results concerning sensitivity analysis are obtained in multiobjective optimization. Several examples are provided to show the results obtained.
1. Introduction
In this paper, we consider a family of parametrized multiobjective optimization problems
(cid:5) ⎧ ⎨ , min f(cid:2)u, x(cid:3) (cid:4) (cid:6) f1(cid:2)u, x(cid:3), f2(cid:2)u, x(cid:3), . . . , fm(cid:2)u, x(cid:3) (cid:2)1.1(cid:3) (cid:2)PVOP(cid:3) ⎩ u ∈ X(cid:2)x(cid:3) ⊆ Rp. s.t.
Here, u is a p-dimensional decision variable, x is an n-dimensional parameter vector, X is a nonempty set-valued map from Rn to Rp, which specifies a feasible decision set, and f is an objective map from Rp × Rn to Rm, where m, n, p are positive integers. The norms of all finite dimensional spaces are denoted by (cid:4) · (cid:4). C is a closed convex pointed cone with nonempty interior in Rm. The cone C induces a partial order ≤C on Rm, that is, the relation ≤C is defined by
∀y, y(cid:6) ∈ Rm. y ≤C y(cid:6) ←→ y(cid:6) − y ∈ C, (cid:2)1.2(cid:3)
2 Fixed Point Theory and Applications
We use the following notion. For any y, y(cid:6) ∈ Rm,
y Based on these notations, we can define the following two sets for a set M in Rm:
(cid:2)i(cid:3) y0 ∈ M is a C-minimal point of M with respect to C if there exists no y ∈ M, such that y ≤C y0, y /(cid:4) y0, (cid:2)ii(cid:3) y0 ∈ M is a weakly C-minimal point of M with respect to C if there exists no y ∈ M, such that y The sets of C-minimal point and weakly C-minimal point of M are denoted by MinCM and
WMinCM, respectively. Let G be a set-valued map from Rn to Rm defined by (cid:8)
. G(cid:2)x(cid:3) (cid:4) (cid:2)1.4(cid:3) (cid:7)
y ∈ Rm | y (cid:4) f(cid:2)u, x(cid:3), for some u ∈ X(cid:2)x(cid:3) G(cid:2)x(cid:3) is considered as the feasible set map. In the vector optimization problem corresponding
to each parameter valued x, our aim is to find the set of C-minimal point of the feasible set
map G(cid:2)x(cid:3). The set-valued map W from Rn to Rm is defined by (cid:2)1.5(cid:3) W(cid:2)x(cid:3) (cid:4) MinCG(cid:2)x(cid:3), for any x ∈ Rn, and call it the perturbation map for (cid:2)PVOP(cid:3). Sensitivity and stability analysis is not only theoretically interesting but also practically
important in optimization theory. Usually, by sensitivity we mean the quantitative analysis,
that is, the study of derivatives of the perturbation function. On the other hand, by stability
we mean the qualitative analysis, that is, the study of various continuity properties of the
perturbation (cid:2)or marginal(cid:3) function (cid:2)or map(cid:3) of a family of parametrized vector optimization
problems. Some interesting results have been proved for sensitivity and stability in optimization
(cid:2)see (cid:5)1–16(cid:6)(cid:3). Tanino (cid:5)5(cid:6) obtained some results concerning sensitivity analysis in vector
optimization by using the concept of contingent derivatives of set-valued maps introduced
in (cid:5)17(cid:6), and Shi (cid:5)8(cid:6) and Kuk et al. (cid:5)7, 11(cid:6) extended some of Tanino’s results. As for
vector optimization with convexity assumptions, Tanino (cid:5)6(cid:6) studied some quantitative
and qualitative results concerning the behavior of the perturbation map, and Shi (cid:5)9(cid:6)
studied some quantitative results concerning the behavior of the perturbation map. Li (cid:5)10(cid:6)
discussed the continuity of contingent derivatives for set-valued maps and also discussed
the sensitivity, continuity, and closeness of the contingent derivative of the marginal map.
By virtue of lower Studniarski derivatives, Sun and Li (cid:5)14(cid:6) obtained some quantitative
results concerning the behavior of the weak perturbation map in parametrized vector
optimization. Higher order derivatives introduced by the higher order tangent sets are very
important concepts in set-valued analysis. Since higher order tangent sets, in general, are
not cones and convex sets, there are some difficulties in studying set-valued optimization
problems by virtue of the higher order derivatives or epiderivatives introduced by the higher Fixed Point Theory and Applications 3 order tangent sets. To the best of our knowledge, second-order contingent derivatives of
perturbation map in multiobjective optimization have not been studied until now. Motivated
by the work reported in (cid:5)5–11, 14(cid:6), we discuss some second-order quantitative results
concerning the behavior of the perturbation map for (cid:2)PVOP(cid:3). The rest of the paper is organized as follows. In Section 2, we collect some important
concepts in this paper. In Section 3, we discuss some relationships between the second-order
contingent derivative of a set-valued map and its profile map. In Section 4, by the second-
order contingent derivative, we discuss the quantitative information on the behavior of the
perturbation map for (cid:2)PVOP(cid:3). In this section, we state several important concepts. Let F : Rn → 2Rm be nonempty set-valued maps. The efficient domain and graph of F are defined by (cid:2)2.1(cid:3) dom(cid:2)F(cid:3) (cid:4) {x ∈ Rn | F(cid:2)x(cid:3) /(cid:4) ∅},
(cid:7)(cid:5) (cid:6) , x, y ∈ Rn × Rm | y ∈ F(cid:2)x(cid:3), x ∈ Rn(cid:8) gph(cid:2)F(cid:3) (cid:4) respectively. The profile map F(cid:8) of F is defined by F(cid:8)(cid:2)x(cid:3) (cid:4) F(cid:2)x(cid:3) (cid:8) C, for every x ∈ dom(cid:2)F(cid:3),
where C is the order cone of Rm. Definition 2.1 (cid:2)see (cid:5)18(cid:6)(cid:3). A base for C is a nonempty convex subset Q of C with 0Rm /∈ clQ,
such that every c ∈ C, c /(cid:4) 0Rm, has a unique representation of the form αb, where b ∈ Q and
α > 0. Definition 2.2 (cid:2)see (cid:5)19(cid:6)(cid:3). F is said to be locally Lipschitz at x0 ∈ Rn if there exist a real number
γ > 0 and a neighborhood U(cid:2)x0(cid:3) of x0, such that (cid:2)2.2(cid:3) F(cid:2)x1(cid:3) ⊆ F(cid:2)x2(cid:3) (cid:8) γ (cid:4)x1 − x2(cid:4)BRm , ∀x1, x2 ∈ U(cid:2)x0(cid:3), where BRm denotes the closed unit ball of the origin in Rm. In this section, let X be a normed space supplied with a distance d, and let A be a subset of
X. We denote by d(cid:2)x, A(cid:3) (cid:4) infy∈Ad(cid:2)x, y(cid:3) the distance from x to A, where we set d(cid:2)x, ∅(cid:3) (cid:4) (cid:8)∞.
Let Y be a real normed space, where the space Y is partially ordered by nontrivial pointed
closed convex cone C ⊂ Y . Now, we recall the definitions in (cid:5)20(cid:6). 4 Fixed Point Theory and Applications Definition 3.1 (cid:2)see (cid:5)20(cid:6)(cid:3). Let A be a nonempty subset X, x0 ∈ cl(cid:2)A(cid:3), and u ∈ X, where cl(cid:2)A(cid:3)
denotes the closure of A. A (cid:2)x0, u(cid:3) of A at (cid:2)x0, u(cid:3) is defined as (cid:2)i(cid:3) The second-order contingent set T (cid:2)2(cid:3) nxn ∈ A (cid:8), xn −→ x, s.t. x0 (cid:8) hnu (cid:8) h2 (cid:10)
. (cid:9)
x ∈ X | ∃hn −→ 0 (cid:2)3.1(cid:3) T (cid:2)2(cid:3)
A (cid:2)x0, u(cid:3) (cid:4) A (cid:2)x0, u(cid:3) of A at (cid:2)x0, u(cid:3) is defined as (cid:2)ii(cid:3) The second-order adjacent set T (cid:4)(cid:2)2(cid:3) nxn ∈ A (cid:8), ∃xn −→ x, s.t. x0 (cid:8) hnu (cid:8) h2 (cid:10)
. (cid:9)
x ∈ X | ∀hn −→ 0 (cid:2)3.2(cid:3) T (cid:4)(cid:2)2(cid:3)
A (cid:2)x0, u(cid:3) (cid:4) Definition 3.2 (cid:2)see (cid:5)20(cid:6)(cid:3). Let X, Y be normed spaces and F : X → 2Y be a set-valued map,
and let (cid:2)x0, y0(cid:3) ∈ gph(cid:2)F(cid:3) and (cid:2)u, v(cid:3) ∈ X × Y . (cid:2)i(cid:3) The set-valued map D(cid:2)2(cid:3)F(cid:2)x0, y0, u, v(cid:3) from X to Y defined by gph(cid:2)F(cid:3) (cid:11) (cid:6) D(cid:2)2(cid:3)F (cid:4) T (cid:2)2(cid:3) (cid:6)
, gph (cid:2)3.3(cid:3) (cid:5)
x0, y0, u, v (cid:5)
x0, y0, u, v is called second-order contingent derivative of F at (cid:2)x0, y0, u, v(cid:3). (cid:2)ii(cid:3) The set-valued map D(cid:4)(cid:2)2(cid:3)F(cid:2)x0, y0, u, v(cid:3) from X to Y defined by gph(cid:2)F(cid:3) (cid:11) (cid:5) (cid:6) D(cid:4)(cid:2)2(cid:3)F (cid:4) T (cid:4)(cid:2)2(cid:3) (cid:6)
, gph (cid:2)3.4(cid:3) (cid:5)
x0, y0, u, v x0, y0, u, v is called second-order adjacent derivative of F at (cid:2)x0, y0, u, v(cid:3). Definition 3.3 (cid:2)see (cid:5)21(cid:6)(cid:3). The C-domination property is said to be held for a subset H of Y if
H ⊂ MinCH (cid:8) C. Proposition 3.4. Let (cid:2)x0, y0(cid:3) ∈ gph(cid:2)F(cid:3) and (cid:2)u, v(cid:3) ∈ X × Y , then D(cid:2)2(cid:3)F (cid:6)
(cid:2)x(cid:3), (cid:6)
(cid:2)x(cid:3) (cid:8) C ⊆ D(cid:2)2(cid:3)(cid:2)F (cid:8) C(cid:3) (cid:2)3.5(cid:3) (cid:5)
x0, y0, u, v (cid:5)
x0, y0, u, v for any x ∈ X. Proof. The conclusion can be directly obtained similarly as the proof of (cid:5)5, Proposition 2.1(cid:6). It follows from Proposition 3.4 that (cid:6)(cid:13) (cid:6)(cid:13)
. (cid:12)
D(cid:2)2(cid:3)F
dom (cid:12)
D(cid:2)2(cid:3)F(cid:8)
⊆ dom (cid:2)3.6(cid:3) (cid:5)
x0, y0, u, v (cid:5)
x0, y0, u, v 5 Fixed Point Theory and Applications Note that the inclusion of (cid:6)
(cid:2)x(cid:3) ⊆ D(cid:2)2(cid:3)F (cid:6)
(cid:2)x(cid:3) (cid:8) C, D(cid:2)2(cid:3)F(cid:8) (cid:2)3.7(cid:3) (cid:5)
x0, y0, u, v (cid:5)
x0, y0, u, v may not hold. The following example explains the case. Example 3.5. Let X (cid:4) R, Y (cid:4) R, and C (cid:4) R(cid:8). Consider a set-valued map F : X → 2Y defined
by (cid:7) (cid:8) ⎧
⎨ y | y ≥ x2 if x ≤ 0, F(cid:2)x(cid:3) (cid:8) (cid:7) (cid:2)3.8(cid:3) ⎩ x2, −1 if x > 0. Let (cid:2)x0, y0(cid:3) (cid:4) (cid:2)0, 0(cid:3) ∈ gph(cid:2)F(cid:3) and (cid:2)u, v(cid:3) (cid:4) (cid:2)1, 0(cid:3), then, for any x ∈ X, D(cid:2)2(cid:3)F (cid:6)
(cid:2)x(cid:3) (cid:4) R, D(cid:2)2(cid:3)F(cid:8) (cid:2)3.9(cid:3) (cid:6)
(cid:2)x(cid:3) (cid:4) {1}. (cid:5)
x0, y0, u, v (cid:5)
x0, y0, u, v Thus, one has x ∈ X, (cid:6)
(cid:2)x(cid:3) (cid:8) C, (cid:6)
(cid:2)x(cid:3) /⊆ D(cid:2)2(cid:3)F D(cid:2)2(cid:3)F(cid:8) (cid:2)3.10(cid:3) (cid:5)
x0, y0, u, v (cid:5)
x0, y0, u, v which shows that the inclusion of (cid:2)3.7(cid:3) does not hold here. Proposition 3.6. Let (cid:2)x0, y0(cid:3) ∈ gph(cid:2)F(cid:3) and (cid:2)u, v(cid:3) ∈ X × Y . Suppose that C has a compact base Q,
then for any x ∈ X, (cid:6)
(cid:2)x(cid:3) ⊆ D(cid:2)2(cid:3)F (cid:6)
(cid:2)x(cid:3). (cid:2)3.11(cid:3) MinCD(cid:2)2(cid:3)F(cid:8) (cid:5)
x0, y0, u, v (cid:5)
x0, y0, u, v Proof. Let x ∈ X. If MinCD(cid:2)2(cid:3)F(cid:8)(cid:2)x0, y0, u, v(cid:3)(cid:2)x(cid:3) (cid:4) ∅, then (cid:2)3.11(cid:3) holds trivially. So, we assume
that MinCD(cid:2)2(cid:3)F(cid:8)(cid:2)x0, y0, u, v(cid:3)(cid:2)x(cid:3) /(cid:4) ∅, and let (cid:5) (cid:6)
(cid:2)x(cid:3). (cid:2)3.12(cid:3) y ∈ MinCD(cid:2)2(cid:3)F(cid:8) x0, y0, u, v Since y ∈ D(cid:2)2(cid:3)F(cid:8)(cid:2)x0, y0, u, v(cid:3)(cid:2)x(cid:3), there exist sequences {hn} with hn → 0(cid:8), {(cid:2)xn, yn(cid:3)} with (cid:2)xn, yn(cid:3) → (cid:2)x, y(cid:3), and {cn} with cn ∈ C, such that nxn (cid:11) (cid:14) (cid:6) ∈ F , (cid:5)
yn − cn for any n. (cid:2)3.13(cid:3) y0 (cid:8) hnv (cid:8) h2
n x0 (cid:8) hnu (cid:8) h2 It follows from cn ∈ C and C has a compact base Q that there exist some αn > 0 and
bn ∈ Q, such that, for any n, one has cn (cid:4) αnbn. Since Q is compact, we may assume without
loss of generality that bn → b ∈ Q. 6 Fixed Point Theory and Applications We now show αn → 0. Suppose that αn (cid:0) 0, then for some ε > 0, we may assume
without loss of generality that αn ≥ ε, for all n, by taking a subsequence if necessary. Let
cn (cid:4) (cid:2)ε/αn(cid:3)cn, then, for any n, cn − cn ∈ C and nxn (cid:11) (cid:14) (cid:6) . (cid:5)
yn − cn ∈ F(cid:8) (cid:2)3.14(cid:3) y0 (cid:8) hnv (cid:8) h2
n x0 (cid:8) hnu (cid:8) h2 Since cn (cid:4) (cid:2)ε/αn(cid:3)cn (cid:4) εbn, for all n, cn → εb /(cid:4) 0Y . Thus, yn − cn → y − εb. It follows from
(cid:2)3.14(cid:3) that (cid:6)
(cid:2)x(cid:3), y − εb ∈ D(cid:2)2(cid:3)F(cid:8) (cid:2)3.15(cid:3) (cid:5)
x0, y0, u, v (cid:2)2(cid:3)F(cid:2)x0, y0, u, v(cid:3)(cid:2)x(cid:3). So, which contradicts (cid:2)3.12(cid:3), since εb ∈ C. Thus, αn → 0 and yn − cn → y. Then, it follows from
(cid:2)3.13(cid:3) that y ∈ D (cid:6)
(cid:2)x(cid:3) ⊆ D(cid:2)2(cid:3)F (cid:6)
(cid:2)x(cid:3), (cid:2)3.16(cid:3) MinCD(cid:2)2(cid:3)F(cid:8) (cid:5)
x0, y0, u, v (cid:5)
x0, y0, u, v and the proof of the proposition is complete. Note that the inclusion of (cid:6)
(cid:2)x(cid:3) ⊆ D(cid:2)2(cid:3)F (cid:6)
(cid:2)x(cid:3), (cid:2)3.17(cid:3) WMinCD(cid:2)2(cid:3)F(cid:8) (cid:5)
x0, y0, u, v (cid:5)
x0, y0, u, v (cid:8). Obviously, C has a compact base. Consider a may not hold under the assumptions of Proposition 3.6. The following example explains the
case. (cid:10) Example 3.7. Let X (cid:4) R, Y (cid:4) R2, and C (cid:4) R2
set-valued map F : X → 2Y defined by
(cid:9)(cid:5) (cid:6) . F(cid:2)x(cid:3) (cid:4) (cid:2)3.18(cid:3) y1, y2 | y1 ≥ x, y2 (cid:4) x2 Let (cid:2)x0, y0(cid:3) (cid:4) (cid:2)0, (cid:2)0, 0(cid:3)(cid:3) ∈ gph(cid:2)F(cid:3) and (cid:2)u, v(cid:3) (cid:4) (cid:2)1, (cid:2)1, 0(cid:3)(cid:3). For any x ∈ X, (cid:5) (cid:6) (cid:7)(cid:5) (cid:8)
, (cid:6)
(cid:2)x(cid:3) (cid:4) D(cid:2)2(cid:3)F(cid:8) x0, y0, u, v (cid:2)3.19(cid:3) | y1 ≥ x, y2 ≥ 1
(cid:6) (cid:8) y1, y2
(cid:7)(cid:5) D(cid:2)2(cid:3)F . (cid:6)
(cid:2)x(cid:3) (cid:4) (cid:5)
x0, y0, u, v y1, 1 | y1 ≥ x Then, for any x ∈ X, WMinCD(cid:2)2(cid:3)F(cid:8)(cid:2)x0, y0, u, v(cid:3)(cid:2)x(cid:3) (cid:4) {(cid:2)y1, 1(cid:3) | y1 ≥ x} ∪ {(cid:2)x, y2(cid:3) | y2 ≥ 1}. So,
the inclusion of (cid:2)3.17(cid:3) does not hold here. (cid:6)
(cid:2)x(cid:3). (cid:2)3.20(cid:3) Proposition 3.8. Let (cid:2)x0, y0(cid:3) ∈ gph(cid:2)F(cid:3) and (cid:2)u, v(cid:3) ∈ X × Y . Suppose that C has a compact
base Q and P (cid:2)x(cid:3) :(cid:4) D(cid:2)2(cid:3)F(cid:8)(cid:2)x0, y0, u, v(cid:3)(cid:2)x(cid:3) satisfies the C-domination property for all x ∈ K :(cid:4)
dom(cid:5)D(cid:2)2(cid:3)F(cid:2)x0, y0, u, v(cid:3)(cid:6), then for any x ∈ K,
(cid:6)
(cid:2)x(cid:3) (cid:4) MinCD(cid:2)2(cid:3)F MinCD(cid:2)2(cid:3)F(cid:8) (cid:5)
x0, y0, u, v (cid:5)
x0, y0, u, v Fixed Point Theory and Applications 7 Proof. From Proposition 3.4, one has D(cid:2)2(cid:3)F (cid:6)
(cid:2)x(cid:3), (cid:6)
(cid:2)x(cid:3) (cid:8) C ⊆ D(cid:2)2(cid:3)F(cid:8) (cid:2)3.21(cid:3) for any x ∈ K. (cid:5)
x0, y0, u, v (cid:5)
x0, y0, u, v It follows from the C-domination property of D(cid:2)2(cid:3)F(cid:8)(cid:2)x0, y0, u, v(cid:3)(cid:2)x(cid:3) and Proposition 3.6 that (cid:2)2(cid:3)F(cid:8) (cid:6)
(cid:2)x(cid:3) (cid:8) C (cid:6)
(cid:2)x(cid:3) ⊆ MinCD(cid:2)2(cid:3)F(cid:8) D (cid:5)
x0, y0, u, v (cid:2)3.22(cid:3) ⊆ D(cid:2)2(cid:3)F (cid:5)
x0, y0, u, v
(cid:6)
(cid:2)x(cid:3) (cid:8) C, for any x ∈ K, (cid:5)
x0, y0, u, v and then D(cid:2)2(cid:3)F (cid:6)
(cid:2)x(cid:3), (cid:6)
(cid:2)x(cid:3) (cid:8) C (cid:4) D(cid:2)2(cid:3)F(cid:8) (cid:2)3.23(cid:3) for any x ∈ K. (cid:5)
x0, y0, u, v (cid:5)
x0, y0, u, v Thus, for any x ∈ K, (cid:6)
(cid:2)x(cid:3), (cid:2)3.24(cid:3) MinCD(cid:2)2(cid:3)F(cid:8) (cid:6)
(cid:2)x(cid:3) (cid:4) MinCD(cid:2)2(cid:3)F (cid:5)
x0, y0, u, v (cid:5)
x0, y0, u, v and the proof of the proposition is complete. The following example shows that the C-domination property of P (cid:2)x(cid:3) in Proposi- tion 3.8 is essential. (cid:8), and let F : X → 2Y be defined by Example 3.9 (cid:2)P (cid:2)x(cid:3) does not satisfy the C-domination property(cid:3). Let X (cid:4) R, Y (cid:4) R2, and
C (cid:4) R2 ⎧
⎨ if x ≤ 0, F(cid:2)x(cid:3) (cid:4) (cid:2)3.25(cid:3) (cid:6)(cid:8) √ ⎩ {(cid:2)0, 0(cid:3)}
(cid:5)
(cid:7)
−x, − x (cid:2)0, 0(cid:3), if x > 0, then ⎧
⎨ if x ≤ 0, F(cid:8)(cid:2)x(cid:3) (cid:4) (cid:8) (cid:6) R2
(cid:8)
(cid:7)(cid:5) (cid:2)3.26(cid:3) √ ⎩ x if x > 0. y1, y2 | y1 ≥ −x, y2 ≥ − Let (cid:2)x0, y0(cid:3) (cid:4) (cid:2)0, (cid:2)0, 0(cid:3)(cid:3) ∈ gph(cid:2)F(cid:3), (cid:2)u, v(cid:3) (cid:4) (cid:2)1, (cid:2)0, 0(cid:3)(cid:3), then, for any x ∈ X, D(cid:2)2(cid:3)F (cid:6)
(cid:2)x(cid:3) (cid:4) R2. P (cid:2)x(cid:3) (cid:4) D(cid:2)2(cid:3)F(cid:8) (cid:2)3.27(cid:3) (cid:6)
(cid:2)x(cid:3) (cid:4) {(cid:2)0, 0(cid:3)}, (cid:5)
x0, y0, u, v (cid:5)
x0, y0, u, v Obviously, P (cid:2)x(cid:3) does not satisfy the C-domination property and (cid:5) (cid:6)
(cid:2)x(cid:3). (cid:2)3.28(cid:3) MinCD(cid:2)2(cid:3)F(cid:8) (cid:6)
(cid:2)x(cid:3) /(cid:4) MinCD(cid:4)(cid:2)2(cid:3)F x0, y0, u, v (cid:5)
x0, y0, u, v 8 Fixed Point Theory and Applications , and let C be the order cone of Rm . The purpose of this section is to investigate the quantitative information on the behavior of
the perturbation map for (cid:2)PVOP(cid:3) by using second-order contingent derivative. Hereafter in
this paper, let x0 ∈ E, y0 ∈ W(cid:2)x0(cid:3), and (cid:2)u, v(cid:3) ∈ Rn × Rm Definition 4.1. We say that G is C-minicomplete by W near x0 if G(cid:2)x(cid:3) ⊆ W(cid:2)x(cid:3) (cid:8) C, (cid:2)4.1(cid:3) ∀x ∈ V (cid:2)x0(cid:3), where V (cid:2)x0(cid:3) is some neighborhood of x0. Remark 4.2. Let C be a convex cone. Since W(cid:2)x(cid:3) ⊆ G(cid:2)x(cid:3), the C-minicompleteness of G by W
near x0 implies that W(cid:2)x(cid:3) (cid:8) C (cid:4) G(cid:2)x(cid:3) (cid:8) C, (cid:2)4.2(cid:3) ∀x ∈ V (cid:2)x0(cid:3). Hence, if G is C-minicomplete by W near x0, then (cid:6) (cid:6)
, D(cid:2)2(cid:3)(cid:2)W (cid:8) C(cid:3) (cid:4) D(cid:2)2(cid:3)(cid:2)G (cid:8) C(cid:3) (cid:2)4.3(cid:3) (cid:5)
x0, y, u, v (cid:5)
x0, y, u, v ∀y ∈ W(cid:2)x0(cid:3). Theorem 4.3. Suppose that the following conditions are satisfied: (cid:2)i(cid:3) G is locally Lipschitz at x0;
(cid:2)ii(cid:3) D(cid:2)2(cid:3)G(cid:2)x0, y0, u, v(cid:3) (cid:4) D(cid:4)(cid:2)2(cid:3)G(cid:2)x0, y0, u, v(cid:3);
(cid:2)iii(cid:3) G is C-minicomplete by W near x0;
(cid:2)iv(cid:3) there exists a neighborhood U(cid:2)x0(cid:3) of x0, such that for any x ∈ U(cid:2)x0(cid:3), W(cid:2)x(cid:3) is a single point set,
then, for all x ∈ Rn, (cid:5) (cid:5) D(cid:2)2(cid:3)W (cid:6)
(cid:2)x(cid:3). (cid:2)4.4(cid:3) (cid:6)
(cid:2)x(cid:3) ⊆ MinCD(cid:2)2(cid:3)G x0, y0, u, v x0, y0, u, v . If D(cid:2)2(cid:3)W(cid:2)x0, y0, u, v(cid:3)(cid:2)x(cid:3) (cid:4) ∅, then (cid:2)4.4(cid:3) holds trivially. Thus, we assume that
Proof. Let x ∈ Rn
D(cid:2)2(cid:3)W(cid:2)x0, y0, u, v(cid:3)(cid:2)x(cid:3) /(cid:4) ∅. Let y ∈ D(cid:2)2(cid:3)W(cid:2)x0, y0, u, v(cid:3)(cid:2)x(cid:3), then there exist sequences {hn} with
hn → 0(cid:8) and {(cid:2)xn, yn(cid:3)} with (cid:2)xn, yn(cid:3) → (cid:2)x, y(cid:3), such that nxn
(cid:14)
, nxn nyn ∈ W
(cid:11)
x0 (cid:8) hnu (cid:8) h2 (cid:11) (cid:14) y0 (cid:8) hnv (cid:8) h2 x0 (cid:8) hnu (cid:8) h2 (cid:2)4.5(cid:3) ⊆ G ∀n. So, y ∈ D(cid:2)2(cid:3)G(cid:2)x0, y0, u, v(cid:3)(cid:2)x(cid:3). Suppose that y /∈ MinCD(cid:2)2(cid:3)G(cid:2)x0, y0, u, v(cid:3)(cid:2)x(cid:3), then there exists y ∈ D(cid:2)2(cid:3)G(cid:2)x0, y0, u, v(cid:3)(cid:2)x(cid:3), such that y − y ∈ C \ {0Y }. (cid:2)4.6(cid:3) Fixed Point Theory and Applications 9 Since D(cid:2)2(cid:3)G(cid:2)x0, y0, u, v(cid:3) (cid:4) D(cid:4)(cid:2)2(cid:3)G(cid:2)x0, y0, u, v(cid:3), for the preceding sequence {hn}, there exists a
sequence {(cid:2)xn, yn(cid:3)} with (cid:2)xn, yn(cid:3) → (cid:2)x, y(cid:3), such that nxn nyn ∈ G (cid:11) ∀n. (cid:14)
, (cid:2)4.7(cid:3) y0 (cid:8) hnv (cid:8) h2 x0 (cid:8) hnu (cid:8) h2 It follows from the locally Lipschitz continuity of G that there exist γ > 0 and a neighborhood V (cid:2)x0(cid:3) of x0, such that (cid:2)4.8(cid:3) G(cid:2)x1(cid:3) ⊆ G(cid:2)x2(cid:3) (cid:8) γ (cid:4)x1 − x2(cid:4)BRm , ∀x1, x2 ∈ V (cid:2)x0(cid:3), where BRm is the closed ball of Rm. From assumption (cid:2)iii(cid:3), there exists a neighborhood V1(cid:2)x0(cid:3) of x0, such that G(cid:2)x(cid:3) ⊆ W(cid:2)x(cid:3) (cid:8) C, (cid:2)4.9(cid:3) ∀x ∈ V1(cid:2)x0(cid:3). Naturally, there exists N > 0, such that nxn, x0 (cid:8) hnu (cid:8) h2 nxn ∈ U(cid:2)x0(cid:3) ∩ V (cid:2)x0(cid:3) ∩ V1(cid:2)x0(cid:3), ∀n > N. (cid:2)4.10(cid:3) x0 (cid:8) hnu (cid:8) h2 Therefore, it follows from (cid:2)4.7(cid:3) and (cid:2)4.8(cid:3) that for any n > N, there exists bn ∈ BRm , such that nxn (cid:11) (cid:14) (cid:6) ∈ G . (cid:2)4.11(cid:3) (cid:5)
yn − γ (cid:4)xn − xn(cid:4)bn y0 (cid:8) hnv (cid:8) h2
n x0 (cid:8) hnu (cid:8) h2 Thus, from (cid:2)4.5(cid:3), (cid:2)4.9(cid:3), and assumption (cid:2)iv(cid:3), one has nyn (cid:11) (cid:14) (cid:5) (cid:6) − yn − γ (cid:4)xn − xn(cid:4)bn y0 (cid:8) hnv (cid:8) h2
n y0 (cid:8) hnv (cid:8) h2 (cid:2)4.12(cid:3) (cid:6) ∈ C, ∀n > N, (cid:4) h2
n (cid:5)
yn − γ (cid:4)xn − xn(cid:4)bn − yn and then it follows from yn − γ (cid:4)xn − xn(cid:4)bn − yn → y − y and C is a closed convex cone that y − y ∈ C, (cid:2)4.13(cid:3) which contradicts (cid:2)4.6(cid:3). Thus, y ∈ MinCD(cid:2)2(cid:3)G(cid:2)x0, y0, u, v(cid:3)(cid:2)x(cid:3) and the proof of the theorem is
complete. (cid:8) | y1 ≥ y2} and The following two examples show that the assumption (cid:2)iv(cid:3) in Theorem 4.3 is essential. Example 4.4 (cid:2)W(cid:2)x(cid:3) is not a single-point set near x0(cid:3). Let C (cid:4) {(cid:2)y1, y2(cid:3) ∈ R2
G : R(cid:8) → 2R2 be defined by (cid:10) (cid:6) (cid:9)(cid:5) , G(cid:2)x(cid:3) (cid:4) C ∪ (cid:2)4.14(cid:3) y1, y2 | y1 ≥ x2 (cid:8) x, y2 ≥ x2 Fixed Point Theory and Applications 10 then (cid:10) (cid:9)(cid:5) (cid:6) . (cid:2)4.15(cid:3) W(cid:2)x(cid:3) (cid:4) {(cid:2)0, 0(cid:3)} ∪ y1, y2 | y1 (cid:4) x2 (cid:8) x, y2 > x2 (cid:8) x Let x0 (cid:4) 0, y0 (cid:4) (cid:2)0, 0(cid:3), and (cid:2)u, v(cid:3) (cid:4) (cid:2)1, (cid:2)1, 1(cid:3)(cid:3), then W(cid:2)x(cid:3) is not a single-point set near x0, and
it is easy to check that other assumptions of Theorem 4.3 are satisfied. For any x ∈ R, one has (cid:7)(cid:5) (cid:6) (cid:8) (cid:7)(cid:5) (cid:6) D(cid:2)2(cid:3)G , (cid:6)
(cid:2)x(cid:3) (cid:4) (cid:5)
x0, y0, u, v y1, y2 y1, y2 (cid:8)
| y1 ≥ 1 (cid:8) x, y2 ∈ R (cid:2)4.16(cid:3) ∪
(cid:6) (cid:5) (cid:8) , D(cid:2)2(cid:3)W | y1 ∈ R, y1 ≥ y2
(cid:7)(cid:5)
(cid:6)
(cid:2)x(cid:3) (cid:4) x0, y0, u, v 1 (cid:8) x, y2 | y2 ≥ 1 (cid:8) x and then (cid:6) (cid:8) (cid:7)(cid:5) . (cid:6)
(cid:2)x(cid:3) (cid:4) (cid:2)4.17(cid:3) MinCD(cid:2)2(cid:3)G (cid:5)
x0, y0, u, v 1 (cid:8) x, y2 | y2 > 1 (cid:8) x (cid:8) | y1 (cid:4) 0} and Thus, for any x ∈ R, the inclusion of (cid:2)4.4(cid:3) does not hold here. Example 4.5 (cid:2)W(cid:2)x(cid:3) is not a single-point set near x0(cid:3). Let C (cid:4) {(cid:2)y1, y2(cid:3) ∈ R2
G : R → 2R2 be defined by ⎧
⎨ C if x (cid:4) 0, (cid:10) G(cid:2)x(cid:3) (cid:4) (cid:15) (cid:9)(cid:5) (cid:6) (cid:2)4.18(cid:3) ⎩ C ∪ 1 (cid:8) |x| if x /(cid:4) 0, y1, y2 | y1 (cid:4) x, y2 ≥ − then ⎧
⎨ if x (cid:4) 0, (cid:14)(cid:10) (cid:11) W(cid:2)x(cid:3) (cid:4) (cid:15) (cid:2)4.19(cid:3) ⎩ x, − if x /(cid:4) 0. 1 (cid:8) |x| {(cid:2)0, 0(cid:3)}
(cid:9)
(cid:2)0, 0(cid:3), Let x0 (cid:4) 0, y0 (cid:4) (cid:2)0, 0(cid:3), and (cid:2)u, v(cid:3) (cid:4) (cid:2)0, (cid:2)0, 0(cid:3)(cid:3), then W(cid:2)x(cid:3) is not a single-point set near x0, and it is easy to check that other assumptions of Theorem 4.3 are satisfied. For any x ∈ R, one has (cid:6) (cid:5) (cid:7)(cid:5) D(cid:2)2(cid:3)G , (cid:6)
(cid:2)x(cid:3) (cid:4) C ∪ x0, y0, u, v y1, y2 (cid:8)
| y1 (cid:4) x, y2 ∈ R (cid:2)4.20(cid:3) (cid:6)
(cid:2)x(cid:3) (cid:4) D(cid:4)(cid:2)2(cid:3)G
(cid:5) D(cid:2)2(cid:3)W (cid:5)
x0, y0, u, v
(cid:6)
(cid:2)x(cid:3) (cid:4) {(cid:2)0, 0(cid:3)}, x0, y0, u, v and then (cid:2)4.21(cid:3) MinCD(cid:2)2(cid:3)G (cid:6)
(cid:2)0(cid:3) (cid:4) ∅. (cid:5)
x0, y0, u, v Thus, for x (cid:4) 0, the inclusion of (cid:2)4.4(cid:3) does not hold here. Fixed Point Theory and Applications 11 (cid:8) and G : R → 2R2 Now, we give an example to illustrate Theorem 4.3. be defined by (cid:10) Example 4.6. Let C (cid:4) R2
(cid:9)(cid:5) (cid:6) , ∀x ∈ R, G(cid:2)x(cid:3) (cid:4) (cid:2)4.22(cid:3) y1, y2 ∈ R2 | x ≤ y1 ≤ x (cid:8) x2, x − x2 ≤ y2 ≤ x then (cid:9)(cid:11) (cid:14)(cid:10) x, x − x2 , ∀x ∈ R. W(cid:2)x(cid:3) (cid:4) (cid:2)4.23(cid:3) Let (cid:2)x0, y0(cid:3) (cid:4) (cid:2)0, (cid:2)0, 0(cid:3)(cid:3) ∈ gph(cid:2)G(cid:3), (cid:2)u, v(cid:3) (cid:4) (cid:2)1, (cid:2)1, 1(cid:3)(cid:3). By directly calculating, for all x ∈ R, one has D(cid:2)2(cid:3)G (cid:6)
(cid:2)x(cid:3) (cid:5)
x0, y0, u, v (cid:5)
x0, y0, u, v
(cid:6) (cid:8) (cid:6)
(cid:2)x(cid:3) (cid:4) D(cid:4)(cid:2)2(cid:3)G
(cid:7)(cid:5) , (cid:4) (cid:2)4.24(cid:3) y1, y2 | x ≤ y1 ≤ x (cid:8) 1, x − 1 ≤ y1 ≤ x (cid:5) D(cid:2)2(cid:3)W (cid:6)
(cid:2)x(cid:3) (cid:4) {(cid:2)x, x − 1(cid:3)}. x0, y0, u, v Then, it is easy to check that assumptions of Theorem 4.3 are satisfied, and the inclusion of
(cid:2)4.4(cid:3) holds. Theorem 4.7. If P (cid:2)x(cid:3) :(cid:4) D(cid:2)2(cid:3)G(cid:8)(cid:2)x0, y0, u, v(cid:3)(cid:2)x(cid:3) fulfills the C-domination property for all x ∈ Ω :(cid:4)
dom(cid:5)D(cid:2)2(cid:3)G(cid:2)x0, y0, u, v(cid:3)(cid:6) and G is C-minicomplete by W near x0, then (cid:6)
(cid:2)x(cid:3) ⊆ D(cid:2)2(cid:3)W (cid:6)
(cid:2)x(cid:3), (cid:2)4.25(cid:3) for any x ∈ Ω. MinCD(cid:2)2(cid:3)G (cid:5)
x0, y0, u, v (cid:5)
x0, y0, u, v Proof. Since C ⊂ Rn, C has a compact base. Then, it follows from Propositions 3.6 and 3.8 and
Remark 4.2 that for any x ∈ Ω, one has (cid:5) (cid:6)
(cid:2)x(cid:3) MinCD(cid:2)2(cid:3)G (cid:6)
(cid:2)x(cid:3) (cid:4) MinCD(cid:2)2(cid:3)G(cid:8) x0, y0, u, v (cid:6)
(cid:2)x(cid:3) (cid:2)4.26(cid:3) (cid:4) MinCD(cid:2)2(cid:3)W(cid:8) ⊆ D(cid:2)2(cid:3)W (cid:5)
x0, y0, u, v
(cid:5)
x0, y0, u, v
(cid:6)
(cid:2)x(cid:3). (cid:5)
x0, y0, u, v Then, the conclusion is obtained and the proof is complete. (cid:8) and Remark 4.8. If the C-domination property of P (cid:2)x(cid:3) is not satisfied in Theorem 4.7, then
Theorem 4.7 may not hold. The following example explains the case. Example 4.9 (cid:2)P (cid:2)x(cid:3) does not satisfy the C-domination property for x ∈ Ω(cid:3). Let C (cid:4) R2
G : R → R2 be defined by ⎧
⎨ if x ≤ 0, G(cid:2)x(cid:3) (cid:4) (cid:2)4.27(cid:3) (cid:6)(cid:8) √ ⎩ {(cid:2)0, 0(cid:3)}
(cid:7)
(cid:5)
−x, − x (cid:2)0, 0(cid:3), if x > 0, 12 Fixed Point Theory and Applications then, ⎧
⎨ if x ≤ 0, G(cid:8)(cid:2)x(cid:3) (cid:4) (cid:2)4.28(cid:3) R2
(cid:8)
(cid:7)(cid:5) (cid:6) (cid:8) √ ⎩ x if x > 0. y1, y2 | y1 ≥ −x, y2 ≥ − Let (cid:2)x0, y0(cid:3) (cid:4) (cid:2)0, (cid:2)0, 0(cid:3)(cid:3) ∈ gph(cid:2)F(cid:3), (cid:2)u, v(cid:3) (cid:4) (cid:2)1, (cid:2)0, 0(cid:3)(cid:3), then, for any x ∈ Ω (cid:4) R, ⎧
⎨ if x ≤ 0, W(cid:2)x(cid:3) (cid:4) (cid:2)4.29(cid:3) {(cid:2)0, 0(cid:3)}
(cid:7)(cid:5) (cid:8) (cid:6) √ ⎩ x if x > 0, y1, y2 | y1 (cid:4) −x, y2 (cid:4) − (cid:5) D(cid:2)2(cid:3)G (cid:6)
(cid:2)x(cid:3) (cid:4) R2, P (cid:2)x(cid:3) (cid:4) D(cid:2)2(cid:3)G(cid:8) for any x ∈ Ω,
(cid:5)
x0, y0, u, v x0, y0, u, v (cid:2)4.30(cid:3) (cid:6)
(cid:2)x(cid:3) (cid:4) {(cid:2)0, 0(cid:3)},
(cid:5) D(cid:2)2(cid:3)W (cid:6)
(cid:2)x(cid:3) (cid:4) ∅. x0, y0, u, v Hence, P (cid:2)x(cid:3) does not satisfy the C-domination property, and MinCD(cid:2)2(cid:3)G(cid:2)x0, y0, u, v(cid:3)(cid:2)x(cid:3) (cid:4)
{(cid:2)0, 0(cid:3)}. Then, (cid:6)
(cid:2)x(cid:3). (cid:6)
(cid:2)x(cid:3) /⊆ D(cid:4)(cid:2)2(cid:3)W (cid:2)4.31(cid:3) MinCD(cid:2)2(cid:3)G (cid:5)
x0, y0, u, v (cid:5)
x0, y0, u, v Theorem 4.10. Suppose that the following conditions are satisfied: (cid:2)i(cid:3) G is locally Lipschitz at x0;
(cid:2)ii(cid:3) D(cid:2)2(cid:3)G(cid:2)x0, y0, u, v(cid:3) (cid:4) D(cid:4)(cid:2)2(cid:3)G(cid:2)x0, y0, u, v(cid:3);
(cid:2)iii(cid:3) G is C-minicomplete by W near x0;
(cid:2)iv(cid:3) there exists a neighborhood U(cid:2)x0(cid:3) of x0, such that for any x ∈ U(cid:2)x0(cid:3), W(cid:2)x(cid:3) is a single- point set; (cid:2)v(cid:3) for any x ∈ Ω :(cid:4) dom(cid:5)D(cid:2)2(cid:3)G(cid:2)x0, y0, u, v(cid:3)(cid:6), D(cid:2)2(cid:3)G(cid:8)(cid:2)x0, y0, u, v(cid:3)(cid:2)x(cid:3) fulfills the C-domi- nation property; then D(cid:2)2(cid:3)W ∀x ∈ Ω. (cid:6)
(cid:2)x(cid:3), (cid:2)4.32(cid:3) (cid:6)
(cid:2)x(cid:3) (cid:4) MinCD(cid:2)2(cid:3)G (cid:5)
x0, y0, u, v (cid:5)
x0, y0, u, v Proof. It follows from Theorems 4.3 and 4.7 that (cid:2)4.32(cid:3) holds. The proof of the theorem is
complete. This research was partially supported by the National Natural Science Foundation of China
(cid:2)no. 10871216 and no. 11071267(cid:3), Natural Science Foundation Project of CQ CSTC and
Science and Technology Research Project of Chong Qing Municipal Education Commission
(cid:2)KJ100419(cid:3). Fixed Point Theory and Applications 13 (cid:5)1(cid:6) A. V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, vol. 165 of Mathematics in Science and Engineering, Academic Press, Orlando, Fla, USA, 1983. (cid:5)2(cid:6) W. Alt, “Local stability of solutions to differentiable optimization problems in Banach spaces,” Journal of Optimization Theory and Applications, vol. 70, no. 3, pp. 443–466, 1991. (cid:5)3(cid:6) S. W. Xiang and W. S. Yin, “Stability results for efficient solutions of vector optimization problems,” Journal of Optimization Theory and Applications, vol. 134, no. 3, pp. 385–398, 2007. (cid:5)4(cid:6) J. Zhao, “The lower semicontinuity of optimal solution sets,” Journal of Mathematical Analysis and Applications, vol. 207, no. 1, pp. 240–254, 1997. (cid:5)5(cid:6) T. Tanino, “Sensitivity analysis in multiobjective optimization,” Journal of Optimization Theory and Applications, vol. 56, no. 3, pp. 479–499, 1988. (cid:5)6(cid:6) T. Tanino, “Stability and sensitivity analysis in convex vector optimization,” SIAM Journal on Control and Optimization, vol. 26, no. 3, pp. 521–536, 1988. (cid:5)7(cid:6) H. Kuk, T. Tanino, and M. Tanaka, “Sensitivity analysis in vector optimization,” Journal of Optimization Theory and Applications, vol. 89, no. 3, pp. 713–730, 1996. (cid:5)8(cid:6) D. S. Shi, “Contingent derivative of the perturbation map in multiobjective optimization,” Journal of Optimization Theory and Applications, vol. 70, no. 2, pp. 385–396, 1991. (cid:5)9(cid:6) D. S. Shi, “Sensitivity analysis in convex vector optimization,” Journal of Optimization Theory and Applications, vol. 77, no. 1, pp. 145–159, 1993. (cid:5)10(cid:6) S. J. Li, “Sensitivity and stability for contingent derivative in multiobjective optimization,” Mathematica Applicata, vol. 11, no. 2, pp. 49–53, 1998. (cid:5)11(cid:6) H. Kuk, T. Tanino, and M. Tanaka, “Sensitivity analysis in parametrized convex vector optimization,” Journal of Mathematical Analysis and Applications, vol. 202, no. 2, pp. 511–522, 1996. (cid:5)12(cid:6) F. Ferro, “An optimization result for set-valued mappings and a stability property in vector problems
with constraints,” Journal of Optimization Theory and Applications, vol. 90, no. 1, pp. 63–77, 1996.
(cid:5)13(cid:6) M. A. Goberna, M. A. L ´opes, and M. I. Todorov, “The stability of closed-convex-valued mappings and
the associated boundaries,” Journal of Mathematical Analysis and Applications, vol. 306, no. 2, pp. 502–
515, 2005. (cid:5)14(cid:6) X. K. Sun and S. J. Li, “Lower Studniarski derivative of the perturbation map in parametrized vector optimization,” Optimization Letters. In press. (cid:5)15(cid:6) T. D. Chuong and J.-C. Yao, “Coderivatives of efficient point multifunctions in parametric vector optimization,” Taiwanese Journal of Mathematics, vol. 13, no. 6A, pp. 1671–1693, 2009. (cid:5)16(cid:6) K. W. Meng and S. J. Li, “Differential and sensitivity properties of gap functions for Minty vector
variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 386–398,
2008. (cid:5)17(cid:6) J.-P. Aubin, “Contingent derivatives of set-valued maps and existence of solutions to nonlinear
inclusions and differential inclusions,” in Mathematical Analysis and Applications, Part A, L. Nachbin,
Ed., vol. 7 of Adv. in Math. Suppl. Stud., pp. 159–229, Academic Press, New York, NY, USA, 1981.
(cid:5)18(cid:6) R. B. Holmes, Geometric Functional Analysis and Its Applications, Graduate Texts in Mathematics, no. 2, Springer, New York, NY, USA, 1975. (cid:5)19(cid:6) J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Pure and Applied Mathematics (cid:2)New York(cid:3), John Wiley & Sons, New York, NY, USA, 1984. (cid:5)20(cid:6) J.-P. Aubin and H. Frankowska, Set-Valued Analysis, vol. 2 of Systems & Control: Foundations & Applications, Biekh¨auser, Boston, Mass, USA, 1990. (cid:5)21(cid:6) D. T. Luc, Theory of Vector Optimization, vol. 319 of Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, 1989.2. Preliminaries
3. Second-Order Contingent Derivatives for Set-Valued Maps
4. Second-Order Contingent Derivative of the Perturbation Maps
Acknowledgments
References
