Vietnam Journal of Mathematics 33:3 (2005) 291–308
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T he Ex ist ence of Solu t ions t o G eneralized
B ilevel Vect or Opt im izat ion P roblem s
N guyen B a M inh and N guyen X uan Tan*
I nsti tute of Mathematics, 18 Hoang Quoc V iet Road, 10307 Hanoi, Vi etnam
Received April 29, 2004
Revised O ct ob er 6, 2005
A bst ract . Generalized bilevel vect or opt imizat ion problems are formulat ed and some
sufficient condit ions on t he exist ence of solut ions for generalized bilevel weakly, P ar et o
and ideal problem s are shown. As sp ecial case, we obt ain result s on t h e exist ence of
solut ions t o generalized bilevel programm ing p roblems given by Lignola an d Morgan.
T hese also include a large numb er of resu lt s con cerning variat ional and qu asi-variat ional
inequ alit ies, equ ilibriu m and quasi-equilibrium prob lems.
1. Int ro d uct io n
Let Dbe a subset of a t opological vect or space Xand Rbe t he space of real
numbers. Given a real funct ion ffrom Dint o R, t he problem of finding ¯x∈D
such t hat
f(¯x)= min
x∈D
f(x)
plays a cent r al role in t he opt imizat ion t heor y. T here is a numb er of b ooks
on opt imizat ion t heory for linear, convex, Lipschit z and, in genera l, cont inuous
problems. Today t his pr oblem is also formulat ed for vect or mult i-valued m ap-
pings. One develop ed t he opt imizat ion t h eory concerning mult i-valued mappings
∗
T he aut h or was par t ially supp ort ed by t he Frit z-T hyssen Foundat ion from Germ any
for t he t h ree m ont hs st ay at t he Inst it ut e of Mat h emat ics of t he Hu mb oldt Universit y
in Berlin and t he Inst it ut e of Mat hemat ics of t he Cologn e Universit y.
292 Nguyen Ba Minh and Nguyen Xuan Tan
wit h t he met hodology a nd t h e applicat ions simila r t o t he ones wit h scalar func-
t ions. Given a cone Cin a topological vect or space Yand a subset A⊂Y,one
can define efficient point s of Awit h r esp ect t o Cby different senses as: Ideal,
P aret o, P rop erly, Weakly, ... (see Definit ion 2 b elow). T he set of t hese efficient
point s is denot ed by αMin (A/ C) for t he case of ideal, P aret o, prop erly, weakly
efficient p oint s, resp ect ively. By 2Ywe denot e t he family of all subset s in Y.For
a given mult i-valued m apping F:D→2Y, we consider t he problem of finding
¯x∈Dsuch that
F(¯x)∩αMin (F(D)/ C)=∅.(GV OP )α
T his is ca lled a general vect or αopt imizat ion problem corresponding t o D
and F. T he set of such p oint s ¯xis denot ed by αS(D , F ;C) a nd is called t he
solut ion set of (GV OP )α.T he element s of αMin (F(D)/ C) a re called opt imal
va lu es of ( GV OP )α.T hese problems have b een st udied by many aut hors, for
examples, Corley [6], Luc [14], Benson [1], J ahn [11], St erna-Karwat [21],...
Now, let X , Y and Zb e t op ological vect or spaces, D⊂X , K ⊂Zbe
nonempt y subset s and C⊂Yb e a cone. Given t he following mult i-valued
mappings
S:D→2D,
T:D→2K,
F:D×K×D→2Y,
we are int erest ed in t he pr oblem of finding ¯x∈D , ¯z∈Ksuch that
¯x∈S(¯x),
¯z∈T(¯x)(GV QOP )α
and
F(¯x, ¯z, ¯x)∩αMin (F(¯x, ¯z, S(¯x)) =∅.
T his is ca lled a general vect or αquasi-optimizat ion problem (αis one of t h e
words: “idea l”, “P ar et o”, “ prop erly”, “weakly”, ..., respect ively ). Such a couple
(¯x, ¯z) is said to be t he solut ion of (GV QOP )α.T he set of such solut ions is said
t o b e t he solut ion set of (GV QOP )αand denot ed by αS(D , K , S, T, F, C). T he
ab ove mult i-valued mappings S, T and Fare called a constraint , pot ential and
ut ility mapping, resp ect ively.
T hese problems cont a in as sp ecial cases, for example, quasi-equilibrium prob-
lems, quasi-var iat ional inequa lit ies, fixed point problems, complement a rit y prob-
lems, as well a s different ot hers t hat have been considered by many m at hemat i-
cians as: P ark [20], Chan and P ang [5], P arida and Sen [19], Fu [9] for qua si-
equilibrium problems, by Blum and Oet t li [3], Minh and Ta n [16], Br owder and
Mint y [17], Ky-Fan [7],..., for equilibrium a nd variat ional inequalit y problems
and by some ot hers for vect or opt imization problems.
Let Y0be anot her t opological vector space wit h a cone C0and f:D×K→
2Y0,we ar e int erest ed in t he p roblem of finding (x∗
,z
∗
)∈αS(D , K , S, T, F, C)
such t hat
Existence of Solutions to Generalized Bilevel Vector Opti mization Problems 293
f(x∗
,z
∗
)∩γMin f(αS(D, K , S, T, F, C))/ C)=∅.(1)(α , γ )
T his is called an (α, γ) bilevel vector opt imizat ion problem. Such a couple
(x∗
,z
∗
) is said t o be a solut ion of (1)(α , γ ).T he set of such solutions is said t o
be the solut ion set of (1)(α , γ )and denot ed by αS2(D , K , S, T, F, f , C). T hese
problems (α, γ is on e of t he words: “ ideal”, “P aret o”, “prop erly”,“ weakly” , ...,
resp ect ively ) cont ain , as a sp ecial case, t he generalized bilevel problem given in
[12] and some ot hers in t he lit erat ure therein.
2. P relim inaries and D e finit ions
T hroughout t his pa per , a s in t he int roduct ion, by X , Y, Z and Y0we denot e real
locally convex t op ological vect or spaces. Given a subset D⊂X , we consider a
mult i-valued mapping F:D→2Y.T he definit ion domain and t he graph of F
are denot ed by
dom F=
x∈D / F (x)=∅
Gr (F)=
(x, y)∈D×Y/ y ∈F(x)
,
resp ect ively. We recall t hat Fis said t o b e a closed mapping if t he graph Gr (F)
cont aining Fis a closed subset in t he product space X×Yand it is said t o be
a compact mapping if t he closure F(D)ofitsrangeF(D)iscompact inY.A
nonempt y t op ologica l space is said t o b e a cyclic if all it s reduced ˇ
Cech homology
group over rat ional vanish. Not e t hat any convex, st ar-shaped, cont ractible set
(see, for example, Definit ion 3.1, Chapt er 6 in [14]) of a t opological vect or space
is acyclic. T he following definit ions can be found in [2]. A mult i-valued mapping
F:D→2Yis said t o be up per sem i-cont inuous (u.s.c) at ¯x∈Dif for each
open set Vcont aining F(¯x), t here exist s an op en set Ucont a ining ¯xsuch that
for each x∈U,F(x)⊂V.Fis said t o be u.s.c on Dif it is u.s.c at all x∈D.
And, Fis said t o be lower semi-cont inuous (l.s.c) at ¯x∈Dif for any open set V
wit h F(¯x)∩V=∅, t here exist s an open set Ucont a ining ¯xsuch that for each
x∈U,F(x)∩V=∅;Fis sa id t o b e l.s.c on Dif it is l.s.c at all x∈D.Fis
said t o be cont inuous on Dif it is at t he same t ime u.s.c and l.s.c on D.Fis
said t o b e acyclic if it is u.s.c wit h compact acyclic values. And, Fis sa id t o be
a compact acyclic mapping if it is a compact mapping and an acyclic mapping
simult a neously.
We also recall t hat a nonempty subset Dof a topological space Xis said t o
be admissible if for every compact subset Qof Dand every neighborhood Vof
t he origin in X, t here is a cont inuous mapping h:Q→Dsuch that x−h(x)∈V
for all x∈Qand h(Q) is cont ained in a finit e dimensional subspace Lof X.
Further, let Ybe a t opological vect or space wit h a cone C.Wedenote
l(C)= C∩(−C). If l(C)= 0wesaythat Cis a point ed cone. We recall t he
following definit ions (see, for example, Definit ion 2.1, Chapt er 2 in [14]).
D e finit ion 1. L e t Abe a n on em pty su bset of Y. W e say that :
294 Nguyen Ba Minh and Nguyen Xuan Tan
1. x∈Ais an ideal effi cien t ( or ideal m in im al) poin t of Awit h respect to C
if y−x∈Cfor every y∈A.
T he set of ideal m in im al poin t s of Ais den ot ed by IM in (A/ C).
2. x∈Ais an effi cien t ( or P aret o–m in im al, or n on dom in an t ) poin t of Aw.r.t.
Cif t here is n o y∈Awit h x−y∈C\l(C).
T he set of effi cien t poin t s of Ais den ot ed by PM in (A/ C).
3. x∈Ais a ( global) proper ly effi cien t poin t of Aw.r.t. Cif t here exists a
co n v e x c o n e ˜
Cwhich is n ot the whole space an d con t ain s C\l(C)in it s
in terior so that x∈PM in
A/ ˜
C
.
T he set of properly effi cien t poin t s of Ais den ot ed by Pr M in (A/ C).
4. S u pposin g t hat in t Cis n on em pt y, x∈Ais a weakly effi cien t poin t of A
w.r.t . Cif x∈PM in (A/ {0} ∪ in t C).
T he set of weakly effi cien t poin t s of Ais den ot ed by WM in (A/ C).
We u se αMin (A/ C) t o denot e one of IMin (A/ C),P Min (A / C),....The
not ions of IMax (A/ C),PMax (A/ C), Pr Max (A/ C), WMax (A/ C)arede-
fined dually.
We have t he following inclusions:
IMin ,(A/ C)⊂Pr Min (A/ C)⊂PMin (A/ C)⊂WMin (A/ C).
Moreover, if IMin (A/ C)=∅,thenIMin (A/ C)= PMin (A/ C) and it is a
point whenever Cis point ed (see P roposit ion 2.2, Chapt er 2 in [14]).
Now, we int roduce new definit ions of t he C-cont inuit ies of a mult i-valued
mapping F:D→2Y.
D e finit ion 2.
1. Fis said t o be upper ( lower ) C–con tin u ou s at ¯x∈dom Fif for an y n eigh-
bo r h ood Vof t he origin in Yt here is a n eighborhood Uof ¯xsu ch that:
F(x)⊂F(¯x)+ V+C
F(¯x)⊂F(x)+ V−C, r espect i vel y
holds for all x∈U∩dom F.
2. If Fis u pper C–con t in u ou s an d lower C–con t in u ou s at ¯xsim ultan eou sly,
we say t hat it is C–con t in u ous at ¯x.
3. If Fis u pper, lower, ... , C–con t in u ou s at an y poin t of dom F, we say t hat
it is upper, lower,.. . con t in u ou s.
In t he sequel if C={0}we shall say that Fis upp er, lower, ..., cont inuous
inst ead of upp er, lower, ..., {0}–continuous.
R e m a r k 1 .
a) If C={0}and F(¯x) is compact , t hen it is easy t o see t hat t he above
definit ions of cont inuit ies coincide wit h the ones given by Berge [2].
b) If Fis upper cont inuous wit h F(x) closed for any x∈D,thenFis closed.
Existence of Solutions to Generalized Bilevel Vector Opti mization Problems 295
c) If Fis compact and F(x)closedforeachx∈D,thenFis u pp er cont inuous
if and only if Fis closed.
d) If F(¯x) is compact , t he t he ab ove definit ions coincide wit h t he ones in [14]
(Definit ion 7.1, Chapt er 1).
In t he sequel, we give some necessary a nd sufficient condit ions on t he upp er
and t he lower C– cont inuit ies .
P ro p osit ion 1 . L e t F:D→2Yan d C⊂Ybe a c lo s ed c o n e .
1) If Fis u pper C–con t in u ous at x0∈dom Fwith F(x0)+ Cclosed, t hen for
an y n et xβ→x0,y
β∈F(xβ)+ C, yβ→y0im ply y0∈F(x0)+ C.
C on versely , if Fis com pact an d for an y n et xβ→x0,y
β∈F(xβ)+ C, yβ→
y0im ply y0∈F(x0)+ C, then Fis u pper C–con t in u ou s at x0.
2) If Fis com pact an d lower C–con t in u ou s at x0∈dom F, then for an y n et
xβ→x0,y
0∈F(x0)+ C, t here is a n et {yβ},y
β∈F(xβ),which has a
con vergen t su bn et {yβγ},y
βγ−y0→c∈C(i .e. yβγ→y0+c∈y0+C).
C on versely , if F(x0)is com pact an d for an y n et xβ→x0,y
0∈F(x0)+ C,
there is a n et {yβ},y
β∈F(xβ),which has a con vergen t su bn et {yβγ},y
βγ−
y0→c∈C, then Fis lower C–con t in u ous at x0.
P roof.
1) Assume first t hat Fis upp er C–cont inuous at x0∈dom Fand xβ→x0,y
β∈
F(xβ)+ C, yβ→y0. We supp ose on t he cont rary t hat y0/∈F(x0)+ C. W e c a n
find a convex and closed neighborhood V0of t he origin in Ysuch that
(y0+V0)∩(F(x0)+ C)= ∅,
or,
(y0+V0/2) ∩(F(x0)+ V0/2+ C)= ∅.
Since yβ→y0,one can find β1≥0 such t hat yβ−y0∈V0/2 for all β≥β1.
T herefore, yβ∈y0+V0/2andFis upp er C–cont inuous at x0,it follows t ha t
one can find a neighborhood Uof x0such that
F(x)⊂(F(x0)+ V0/2+ C) for all x∈U∩dom F.
Since xβ→x0,one can find β2≥0 such t hat xβ∈Uand
yβ∈F(xβ)+ C⊂(F(x0)+ V0/2+ C) for all x∈U∩dom F.
T his im plies t hat
yβ∈(y0+V0/2) ∩(F(x0)+ V0/2+ C)= ∅for all β≥max{β1,β
2}.
and we have a cont radict ion. T hus, we conclude y0∈F(x0)+ C. Now, assume
t hat Fis compact and for any net xβ→x0,y
β∈F(xβ)+ C, yβ→y0imply
y0∈F(x0)+ C. On t he cont rary, we assume that Fis not upper C–continuous
at x0.T his im plies t ha t t here is a neighb orhood Vof the origin in Ysuch that
for a ny n eighb or hood Uβof x0one can find xβ∈Uβsuch that