Volume 2008, Article ID 678014,7pages doi:10.1155/2008/678014
Research Article
Existence of Solutions for Nonconvex and Nonsmooth Vector Optimization Problems
Zhi-Bin Liu,1 Jong Kyu Kim,2 and Nan-Jing Huang3
1Department of Applied Mathematics, Southwest Petroleum University, Chengdu, Sichuan 610500, China
2Department of Mathematics, Kyungnam University, Masan, Kyungnam 631701, South Korea
3Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
Correspondence should be addressed to Jong Kyu Kim,[email protected] Received 9 January 2008; Accepted 4 April 2008
Recommended by R. P. Gilbert
We consider the weakly efficient solution for a class of nonconvex and nonsmooth vector optimiza- tion problems in Banach spaces. We show the equivalence between the nonconvex and nonsmooth vector optimization problem and the vector variational-like inequality involving set-valued map- pings. We prove some existence results concerned with the weakly efficient solution for the noncon- vex and nonsmooth vector optimization problems by using the equivalence and Fan-KKM theorem under some suitable conditions.
Copyrightq2008 Zhi-Bin Liu et al. 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.
1. Introduction
The concept of vector variational inequality was first introduced by Giannessi1in 1980. Since then, existence theorems for solution of general versions of the vector variational inequality have been studied by many authorssee, e.g.,2–9and the references therein. Recently, vec- tor variational inequalities and their generalizations have been used as a tool to solve vector optimization problemssee7,10–14. Chen and Craven11obtained a sufficient condition for the existence of weakly efficient solutions for differentiable vector optimization problems involving differentiable convex functions by using vector variational inequalities for vector valued functions. Kazmi12proved a sufficient condition for the existence of weakly efficient solutions for vector optimization problems involving differentiable preinvex functions by us- ing vector variational-like inequalities. For the nonsmooth case, Lee et al.7established the existence of the weakly efficient solution for nondifferentiable vector optimization problems by using vector variational-like inequalities for set-valued mappings. Similar results can be found in10. It is worth mentioning that Lee et al.7and Ansari and Yao10obtained their
existence results under the assumption thatRm ⊂Cxfor allx∈Rn, whereCxis a convex cone inRm. However, this condition is restrict and it does not hold in general.
In this paper, we consider the weakly efficient solution for a class of nonconvex and nonsmooth vector optimization problems in Banach spaces. We show the equivalence between the nonconvex and nonsmooth vector optimization problem and the vector variational-like inequality involving set-valued mappings. We prove some existence results concerned with the weakly efficient solution for the nonconvex and nonsmooth vector optimization problems by using the equivalence and Fan-KKM theorem without the restrict conditionRm ⊂Cxfor allx∈Rn. Our results generalize and improve the results obtained by Lee et al.7and Ansari and Yao10.
2. Preliminaries
LetXbe a real Banach space endowed with a norm·andX∗its dual space, we denote by·,· the dual pair betweenXandX∗. LetRmbe them-dimensional Euclidean space, letS⊂Xbe a nonempty subset, and letK⊂Rmbe a nonempty closed convex cone with intK /∅, where int denotes interior.
Definition 2.1. A real valued functionh:X→Ris said to be locally Lipschitz at a pointx∈Xif there exists a numberL >0 such that
|hy−hz| ≤Ly−z 2.1
for ally, zin a neighborhood ofx.his said to be locally Lipschitz onXif it is locally Lipschitz at each point ofX.
Definition 2.2. Leth:X→Rbe a locally Lipschitz function. Clarke15generalized directional derivative ofhatx∈Xin the directionv, denoted byh◦x;v, is defined by
h◦x;v lim sup
y→x, t↓0
hytv−hy
t . 2.2
Clarke15generalized gradient ofhatx∈X, denoted by∂hx, is defined by
∂hx
ξ∈X∗:h◦x;v≥ ξ, d ∀v∈X
. 2.3
Letf : X→Rmbe a vector valued function given byf f1, f2, . . . , fm, where eachfi,i 1,2, . . . , m, is a real valued function defined onX. Thenfis said to be locally Lipschitz onXif eachfiis locally Lipschitz onX.
The generalized directional derivative of a locally Lipschitz functionf:X→Rmatx∈X in the directionvis given by
f◦x;v
f1◦x;v, f2◦x;v, . . . , fm◦x;v
. 2.4
The generalized gradient ofhatxis the set
∂fx ∂f1x×∂f2x× · · · ×∂fmx, 2.5 where∂fixis the generalized gradient offiatxfori 1,2, . . . , m.
Every elementA ξ1, ξ2, . . . , ξm∈∂fxis a continuous linear operator fromXtoRm and
Ay
ξ1, y ,
ξ2, y , . . . ,
ξm, y
∈Rm, ∀y∈X. 2.6
Definition 2.3. Letf:X→Rmbe a locally Lipschitz function.
ifis said to beK-invex with respect toηatu∈X, if there existsη:X×X→Xsuch that for allx∈XandA∈∂fu,
fx−fu−
A, ηx, u
∈K. 2.7
iif is said to beK-pseudoinvex with respect toηatu∈Xif there existsη :X×X→X such that for allx∈XandA∈∂fu,
fx−fu∈ −intK ⇒
A, ηx, u
∈ −intK. 2.8
In this paper, we consider the following nonsmooth vector optimization problem:
K-minimize fx,
subject to x∈S, VOP
wheref f1, f2, . . . , fm,fi:X→R,i 1,2, . . . , m, are locally Lipschitz functions.
Definition 2.4. A pointx0∈Sis said to be a weakly efficient solution offif there exists noy∈S such that
fy−fx∈ −intK. 2.9
In order to prove our main results, we need the following definition and lemmas.
Definition 2.5 see 16. A multivalued mappingG : X→2X is called KKM-mapping if for any finite subset{x1, x2, . . . , xn}ofX, co{x1, x2, . . . , xn}is contained inn
i 1Gxi, where coA denotes the convex hull of the setA.
Lemma 2.6see16. LetMbe a nonempty subset of a Hausdorfftopological vector spaceX. Let G:M→2Xbe a KKM-mapping such thatGxis closed for anyx∈Mand is compact for at least one x∈M. Then y∈MGy/∅.
Lemma 2.7see2. LetKbe a convex cone of topological vector spaceX. Ify−x∈Kandx /∈ −intK, theny /∈ −intKfor anyx, y∈X.
3. Main results
In order to obtain our main results, we introduce the following vector variational-like inequal- ity problem, which consists in findingx0∈Ssuch that for allA∈∂fx0,
A, η y, x0
/∈ −intK, ∀y∈S. VVIP
First, we establish the following relations betweenVOPandVVIP.
Lemma 3.1. Letf : X→Rm be a locally Lipschitz function andη : S×S→X. Then the following arguments hold.
iSuppose thatfisK-invex with respect toη. Ifx0is a solution ofVVIP, thenx0is a weakly efficient solution of VOP.
iiSuppose thatfisK-pseudoinvex with respect toη. Ifx0is a solution of VVIP, thenx0is a weakly efficient solution of VOP.
iiiSuppose thatf is−K-invex with respect toη. Ifx0is a weakly efficient solution of VOP, thenx0is a solution of VVIP.
Proof. iLetx0be a solution ofVVIP. Then A, η
y, x0
/∈ −intK, ∀A∈∂f x0
, y∈S. 3.1
By theK-invexity offwith respect toη, we get fy−f
x0
− A, η
y, x0
∈K, ∀A∈∂f x0
, y∈S. 3.2
From3.1,3.2andLemma 2.7, we obtain fy−f
x0
/∈ −intK, ∀y∈S. 3.3
Therefore,x0is a weakly efficient solution ofVOP.
iiLetx0be a solution ofVVIP. Suppose thatx0is not a weakly efficient solution of VOP. Then, there existsy∈Ssuch that
fy−f x0
∈ −intK. 3.4
SincefisK-pseudoinvex with respect toη, then A, η
y, x0
∈ −intK, ∀A∈∂f x0
, 3.5
which contradicts the fact thatx0is a solution ofVVIP.
iiiAssume thatx0is a weakly efficient solution ofVOP. Then, fy−f
x0
/∈ −intK, ∀y∈S. 3.6
Sincefis−K-invex with respect toη, then
fy−fx0− A, ηy, x0 ∈ −K, ∀A∈∂fx0, y∈S. 3.7 It follows fromLemma 2.7that
A, η y, x0
/∈ −intK, ∀A∈∂f x0
, y∈S. 3.8 Therefore,x0is a solution ofVVIP.
Now we establish the following existence theorem.
Theorem 3.2. LetS ⊂ X be a nonempty convex set andη :S×S→X. Letf :X→Rm be a locally LipschitzK-pseudoinvex function. Assume that the following conditions hold
iηx, x 0 for anyx∈S,ηy, xis affine with respect toyand continuous with respect tox;
iithere exist a compact subsetDofSandy0∈Dsuch that A, η
y0, x
∈ −intK, ∀x∈S\D, A∈∂fx. 3.9 ThenVOPhas a weakly efficient solution.
Proof. ByLemma 3.1ii, it suffices to prove thatVVIPhas a solution. DefineG:S→2Sby Gy
x∈S:
A, ηy, x
/∈ −intK,∀A∈∂fx
, ∀y∈S. 3.10 First we show thatG is a KKM-mapping. By conditioni, we get y ∈ Gy. Hence, Gy/∅for ally ∈ S. Suppose that there exists a finite subset{x1, x2, . . . , xm} ⊆ S and that αi ≥0,i 1,2, . . . , m, withm
i 1αi 1 such thatx m
i 1αixi/∈m
i 1Gxi. Then,x /∈Gxifor all i 1,2, . . . , m. It follows that there existsA∈∂fxsuch that
A, η xi, x
∈ −intK, i 1,2, . . . , m. 3.11 SinceKis a convex cone andηis affine with respect to the first argument,
A, ηx, x
∈ −intK. 3.12
which gives 0 ∈ −intK. This is a contradiction since 0/∈ −intK. Therefore, G is a KKM- mapping.
Next, we show thatGyis a closed set for anyy∈S. In fact, let{xn}be a sequence of Gywhich converges to somex0∈S. Then for allAn∈∂fxn, we have
An, η y, xn
/∈ −intK. 3.13
Sincef is locally Lipschitz, then there exists a neighborhoodNx0ofx0andL >0 such that for anyx, y∈Nx0,
fx−fy≤Lx−y. 3.14 It follows that for anyx∈Nx0and anyA∈∂fx,A ≤L. Without loss of generality, we may assume thatAnconverges toA0. Since the set-valued mappingx→∂fxis closedsee 15, page 29andAn ∈∂fxn,A0 ∈∂fx0. By the continuity ofηy, xwith respect to the second argument, we have
An, η y, xn
−→
A0, η y, x0
. 3.15 SinceRm\ −intKis closed, one has
A0, η y, x0
/∈ −intK. 3.16
Hence,Gyis a closed set for anyy∈S.
By condition ii, we haveGy0 ⊂ D. AsGy0is closed andD is compact, Gy0is compact. Therefore, byLemma 2.6, we have that there existsx∗∈Ssuch that
x∗∈
y∈SGy, 3.17
or equivalently,
A, η y, x∗
/∈ −intK, ∀A∈∂f x∗
, y∈S. 3.18
That is,x∗is a solution ofVVIP. This completes the proof.
Corollary 3.3. LetS ⊂ Xbe a nonempty convex set andη : S×S→X. Letf : X→Rmbe a locally LipschitzK-invex function. Assume that the following conditions hold:
iηx, x 0 for anyx∈S,ηy, xis affine with respect toyand continuous with respect tox;
iithere exist a compact subsetDofSandy0∈Dsuch that A, η
y0, x
∈ −intK, ∀x∈S\D, A∈∂fx. 3.19
ThenVOPhas a weakly efficient solution.
Proof. Since aK-invex function isK-pseudoinvex, byTheorem 3.2, we obtain the result.
Acknowledgments
This work was supported by the National Natural Science Foundation of China10671135, the Specialized Research Fund for the Doctoral Program of Higher Education20060610005 and the Open FundPLN0703of State Key Laboratory of Oil and Gas Reservoir Geology and ExploitationSouthwest Petroleum University. And J. K. Kim was supported by the Korea Research Fundation Grant funded by the Korean GovermentMOEHRD, Basic Research Pro- motion FundKRF-2006-311-C00201.
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