Existence of Solutions for Nonconvex and Nonsmooth Vector Optimization Problems

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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 Huang}3

1_{Department of Applied Mathematics, Southwest Petroleum University, Chengdu,}

Sichuan 610500, China

2_{Department of Mathematics, Kyungnam University, Masan, Kyungnam 631701, South Korea} 3_{Department 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

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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∗. LetRm_{be the}_m_{-dimensional Euclidean space, let}_S_⊂_X_{be a}

nonempty subset, and letK⊂Rmbe a nonempty closed convex cone with int_{K /}∅, 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

Let f : X→Rm _{be a vector valued function given by} _f _{f1, f2, . . . , f}

m, where each fi, 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→Rm_at_x_∈_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

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Definition 2.3. Letf :X→Rm_{be 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 mapping G : X→2X is called KKM-mapping if for

any finite subset {x1, x2, . . . , xn}of X, co{x1, x2, . . . , xn} is contained in

_n

i 1Gxi, where coA

denotes the convex hull of the setA.

Lemma 2.6see16. Let M be a nonempty subset of a Hausdorﬀtopological vector space X. Let G:M→2X_{be a KKM-mapping such that}_G_x_{is closed for any}_x_∈_M_{and 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, ηy, x0/∈ −intK, ∀y∈S. VVIP

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Lemma 3.1. Let f : X→Rm _{be a locally Lipschitz function and}_η _: _S_×_S_→_X_{. Then the following}

arguments hold.

iSuppose thatf isK-invex with respect toη. Ifx0is a solution of VVIP, 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 that f 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, ηy, x0/∈ −intK, ∀A∈∂fx0, y∈S. 3.1

By theK-invexity offwith respect toη, we get

fy−fx0−A, ηy, x0∈K, ∀A∈∂fx0, y∈S. 3.2

From3.1,3.2andLemma 2.7, we obtain

fy−fx0/∈ −intK, ∀y∈S. 3.3

Therefore,x0is a weakly eﬃcient solution ofVOP.

iiLetx0 be a solution ofVVIP. Suppose thatx0 is not a weakly eﬃcient solution of

VOP. Then, there existsy∈Ssuch that

fy−fx0∈ −intK. 3.4

Sincef isK-pseudoinvex with respect toη, then

A, ηy, x0∈ −intK, ∀A∈∂fx0, 3.5

which contradicts the fact thatx0is a solution ofVVIP.

iiiAssume thatx0is a weakly eﬃcient solution ofVOP. Then,

fy−fx0/∈ −intK, ∀y∈S. 3.6

Sincef is−K-invex with respect toη, then

fy−fx0− A, ηy, x0 ∈ −K, ∀A∈∂fx0, y∈S. 3.7

It follows fromLemma 2.7that

A, ηy, x0/∈ −intK, ∀A∈∂fx0, y∈S. 3.8

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Now we establish the following existence theorem.

Theorem 3.2. Let S ⊂ X be a nonempty convex set and η : S×S→X. Let f : X→Rm_{be a locally}

LipschitzK-pseudoinvex function. Assume that the following conditions hold

iηx, x 0for anyx∈S,ηy, xis aﬃne with respect toyand continuous with respect tox;

iithere exist a compact subsetDofSandy0∈Dsuch that

A, ηy0, 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→2S_by

Gy x∈S:A, ηy, x∈ −/ intK, ∀A∈∂fx, ∀y∈S. 3.10

First we show that G is a KKM-mapping. By condition i, 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, withmi 1αi 1 such thatx mi 1αixi/∈mi 1Gxi. Then,x /∈Gxifor all

i 1,2, . . . , m. It follows that there existsA∈∂fxsuch that

A, ηxi, 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 neighborhoodNx0of x0andL > 0 such that for anyx, y ∈Nx0,

_f_x₋_f_y_≤_L_x₋_y_. _3.14

It follows that for anyx∈Nx0and any A ∈∂fx,A ≤L. Without loss of generality, we may assume thatAn converges 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, ηy, x0. 3.15

SinceRm_{\ −}_int_K_{is closed, one has}

A0, ηy, x0/∈ −intK. 3.16

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By condition ii, we have Gy0 ⊂ D. As Gy0is closed and D is compact, Gy0 is compact. Therefore, byLemma 2.6, we have that there existsx∗∈Ssuch that

x∗∈

y∈S

Gy, 3.17

or equivalently,

A, ηy, x∗/∈ −intK, ∀A∈∂fx∗, y∈S. 3.18

That is,x∗is a solution ofVVIP. This completes the proof.

Corollary 3.3. LetS ⊂ X be a nonempty convex set andη : S×S→X. Letf : X→Rm _{be a locally}

LipschitzK-invex function. Assume that the following conditions hold:

iηx, x 0for anyx∈S,ηy, xis aﬃne with respect toyand continuous with respect tox;

iithere exist a compact subsetDofSandy0∈Dsuch that

A, ηy0, 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 China 10671135, 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 Exploitation Southwest 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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