 Advances in Pure Mathematics, 2011, 1, 184-186 doi:10.4236/apm.2011.14032 Published Online July 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Well-Posedness for Tightly Proper Efficiency in Set-Valued Optimization Problem* Yangdong Xu#, Pingping Zhang College of Mathematics and Statistics, Chongqing University, Chongqing, China E-mail: #xyd04010241@126.co m, zhpp0 4010248@163 .com Received April 3, 2011; revised April 30, 2011; accepted May 10, 2011 Abstract In this paper, a characterization of tightly properly efficient solutions of set-valued optimization problem is obtained. The concept of the well-posedness for a special scalar problem is linked with the tightly properly efficient solutions of set-valued optimization problem. Keywords: Set-Valued Optimization Problem, Tightly Proper Efficiency, Well-Posedness 1. Introduction One important problem in vector optimization is to find the efficient points of a set. As observed by Kuhn, Tucker and later by Geoffrion, some efficient points ex-hibit certain abnormal properties. To eliminate such ab-normal efficient points, various concept of proper effi-ciency have been introduced. The original concept was introduced by Kuhn and Tucker  and Geoffrion , and later modified and formulated in a more generalized framework by Borwein , Hartley , Benson , Henig , Borwein and Zhuang ; also see the refer-ences there in. Particularly, the concept of tightly proper efficiency was introduced by Zaffaroni , and he used a special scalar function to characterize the tightly proper efficiency, and obtained some properties of tightly proper efficiency. In this paper, we study the characterization and well- posedness for tightly proper efficiency in set-valued vector optimization problem. The paper is organized as follows. In Section 2, some concepts of tightly proper efficiency and some preliminary results are given. In Section 3, the characterization and well-posedness for tightly proper efficiency in set-valued vector optimiza-tion problem is discussed. 2. Preliminaries Throughout this paper, let X be a linear space, and YZ be two finite dimensional, with topological dual spaces and *Y*Z. For a set AY, , , clA intAA, and cA denote the closure, the interior, the boundary and the complement of A, respectively. Moreover, we will denote with the closed unit ball of Y. A set is said to be a cone if BCYcC for any cC and λ ≥ 0. A cone C is said to be convex if CC C, and it is said to be pointed if . In the sequel we suppose that is a convex, closed, pointed cone with nonempty interior. We say that the set CC=0YCY is a base for if is convex with 0CclC and ,\0=Cc=e: =Yy, >0ony. Definition 2.1: A point ySY is said to be efficient with respect to (denoted C,yESC) if =0YCSy Definition 2.2:  The point ySY is called tightly proper efficient with respect to C (denoted ,yTPES CK) if there exists an open convex set Y with 0YK satisfying Sy=K and there exists >0 such that cKBC B It is easy to verify that ,,SS C0yTPclSE CES Definition 2.3:  Let be a nonempty subset of . The contingent cone (or the Bouligand tangent cone) to at YS is the set  00with= limnnnyy v yy0,: ,y YR :,Y nnynTS v*This research was partially supported by the National Natural Science Foundation of China (Grant number: 10871216). Y. D. XU ET AL. 185Jahn  have gotten the following proposition on the contingent cone to at S0yclS. Proposition 2.1 a nonempty convex subset of a real normed space. Then :  Let S be,=TS CclconeSy 3. Tightly Proper Efficiency and Well-Posedness onsider the following vector optimization problem with Cset-valued maps: (VP) minFx,  s.t. Gx D,xX, where :2YFX and :2ZGX are set-valued maps, respectively. D is ax, pointed cone of closed,conve Z. Denuote the feasible soltion set of (VP) by thimage of := :9AxXGx D ande A under F by  =xAFAFx Definition 3.1: A point x is said to proper efficient solution (VP)be a tightly of , if there exists yFx such that ,yTPEFAC, and the point ,xy of is said to be a tightly properly efficient minimizer e define(VP). Definition 3.2: For a set SY. Let the function R b \=SSYS:SY d as ydy dy where =inf :Sdy sys ith =dyS w. S was introduceThe function d inin properties are ged together in the following pro- position. osition 3.1:  Letbee wi, ather , its maProp a convex conth no C nempty interiorthen the function C is convex, positively homogenous and lipschitzian. Moreover, this function is negative on the interior of C null on C, and positive on int cC. We consider the parameterized scalar problem: yP min Cyy s.t. yFA where yY. Definition 3.3: Let yY , the parameterized scalar optimization problemyP is Tikhonov well-posed if 1)=>0CCyy dy for all yyFA with yy; 2) for all nyFA with 0Cndyy implies that nyy. denote by We=yWPVP Fyov weLemma 3.1onsid llowingAPis Tikhonllposed. :  Cer the fo statements: (a) the point y is a tightly properly efficient point in (b) there exists an open convex set such that S; KY\0YCK and =Sy K; (c) ,=0YTS CC. If Yyn it holds thatfinite dimall stateents are equivalent. eorem ccterize the relation betweenparameterized scro is a any normed space, the ). If is ensional, we have () and ()) (abcalso that()ca followi( Y hmThe harang t tightly properly efficient points of(VP) and the alar pblem yP. Theore et m 3.1: LxA, yFx. The ,xy is a tightly properly efficient minimizer of (VP), then y is a solution of yP. Proof. We show that y is a solution of the scalar problem yP. Indeed3.1 and Lemma 3.1(b), we have , by Proposition =0,CCyyd y yFA Noting y that=0Cdyy, thus we have that y is just the solution ofyP. the problem  otRemark 3.1: The converse of Theorem 3.1 mabe n valid, the following example can illustrate the case. 1: Let y Example 3.=XR, 2=YR and =ZR. Given 2=CR, =DR.   [0,1],,F2,111,xyR R yxifx=x xyRR otherwise =, 1,foranygxxx xX Thus, the feasible set of (VP)  =| =[0,AxXGx R)  FA can see Figure 1. The set of FA. Figure 1. The set of Copyright © 2011 SciRes. APM Y. D. XU ET AL. Copyright © 2011 SciRes. APM 186 , but 1, 1,0 (VP). Th1,0yY is a solution of yP is not a tightly properly efficient minimizer ofus, the converse of Theorem 3.1 is not valid. Theorem 3.2: Let , yFxxA. Then, the ,xy is a tightly properly efficient minimizer of yP if and only if the scalar problem yP is Tikhonnov well-posed. Proof. If ,xy is a tightly properly efficient minimizer ofyP, let us show that the scalar problem yP is Tik well-posed. We argue by contr- diction: if the conclusion is false, then there exists a sequence honnovyanFA such that nyyB for some >0 and 0nyy, which means that there exists a sequence ncC with Cd0nnycy and 2B. Since we ay=nnn c can alws writenc with n, it follows that n does not coverge to zero, i.e., there exists a subsequence (we agall it nnain c) with >n for som>e 0. Now take 12=nnnycy , since n is bounded away from zero and 0n, thus we have