Lower Convergence of Minimal Sets in Star-Shaped Vector Optimization Problems

Abstract

Let be a sequence of nonempty star-shaped sets. By using generalized domination property, we study the lower convergence of minimal sets . The distinguishing feature of our results lies in disuse of convexity assumptions (only using star-shapedness).

1. Introduction

Stability analysis is one of the most important and interesting subjects and its role has been widely recognized in the theory of optimization. In the literature, two classical approaches can be found to study stability in vector optimization. One is to investigate continuity properties of the optimal multifunctions [1–3]. Another is to study the set-convergence of minimal sets of perturbed sets converging to a given set [4–6]. Bednarczuk [1, 2] obtained some stability results by investigating the Hölder continuity of minimal point functions in vector optimization problems. Bednarczuk [3] established the stability by investigating the lower semicontinuity of minimal points in vector optimization. Luc et al. [4] investigated the stability of vector optimization in terms of the convergence of the efficient sets. Miglierina and Molho [5] obtained some results on stability of convex vector optimization problems by considering the convergence of minimal sets. Convexity is a very common assumption and plays important roles in stability analysis in vector optimization. By using convexity assumptions, Tanino [7] considered the stability of the efficient set in vector optimization. Bednarczuk [8] investigated the stability of Pareto points to finite-dimension parametric convex vector optimization. In [5, 9], the authors used convexity to establish Kuratowski-Painlevé and Attouch-Wets convergence of minimal sets. For more results concerning use of convexity in stability analysis, we refer readers to [10, 11].

However, many practical problems can only be modelled as nonconvex optimization problems. So it is interesting and important to weaken convexity assumption. Star-shapedness is one of the most important generalizations of convexity. Crespi et al. [12, 13] used star-shapedness to study scalar Minty variational inequalities and scalar optimization problems. Fang and Huang [14] used star-shapedness to study the well-posedness of vector optimization problems. Shveidel [15] studied the separability and its application to an optimization problem. In this paper, following the ideas of [5, 9], we investigate the lower convergence of minimal sets in star-shaped vector optimization problems.

2. Preliminaries and Notations

In what follows, unless otherwise specified, we always suppose that is a normed linear space with dual space and is the closed ball centered at with radius . Let be nonempty subsets of , let be a sequence of nonempty subsets of , and let be a pointed, closed, and convex cone with , where denotes the interior of . We say that is a base of if and only if is convex, , and , where and denote the closure and cone hull of , respectively.

Definition 1. A point is called a minimal point of (with respect to ) if and only if . Denote by the set of all minimal points of . A point is called a weakly minimal point of if and only if . Denote by the set of all weakly minimal points of . A point is called strictly minimal point (see [3]) of if and only if for every there exists such that . Denote by the set of all strictly minimal points of . Obviously .

Definition 2. The generalized domination property holds for if and only if .

Remark 3. (i) Clearly the domination property (see [11]) implies the generalized domination property . (ii) The containment property ([16]) implies the generalized domination property . (iii) The weak containment property ([1]) implies the generalized domination property . For more details on relationship among the domination property, the containment property, and the weak containment property, we refer readers to [8].

Definition 4. Given a set and a sequence of subsets of , the Kuratowski-Painlevé lower and upper limits are defined as follows: We say that converges to in the sense of Kuratowski-Painlevé if and only if . When we consider the limits in the weak topology on rather than the original norm topology, we denote the lower and upper limits above by and , respectively. When , we say that converges to (denoted by ) in the sense of Kuratowski-Painlevé with respect to weak topology. We say that converges to in the sense of Mosco if and only if .

Definition 5. Give two nonempty subsets and of , and define where . One says that a sequence of subsets of converges to in the sense of Attouch-Wets if and only if for all . One says that is upper (or lower) limit of in the sense of Attouch-Wets if and only if for all .

Remark 6. When is finite-dimensional, the notions of set-convergence in Definitions 4 and 5 coincide whenever we consider a sequence of closed sets. For more relationship between the various concepts of set-convergence, we refer readers to [6].

Definition 7. Given a set , the kernel of is defined by A set is called star-shaped if and only if or . Obviously every convex set is star-shaped and the converse is not true in general.

3. Main Results

In this section, we investigate the lower convergence of minimal sets in star-shaped vector optimization.

The following proposition shows that the limit set of a Kuratowski-Painlevé converging sequence of star-shaped sets is star-shaped.

Proposition 8. Let be a normed linear space and let be a sequence of nonempty star-shaped subsets of . Then .

Proof. By the definition of , we get . Suppose to the contrary that there exists such that . Then there exist and such that . Since and , there exist sequences and such that It follows that This contradicts .

Remark 9. Let be a sequence of nonempty star-shaped subsets of and and . By Proposition 8, . It is known that the limit set of a Kuratowski-Painlevé converging sequence of convex sets is convex (see Proposition 3.1 of [17]). In this sense, Proposition 8 generalizes Proposition 3.1 of [17] to star-shaped case.

Theorem 10. Let be a normed linear space, let be a pointed, closed, and convex cone with a sequentially weakly compact base , and let be a sequence of nonempty subsets of . Assume that (i) is closed and star-shaped for all ;(ii)the generalized domination property holds for all ;(iii). Then .

Proof. If , then the conclusion holds trivially. Let . Suppose to the contrary that there exists such that . Without loss of generality, we can assume that . Since , there exists a sequence of such that and , for all sufficiently large . Let , where is the set of all natural numbers. can be regarded as a subsequence of since . By the generalized domination property for , for every , there exist and such that . Consider the following two cases. (I) converges to . Since , there exists a sequence such that as . Then there exists a strictly increasing function such that . Let It is easy to see that and for all sufficiently large . Thus, , a contradiction. (II) does not converge to . By the closedness of , we have . Since is star-shaped, for all sufficiently large . Since is a base of , for every , there exist and such that . If , then , a contradiction. Then there exists such that, up to a subsequence, for all . Take and . It follows that for all sufficiently large . By the sequentially weak compactness of , up to a subsequence, converges weakly to . In another word, admits a subsequence converging weakly to . We have , since and . It contradicts the minimality of .

Theorem 11. Let be a normed linear space, let be a pointed, closed, and convex cone with a sequentially weakly compact base , and let be a sequence of nonempty subsets of . Assume that (i) is closed and star-shaped for all ;(ii)the generalized domination property holds for all ;(iii).Then .

Proof. The conclusion follows from almost the same arguments as in Theorem 10.

Remark 12. Theorems 10 and 11 generalize Theorems 3.1 and 3.2 of [5], respectively.

Remark 13. Note that if , the results of Theorems 10 and 11 are trivial. In the sequel we present some conditions under which the intersection is nonempty. We first recall some concepts and results.

Definition 14. Let and . The set is called a section of at and denoted by .

Definition 15. A nonempty convex set is said to be rotound when its boundary does not contain line segments.

Proposition 16. If is nonempty, closed, and star-shaped, then is closed and convex.

Proof. Let and . For any and any , it follows that Let Clearly . It follows that Since , we have and Therefore, for all and so is convex.
Let such that . Obviously since is closed. We will prove . For any and any , we have . Letting , we have since is closed. Thus, is closed.

Remark 17. Let be a sequence of nonempty closed and star-shaped subsets of and and . By Proposition 3.1 of [17] and Propositions 8 and 16, is a closed convex subset of .

The following proposition presents some conditions under which the intersection is nonempty.

Proposition 18. Let be a normed linear space, let be a pointed, closed, and convex cone with , and let be a sequence of nonempty closed and star-shaped subsets of . Let and be nonempty subsets of . Assume that (i) and ;(ii);(iii) is rotound and for some , where is the section of at (see Definition 14). Then .

Proof. Since , it follows from Propositions 2.6 and 2.8 of Luc [11] that This yields Taking into account the assumptions from Theorem 4.4 of Miglierina and Molho [5], we get By Proposition 2.6 of Luc [11], It follows that

The following example further illustrates the results of Theorems 10 and 11.

Example 19. Let , and Then has a compact base, is closed and star-shaped, and the generalized domination property holds for . By Theorem 10, we have Indeed, it is easy to see that Therefore,

The following example shows that the sequentially weak compactness of is essential in Theorems 10 and 11.

Example 20. Let be endowed with the usual norm; let be the nonnegative orthant. Let be the canonical orthonormal base of and It is easy to see that is not convex but star-shaped and . Further we have

Theorem 21. Let be a normed linear space, let , be a pointed, closed, and convex cone, and let be a sequence of nonempty subsets of . Assume that (i) is closed and star-shaped for all ;(ii)the generalized domination property holds for all ;(iii) and for all . Then .

Proof. If , then the conclusion holds trivially. Let . Suppose to the contrary that there exists such that . Without loss of generality, we can suppose that . Then there exists a sequence of such that and , for all sufficiently large . Let . can be regarded as a subsequence of since . Since the generalized domination property holds for , there exists such that , for all . The closedness of implies . It follows from the star-shapedness of that for all sufficiently large . By assumption (iii), for any and for any , we have Since , it follows from (25) and (26) that, for any , Now we prove that the following property holds: for any there exists such that If it is not the case, then , for all , , such that Since , there exists such that We can choose in (29) such that . It follows that This together with (30) implies that . But from (29), one has a contradiction. Thus, (28) holds. It follows from (27) and (28) that, for any , This arrives at a contradiction since and does not converge to .

Remark 22. Theorem 21 generalizes Theorem 3.5 of [5] to the star-shaped case.

Example 23. Let , It is easy to see that all assumptions of Theorem 21 hold. By Theorem 21, we have Indeed, it is easily seen that Thus,

Theorem 24. Let be a normed linear space, let be a pointed, closed, and convex cone, and let be a sequence of nonempty subsets of . Assume that (i) is closed and star-shaped for all ;(ii)the generalized domination property holds for all ;(iii) and for all ;(iv) is relatively compact for every . Then, for each , .

Proof. If , then the conclusion holds trivially. Let . Suppose on the contrary that the conclusion of the theorem does not hold. Then there exist , and a subsequence of such that This yields that, for every , there exists such that Since is relatively compact, up to a subsequence, . By Theorem 21, for each , Then there exists a sequence such that as and , for all sufficiently large . We can choose a strictly increasing function such that as . Thus, . It follows that a contradiction.

Remark 25. Theorem 24 generalizes Theorem 3.7 of [5] to the star-shaped case.

Proposition 26. Let be a normed linear space, let be a pointed, closed, and convex cone, and let be a nonempty, closed, and star-shaped subset of . Assume that, for every , there exists a nondecreasing function satisfying , for all and Then

Proof. It is sufficient to prove . Suppose on the contrary that there exists such that . By the definition of , there exist , such that , and , for all . Take . Then . By the minimality of , we have . It follows that By the assumption, there exists a nondecreasing function satisfying and for all such that This implies that contradicting (44).

Remark 27. Proposition 26 is inspired by Proposition 3.9 of [5].

Corollary 28. Let be a normed linear space, let be a pointed, closed, and convex cone, and let be a sequence of nonempty subsets of . Assume that (i) is closed and star-shaped for all ;(ii)the generalized domination property holds for all ;(iii) and for all ;(iv)for every , there exists a nondecreasing function satisfying for all and Then

Proof. From Proposition 8, . By Theorem 21, . By using assumption (iv), from Proposition 26, we have The proof is complete.

Corollary 29. Let be a normed linear space, let be a pointed, closed, and convex cone, and let be a sequence of nonempty subsets of . Assume that (i) is closed and star-shaped for all ;(ii)the generalized domination property holds for all ;(iii) and for all ;(iv) is relatively compact for all ;(v)for every , there exists a nondecreasing function satisfying for all and Then, for each ,

Proof. The conclusion follows from Propositions 8 and 26 and Theorem 24.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author thanks the referees for their helpful comments and suggestions which lead to improvements of this paper. The author thanks Dr. Y. P. Fang for his helpful discussion when preparing this paper. This work was partially supported by the National Science Foundation of China (11201042) and the Scientific Research Foundation of CUIT (J201216 and KYTZ201128).