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

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


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][2][3]. Another is to study the set-convergence of minimal sets of perturbed sets converging to a given set [4][5][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 starshaped vector optimization problems.

Preliminaries and Notations
In what follows, unless otherwise specified, we always suppose that is a normed linear space with dual space * and B is the closed ball centered at 0 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 int ̸ = 0, where int denotes the interior of . We say that ⊂ is a base of if and only if is convex, 0 ∉ cl , and = cone , where cl and cone denote the closure and cone hull of , respectively.
We say that { } converges to in the sense of Kuratowski-Painlevé if and only if Ls ⊂ ⊂ Li . 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 − Li and − Ls , respectively. When − Ls ⊂ ⊂ − Li , 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 − Ls ⊂ ⊂ Li .  for all > 0.

Remark 6.
When is finite-dimensional, the notions of setconvergence 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 ker of is defined by A set is called star-shaped if and only if ker ̸ = 0 or = 0. Obviously every convex set is star-shaped and the converse is not true in general.

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.
Proof. If ∩ Min = 0, then the conclusion holds trivially. Let ∩ Min ̸ = 0. Suppose to the contrary that there exists ∈ ∩ Min such that ∉ Li(Min ). Without loss of generality, we can assume that = 0. Since = Li(ker ), there exists a sequence { } of such that → = 0 and ∈ ker , for all sufficiently large . Let = { ∈ : ∈ \ Min }, where is the set of all natural numbers. can be regarded as a subsequence of since 0 ∉ Li(Min ). By the generalized domination property (GDP) for , for every ∈ , there exist ∈ clMin and ∈ such that = + . Consider the following two cases.
Then there exists a strictly increasing function : → such that It is easy to see that̃→ = 0 and̃∈ Min for all sufficiently large . Thus, ∈ Li(Min ), a contradiction.
Proof. The conclusion follows from almost the same arguments as in Theorem 10.
Remark 13. Note that if ∩Min = 0, 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.
Remark 17. Let { } be a sequence of nonempty closed and star-shaped subsets of and → and ker → . By Proposition 3.1 of [17] and Propositions 8 and 16, is a closed convex subset of ker .
The following proposition presents some conditions under which the intersection ∩ Min is nonempty.

Proposition 18. Let be a normed linear space, let be a pointed, closed, and convex cone with int ̸ = 0, and let { } be a sequence of nonempty closed and star-shaped subsets of . Let and be nonempty subsets of . Assume that
(i) → and ker → ; (ii) Ls(ker ∩ Min ) ̸ = 0; (iii) is rotound and = for some ∈ , where is the section of at (see Definition 14).
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 Ls Min (ker ) ⊂ Min = ∩ Min .
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 = 2 , = 2 + , and Then has a compact base, is closed and star-shaped, and the generalized domination property (GDP) holds for . By Theorem 10, we have The following example shows that the sequentially weak compactness of is essential in Theorems 10 and 11.
Example 20. Let = 2 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 ker = {0}. Proof. If ∩StMin = 0, then the conclusion holds trivially. Let ∩ StMin ̸ = 0. Suppose to the contrary that there exists ∈ ∩ StMin such that ∉ LiMin . Without loss of generality, we can suppose that = 0. Then there exists a sequence { } of such that → = 0 and ∈ ker , for all sufficiently large . Let = { ∈ : ∉ Min }. can be regarded as a subsequence of since ∉ LiMin . Since the generalized domination property (GDP) holds for , there exists ∈ clMin 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 > 0 and for any > 0, we have Since → 0, it follows from (25) and (26) that, for any > 0, Now we prove that the following property holds: for any > 0 there exists > 0 such that If it is not the case, then ∃ 0 > 0, for all > 0, ∃ ∈ , , ∈ B , ∈ , such that Since 0 ∈ StMin , there exists 0 > 0 such that We can choose in (29) such that < min{ 0 /2, 0 /2}. It follows that (31) This together with (30) implies that ‖ ‖ ≤ 0 /2. But from (29), one has a contradiction. Thus, (28) holds. It follows from (27) and (28) that, for any > 0, This arrives at a contradiction since 0 ∉ Li(Min ) and { } ∈ does not converge to 0.
It is easy to see that all assumptions of Theorem 21 hold. By Theorem 21, we have Indeed, it is easily seen that   (ii) the generalized domination property (GDP) holds for all ; (iii) = Li(ker ) and ( , ) → 0 for all > 0; (iv) StMin ∩B ∩ is relatively compact for every > 0.
Proof. If ∩ StMin = 0, then the conclusion holds trivially. Let ∩ StMin ̸ = 0. Suppose on the contrary that the conclusion of the theorem does not hold. Then there exist > 0, > 0, and a subsequence { } of { } such that This yields that, for every , there exists ∈ B ∩ ∩StMin such that Since B ∩ ∩ StMin is relatively compact, up to a subsequence, → ∈ cl(B ∩ ∩ StMin ). By Theorem 21, for each , Then there exists a sequence { } ∈ such that → as → ∞ and ∈ Min , for all sufficiently large . We can choose a strictly increasing function : → such that ( ) → as → ∞. Thus, ( , Min ) → 0. It follows that Then This implies that contradicting (44).
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  The proof is complete.