Let An be a sequence of nonempty star-shaped sets. By using generalized domination property, we study the lower convergence of minimal sets MinAn. 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 X is a normed linear space with dual space X* and Bρ is the closed ball centered at 0 with radius ρ. Let A,B be nonempty subsets of X, let {An} be a sequence of nonempty subsets of X, and let K⊂X be a pointed, closed, and convex cone with intK≠∅, where intK denotes the interior of K. We say that G⊂X is a base of K if and only if G is convex, 0∉clG, and K=coneG, where clG and coneG denote the closure and cone hull of G, respectively.

Definition 1.

A point a∈A is called a minimal point of A (with respect to K) if and only if A∩(a-K)={a}. Denote by MinA the set of all minimal points of A. A point a∈A is called a weakly minimal point of A if and only if A∩(a-intK)=∅. Denote by WMinA the set of all weakly minimal points of A. A point a∈A is called strictly minimal point (see [3]) of A if and only if for every ϵ>0 there exists δ>0 such that (A-a)∩(Bδ-K)⊂Bϵ. Denote by StMinA the set of all strictly minimal points of A. Obviously StMinA⊂MinA.

Definition 2.

The generalized domination property (GDP) holds for A if and only if A⊂clMinA+K.

Remark 3.

(i) Clearly the domination property (DP) (see [11]) implies the generalized domination property (GDP). (ii) The containment property (CP) ([16]) implies the generalized domination property (GDP). (iii) The weak containment property (WCP) ([1]) implies the generalized domination property (GDP). 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 A and a sequence {An} of subsets of X, the Kuratowski-Painlevé lower and upper limits are defined as follows:
(1)LiAn={x∈X:x=limn→∞xn,xn∈An,mmm∀sufficientlylargenx∈X:x=limn→∞xn,xn∈An},LsAn={x∈X:x=lims→∞xs,xs∈Ans,{ns}mmmisasubsequenceof{n}x∈X:x=limn→∞xn,xn∈An}.
We say that {An} converges to A in the sense of Kuratowski-Painlevé if and only if LsAn⊂A⊂LiAn. When we consider the limits in the weak topology on X rather than the original norm topology, we denote the lower and upper limits above by w-LiAn and w-LsAn, respectively. When w-LsAn⊂A⊂w-LiAn, we say that {An} converges to A (denoted by An→A) in the sense of Kuratowski-Painlevé with respect to weak topology. We say that {An} converges to A in the sense of Mosco if and only if w-LsAn⊂A⊂LiAn.

Definition 5.

Give two nonempty subsets A and B of X, and define
(2)e(A,B)=supa∈Ad(a,B),eρ(A,B)=e(A∩Bρ,B),Hρ(A,B)=max{eρ(A,B),eρ(B,A)},
where d(x,A)=infa∈A∥x-a∥. One says that a sequence {An} of subsets of X converges to A in the sense of Attouch-Wets if and only if limn→∞Hρ(An,A)→0 for all ρ>0. One says that A is upper (or lower) limit of {An} in the sense of Attouch-Wets if and only if
(3)limn→∞eρ(An,A)⟶0(orlimn→∞eρ(A,An)⟶0)
for all ρ>0.

Remark 6.

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

Definition 7.

Given a set A, the kernel kerA of A is defined by
(4)kerA={a∈A:a+λ(x-a)∈A,∀x∈A,∀λ∈[0,1]}.
A set A is called star-shaped if and only if kerA≠∅ or A=∅. 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 X be a normed linear space and let {An} be a sequence of nonempty star-shaped subsets of X. Then Li(kerAn)⊂ker(LiAn).

Proof.

By the definition of Li(kerAn), we get Li(kerAn)⊂LiAn. Suppose to the contrary that there exists b∈Li(kerAn) such that b∉ker(LiAn). Then there exist a∈LiAn and λ∈[0,1) such that b+λ(a-b)∉LiAn. Since a∈LiAn and b∈Li(kerAn), there exist sequences {an} and {bn} such that
(5)an⟶a,bn⟶b,an∈An,bn∈kerAnffffffffffffffffffffffffffforallsufficientlylargen.
It follows that
(6)bn+λ(an-bn)⟶b+λ(a-b),bn+λ(an-bn)∈Anmmmmimm∀sufficientlylargen.
This contradicts b+λ(a-b)∉LiAn.

Remark 9.

Let {An} be a sequence of nonempty star-shaped subsets of X and An→A and kerAn→B. By Proposition 8, B⊂kerA⊂A. 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 X be a normed linear space, let K be a pointed, closed, and convex cone with a sequentially weakly compact base G, and let {An} be a sequence of nonempty subsets of X. Assume that

An is closed and star-shaped for all n;

the generalized domination property (GDP) holds for all An;

A=w-LsAn,B=Li(kerAn).

Then B∩MinA⊂Li(MinAn).
Proof.

If B∩MinA=∅, then the conclusion holds trivially. Let B∩MinA≠∅. Suppose to the contrary that there exists a∈B∩MinA such that a∉Li(MinAn). Without loss of generality, we can assume that a=0. Since B=Li(kerAn), there exists a sequence {an} of X such that an→a=0 and an∈kerAn, for all sufficiently large n. Let J={j∈N:aj∈Aj∖MinAj}, where N is the set of all natural numbers. J can be regarded as a subsequence of N since 0∉Li(MinAn). By the generalized domination property (GDP) for An, for every j∈J, there exist bj∈clMinAj and kj∈K such that aj=bj+kj. Consider the following two cases.

{bj}j∈J converges to a=0. Since bj∈clMinAj, there exists a sequence {bjk}⊂MinAj such that bjk→bj as k→∞. Then there exists a strictly increasing function ϕ:J→J such that bjϕ(j)→a=0. Let
(7)a~n={an,n∈N∖J,bnϕ(n),n∈J.

It is easy to see that a~n→a=0 and a~n∈MinAn for all sufficiently large n. Thus, a∈Li(MinAn), a contradiction.

{bj}j∈J does not converge to a=0. By the closedness of Aj, we have bj∈Aj. Since Aj is star-shaped,
(8)[aj,bj]={x∈X:x=aj+(1-α)(bj-aj),α∈[0,1]}⊂Aj∩(aj-K)

for all sufficiently large j∈J. Since G is a base of K, for every j∈J, there exist gj∈G and λj>0 such that bj=aj-λjgj. If λj→0, then bj→0, a contradiction. Then there exists ϵ>0 such that, up to a subsequence, λj>ϵ for all j∈J. Take λ~j=λj/ϵ and g~j=ϵgj∈ϵG. It follows that
(9)cj∶=aj+(1-(1-1λ~j))(bj-aj)=aj-g~j∈Aj∩(aj-K)
for all sufficiently large j∈J. By the sequentially weak compactness of G, up to a subsequence, g~j converges weakly to g≠0. In another word, {cj}j∈J admits a subsequence converging weakly to -g≠0. We have -g∈A∩(-K), since cj∈Aj∩(aj-K) and A=w-LsAn. It contradicts the minimality of a=0.
Theorem 11.

Let X be a normed linear space, let K be a pointed, closed, and convex cone with a sequentially weakly compact base G, and let {An} be a sequence of nonempty subsets of X. Assume that

An is closed and star-shaped for all n;

the generalized domination property (GDP) holds for all An;

A=w-LsAn,B=w-Li(kerAn).

Then B∩MinA⊂w-Li(MinAn).
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 B∩MinA=∅, 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 D⊂X and x∈X. The set D∩(x-K) is called a section of D at x and denoted by Dx.

Definition 15.

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

Proposition 16.

If D is nonempty, closed, and star-shaped, then kerD is closed and convex.

Proof.

Let x,y∈kerD and x(t)=ty+(1-t)x,forallt∈[0,1]. For any z∈D and any λ∈[0,1], it follows that
(10)x(t)+λ(z-x(t))=x+[1-(1-t)(1-λ)]×{[y+λ1-(1-t)(1-λ)(z-y)]-x}.
Let
(11)α=1-(1-t)(1-λ),β=λ1-(1-t)(1-λ),y(β)=y+β(z-y).
Clearly α,β∈[0,1]. It follows that
(12)x(t)+λ(z-x(t))=x+α(y(β)-x).
Since {x,y}⊂kerD, we have y(β)∈D and
(13)x(t)+λ(z-x(t))∈D,∀λ∈[0,1].
Therefore, x(t)∈kerD for all t∈[0,1] and so kerD is convex.

Let {xn}⊂kerD such that xn→x*. Obviously x*∈D since D is closed. We will prove x*∈kerD. For any z∈D and any t∈[0,1], we have xn+t(z-xn)∈D. Letting n→∞, we have x*+t(z-x*)∈D since D is closed. Thus, kerD is closed.

Remark 17.

Let {An} be a sequence of nonempty closed and star-shaped subsets of X and An→A and kerAn→B. By Proposition 3.1 of [17] and Propositions 8 and 16, B is a closed convex subset of kerA.

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

Proposition 18.

Let X be a normed linear space, let K be a pointed, closed, and convex cone with intK≠∅, and let {An} be a sequence of nonempty closed and star-shaped subsets of X. Let A and B be nonempty subsets of X. Assume that

An→A and kerAn→B;

Ls(kerAn∩MinAn)≠∅;

B is rotound and B=Ax for some x∈X, where Ax is the section of A at x (see Definition 14).

Then B∩MinA≠∅.
Proof.

Since B=Ax, it follows from Propositions 2.6 and 2.8 of Luc [11] that
(14)B∩MinA⊂MinB⊂MinA.
This yields
(15)B∩MinA=MinB.
Taking into account the assumptions from Theorem 4.4 of Miglierina and Molho [5], we get
(16)LsMin(kerAn)⊂MinB=B∩MinA.
By Proposition 2.6 of Luc [11],
(17)kerAn∩MinAn⊂Min(kerAn).
It follows that
(18)∅≠Ls(kerAn∩MinAn)⊂LsMin(kerAn)⊂B∩MinA.

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

Example 19.

Let X=R2,K=R+2, and
(19)An={(x,y):x+y≥-1n,-1n≤x≤0,-1n≤y≤0}∪{(x,0):-2n≤x≤-1n},∀n∈N.
Then K has a compact base, An is closed and star-shaped, and the generalized domination property (GDP) holds for An. By Theorem 10, we have
(20)B∩MinA⊂Li(MinAn).
Indeed, it is easy to see that
(21)kerAn={(x,0):-1n≤x≤0},MinAn={(-2n,0)}∪{(x,y):x+y=-1n,-1n<x≤0},B=Li(kerAn)={(0,0)},A=LsAn={(0,0)},MinA={(0,0)}.
Therefore,
(22)B∩MinA⊂Li(MinAn)={(0,0)}.

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

Example 20.

Let X=l2 be endowed with the usual norm; let K be the nonnegative orthant. Let {en}n∈N be the canonical orthonormal base of K and
(23)An=[-nen,0]∪[-nen+1,0].
It is easy to see that An is not convex but star-shaped and kerAn={0}. Further we have
(24)A=w-Ls(An)={0},B=Li(kerAn)={0},MinAn={-nen,-nen+1},A=MinA={0},w-Li(MinAn)=∅.

Theorem 21.

Let X be a normed linear space, let A⊂X, K⊂X be a pointed, closed, and convex cone, and let {An} be a sequence of nonempty subsets of X. Assume that

An is closed and star-shaped for all n;

the generalized domination property (GDP) holds for all An;

B=Li(kerAn) and eρ(An,A)→0 for all ρ>0.

Then B∩
StMin
A⊂Li(MinAn).
Proof.

If B∩StMinA=∅, then the conclusion holds trivially. Let B∩StMinA≠∅. Suppose to the contrary that there exists a∈B∩StMinA such that a∉LiMinAn. Without loss of generality, we can suppose that a=0. Then there exists a sequence {an} of X such that an→a=0 and an∈kerAn, for all sufficiently large n. Let J={j∈N:aj∉MinAj}. J can be regarded as a subsequence of N since a∉LiMinAn. Since the generalized domination property (GDP) holds for Aj, there exists bj∈clMinAj such that bj∈aj-K, for all j∈J. The closedness of Aj implies bj∈Aj. It follows from the star-shapedness of Aj that
(25)[aj,bj]⊂Aj∩(aj-K)
for all sufficiently large j∈J. By assumption (iii), for any ρ>0 and for any ϵ>0, we have
(26)An∩Bρ⊂A+Bϵ∀sufficientlylargen.
Since an→0, it follows from (25) and (26) that, for any ϵ>0,
(27)[aj,bj]∩Bρ⊂(A+Bϵ)∩(Bϵ-K)dddddddddddd∀sufficientlylargej∈J.
Now we prove that the following property holds: for any ϵ>0 there exists η>0 such that
(28)(A+Bη)∩(Bη-K)⊂Bϵ.
If it is not the case, then ∃ϵ0>0, for all η>0, ∃x∈A,bη,bη′∈Bη,k∈K, such that
(29)x+bη=bη′-k,∥x+bη∥>ϵ0.
Since 0∈StMinA, there exists η0>0 such that
(30)A∩(Bη0-K)⊂Bϵ0/2.
We can choose η in (29) such that η<min{η0/2,ϵ0/2}. It follows that
(31)x=bη′-bη-k∈(B2η-K)∩A⊂(Bη0-K)∩A.
This together with (30) implies that ∥x∥≤ϵ0/2. But from (29), one has
(32)∥x∥≥ϵ0-∥bη∥>ϵ0-ϵ02=ϵ02,
a contradiction. Thus, (28) holds. It follows from (27) and (28) that, for any ϵ>0,
(33)[aj,bj]∩Bρ⊂Bϵ∀sufficientlylargej∈J.
This arrives at a contradiction since 0∉Li(MinAn) and {bj}j∈J does not converge to 0.

Remark 22.

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

Example 23.

Let X=R2,K=R+2,
(34)An={(x,y):x+y=1,0≤x≤1}∪{(x,0):1≤x≤2+1n},mmmmmmmmmmmm∀n∈N,A={(x,y):x+y=1,0≤x≤1}∪{(x,0):1≤x≤2}.
It is easy to see that all assumptions of Theorem 21 hold. By Theorem 21, we have
(35)B∩StMinA⊂Li(MinAn).
Indeed, it is easily seen that
(36)kerAn={(1,0)},MinAn={(x,y):x+y=1,0≤x≤1},B=Li(kerAn)={(1,0)},MinA=StMinA={(x,y):x+y=1,0≤x≤1}.
Thus,
(37){(1,0)}=B∩StMinA⊂Li(MinAn)={(x,y):x+y=1,0≤x≤1}.

Theorem 24.

Let X be a normed linear space, let K⊂X be a pointed, closed, and convex cone, and let {An} be a sequence of nonempty subsets of X. Assume that

An is closed and star-shaped for all n;

the generalized domination property (GDP) holds for all An;

B=Li(kerAn) and eρ(An,A)→0 for all ρ>0;

StMin
A∩Bρ∩B is relatively compact for every ρ>0.

Then, for each ρ>0, limn→∞eρ(B∩
StMin
A,MinAn)=0.
Proof.

If B∩StMinA=∅, then the conclusion holds trivially. Let B∩StMinA≠∅. Suppose on the contrary that the conclusion of the theorem does not hold. Then there exist ρ>0,ϵ>0, and a subsequence {Ank} of {An} such that
(38)eρ(B∩StMinA,MinAnk)>2ϵ,∀k.
This yields that, for every k, there exists mk∈Bρ∩B∩StMinA such that
(39)d(mk,MinAnk)>2ϵ.
Since Bρ∩B∩StMinA is relatively compact, up to a subsequence, mk→a∈cl(Bρ∩B∩StMinA). By Theorem 21, for each k,
(40)mk∈B∩StMinA⊂Li(MinAn).
Then there exists a sequence {aks}s∈N such that aks→mk as s→∞ and aks∈MinAs, for all sufficiently large s. We can choose a strictly increasing function ϕ:N→N such that akϕ(k)→a as k→∞. Thus, d(a,MinAnk)→0. It follows that
(41)d(a,MinAnk)≥d(mk,MinAnk)-d(mk,a)>ϵsssssssssssssssssssssssssssss∀sufficientlylargek,
a contradiction.

Remark 25.

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

Proposition 26.

Let X be a normed linear space, let K⊂X be a pointed, closed, and convex cone, and let A be a nonempty, closed, and star-shaped subset of X. Assume that, for every x0∈kerA∩MinA, there exists a nondecreasing function δx0:[0,+∞)→[0,+∞) satisfying δx0(0)=0,δx0(t)>0, for all t>0 and
(42)12(x0+x)+Bδx0(∥x0-x∥)⊂A,∀x∈A.
Then
(43)kerA∩MinA=kerA∩
StMin
A.

Proof.

It is sufficient to prove kerA∩MinA⊂kerA∩StMinA. Suppose on the contrary that there exists a∈kerA∩MinA such that a∉StMinA. By the definition of StMinA, there exist δ>0,{qn}⊂X,{kn}⊂K, such that qn→0,qn-kn+a∈A, and ∥an-kn∥>η, for all n. Take xn=qn-kn+a. Then d((xn-a)/2,-K)→0. By the minimality of a, we have (-K)∩(A-a)={0}. It follows that
(44)d(xn-a2,X∖(A-a))⟶0.
By the assumption, there exists a nondecreasing function δa:[0,+∞)→[0,+∞) satisfying δa(0)=0 and δa(t)>0 for all t>0 such that
(45)xn+a2+Bδa(∥xn-a∥)⊂A.
This implies that
(46)d(xn-a2,X∖(A-a))≥δa(∥xn-a∥),
contradicting (44).

Remark 27.

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

Corollary 28.

Let X be a normed linear space, let K be a pointed, closed, and convex cone, and let {An} be a sequence of nonempty subsets of X. Assume that

An is closed and star-shaped for all n;

the generalized domination property (GDP) holds for all An;

B=Li(kerAn) and eρ(An,A)→0 for all ρ>0;

for every x0∈kerA∩MinA, there exists a nondecreasing function δx0:[0,+∞)→[0,+∞) satisfying δx0(0)=0,δx0(t)>0 for all t>0 and
(47)12(x0+x)+Bδx0(∥x0-x∥)⊂A,∀x∈A.

Then
(48)B∩MinA⊂Li(MinAn). Proof.

From Proposition 8, B⊂kerA. By Theorem 21, B∩StMinA⊂Li(MinAn). By using assumption (iv), from Proposition 26, we have
(49)B∩StMinA=B∩kerA∩StMinA=B∩kerA∩MinA=B∩MinA⊂Li(MinAn).
The proof is complete.

Corollary 29.

Let X be a normed linear space, let K be a pointed, closed, and convex cone, and let {An} be a sequence of nonempty subsets of X. Assume that

An is closed and star-shaped for all n;

the generalized domination property (GDP) holds for all An;

B=Li(kerAn) and eρ(An,A)→0 for all ρ>0;

StMin
A∩B∩Bρ is relatively compact for all ρ>0;

for every x0∈kerA∩MinA, there exists a nondecreasing function δx0:[0,+∞)→[0,+∞) satisfying δx0(0)=0,δx0(t)>0 for all t>0 and
(50)12(x0+x)+Bδx0(∥x0-x∥)⊂A,∀x∈A.

Then, for each ρ>0,
(51)limn→∞eρ(B∩MinA,MinAn)=0.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).

BednarczukE. M.Hŏlder-like properties of minimal points in vector optimizationBednarczukE. M.Upper Hölder continuity of minimal pointsBednarczukE. M.A note on lower semicontinuity of minimal pointsLucD. T.LucchettiR.MalivertiC.Convergence of the efficient sets. Set convergence in nonlinear analysis and optimizationMiglierinaE.MolhoE.Convergence of minimal sets in convex vector optimizationSonntagY.ZalinescuC.Set convergences: a survey and a classificationTaninoT.Stability and sensitivity analysis in convex vector optimizationBednarczukE. M.Continuity of minimal points with applications to parametric multiple objective optimizationLucchettiR. E.MiglierinaE.Stability for convex vector optimization problemsGöpfertA.RiahiH.TammerC.ZălinescuC.LucD. T.CrespiG. P.GinchevI.RoccaM.Minty variational inequalities, increase-along-rays property and optimizationCrespiG. P.GinchevI.RoccaM.Existence of solutions and star-shapedness in Minty variational inequalitiesFangY.-P.HuangN.-J.Increasing-along-rays property, vector optimization and well-posednessShveidelA.Separability of star-shaped sets and its application to an optimization problemBednarczukE. M.LakshmikanthamV.Some stability results for vector optimization problems in partially ordered topological vector sapcesProceedings of the 1st World Congress of Nonlinear Analysis1996Berlin, GermanyWalter de Gruyter23712382AttouchH.RiahiH.Stability results for Ekeland's ε-variational principle and cone extremal solutions