Abstract

Some new properties are obtained for generalized second-order contingent (adjacent) epiderivatives of set-valued maps. By employing the generalized second-order adjacent epiderivatives, necessary and sufficient conditions of Benson proper efficient solutions are given for set-valued optimization problems. The results obtained improve the corresponding results in the literature.

1. Introduction

The investigation of the optimality conditions is one of the most attractive topics of optimization theory since many optimization problems encountered in economics, engineering, and other fields involve vector-valued maps (or set-valued maps) as constraints and objectives (see [1โ€“3]). With the concept of contingent derivative for a set-valued map (see [3]), Corley [4] investigated optimality conditions for general set-valued optimization problems. But it turns out that the optimality conditions do not coincide under standard assumptions. Jahn and Rauh [5] introduced the contingent epiderivative of a set-valued map and then obtained unified necessary and sufficient optimality conditions. The essential differences between the definitions of the contingent derivative and the contingent epiderivative are that the graph is replaced by the epigraph and the derivative is single-valued. But the existence of the contingent epiderivative of a set-valued map in a general setting is an open question. To overcome the difficulty, Chen and Jahn [6] introduced a generalized contingent epiderivative of a set-valued map which is a set-valued map. They proved that the generalized contingent epiderivative exists under standard assumptions and obtained a unified necessary and sufficient condition in terms of the generalized contingent epiderivative. As to other concepts of epiderivatives for set-valued maps and applications to optimality conditions, one can refer to [7โ€“11].

Since higher-order tangent sets introduced in [3], in general, are not cones and convex sets, there are some difficulties in studying higher-order optimality conditions and duality for general set-valued optimization problems. Until now, there are only a few papers to deal with higher-order optimality conditions and duality of set-valued optimization problems by virtue of the higher-order derivatives or epiderivatives introduced by the higher-order tangent sets. Jahn et al. [12] introduced second-order contingent epiderivative and generalized contingent epiderivative for a set-valued map and obtained some second-order optimality conditions based on these concepts. In [13], Li et al. studied some properties of higher-order tangent sets and higher-order derivatives introduced in [3] and then obtained higher-order Fritz John type necessary and sufficient optimality conditions for set-valued optimization problems in terms of the higher-order derivatives. By using these concepts, they also discussed higher-order Mond-Weir duality for a set-valued optimization problem in [14]. In general, since the epigraph of a set-valued map has nicer properties than the graph of a set-valued map, it is advantageous to employ the epiderivatives in set-valued optimization. Li and Chen [15] introduced the definitions of higher-order generalized contingent epiderivative and higher-order generalized adjacent epiderivative and obtained higher-order optimality conditions for Henig properly efficient solutions of a set-valued optimization problem with constraints based on the higher-order generalized adjacent epiderivative and contingent epiderivative. Chen et al. [16] introduced the concepts of higher-order weak contingent epiderivative and higher-order weak adjacent epiderivative for set-valued maps and obtained higher order Mond-Weir type duality, higher-order Wolfe type duality, and higher-order Kuhn-Tucker type optimality conditions to a constrained set-valued optimization problem based on the higher-order weak adjacent (contingent) epiderivatives and Henig efficiency. Since the existence of the higher-order contingent (adjacent) derivative and the higher-order generalized contingent (adjacent) epiderivative of a set-valued map in a general setting may not exist, Wang and Li [17] introduced the generalized higher-order contingent (adjacent) epiderivative of a set-valued map and obtained the optimality conditions for Henig efficient solutions to set-valued optimization problems. Wang et al. [18] introduced the generalized higher-order contingent (adjacent) derivatives of set-valued maps, and necessary and sufficient optimality conditions are obtained for weakly efficient solutions of set-valued optimization problems under no convexity assumptions.

To the best of our knowledge, by virtue of the second-order derivatives or epiderivatives introduced by the higher-order tangent sets, the second-order optimality conditions have not been studied for Benson proper efficient solutions of set-valued optimization problems until now. This paper aims to investigate the second-order optimality conditions for Benson proper efficient solutions of set-valued optimization problems by virtue of the generalized second-order epiderivatives.

The rest of the paper is organized as follows. In Section 2, we collect some of the concepts and introduce two kinds of set-valued optimization problem models. In Section 3, we recall the concepts and properties of generalized second-order contingent (adjacent) epiderivatives of set-valued maps and discuss some new properties of them. In Section 4, we establish second-order necessary and sufficient conditions for Benson proper efficient solutions to a set-valued optimization problem, whose constraint set is determined by a fixed set. In Section 5, we establish second-order Kuhn-Tuck type optimality conditions for Benson proper efficient solutions to a set-valued optimization problem, whose constraint set is determined by a set-valued map.

2. Preliminaries

Throughout this paper, let ๐‘‹,๐‘Œ, and ๐‘ be three real normed spaces, where the spaces ๐‘Œ and ๐‘ are partially ordered by nontrivial closed convex pointed cones ๐ถโŠ‚๐‘Œ and ๐ทโŠ‚๐‘ with int๐ทโ‰ โˆ…, respectively. Denote by ๐‘Œโˆ— the topological dual space of ๐‘Œ and by 0๐‘Œ the zero element in ๐‘Œ. The dual cone of ๐ถ is defined as๐ถ+=๎€ฝ๐‘“โˆˆ๐‘Œโˆ—๎€พโˆฃ๐‘“(๐‘)โ‰ฅ0,โˆ€๐‘โˆˆ๐ถ.(2.1) The quasi-interior ๐ถ+๐‘– of ๐ถ+ is the set๐ถ+๐‘–=๎€ฝ๐‘“โˆˆ๐‘Œโˆ—๎€ฝ0โˆฃ๐‘“(๐‘)>0,โˆ€๐‘โˆˆ๐ถโงต๐‘Œ๎€พ๎€พ.(2.2) Recall that a base of a cone ๐ถ is a convex subset ๐ต of ๐ถ such that0๐‘Œโˆ‰cl๐ต,๐ถ=cone๐ต.(2.3) Of course, ๐ถ is pointed whenever ๐ถ has a base. Furthermore, if ๐ถ is a nonempty closed convex pointed cone in ๐‘Œ, then ๐ถ+๐‘–โ‰ โˆ… if and only if ๐ถ has a base.

Let ๐นโˆถ๐ธโ†’2๐‘Œ and ๐บโˆถ๐ธโ†’2๐‘ be set-valued maps. The effective domain, the graph, and the epigraph of ๐น are defined by dom(๐น)={๐‘ฅโˆˆ๐ธโˆฃ๐น(๐‘ฅ)โ‰ โˆ…},gph(๐น)={(๐‘ฅ,๐‘ฆ)โˆˆ๐ธร—๐‘Œโˆฃ๐‘ฅโˆˆ๐ธ,๐‘ฆโˆˆ๐น(๐‘ฅ)} and epi(๐น)={(๐‘ฅ,๐‘ฆ)โˆˆ๐ธร—๐‘Œโˆฃ๐‘ฅโˆˆ๐ธ๐‘ฆโˆˆ๐น(๐‘ฅ)+๐ถ}, respectively. Denote โ‹ƒ๐น(๐ธ)=๐‘ฅโˆˆ๐ธ๐น(๐‘ฅ). Let (๐‘ฅ0,๐‘ฆ0)โˆˆgph(๐น),๐‘ง0โˆˆ๐บ(๐‘ฅ0)โˆฉ(โˆ’๐ท).

Definition 2.1 (see [19]). ๐น is called ๐ถ-convex on a nonempty convex subset ๐ธ, if, for any ๐‘ฅ1,๐‘ฅ2โˆˆ๐ธ and ๐œ†โˆˆ(0,1), such that ๐œ†๐น(๐‘ฅ1)+(1โˆ’๐œ†)๐น(๐‘ฅ2)โŠ†๐น(๐œ†๐‘ฅ1+(1โˆ’๐œ†)๐‘ฅ2)+๐ถ.

Definition 2.2 (see [20]). Let ๐นโˆถ๐ธโ†’2๐‘Œ be a set-valued map, ๐‘ฅ0โˆˆ๐ธ,(๐‘ฅ0,๐‘ฆ0)โˆˆgph(๐น). ๐น is said to be generalized ๐ถ-convex at (๐‘ฅ0,๐‘ฆ0) on ๐ธ, if cone(epi๐นโˆ’{(๐‘ฅ0,๐‘ฆ0)}) is convex.

Remark 2.3 (see [20]). If ๐น is ๐ถ-convex on convex set ๐ธ, then ๐น is generalized ๐ถ-convex at (๐‘ฅ0,๐‘ฆ0)โˆˆgph(๐น) on ๐ธ. But the converse may not hold.

Definition 2.4 (see [6]). Let ๐‘€ be a subset of ๐‘Œ. ๐‘ฆ0โˆˆ๐‘€ is said to be a Benson proper efficient point of ๐‘€ if (โˆ’๐ถ)โˆฉclcone(๐‘€+๐ถโˆ’{๐‘ฆ0})={0๐‘Œ}. The set of all Benson proper efficient points of ๐‘€ is denoted by ๐‘ƒmin[๐‘€,๐ถ].

Definition 2.5 (see [21]). Let ๐‘Œ be a real normed space, ๐ต be a base of ๐ถ, and ๐‘€โŠ‚๐‘Œ be a nonempty subset. ๐‘ฆโˆˆ๐‘€ is said to be a Henig efficient point of ๐‘€, written as ๐‘ฆโˆˆ๐ป๐ธ(๐‘€,๐ต), if for some ๐œ–โˆˆ(0,๐›ฟ), (๐‘€โˆ’{๐‘ฆ})โˆฉโˆ’int(cone(๐œ–๐‘ˆ+๐ต))=โˆ…,(2.4) where ๐›ฟ=inf{โ€–๐‘โ€–โˆถ๐‘โˆˆ๐ต} and ๐‘ˆ is the closed unit ball of ๐‘Œ.

Lemma 2.6 (see [22]). Let ๐‘ƒ and ๐ถ be two closed convex cones in a locally convex vector space, and let ๐ถ be pointed and have a compact base. If ๐‘ƒโˆฉ(โˆ’๐ถ)={0๐‘Œ}, then there exists โ„Žโˆˆ๐ถ+๐‘– such that โ„Žโˆˆ๐‘ƒ+.

Definition 2.7 (see [6]). A set-valued map ๐ปโˆถ๐‘‹โ†’2๐‘Œ is said to be (i)strictly positive homogeneous if ๐ป(๐›ผ๐‘ฅ)=๐›ผ๐ป(๐‘ฅ),โˆ€๐›ผ>0,โˆ€๐‘ฅโˆˆ๐‘‹,(2.5)(ii)subadditive if ๐ป๎€ท๐‘ฅ1๎€ธ๎€ท๐‘ฅ+๐ป2๎€ธ๎€ท๐‘ฅโŠ†๐ป1+๐‘ฅ2๎€ธ+๐ถ.(2.6)

In this paper, consider the following set-valued optimization problem:min๐น(๐‘ฅ),s.t.๐‘ฅโˆˆ๐ธ.(OP)

that is, to find a pair (๐‘ฅ0,๐‘ฆ0) with ๐‘ฅ0โˆˆ๐พ and ๐‘ฆ0โˆˆ๐น(๐‘ฅ0) is called a Benson proper efficient element of problem (OP) if ๐‘ฆ0โˆˆ๐‘ƒmin[๐น(๐ธ),๐ถ]. We also consider the following constraint set-valued optimization problem:min๐น(๐‘ฅ),s.t.๐บ(๐‘ฅ)โˆฉ(โˆ’๐ท)โ‰ โˆ…,๐‘ฅโˆˆ๐ธ.(SOP)

Set ๐พโˆถ={๐‘ฅโˆˆ๐ธโˆฃ๐บ(๐‘ฅ)โˆฉ(โˆ’๐ท)โ‰ โˆ…}. A pair (๐‘ฅ0,๐‘ฆ0) with ๐‘ฅ0โˆˆ๐พ and ๐‘ฆ0โˆˆ๐น(๐‘ฅ0) is called a Benson proper efficient element of problem (SOP) if ๐‘ฆ0โˆˆ๐‘ƒmin[๐น(๐พ),๐ถ].

3. Properties of Generalized Second-Order Contingent (Adjacent) Epiderivatives

In this section, we discuss some new properties of generalized second-order contingent and adjacent epiderivatives. Moreover, we give an example to show these properties. Let ๐น be a set-valued map from ๐ธโŠ†๐‘‹ to ๐‘Œ.

Definition 3.1 (see [17]). Let ๐น be a set-valued map from ๐ธโŠ†๐‘‹ to ๐‘Œ, (๐‘ฅ0,๐‘ฆ0)โˆˆgph(๐ป),๐‘ขโˆˆ๐‘‹,๐‘ฃโˆˆ๐‘Œ.(i)The generalized second-order contingent epiderivative ๐บ-๐ท(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ) of ๐น at (๐‘ฅ0,๐‘ฆ0) for vectors (๐‘ข,๐‘ฃ) is the set-valued map from ๐‘‹ to ๐‘Œ defined by ๎€ทgph๐บ-๐ท(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ๎€ธ๎€ธ=๐บ-๐‘‡(2)epi(๐น)๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ.,๐‘ข,๐‘ฃ(3.1)(ii)The generalized second-order adjacent epiderivative ๐บ-๐ทโ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ) of ๐น at (๐‘ฅ0,๐‘ฆ0) for vectors (๐‘ข,๐‘ฃ) is the set-valued map from ๐‘‹ to ๐‘Œ defined by ๎‚€gph๐บ-๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ๎‚,๐‘ข,๐‘ฃ=๐บ-๐‘‡โ™ญ(2)epi(๐น)๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ,๐‘ข,๐‘ฃ.(3.2)

Remark 3.2. If (๐‘ข,๐‘ฃ)=(0๐‘‹,0๐‘Œ), then ๐บ-๐ท(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ๎€ท๐‘ฅ,๐‘ข,๐‘ฃ=๐บ-๐ท๐น0,๐‘ฆ0๎€ธ,๐บ-๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ,๐‘ข,๐‘ฃ=๐บ-๐ทโ™ญ๐น๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ.(3.3)

From [17, Proposition 3.9 and Corollary 3.11], we have the following result.

Proposition 3.3. Let ๐‘ฅ,๐‘ฅ0โˆˆ๐ธ, ๐‘ฆ0โˆˆ๐น(๐‘ฅ0), (๐‘ข,๐‘ฃ)โˆˆ{0๐‘‹}ร—๐ถ. Then, (i)๐น(๐‘ฅ)+๐ถโˆ’{๐‘ฆ0}โŠ‚๐บ-๐ท(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ)(๐‘ฅโˆ’๐‘ฅ0); (ii)๐น(๐‘ฅ)+๐ถโˆ’{๐‘ฆ0}โŠ‚๐บ-๐ทโ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ)(๐‘ฅโˆ’x0).

Proposition 3.4. Let ๐ธ be a nonempty subset of ๐‘‹, ๐‘ฅ0โˆˆ๐ธ, ๐‘ฆ0โˆˆ๐น(๐‘ฅ0). Let ๐น be generalized ๐ถ-convex at (๐‘ฅ0,๐‘ฆ0) on ๐ธ, ๐‘ขโˆˆ๐ธ,๐‘ฃโˆˆ๐น(๐‘ข)+๐ถ. Then, ๎€ฝ๐‘ฆ๐น(๐‘ฅ)+๐ถโˆ’0๎€พโŠ‚๐บ-๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0,๐‘ขโˆ’๐‘ฅ0,๐‘ฃโˆ’y0๎€ธ๎€ท๐‘ฅโˆ’๐‘ฅ0๎€ธ,forany๐‘ฅโˆˆ๐ธ.(3.4)

Proof. Take a arbitrary sequence {โ„Ž๐‘›} with โ„Ž๐‘›โ†’0+, any ๐‘โˆˆ๐ถ, ๐‘ฅโˆˆ๐ธ and ๐‘ฆโˆˆ๐น(๐‘ฅ). Since ๐ป is generalized ๐ถ-convex at (๐‘ฅ0,๐‘ฆ0) on ๐ธ, cone(epi(๐น)โˆ’{(๐‘ฅ0,๐‘ฆ0)}) is convex, and then โ„Ž๐‘›๎€ท๐‘ขโˆ’๐‘ฅ0,๐‘ฃโˆ’๐‘ฆ0๎€ธ๎€ท๐‘ฅโˆˆconeepi๐นโˆ’๎€ฝ๎€ท0,๐‘ฆ0๎€ธ๎€พ๎€ธ.(3.5) It follows from โ„Ž๐‘›>0, and cone(epi๐นโˆ’{(๐‘ฅ0,๐‘ฆ0)}) is a convex cone that ๎€ท๐‘ฅ๐‘›,๐‘ฆ๐‘›๎€ธโˆถ=โ„Ž๐‘›๎€ท๐‘ขโˆ’๐‘ฅ0,๐‘ฃโˆ’๐‘ฆ0๎€ธ+โ„Ž2๐‘›๎€ท๐‘ฅโˆ’๐‘ฅ0,๐‘ฆ+๐‘โˆ’๐‘ฆ0๎€ธ๎€ท๐‘ฅโˆˆconeepi๐นโˆ’๎€ฝ๎€ท0,๐‘ฆ0.๎€ธ๎€พ๎€ธ(3.6) Then, ๎€ท๐‘ฅโˆ’๐‘ฅ0,๐‘ฆ+๐‘โˆ’๐‘ฆ0๎€ธ=๎€ท๐‘ฅ๐‘›,๐‘ฆ๐‘›๎€ธโˆ’โ„Ž๐‘›๎€ท๐‘ขโˆ’๐‘ฅ0,๐‘ฃโˆ’๐‘ฆ0๎€ธโ„Ž2๐‘›,(3.7) Thus, ๎€ฝ๐‘ฆ๐น(๐‘ฅ)+๐ถโˆ’0๎€พโŠ‚๐บ-๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0,๐‘ขโˆ’๐‘ฅ0,๐‘ฃโˆ’๐‘ฆ0๎€ธ๎€ท๐‘ฅโˆ’๐‘ฅ0๎€ธ,(3.8) and the proof of the proposition is complete.

Corollary 3.5. Let ๐ธ be a nonempty subset of ๐‘‹, ๐‘ฅ0โˆˆ๐ธ, ๐‘ฆ0โˆˆ๐น(๐‘ฅ0). If ๐น is generalized ๐ถ-convex at (๐‘ฅ0,๐‘ฆ0) on ๐ธ, ๐‘ขโˆˆ๐ธ,๐‘ฃโˆˆ๐น(๐‘ข)+๐ถ, then ๎€ฝ๐‘ฆ๐น(๐‘ฅ)+๐ถโˆ’0๎€พโŠ‚๐บ-๐ท(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0,๐‘ขโˆ’๐‘ฅ0,๐‘ฃโˆ’๐‘ฆ0๎€ธ๎€ท๐‘ฅโˆ’๐‘ฅ0๎€ธ,forany๐‘ฅโˆˆ๐ธ.(3.9)

Proposition 3.6. Let (๐‘ฅ0,๐‘ฆ0)โˆˆgph(๐น),(๐‘ข,๐‘ฃ)โˆˆ๐‘‹ร—๐‘Œ. Then, (i)๐บ-๐ทโ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ) is strictly positive homogeneous. Moreover, if ๐น is generalized ๐ถ-convex at (๐‘ฅ0,๐‘ฆ0) on ๐ธ, then (ii)๐บ-๐ทโ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ) is subadditive.

Proof. (i) Since ๐บ-๐‘‡โ™ญ(2)epi(๐น)๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ,๐‘ข,๐‘ฃ=๐‘‡โ™ญ(2)cone(epi(๐น)โˆ’{(๐‘ฅ0,๐‘ฆ0)})๎€ท0๐‘‹,0๐‘Œ๎€ธ,๐‘ข,๐‘ฃ=๐‘‡โ™ญcone(epi(๐น)โˆ’{(๐‘ฅ0,๐‘ฆ0)})(๐‘ข,๐‘ฃ),(3.10) โ€‰๐บ-๐ทโ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ) is strictly positive homogeneous.
(ii) Let ๐‘ฅ1,๐‘ฅ2โˆˆ๐‘‹, ๐‘ฆ1โˆˆ๐บ-๐ทโ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ)(๐‘ฅ1), ๐‘ฆ2โˆˆ๐บ-๐ทโ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ)(๐‘ฅ2). Then one has (๐‘ฅ1,๐‘ฆ1),(๐‘ฅ2,๐‘ฆ2)โˆˆ๐บ-๐‘‡โ™ญ(2)epi(๐น)(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ). Since ๐น is generalized ๐ถ-convex at (๐‘ฅ0,๐‘ฆ0) on ๐ธ, cone(epi๐นโˆ’{(๐‘ฅ0,๐‘ฆ0)}) is convex, and then it follows from the proof of [13, Proposition 3.2], that ๐บ-๐‘‡โ™ญ(2)epi(๐น)(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ) is convex. Thus, we have 12๎€ท๐‘ฅ1,๐‘ฆ1๎€ธ+12๎€ท๐‘ฅ2,๐‘ฆ2๎€ธ=12๎€ท๐‘ฅ1+๐‘ฅ2,๐‘ฆ1+๐‘ฆ2๎€ธโˆˆ๐บ-๐‘‡โ™ญ(2)epi(๐น)๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ,,๐‘ข,๐‘ฃ(3.11) and then it follows from (i) that ๐‘ฆ1+๐‘ฆ2โˆˆ๐บ-๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0๐‘ฅ,๐‘ข,๐‘ฃ๎€ธ๎€ท1+๐‘ฅ2๎€ธ,(3.12) which implies ๐บ-๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0๐‘ฅ,๐‘ข,๐‘ฃ๎€ธ๎€ท1๎€ธ+๐บ-๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0๐‘ฅ,๐‘ข,๐‘ฃ๎€ธ๎€ท2๎€ธโŠ†๐บ-๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0๐‘ฅ,๐‘ข,๐‘ฃ๎€ธ๎€ท1+๐‘ฅ2๎€ธ+๐ถ.(3.13) The proof of the proposition is complete.

For the sake of comparison, we recall some notions in [13, 15, 16].

Definition 3.7 (see [13]). The second-order adjacent derivative ๐ทโ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ) of ๐น at (๐‘ฅ0,๐‘ฆ0) for vector (๐‘ข,๐‘ฃ) is the set-valued map from ๐‘‹ to ๐‘Œ defined by ๎‚€๐ทgphโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ๎‚,๐‘ข,๐‘ฃ=๐‘‡โ™ญ(2)Graph(๐น)๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ,๐‘ข,๐‘ฃ.(3.14)

Definition 3.8 (see [15]). The second-order generalized adjacent epiderivative ๐ท๐‘”โ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ) of ๐น at (๐‘ฅ0,๐‘ฆ0) for vectors (๐‘ข,๐‘ฃ) is the set-valued map from ๐‘‹ to ๐‘Œ defined by ๐ท๐‘”โ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ,๐‘ข,๐‘ฃ(๐‘ฅ)=Min๐ถ๎‚†๐‘ฆโˆˆ๐‘Œโˆถ(๐‘ฅ,๐‘ฆ)โˆˆ๐‘‡โ™ญ(2)epi(๐น)๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ๎‚‡,๐‘ข,๐‘ฃ.(3.15)

Definition 3.9 (see [16]). The second-order weak adjacent epiderivative ๐ท๐‘”โ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ) of ๐น at (๐‘ฅ0,๐‘ฆ0) for vector (๐‘ข,๐‘ฃ) is the set-valued map from ๐‘‹ to ๐‘Œ defined by ๐ท๐‘คโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ,๐‘ข,๐‘ฃ(๐‘ฅ)=๐‘ŠMin๐ถ๎‚†๐‘ฆโˆˆ๐‘Œโˆถ(๐‘ฅ,๐‘ฆ)โˆˆ๐‘‡โ™ญ(2)epi(๐น)๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ๎‚‡,๐‘ข,๐‘ฃ.(3.16)

Remark 3.10. Notice that in Proposition 3.6, we establish a special property of ๐บ-๐ทโ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ), which is similar to the corresponding property of the generalized contingent epiderivative in [6, Theorem 1]. But we cannot obtain the similar property for ๐ทโ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ), ๐ท๐‘”โ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ), and ๐ท๐‘คโ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ), even though ๐น is a ๐ถ-convex map. The following example explains the case.

Example 3.11. Let ๐‘‹=๐‘…,๐‘Œ=๐‘…2,๐ถ=๐‘…2+, ๐น(๐‘ฅ)={(๐‘ฆ1,๐‘ฆ2)โˆˆ๐‘…2โˆฃ๐‘ฆ1โ‰ฅ๐‘ฅ,๐‘ฆ2โ‰ฅ๐‘ฅ2},โˆ€๐‘ฅโˆˆ๐‘‹. Take (๐‘ฅ0,๐‘ฆ0)=(0,(0,0))โˆˆgph(๐น) and (๐‘ข,๐‘ฃ)=(1,(1,0)). Then, ๐‘‡โ™ญ(2)gph(๐น)๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ,๐‘ข,๐‘ฃ=๐‘‡โ™ญ(2)epi(๐น)๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ=๎š,๐‘ข,๐‘ฃ๐‘ฅโˆˆ๐‘…[[({๐‘ฅ}ร—๐‘ฅ,+โˆž)ร—1,+โˆž)),๐บ-๐‘‡โ™ญ(2)epi(๐น)๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ=๎š,๐‘ข,๐‘ฃ๐‘ฅโˆˆ๐‘…๎€ท{[๐‘ฅ}ร—๐‘ฅ,+โˆž)ร—๐‘…+๎€ธ.(3.17) Therefore, for any ๐‘ฅโˆˆ๐‘‹, one has ๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ[[๐ท,๐‘ข,๐‘ฃ(๐‘ฅ)=๐‘ฅ,+โˆž)ร—1,+โˆž),๐‘”โ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ๐ท,๐‘ข,๐‘ฃ(๐‘ฅ)={(๐‘ฅ,1)},๐‘คโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ,๐‘ข,๐‘ฃ(๐‘ฅ)=๎€ฝ๎€ท๐‘ฅ,๐‘ฆ2๎€ธ๐‘ฆ2๎€พโˆช๐‘ฆโ‰ฅ1๎€ฝ๎€ท1๎€ธ,1โˆฃ๐‘ฆ1๎€พ,โ‰ฅ๐‘ฅ๐บ-๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ[,๐‘ข,๐‘ฃ(๐‘ฅ)=๐‘ฅ,+โˆž)ร—๐‘…+.(3.18) Naturally, ๐ทโ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ), ๐ท๐‘”โ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ), and ๐ท๐‘คโ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ) are not strictly positive homogeneous, but ๐บ-๐ทโ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ) is strictly positive homogeneous here.

It follows from [13, Corollary 3.1], and the proof of Proposition 3.6 that the following result holds.

Corollary 3.12. Let (๐‘ฅ0,๐‘ฆ0)โˆˆgph(๐น),(๐‘ข,๐‘ฃ)โˆˆ๐‘‹ร—๐‘Œ. Then (i)๐บ-๐ท(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ) is strictly positive homogeneous. Moreover, if ๐น is generalized ๐ถ-convex at (๐‘ฅ0,๐‘ฆ0) on ๐ธ, and (๐‘ข,๐‘ฃ)โˆˆcone(epi(๐ป)โˆ’{(๐‘ฅ0,๐‘ฆ0)}), then (ii)๐บ-๐ท(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ) is subadditive.

Remark 3.13. Under the assumption of cone-convex maps, Chen and Jahn [6, Theorem 1] obtained the subadditivity of ๐ท๐‘”๐น(๐‘ฅ0,๐‘ฆ0). However, it is under the assumption of generalized cone-convex maps that we obtain the subadditivity of ๐บ-๐ท(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ) in Corollary 3.12.

By Propositions 3.3โ€“3.6 and Corollary 3.12, we obtain the following results.

Corollary 3.14. Let (๐‘ฅ0,๐‘ฆ0)โˆˆgph(๐น),(๐‘ข,๐‘ฃ)โˆˆ๐‘‹ร—๐‘Œ. Then, ๎€ท0๐‘‹,0๐‘Œ๎€ธ๎€ทโˆˆgph๐บ-๐ท(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0,๎€ท0,๐‘ข,๐‘ฃ๎€ธ๎€ธ๐‘‹,0๐‘Œ๎€ธ๎‚€โˆˆgph๐บ-๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ๎‚.,๐‘ข,๐‘ฃ(3.19)

Corollary 3.15. Let (๐‘ฅ0,๐‘ฆ0)โˆˆgph(๐น) and (๐‘ข,๐‘ฃ)โˆˆ๐‘‹ร—๐‘Œ. If ๐น is generalized ๐ถ-convex at (๐‘ฅ0,๐‘ฆ0) on ๐ธ, then(i)gph(๐บ-๐ทโ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ)) is a closed convex cone. Moreover, if (๐‘ข,๐‘ฃ)โˆˆ๐‘๐‘œ๐‘›๐‘’(๐‘’๐‘๐‘–(๐ป)โˆ’{(๐‘ฅ0,๐‘ฆ0)}), then (ii)gph(๐บ-๐ท(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ)) is a closed convex cone.

4. Second-Order Optimality Conditions of (OP)

In this section, by employing the generalized second-order adjacent epiderivative, we will discuss the second-order optimality conditions for Benson proper efficient solutions of (OP).

Theorem 4.1. Let ๐‘ฅ0โˆˆ๐ธ,๐‘ฆ0โˆˆ๐น(๐‘ฅ0), and (๐‘ข,๐‘ฃ)โˆˆ๐‘‹ร—(โˆ’๐ถ). If (๐‘ฅ0,๐‘ฆ0) is a Benson proper efficient element of (OP), then, for every ๐‘ฅโˆˆฮฉโˆถ=dom[๐บ-๐ทโ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ)], one has ๐บ-๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ๎€ท๎€ฝ0,๐‘ข,๐‘ฃ(๐‘ฅ)โŠ‚๐‘Œโงตโˆ’๐ถโงต๐‘Œ.๎€พ๎€ธ(4.1)

Proof. Since (๐‘ฅ0,๐‘ฆ0) is a Benson proper efficient element of (OP), ๎€ท๐น๎€ฝ๐‘ฆclcone(๐ธ)+๐ถโˆ’0โˆฉ๎€ฝ0๎€พ๎€ธ(โˆ’๐ถ)=๐‘Œ๎€พ.(4.2)
Assume that there exists an ๐‘ฅโˆˆฮฉ such that (4.1) does not hold, that is, there exists ๐‘ฆโˆˆ๐บ-๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ๎€ธ๎€ท๐‘ฅ๎€ธโˆฉ๎€ท๎€ฝ0โˆ’๐ถโงต๐‘Œ,๎€พ๎€ธ(4.3) and then, by the definition of generalized second-order adjacent epiderivatives, for arbitrary sequence {โ„Ž๐‘›} with โ„Ž๐‘›โ†’0+, there exists a sequence ๐‘ฅ๎€ฝ๎€ท๐‘›,๐‘ฆ๐‘›๎€ท๐‘ฅ๎€ธ๎€พโˆˆconeepi๐นโˆ’๎€ฝ๎€ท0,๐‘ฆ0๎€ธ๎€พ๎€ธ(4.4) such that ๎€ท๐‘ฅ๐‘›,๐‘ฆ๐‘›๎€ธโˆ’โ„Ž๐‘›(๐‘ข,๐‘ฃ)โ„Ž2๐‘›โŸถ๎€ท๐‘ฅ,๐‘ฆ๎€ธ.(4.5)
Since ๐‘ฃโˆˆโˆ’๐ถ,โ„Ž๐‘›>0 and ๐ถ is a convex cone, โ„Ž๐‘›๐‘ฃโˆˆโˆ’๐ถ. Then, by (4.4), we get ๐‘ฆ๐‘›โˆ’โ„Ž๐‘›๎€ท๐น๎€ฝ๐‘ฆ๐‘ฃโˆˆcone(๐ธ)+๐ถโˆ’0๎€พ๎€ธ.(4.6) Thus, from (4.5) and (4.3), one obtains ๎€ท๐น๎€ฝ๐‘ฆ๐‘ฆโˆˆclcone(๐ธ)+๐ถโˆ’0โˆฉ๎€ท๎€ฝ0๎€พ๎€ธโˆ’๐ถโงต๐‘Œ๎€พ๎€ธ,(4.7) which contradicts (4.2). So (4.1) holds and the proof is complete.

Theorem 4.2. Let (๐‘ข,๐‘ฃ)โˆˆ๐‘‹ร—(โˆ’๐ถ), ๐‘ฅ0โˆˆ๐ธ,๐‘ฆ0โˆˆ๐น(๐‘ฅ0) and let ๐ถ be locally compact. Suppose that the following conditions are satisfied: (i)๐น is generalized ๐ถ-convex at (๐‘ฅ0,๐‘ฆ0) on ๐ธ; (ii)the pair (๐‘ฅ0,๐‘ฆ0) is a Benson proper efficient element of (OP).
Then, there exists ๐œ™โˆˆ๐ถ+๐‘– such that โŽงโŽชโŽจโŽชโŽฉ๎šinf๐œ™(๐‘ฆ)โˆฃ๐‘ฆโˆˆ๐‘ฅโˆˆฮฉ๐บ-๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ(โŽซโŽชโŽฌโŽชโŽญ,๐‘ข,๐‘ฃ๐‘ฅ)=0,(4.8) where ฮฉโˆถ=dom[๐บโˆ’๐ทโ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ)].

Proof. Define ๎š๐‘€=๐‘ฅโˆˆฮฉ๐บ-๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ,๐‘ข,๐‘ฃ(๐‘ฅ).(4.9)
By the similar line of proof for convexity of ๐‘€ in [15], Theorem 5.1, we obtain that ๐‘€ is a convex set. It follows from Theorem 4.1, that ๎€ฝ0๐‘€โˆฉ(โˆ’๐ถ)=๐‘Œ๎€พ.(4.10)
Thus, by Lemma 2.6, there exists ๐œ™โˆˆ๐ถ+๐‘– such that ๐œ™โˆˆ๐‘€+. So, we have ๐œ™(๐‘ฆ)โ‰ฅ0,โˆ€๐‘ฆโˆˆ๐‘€.(4.11) It follows from Corollary 3.15 that 0๐‘Œโˆˆ๐‘€, so โŽงโŽชโŽจโŽชโŽฉ๎šinf๐œ™(๐‘ฆ)โˆฃ๐‘ฆโˆˆ๐‘ฅโˆˆฮฉ๐บ-๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0๎€ธ(โŽซโŽชโŽฌโŽชโŽญ,๐‘ข,๐‘ฃ๐‘ฅ)=0,(4.12) and the proof of the theorem is complete.

Theorem 4.3. Let ๐‘ฅ0โˆˆ๐ธ,๐‘ฆ0โˆˆ๐น(๐‘ฅ0) and (๐‘ข,๐‘ฃ)โˆˆ{0๐‘‹}ร—๐ถ. If there exists ๐œ™โˆˆ๐ถ+๐‘– such that ๐œ™(๐‘ฆ)โ‰ฅ0,โˆ€๐‘ฆโˆˆ๐บ-๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ๎€ธ๎€ท๐‘ฅโˆ’๐‘ฅ0๎€ธ,๐‘ฅโˆˆ๐ธ,(4.13) then the pair (๐‘ฅ0,๐‘ฆ0) is a Benson proper efficient solution of (OP).

Proof. It follows from Proposition 3.3 that ๎€ฝ๐‘ฆ๐น(๐‘ฅ)+๐ถโˆ’0๎€พโŠ‚๐บ-๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ๎€ธ๎€ท๐‘ฅโˆ’๐‘ฅ0๎€ธ,(4.14) for all ๐‘ฅโˆˆ๐ธ,๐‘ฆโˆˆ๐น(๐‘ฅ). Then, by (4.13), (4.14), and ๐œ™โˆˆ๐ถ+๐‘–, we get ๐œ™๎€ท๐‘ฆ(๐‘ฆ)โ‰ฅ๐œ™0๎€ธ,โˆ€๐‘ฆโˆˆ๐น(๐ธ).(4.15)
Thus, by the sufficient condition of [23, Theorem 4.1], we get that (๐‘ฅ0,๐‘ฆ0) is a Benson proper efficient solution of (OP), and the proof of the theorem is complete.

Remark 4.4. In Theorem 4.3, no assumption of generalized convexity is imposed.

Remark 4.5. If we use ๐ทโ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ) instead of ๐บ-๐ทโ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ) in Theorem 4.3, then the corresponding result for ๐ทโ™ญ(2)๐น(๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ) may not hold. The following example explains the case.

Example 4.6. Suppose that ๐‘‹=๐‘Œ=๐ธ=๐‘…, ๐ถ=๐‘…+. Let ๐นโˆถ๐ธโ†’2๐‘Œ be a set-valued map with โŽงโŽชโŽจโŽชโŽฉ๎‚†1๐น(๐‘ฅ)=๐‘ฆโˆˆ๐‘…โˆถ๐‘ฆโ‰ฅ|๐‘ฅ|+1,๐‘ฅโ‰ค2๎‚‡,๎‚†1๐‘ฆโˆˆ๐‘…โˆถ๐‘ฆโ‰ฅโˆ’๐‘ฅ+1,๐‘ฅ>2๎‚‡.(4.16) Consider the following constrained set-valued optimization problem (4.1): min๐น(๐‘ฅ),s.t.๐‘ฅโˆˆ๐ธ.(4.17) Take (๐‘ฅ0,๐‘ฆ0)=(0,1)โˆˆgph(๐น),(๐‘ข,๐‘ฃ)=(0,0)โˆˆ{0}ร—๐ถ. We have ๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ๎€ธ๎€ท๐‘ฅโˆ’๐‘ฅ0๎€ธ={๐‘ฆโˆˆ๐‘…โˆถ๐‘ฆโ‰ฅ|๐‘ฅ|},โˆ€๐‘ฅโˆˆ๐ธ.(4.18) Then, for any ๐œ™โˆˆ๐ถ+๐‘–, we have ๐œ™(๐‘ฆ)โ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0,๐‘ข,๐‘ฃ๎€ธ๎€ท๐‘ฅโˆ’๐‘ฅ0๎€ธ,๐‘ฅโˆˆ๐ธ.(4.19) Naturally, the pair (๐‘ฅ0,๐‘ฆ0) is not a Benson proper efficient solution of (4.1).

From Proposition 3.4, we know that the following theorem holds.

Theorem 4.7. Let ๐‘ฅ0โˆˆ๐ธ,๐‘ฆ0โˆˆ๐น(๐‘ฅ0). Suppose that the following conditions are satisfied: (i)๐‘ขโˆˆ๐ธ,๐‘ฃโˆˆ๐น(๐‘ข)+๐ถ, (ii)๐น is generalized ๐ถ-convex at (๐‘ฅ0,๐‘ฆ0) on ๐ธ, (iii)there exists ๐œ™โˆˆ๐ถ+๐‘– such that ๐œ™(๐‘ฆ)โ‰ฅ0,โˆ€๐‘ฆโˆˆ๐บ-๐ทโ™ญ(2)๐น๎€ท๐‘ฅ0,๐‘ฆ0,๐‘ขโˆ’๐‘ฅ0,๐‘ฃโˆ’๐‘ฆ0๎€ธ๎€ท๐‘ฅโˆ’๐‘ฅ0๎€ธ,๐‘ฅโˆˆ๐ธ.(4.20)
Then the pair (๐‘ฅ0,๐‘ฆ0) is a Benson proper efficient solution of (OP).

5. Second-Order Kuhn-Tuck Type Optimality Conditions of (SOP)

In this section, by employing the generalized second-order adjacent epiderivative, we will discuss the second-order Kuhn-Tuck type optimality conditions for Benson proper efficient solutions of (SOP). The notation (๐น,๐บ)(๐‘ฅ) is used to denote ๐น(๐‘ฅ)ร—๐บ(๐‘ฅ). Let ๐‘ฅ0โˆˆ๐พ,๐‘ฆ0โˆˆ๐น(๐‘ฅ0). Firstly, we recall a result in [24].

The interior tangent cone of ๐พ at ๐‘ฅ0 is defined as IT๐พ๎€ท๐‘ฅ0๎€ธ=๎€ฝ๐‘ขโˆˆ๐‘‹โˆƒ๐œ†>0,โˆ€๐‘กโˆˆ(0,๐œ†),โˆ€๐‘ข๎…žโˆˆ๐ต๐‘‹(๐‘ข,๐œ†),๐‘ฅ0+๐‘ก๐‘ข๎…ž๎€พโˆˆ๐พ,(5.1) where ๐ต๐‘‹(๐‘ข,๐œ†) stands for the closed ball centered at ๐‘ขโˆˆ๐‘‹ and of radius ๐œ†.

Lemma 5.1 (see [24]). If ๐พโŠ‚๐‘‹ is convex, ๐‘ฅ0โˆˆ๐พ and int๐พโ‰ โˆ…, then ITint๐พ๎€ท๐‘ฅ0๎€ธ๎€ท๎€ฝ๐‘ฅ=intcone๐พโˆ’0๎€พ๎€ธ.(5.2)

Theorem 5.2. Let (๐‘ข,๐‘ฃ,๐‘ค)โˆˆ๐‘‹ร—(โˆ’๐ถ)ร—(โˆ’๐ท), ๐‘ง0โˆˆ๐บ(๐‘ฅ0)โˆฉ(โˆ’๐ท). If (๐‘ฅ0,๐‘ฆ0) is a Benson proper efficient element of (๐ถ๐‘ƒ), then ๎‚ƒ๐บ-๐ทโ™ญ(2)๎€ท๐‘ฅ(๐น,๐บ)0,๐‘ฆ0,๐‘ง0,๐‘ข,๐‘ฃ,๐‘ค1+๐‘ง0๎€ธ๎‚„๎€ทโˆ’๎€ฝ0(๐‘ฅ)โŠ‚(๐‘Œร—๐‘)โงต๎€ท๎€ท๐ถโงต๐‘Œร—๎€ท๎€พ๎€ธint๐ท+๐‘ง0๎€ธ๎€ธ๎€ธ,(5.3) for all ๐‘ฅโˆˆฮฉโˆถ=dom[๐บ-๐ทโ™ญ(2)(๐น,๐บ)(๐‘ฅ0,๐‘ฆ0,๐‘ง0,๐‘ข,๐‘ฃ,๐‘ค1+๐‘ง0)].

Proof. Since (๐‘ฅ0,๐‘ฆ0) is a Benson proper efficient element of (๐ถ๐‘ƒ), ๎€ท๐น๎€ฝ๐‘ฆclcone(๐พ)+๐ถโˆ’0๎€ฝ0๎€พ๎€ธโˆฉโˆ’๐ถ=๐‘Œ๎€พ.(5.4)
Suppose that there exists some ๐‘ฅโˆˆฮฉ such that (5.3) does not hold, that is, there exists some ๎€ท๐‘ฆ,๐‘ง๎€ธโˆˆ๐บ-๐ทโ™ญ(2)๎€ท๐‘ฅ(๐น,๐บ)0,๐‘ฆ0,๐‘ง0,๐‘ข,๐‘ฃ,๐‘ค+๐‘ง0๎€ธ๎€ท๐‘ฅ๎€ธ๎€ท,(5.5)๐‘ฆ,๐‘ง๎€ธ๎€ฝ0โˆˆโˆ’๎€ท๎€ท๐ถโงต๐‘Œร—๎€ท๎€ฝ๐‘ง๎€พ๎€ธint๐ท+0๎€พ๎€ธ๎€ธ.(5.6) It follows from (5.5) and the definition of generalized second-order contingent epiderivatives that there exist sequences {โ„Ž๐‘›} with โ„Ž๐‘›โ†’0+ and ๐‘ฅ๎€ฝ๎€ท๐‘›,๐‘ฆ๐‘›,๐‘ง๐‘›๎€ท๐‘ฅ๎€ธ๎€พโˆˆconeepi(๐น,๐บ)โˆ’๎€ฝ๎€ท0,๐‘ฆ0,๐‘ง0๎€ธ๎€พ๎€ธ(5.7) such that ๎€ท๐‘ฅ๐‘›,๐‘ฆ๐‘›,๐‘ง๐‘›๎€ธโˆ’โ„Ž๐‘›๎€ท๐‘ข,๐‘ฃ,๐‘ค+๐‘ง0๎€ธโ„Ž2๐‘›โŸถ๎€ท๐‘ฅ,๐‘ฆ,๐‘ง๎€ธ.(5.8)
From (5.6) and (5.8), there exists a sufficiently large natural number ๐‘1 such that ๐‘ง๐‘›๐‘งโˆถ=๐‘›โˆ’โ„Ž๐‘›๎€ท๐‘ค+๐‘ง0๎€ธโ„Ž2๐‘›=1โ„Ž๐‘›๎‚ต๐‘ง๐‘›โˆ’โ„Ž๐‘›๐‘คโ„Ž๐‘›โˆ’๐‘ง0๎‚ถ๎€ทโˆˆโˆ’int๐ท+๐‘ง0๎€ธ๎€ท๎€ฝ๐‘งโŠ‚โˆ’intcone๐ท+0๎€พ๎€ธ,โˆ€๐‘›>๐‘1.(5.9)
According to (5.9) and Lemma 5.1, we obtain โˆ’๐‘งโˆˆITint๐ท(โˆ’๐‘ง0). Then, it follows from the definitions of ITint๐ท(โˆ’๐‘ง0) that โˆƒ๐œ†>0, โˆ€๐‘กโˆˆ(0,๐œ†),โˆ€๐‘ข๎…žโˆˆ๐ต๐‘(โˆ’๐‘ง,๐œ†),โˆ’๐‘ง0+๐‘ก๐‘ข๎…žโˆˆint๐ท. Since โ„Ž๐‘›โ†’0+ and (5.9), there exists a sufficiently large natural number ๐‘2 such that โ„Ž๐‘›โˆˆ(0,๐œ†),for ๐‘›>๐‘2 and โˆ’๐‘ง0+โ„Ž๐‘›(โˆ’๐‘ง๐‘›)โˆˆint๐ท, for ๐‘›>๐‘2, that is, ๐‘ง๐‘›โˆ’โ„Ž๐‘›๐‘คโ„Ž๐‘›โˆˆโˆ’int๐ท,for๐‘›>๐‘2.(5.10) It follows from โ„Ž๐‘›>0,๐‘คโˆˆโˆ’๐ท and ๐ท is a convex cone that ๐‘ง๐‘›โˆˆโˆ’int๐ท,for๐‘›>๐‘2.(5.11) Then from (5.7), ๐‘ง๐‘›โˆˆcone(๐บ(๐‘ฅ๐‘›)+๐ทโˆ’{๐‘ง0}), there exist ๐œ†๐‘›>0, ฬƒ๐‘ฅ๐‘›โˆˆ๐ธ, ฬƒ๐‘ง๐‘›โˆˆ๐บ(ฬƒ๐‘ฅ๐‘›), ฬƒ๐‘ฆ๐‘›โˆˆ๐น(ฬƒ๐‘ฅ๐‘›), ๐‘๐‘›โˆˆ๐ถ and ๐‘‘๐‘›โˆˆ๐ท such that ๐‘ฆ๐‘›=๐œ†๐‘›(ฬƒ๐‘ฆ๐‘›+๐‘๐‘›โˆ’๐‘ฆ0)) and ๐‘ง๐‘›=๐œ†๐‘›(ฬƒ๐‘ง๐‘›+๐‘‘๐‘›โˆ’๐‘ง0)). It follows from (5.11) that ฬƒ๐‘ง๐‘›โˆˆ๐บ(ฬƒ๐‘ฅ๐‘›)โˆฉ(โˆ’๐ท), for ๐‘›>๐‘2, and then ฬƒ๐‘ฅ๐‘›โˆˆ๐พ,for๐‘›>๐‘2.(5.12)
Since ๐‘ฃโˆˆโˆ’๐ถ,โ„Ž๐‘›>0 and ๐ถ is a convex cone, โ„Ž๐‘›๐‘ฃโˆˆโˆ’๐ถ. Then by (5.7), (5.12), and (5.8), we get ๎€ท๐น๎€ฝ๐‘ฆ๐‘ฆโˆˆclcone(๐พ)+๐ถโˆ’0๎€ฝ๐‘๎€พ๎€ธ,for๐‘›>max1,๐‘2๎€พ.(5.13) Then it follows from (5.6) that ๎€ท๐น๎€ฝ๐‘ฆ๐‘ฆโˆˆclcone(๐พ)+๐ถโˆ’0๎€ท๎€ฝ0๎€พ๎€ธโˆฉโˆ’๐ถโงต๐‘Œ๎€ฝ๐‘๎€พ๎€ธ,for๐‘›>max1,๐‘2๎€พ,(5.14) which contradicts (5.4). Thus, (5.3) holds and the proof of the theorem is complete.

Theorem 5.3. Let (๐‘ข,๐‘ฃ,๐‘ค)โˆˆ๐‘‹ร—(โˆ’๐ถ)ร—(โˆ’๐ท), ๐‘ฅ0โˆˆ๐ธ,(๐‘ฆ0,๐‘ง0)โˆˆ(๐น,๐บ)(๐‘ฅ0). Suppose that the following conditions are satisfied: (i)๐ถ has a compact base; (ii)(๐น,๐บ) is generalized ๐ถร—๐ท-convex at (๐‘ฅ0,๐‘ฆ0,๐‘ง0) on ๐ธ; (iii)the pair (๐‘ฅ0,๐‘ฆ0) is a Benson proper efficient element of (SOP).
Then, there exist ๐œ™โˆˆ๐ถ+๐‘– and ๐œ“โˆˆ๐ทโˆ— such that ๎€ท๐‘งinf{๐œ™(๐‘ฆ)+๐œ“(๐‘ง)โˆฃ(๐‘ฆ,๐‘ง)โˆˆฮ”}=0,๐œ“0๎€ธ=0,(5.15) where โ‹ƒฮ”โˆถ=๐‘ฅโˆˆฮฉ๐บ-๐ทโ™ญ(2)(๐น,๐บ)(๐‘ฅ0,๐‘ฆ0,๐‘ง0,๐‘ข,๐‘ฃ,๐‘ค+๐‘ง0)(๐‘ฅ) and ฮฉโˆถ=dom[๐บ-๐ทโ™ญ(2)(๐น,๐บ)(๐‘ฅ0,๐‘ฆ0,๐‘ง0,๐‘ข,๐‘ฃ,๐‘ค+๐‘ง0)].

Proof. Define ๎š๐‘€=๐‘ฅโˆˆฮฉ๎‚ƒ๐บ-๐ทโ™ญ(2)๎€ท๐‘ฅ(๐น,๐บ)0,๐‘ฆ0,๐‘ง0,๐‘ข,๐‘ฃ,๐‘ค+๐‘ง0๎€ธ๎€ท0(๐‘ฅ)+๐‘Œ,๐‘ง0๎€ธ๎‚„.(5.16)
By the similar line of proof for convexity of ๐‘€ in [15, Theorem 5.1], we obtain that ๐‘€ is a convex set. It follows from Theorem 5.2 that ๎€ทโˆ’๎€ฝ0๐‘€โˆฉ๎€ท๎€ท๐ถโงต๐‘Œ๎€พ๎€ธร—int๐ท๎€ธ๎€ธ=โˆ….(5.17) Thus, by employing Corollary 3.15, it follows from a standard separation theorem of convex sets and the similar proof method of [8, Theorem 1] that there exist ๐œ™โˆˆ๐ถ+๐‘– and ๐œ“โˆˆ๐ทโˆ— such that ๎€ท๐‘งinf{๐œ™(๐‘ฆ)+๐œ“(๐‘ง)โˆฃ(๐‘ฆ,๐‘ง)โˆˆฮ”}=0,๐œ“0๎€ธ=0.(5.18) The proof is complete.

Remark 5.4. It follows from Remarks 2.3 and 3.2 that the necessary optimality condition in Theorem 5.3 is obtained under weaker assumptions than those assumed of [8, Theorem 1].

Now we give an example to illustrate the necessary optimality conditions for generalized second-order contingent epiderivatives.

Example 5.5. Suppose that ๐‘‹=๐‘Œ=๐‘=๐ธ=๐‘…, ๐ถ=๐ท=๐‘…+. Let ๐นโˆถ๐ธโ†’2๐‘Œ be a set-valued map with ๎€ฝ๐น(๐‘ฅ)=๐‘ฆโˆˆ๐‘…โˆฃ๐‘ฆโ‰ฅ๐‘ฅ2/3๎€พ,๐‘ฅโˆˆ๐ธ,(5.19) and ๐บโˆถ๐ธโ†’๐‘ be a set-valued map with ๐บ(๐‘ฅ)={๐‘งโˆˆ๐‘…โˆฃ๐‘งโ‰ฅโˆ’๐‘ฅ},๐‘ฅโˆˆ๐ธ.(5.20)
Consider the following constrained set-valued optimization problem (4.2): min๐น(๐‘ฅ),s.t.๐‘ฅโˆˆ๐ธ,๐บ(๐‘ฅ)โˆฉ(โˆ’๐ท)โ‰ โˆ….(5.21) Take (๐‘ฅ0,๐‘ฆ0,๐‘ง0)=(0,0,0)โˆˆgph(๐น,๐บ) and (๐‘ข,๐‘ฃ,๐‘ค)=(1,0,โˆ’1)โˆˆ๐‘‹ร—(โˆ’๐ถ)ร—(โˆ’๐ท). Naturally, (๐น,๐บ) is generalized ๐ถร—๐ท-convex at (๐‘ฅ0,๐‘ฆ0,๐‘ง0) on ๐ธ, and (๐‘ฅ0,๐‘ฆ0) is a Benson proper efficient solution of (4.2). By directly calculation, we have ๐บ-๐ทโ™ญ(2)๎€ท๐‘ฅ(๐น,๐บ)0,๐‘ฆ0,๐‘ง0,๐‘ข1,๐‘ฃ1,๐‘ค1+๐‘ง0๎€ธ๎€ฝ(๐‘ฅ)=(๐‘ฆ,๐‘ง)โˆˆ๐‘…2๎€พโˆถ๐‘ฆโ‰ฅ0,๐‘งโ‰ฅโˆ’๐‘ฅ,โˆ€๐‘ฅโˆˆ๐ธ.(5.22) Take ๐œ™=1โˆˆ๐ถ+๐‘– and ๐œ‘=0โˆˆ๐ทโˆ—. Naturally, necessary optimality condition of Theorem 5.3 holds here.

Theorem 5.6. Let ๐‘ฅ0โˆˆ๐ธ,(๐‘ฆ0,๐‘ง0)โˆˆ(๐น,๐บ)(๐‘ฅ0). Suppose that the following conditions are satisfied: (i)(๐‘ข,๐‘ฃ,๐‘ค)โˆˆ{0๐‘‹}ร—๐ถร—๐ท, (ii) there exist ๐œ™โˆˆ๐ถ+๐‘– and ๐œ“โˆˆ๐ทโˆ— such that ๎€ท๐‘งinf{๐œ™(๐‘ฆ)+๐œ“(๐‘ง)โˆฃ(๐‘ฆ,๐‘ง)โˆˆฮ˜}=0,๐œ“0๎€ธ=0,(5.23) where โ‹ƒฮ˜โˆถ=๐‘ฅโˆˆ๐พ๐บ-๐ทโ™ญ(2)(๐น,๐บ)(๐‘ฅ0,๐‘ฆ0,๐‘ง0,๐‘ข,๐‘ฃ,๐‘ค)(๐‘ฅโˆ’๐‘ฅ0).
Then, the pair (๐‘ฅ0,๐‘ฆ0) is a Benson proper efficient solution of (SOP).

Proof. It follows from Proposition 3.3 that ๎€ท๐‘ฆโˆ’๐‘ฆ0,๐‘งโˆ’๐‘ง0๎€ธโˆˆ๐บ-๐ทโ™ญ(2)๎€ท๐‘ฅ(๐น,๐บ)0,๐‘ฆ0,๐‘ง0,๐‘ข,๐‘ฃ,๐‘ค๎€ธ๎€ท๐‘ฅโˆ’๐‘ฅ0๎€ธ,(5.24) for all ๐‘ฆโˆˆ๐น(๐‘ฅ),๐‘งโˆˆ๐บ(๐‘ฅ),๐‘ฅโˆˆ๐พ. Then, by assumption (ii), we have ๐œ™๎€ท๐‘ฆโˆ’๐‘ฆ0๎€ธ๎€ท+๐œ“๐‘งโˆ’๐‘ง0๎€ธโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐น(๐พ),๐‘งโˆˆ๐บ(๐พ).(5.25) Thus, there exists a ๐‘งโˆˆ๐บ(๐พ) with ๐‘งโˆˆโˆ’๐ท such that ๐œ“(๐‘ง)โ‰ค0. It follows from ๐œ“(๐‘ง0)=0 and (5.25) that ๐œ™๎€ท๐‘ฆ(๐‘ฆ)โ‰ฅ๐œ™0๎€ธ,โˆ€๐‘ฆโˆˆ๐น(๐พ).(5.26) It follows from the sufficient condition of [23, Theorem 4.1], that (๐‘ฅ0,๐‘ฆ0) is a Benson proper efficient solution of (SOP) and the proof of the theorem is complete.

Remark 5.7. Since Theorem 5.6 does not involve the assumption of convexity, it improves and generalizes [8, Theorem 2].

From Proposition 3.4 and the proof of Theorem 5.6, we easily obtain that the following Theorem holds.

Theorem 5.8. Let ๐‘ฅ0โˆˆ๐ธ,(๐‘ฆ0,๐‘ง0)โˆˆ(๐น,๐บ)(๐‘ฅ0). Suppose that the following conditions are satisfied: (i)๐‘ขโˆˆ๐พ,๐‘ฃโˆˆ๐น(๐‘ข)+๐ถ,๐‘คโˆˆ๐บ(๐‘ข)+๐ท, (ii)(๐น,๐บ) is generalized ๐ถร—๐ท-convex at (๐‘ฅ0,๐‘ฆ0,๐‘ง0) on ๐ธ;(iii) there exist ๐œ™โˆˆ๐ถ+๐‘– and ๐œ“โˆˆ๐ทโˆ— such that inf(๐‘ฆ,๐‘ง)โˆˆฮ˜๎€ท๐‘ง{๐œ™(๐‘ฆ)+๐œ“(๐‘ง)}=0,๐œ“0๎€ธ=0,(5.27) where โ‹ƒฮ˜โˆถ=๐‘ฅโˆˆ๐พ๐บ-๐ทโ™ญ(2)(๐น,๐บ)(๐‘ฅ0,๐‘ฆ0,๐‘ง0,๐‘ขโˆ’๐‘ฅ0,๐‘ฃโˆ’๐‘ฆ0,๐‘คโˆ’๐‘ง0)(๐‘ฅโˆ’๐‘ฅ0).
Then, the pair (๐‘ฅ0,๐‘ฆ0) is a Benson proper efficient solution of (SOP).

Acknowledgments

The authors would like to thank anonymous referees for their valuable comments and suggestions, which helped to improve the paper. This research was partially supported by the National Natural Science Foundation of China (10901004, 11171362, and 11071267), Natural Science Foundation Project of CQ CSTC (cstcjjA00019), and Science and Technology Research Project of Chong Qing Municipal Education Commission (KJ100419).