Characterizations of Benson proper efficiency of set-valued optimization in real linear spaces

Hongwei Liang; Zhongping Wan

doi:10.1515/math-2019-0104

Article Open Access

Characterizations of Benson proper efficiency of set-valued optimization in real linear spaces

Hongwei Liang and Zhongping Wan

Published/Copyright: October 18, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

A new class of generalized convex set-valued maps termed relatively solid generalized cone-subconvexlike maps is introduced in real linear spaces not equipped with any topology. This class is a generalization of generalized cone-subconvexlike maps and relatively solid cone-subconvexlike maps. Necessary and sufficient conditions for Benson proper efficiency of set-valued optimization problem are established by means of scalarization, Lagrange multipliers, saddle points and duality. The results generalize and improve some corresponding ones in the literature. Some examples are afforded to illustrate our results.

Keywords: set-valued maps; relative algebraic interior; vector closure; Benson proper efficiency; generalized cone subconvexlikeness

MSC 2010: 90C26; 90C30; 90C46

1 Introduction

It is well known that the notion of (weak) efficiency plays a very critical role in vector optimization problem. Since the range of the set of (weakly) efficient solutions is often too large, various concepts of proper efficiency have been proposed in the literature [1, 2, 3, 4]. By virtue of weak or proper efficiency, some scholars have involved many research areas of set-valued optimization problems [5, 6, 7, 8, 9, 10, 11, 12]. Meanwhile, in the past few years, the authors [13, 14, 15, 16, 17, 18] introduced generalized convexity of set-valued maps to characterize weak efficiency and proper efficiency. The usual framework of the above papers is that of topological linear spaces or locally convex spaces. As we know, the linear spaces are wider than the topological spaces or locally convex spaces, how to extend the concepts of weak or proper efficiency and generalized convexity from the topological spaces or locally convex spaces to the linear spaces is a meaningful topic.

Through the concept of algebraic interior, Li [19] defined the notions of weakly efficient solutions of vector optimization and cone subconvexlike set-valued maps in real linear spaces. Some optimality conditions had been obtained. To avoid the algebraic interior is empty, in real linear spaces, by virtue of relative algebraic interior and vector closure, Adán and Novo [20, 21, 22] also defined weakly efficient solutions, some kinds of properly efficient solutions of vector optimization problems and generalized convexlikeness of vector-valued maps in real linear spaces. Scalarization, multiplier rules, and saddle-point theorems had been also obtained. Adán and Novo [23] established Lagrangian type duality theorems and saddle-points theorems for weak and Benson proper efficient solutions of vector optimization problems with generalized convexity introduced in [22].

Under the assumption of generalized cone subconvexlikeness refering to relative algebraic interior, Hernández et al. [24] and Zhou et al. [25] obtained Fritz-John and Kuhn-Tucker optimality conditions for weakly efficient solutions of set-valued optimization problem in linear spaces. Some approximate solution concepts had been generalized to the algebraic setting and their characterizations had been derived via linear scalarizations (see [26, 27]), Lagrangian optimality conditions and saddle point theorems (see [28]).

It is well-known that many maps are not generalized convex set-valued maps introduced in [19, 21, 22, 24, 25], the reason is that the conditions for these generalized convex maps are too strong. Hence, in this work, we introduce a more general generalized convex set-valued maps and extend scalarization theorems and Lagrange multiplier theorems under this generalized convexity hypotheses in real linear spaces. Meanwhile, to our knowledge, up to now there are few works focus on Benson proper saddle points and duality of set-valued optimization problems in linear spaces. In this paper, we further investigate Benson proper efficiency of the set-valued optimization problem in terms of saddle points theorems and duality theorems in linear spaces. Moreover, the duality theorems is important because of the several applications to, for instance, population dynamics as dynamics vaccination. Results in this line had been derived in [14, 22, 23, 24, 29].

The remaining of this article is organized as follows. Section 2 provide some basic definitions we need in this paper. In Section 3, a new class generalized cone subconvexlike of set-valued maps is introduced in real linear spaces. We state the major results for the characterizations of Benson proper efficiency of the set-valued optimization problem in Sections 4-7.

2 Preliminaries

Throughout this paper, we always assume that X, Y and Z are three real linear spaces. Let 0 denote the zero element of every space. Let K ⊂ Y be nonempty. We denote by

cone(K):={λk|k∈K, λ≥0},aff(K):={∑i=1nλiki:∀i∈{1,…,n}, ki∈K, ∑i=1nλi=1},

the generated cone, affine hull of K, respectively. K is called a cone if λ K ⊆ K for any λ ≥ 0. A cone K is convex if K + K ⊆ K. A cone K is called pointed if K ∩ (−K) = {0}. A cone K ⊆ Y is called nontrivial if {0} ≠ K ≠ Y. λ₁ k₁ + λ₂ k₂ ∈ K, ∀ λ₁, λ₂ ≥ 0, ∀ k₁, k₂ ∈ K Let Y^∗ and Z^∗ stand for the algebraic dual spaces of Y and Z, respectively. From now on, let C and D be nontrivial pointed convex cones in Y and Z, respectively. The algebraic dual cone C⁺ and strictly algebraic dual cone C⁺ⁱ of C are defined as

C+:={y∗∈Y∗∣y∗(y)≥0, ∀y∈C}

and

C+i:={y∗∈Y∗∣y∗(y)>0, ∀y∈C∖{0}}.

The meaning of D⁺ is similar to that of C⁺.

In the sequel, Y is assumed to be ordered through the following quasi order:

y1,y2∈Y, y1≤Cy2⟺y2−y1∈C.

Definition 2.1

[30] Let K be a nonempty subset in Y. The algebraic interior of K is the set

cor(K):={k∈K∣∀v∈Y,∃λ′>0,∀λ∈[0,λ′], s.t. k+λv∈K}.

We say that K is algebraic open, if cor(K) = K.

Definition 2.2

[31] Let K be a nonempty subset in Y. The relative algebraic interior of K is the set

icr(K):={k∈K∣∀v∈aff(K)−k,∃λ′>0,∀λ∈[0,λ′], s.t. k+λv∈K}.

We say that K is solid (relatively solid) if cor(K) ≠ ∅ (icr(K) ≠ ∅). Clearly, cor(K) ⊆ icr(K), if cor K ≠ ∅, then cor(K) = icr(K) i.e., if K is solid, then K is relatively solid.

Remark 2.1

Let C be a nontrivial pointed convex cones in Y. Then, ∀ φ ∈ C⁺ⁱ, φ(y) > 0, ∀ y ∈ icr(C).

Proof

Since C be a nontrivial pointed cone in Y, by Remark 2.3 of [25], 0 ∉ icr(C). Hence, icr(C) ⊂ C ∖ {0}. It follows from the Definition of C⁺ⁱ that the conclusion holds.

Definition 2.3

[21] Let K be a nonempty subset in Y. The vector closure of K is the set

vcl(K):={k∈Y∣∃v∈Y,∀λ′>0,∃λ∈[0,λ′], s.t. k+λv∈K}={k∈Y∣∃v∈Y, ∃λn→0+, s.t. k+λnv∈K, ∀n∈N}.

The set K is called vectorially closed (v-closed) if K = vcl (K).

Remark 2.2

Shi [31] (p17) introduced a concept that equivalent to Definition 2.3 called algebraic closure and denoted by K^c.

Lemma 2.1

[31] Let K ⊂ Y is convex. Then K^c is convex, i.e., vcl (K) is convex.

Lemma 2.2

[31] (Strong Separation Theorem) Let Y be a linear space and S, T ⊂ Y be two nonempty sets with S − T is a convex set in Y. If icr(S − T) ≠ ∅ and 0 ∉ (S − T)^c, then there exists y^∗ ∈ Y^∗ ∖ {0} such that sup_t∈T y^∗(t) < inf_s∈S y^∗(s).

Lemma 2.3

[20] Let K₁ and K₂ be two nontrivial convex cones in Y₁ and Y₂, respectively. If icr(K₁) ≠ ∅ and icr(K₂) ≠ ∅, then icr(K₁ × K₂) = icr(K₁) × icr(K₂).

The following lemma was stated in [23, Propositions 5(iii) and 6(i)].

Lemma 2.4

[21] Let K ⊂ Y and C ⊂ Y be a nontrivial convex cone. Then,

vcl(cone (K) + C) = vcl (cone(K + C)).
If C is relatively solid, then vcl(K + C) = vcl(K + icr (C)).

Lemma 2.5

[23] Let a ∈ Y and C be a relatively solid convex cone of Y. Then {a} + icr (C) = icr({a} + icr (C)).

Lemma 2.6

[22] If C ⊂ Y is a convex cone, then C is relatively solid if and only if C⁺ is relatively solid.

The following cone separation theorem is due to Adán and Novo [24, Theorem 2.2]. By Lemma 2.6, the assumption on the relative solidness of K supposed in [24, Theorem 2.2] can be removed.

Lemma 2.7

[22] (Separation Theorem) Let M, K be two v-closed convex cones in Y. Let M be relatively solid and K⁺ be solid. If M ∩ K = {0}, then there exists a linear functional l ∈ Y^∗ ∖ {0} such that ∀ k ∈ K, m ∈ M,

l(k)≥0≥l(m),

and furthermore, ∀ k ∈ K ∖ {0}, l(k) > 0.

Lemma 2.8

[24] Let K be a relatively solid convex subset of Y and φ : Y → Z be a linear map. Then φ (icr(K)) = icr(φ(K)).

Lemma 2.9

[24] Let K be a relatively solid convex subset of Y. Then icr(K) ⊂ icr(cone(K)).

Definition 2.4

[24] Let B ⊂ Y, y ∈ B be called a Benson proper minimal point of B (denoted by y ∈ PMin(B, C)) if vcl(cone(B − y + C)) ∩ (−C) = {0}.

By Definition 2.4, we can get the following Definition.

Definition 2.5

Let B ⊂ Y, y ∈ B be called a Benson proper maximal point of B (denoted by y ∈ PMax(B, C)) if vcl(cone(B − y − C)) ∩ C = {0}.

3 Generalized cone subconvexlike set-valued maps

In this section, we introduce some classes of generalized convex set-valued maps and discuss their relationships. From now on, suppose that A is a nonempty subset of X and F : A ⇉ Y is a set-valued map on A. Let F(A) = ⋃_x∈A F(x).

Definition 3.1

[24] Let icr(C) ≠ ∅. The set-valued map F : A ⇉ Y is said to be C-subconvexlike on A iff F(A) + icr (C) is a convex set in Y.

Definition 3.2

[24] Let icr(C) ≠ ∅. The set-valued map F : A ⇉ Y is said to be relatively solid C-subconvexlike on A iff F is C-subconvexlike on A and F(A) + icr(C) is relatively solid.

Definition 3.3

[25] Let icr(C) ≠ ∅. The set-valued map F : A ⇉ Y is said to be generalized C-subconvexlike on A iff cone (F(A)) + icr (C) is a convex set in Y.

Lemma 3.1

[25] If the set-valued map F : A ⇉ Y is C-subconvexlike on A, then F is generalized C-subconvexlike on A.

By Definition 3.2 and Definition 3.3, we give the following definition.

Definition 3.4

Let icr(C) ≠ ∅. The set-valued map F : A ⇉ Y is said to be relatively solid generalized C-subconvexlike on A, if F is generalized C-subconvexlike on A and cone (F(A)) + icr (C) is relatively solid.

Proposition 3.1

If the set-valued map F : A ⇉ Y is relatively solid C-subconvexlike on A, then F is relatively solid generalized C-subconvexlike on A.

Proof

From Lemma 3.1, F is generalized C-subconvexlike on A is valid. We need to prove the following expression holds:

icr[cone(F(A))+icr(C)]≠∅. (3.1)

Since F(A) + icr (C) is convex and icr (F(A) + icr (C)) ≠ ∅. It follows from Lemma 2.9 that cone (F(A) + icr (C)) is convex and icr [cone (F(A) + icr (C))] ≠ ∅. Due to Proposition 4(i) of [21], one obtains

icr[vcl(cone(F(A)+icr(C)))]≠∅.

As cone (F(A) + icr (C)) = cone (F(A) + C), then,

icr[vcl(cone(F(A)+C))]≠∅. (3.2)

On the other hand, again, by Proposition 4(i) of [21], we get

icr[cone(F(A))+icr(C)]=icr[vcl(cone(F(A))+icr(C))]. (3.3)

By Lemma 2.4, we obtain

vcl(cone(F(A))+icr(C))=vcl(cone(F(A))+C))=vcl(cone(F(A)+C)). (3.4)

It follows from (3.3) and (3.4) that

icr[cone(F(A))+icr(C)]=icr[vcl(cone(F(A)+C))],

which together with (3.2), (3.1) holds.

Remark 3.1

The example below illustrates relatively solid generalized C-subconvexlike set-valued maps not necessarily imply relatively solid C-subconvexlike. Thus, the relatively solid generalized C-subconvexlike maps is more extensive than relatively solid C-subconvexlikeness. So, in this paper, it makes more sense for us to study relevant issues in virtue of relatively solid generalized C-subconvexlikeness.

Example 3.1

Let X = Y = ℝ², A = {(2, 0), (1, 1)} and C = {(y₁, 0) | y₁ ≥ 0}. Consider the set-valued map F: A ⇉ Y defined by

F(2,0)={y1,y2)∣32≤y2≤−y1+3, y1≥0},F(1,1)={y1,y2)∣0≤y2≤−y1+3, y1≥32}.

Obviously, cone (F(A)) + icr (C) is a convex set in Y and icr(cone (F(A)) + icr (C)) = {y₁, y₂) | y₁ > 0, y₂ > 0} ≠ ∅, i.e., F is relatively solid generalized C-subconvexlike on A. However, F(A) + icr (C) is not a convex set in Y, so F is not C-subconvexlike on A. By definition 3.3, F is not relatively solid C-subconvexlike on A.

Consider the following two systems:

System 1. There exists x₀ ∈ A such that F(x₀) ∩ (−icr (C)) ≠ ∅.

System 2. There exists φ ∈ C⁺ ∖ {0} such that φ(y) ≥ 0, ∀ y ∈ F(x).

Lemma 3.2

[25] Let C be pointed convex cone with nonempty relative algebraic interior in Y.

uppose that F is relatively solid generalized C-subconvexlike on A. If System 1 has no solutions, then System 2 has a solution.
If φ ∈ C⁺ⁱ is a solution of System 2, then System 1 has no solutions.

4 Scalarization

Now, we consider the following unconstrainted vector optimization problem with set-valued maps:

(UVP) Min F(x)s.t. x∈A

and its scalar problem

(UVP)φ Min(φ∘F)(x) s.t. x∈A,

where φ ∈ Y^∗ ∖ {0_Y^∗}.

Definition 4.1

A point x̄ ∈ A is called Benson proper efficient solution of (UVP) if F(x̄) ∩ PMin (F(A), C) ≠ ∅. A pair (x̄, ȳ) is called a Benson proper efficient element of (UVP) if ȳ ∈ F(x̄) ∩ PMin (F(A), C).

Definition 4.2

x̄ ∈ A is called a minimal solution of (UVP)_φ if there exists ȳ ∈ F(x̄) such that

φ(y¯)≤φ(y), ∀y∈F(x).

The pair (x̄, ȳ) is called a minimizer element of (UVP)_φ.

Theorem 4.1

Let φ ∈ C⁺ⁱ. If (x̄, ȳ) is a minimizer element of (UVP)_φ, then (x̄, ȳ) is a Benson proper efficient element of (UVP).

Proof

Let c ∈ vcl(cone(F(A) − ȳ + C)) ∩ (−C). Then, c ∈ vcl(cone(F(A) − ȳ+C)). By Definition 2.3, there exist v ∈ Y and a sequence λ_n → 0⁺, such that c + λ_n v ∈ cone(F(A) − ȳ + C) for all n ∈ ℕ. Therefore, there exist sequences {μ_n} ⊂ ℝ₊, {c_n} ⊂ C and {y_n} ⊂ F(A) such that

c+λnv=μn(yn+cn−y¯).

Hence,

φ(c)+λnφ(v)=μn(φ(yn)+φ(cn)−φ(y¯)). (4.1)

As (x̄, ȳ) is a minimizer element of (UVP)_φ and φ ∈ C⁺ⁱ, we have φ (y_n) ≥ φ (ȳ) and φ (c_n) ≥ 0 for all n ∈ ℕ. Therefore, the right-hand side of (4.1) is nonnegative, and so

φ(c)+λnφ(v)≥0. (4.2)

Taking λ_n → 0⁺ in (4.2), we have

φ(c)≥0.

On the other hand, since c ∈ − C, we obtain

φ(c)≤0.

Thus,

φ(c)=0,

then c = 0 and

vcl(cone(F(S)−y¯+C))∩(−C)=0,

which together with ȳ ∈ F(x̄), it results

y¯∈F(x¯)∩PMin(F(A),C).

Hence, (x̄, ȳ) is a Benson proper efficient element of (UVP).

Remark 4.1

The proof of Theorem 4.1 is different from Theorem 4.1 of [22] and Theorem 5.3 of [24].

Theorem 4.2

Let C be v-closed and C⁺ be solid. Let x̄ ∈ A and ȳ ∈ F(x̄). Suppose that the following conditions hold:

(x̄, ȳ) is a Benson proper efficient element of (UVP);
F − ȳ is relatively solid generalized C-subconvexlike on A.

Then, there exists φ ∈ C⁺ⁱ such that (x̄, ȳ) is a minimizer element of (UVP)_φ.

Proof

By condition (i). Then,

vcl(cone(F(A)−y¯+C))∩(−C)={0}. (4.3)

It follows from condition (ii) that cone(F(A) − ȳ) + icr (C) is a relatively solid, convex set in Y. Then, according to Proposition 3(iii)-(iv) and Proposition 4(i) in [21], we conclude that vcl (cone(F(A) − ȳ) + icr (C)) is a relatively solid, v-closed, convex set in Y. Furthermore, it follows from (i) and (ii) of Lemma 2.4 that

vcl(cone(F(A)−y¯)+icr(C))=vcl(cone(F(A)−y¯)+C)=vcl(cone(F(A)−y¯+C)).

Hence, by Lemma 2.5, there exists a linear functional φ ∈ C⁺ⁱ such that

φ(y)≥0, ∀y∈vcl(cone(F(A)−y¯+C)).

Clearly, F(A) − ȳ + C ⊂ vcl (cone (F(A) − ȳ+ C)), then

φ(y)−φ(y¯)+φ(c)≥0, ∀y∈F(A), ∀c∈C. (4.4)

Taking c = 0 in (4.4), this yields

φ(y)≥φ(y¯), ∀y∈f(A).

Thus, (x̄, ȳ) is a minimizer element of (UVP)_φ.

Remark 4.2

Theorem 4.2 extends Theorem 4.2 of [22] and Theorem 5.4 of [24].

Corollary 4.1

Let C be v-closed and C⁺ be solid, x̄ ∈ A and ȳ ∈ F(x̄). Let F − ȳ be relatively solid generalized C-subconvexlike on A. Then (x̄, ȳ) is a Benson proper efficient element of (UVP) if and only if (x̄, ȳ) is a minimizer element of (UVP)_φ for φ ∈ C⁺ⁱ.

5 Lagrange multiplier rules

Let F : A ⇉ Y and G : A ⇉ Z be two set-valued maps on A. We suppose that icr (C) × icr (D) ≠ ∅.

(CVP) Min F(x) s.t. G(x)∩(−D)≠∅, x∈A.

Denote the feasible solution set of (CVP) by

S={x∈A∣G(x)∩(−D)≠∅}

and write F(S) = ⋃_x∈S F(x).

Definition 5.1

A point x̄ is called Benson proper efficient solution of (CVP) if x̄ ∈ S and F(x̄) ∩ PMin (F(S), C) = ∅. A pair (x̄, ȳ) is called a Benson proper efficient element of (CVP) if ȳ ∈ F(x̄) ∩ PMin (F(S), C).

We denote by L(Z, Y) the set of all linear operators from Z to Y and by L⁺(Z, Y) the set of positive linear operators defined as

L+(Z,Y)={T∈L(Z,Y)∣T(D)⊂C}.

Denote by (F, G) the set-valued map from A to Y × Z defined by (F, G)(x) = F(x) × G(x). where Y × Z is an linear space with nontrivial pointed convex cone C × D.

By Definition 3.4, (F, G)(x) is relatively solid generalized C × D-subconvexlike on A if cone ((F, G)(A)) + icr (C × D) is a relatively solid convex set in Y × Z.

The Lagrangian set-valued map of (CVP) is defined by

L(x,T)=F(x)+T(G(x)), ∀(x,T)∈A×L+(Z,Y).

Consider the following unconstrained set-valued optimization problem:

(UVP)T Min L(x,T) s.t. (x,T)∈A×L+(Z,Y).

Proposition 5.1

Let (F, G) : A ⇉ Y × Z be generalized C × D-subconvexlike on A.

If φ ∈ C⁺, then (φ ∘ F, G) is generalized ℝ₊ × D-subconvexlike on A.
If T ∈ L⁺(Z, Y), then F + T ∘ G is generalized K-subconvexlike on A.

Furthermore, if (F, G) is relatively solid, then (φ ∘ F, G) is relatively solid in part (i) and F + T ∘ G is relatively solid in part (ii).

Proof

Denote by i the identity map in Y. A linear map (φ, i) : Y × Z → ℝ × Z be defined by (φ, i)(y, z) = (φ(y), z). Because cone ((F, G)(A)) + icr (C × D) is a convex set in Y × Z. By the linearity of φ, the image set (φ, i)(cone ((F, G)(A)) + icr (C × D)) is convex. Because of Lemma 2.3 and Lemma 2.8, then,

(φ,i)(cone((F,G)(A))+icr(C×D))=cone((φ∘F,G)(A))+φ(icr(C))×icr(D)=cone((φ∘F,G)(A))+icr(φ(C))×icr(D)=cone((φ∘F,G)(A))+icr(R+)×icr(D)=cone((φ∘F,G)(A))+icr(R+×D).

As a consequence, cone ((φ ∘ F, G)(A)) + icr (ℝ₊ × D) is convex, i.e., (φ ∘ F, G) is generalized ℝ₊ × D-subconvexlike on A.
Define (i, T) : Y × Z → Y by (i, T)(y, z) = y + T(z). Similar to (i), (i, T)(cone ((F, G)(A)) + icr (C × D)) is convex. Moreover,

(i,T)(cone((F,G)(A)))+icr(C×D))=(i,T)cone((F,G)(A))+(i,T)(icr(C)×icr(D))=cone(F(A)+T(G(A)))+icr(C)+T(icr(D))=cone(F(A)+T(G(A)))+icr(C).

Therefore, F + T ∘ G is generalized C-subconvexlike on A.

Finally, since icr[cone ((F, G)(A)) + icr (C × D)] ≠ ∅, then,

∅≠(φ,i)[icr(cone((F,G)(A)+icr(C×D))]=icr[cone((φ∘F,G)(A))+φ(icr(C))×icr(D)]=icr[cone((φ∘F,G)(A))+icr(φ(C))×icr(D)]=icr[cone((φ∘F,G)(A))+icr(R+)×icr(D)]=icr[cone((φ∘F,G)(A))+icr(R+×D)].

Thus, (φ ∘ F, G) is relatively solid. By the proof of above, it is easy to cheak that F + T ∘ G is relatively solid.

Definition 5.2

We say that the optimization problem (CVP) satisfies the generalized Slater constraint qualification if there exists x ∈ S such that G(x) ∩ − icr (D) ≠ ∅.

Theorem 5.1

Let C is v-closed and C⁺ is solid. Let x̄ ∈ S and ȳ ∈ F(x̄). Assume that the following conditions hold:

(x̄, ȳ) is a Benson proper efficient element of (CVP) and F − ȳ is relatively solid generalized C-subconvexlike on S;
(F − ȳ, G) is relatively solid generalized C × D-subconvexlike on A;
(CVP) satisfies the generalized Slater constraint condition;
icr (D) ⊂ icr[cone (G(A)) + icr (D)].

Then, there exists T ∈ L⁺(Z, Y) such that 0 ∈ T(G(x̄) ∩ − D) and (x̄, ȳ) is a Benson proper efficient element of the unconstrained problem (UVP)_T.

Proof

By assumption (i), we can apply Theorem 4.2 to problem (CVP), then there exists φ ∈ C⁺ⁱ such that

φ(y)≥φ(y¯), ∀y∈f(S). (5.1)

Define set-valued map I : A ⇉ ℝ × Z by

I(x):=(φ∘(F−y¯),G)(x)=[φ(F(x)−y¯)]×G(x)=φ(F(x))×G(x)−(φ(y¯),0).

By assumption (ii) and Proposition 5.1(ii), we deduce that I(x) is relatively solid generalized ℝ₊ × D-subconvexlike on A. Together with (5.1), it is easy to check that

I(A)∩−icr(R+×D)≠∅. (5.2)

By Lemma 3.2, there exists (r, ψ) ∈ ℝ₊ × D⁺ ∖ {(0, 0)} such that

r(φ(F(x)−y¯))+ψ(G(x))≥0, ∀x∈A, (5.3)

and

(r,ψ)(y1,y2)>0, ∀(y1,y2)∈icr[cone((φ∘(F−y¯),G)(A))+icr(R+×D)]. (5.4)

We note that r > 0. Otherwise, if r = 0, in this case, (5.3) can be written as

ψ(G(x))≥0, ∀x∈A, (5.5)

and (5.4) becomes

ψ(icr[cone(G(A))+icr(D)])>0. (5.6)

It follows from assumption (iv) and (5.6) that

ψ(icr(D))>0. (5.7)

By assumption (iii), there exists x₀ ∈ A, such that G(x₀) ∈ − icr (D), by (5.7), ψ(G(x₀)) < 0. Which contradicts (5.5). So r > 0.

Since x̄ ∈ S and ψ ∈ D⁺, there exists z̄ ∈ G(x̄) ∩ − D such that ψ(z̄) ≤ 0. Taking x = x̄, ȳ ∈ F(x̄) and z̄ ∈ G(x̄) in (5.3), we obtain ψ(z̄) ≥ 0, thus, ψ(z̄) = 0. Hence,

0∈ψ(G(x¯)∩−D). (5.8)

From r > 0 and φ ∈ C⁺ⁱ, we can select c ∈ C such that

rφ(c)=1. (5.9)

Define a linear operator T : Z → Y as

T(z)=ψ(z)c. (5.10)

Obviously, T(D) ⊂ C, i.e., T ∈ L⁺(Z, Y) and by (5.8), we have

0∈T(G(x¯)∩−D). (5.11)

From (5.3), (5.9) and (5.10), we obtain

rφ(F(x)+T(G(x)))=rφ(F(x))+ψ(G(x))rφ(c)=rφ(F(x))+ψ(G(x))≥rφ(y¯), ∀x∈A.

Dividing the above inequality by r > 0 leads to

φ(y¯)≤φ(F(x)+T(G(x)).

Since ȳ ∈ F(x̄) ⊂ F(x̄) + T(G(x̄)) = L(x̄, T), it means that (x̄, ȳ) is a minimizer element of the following scalar optimization problem

Min φ(L(x,T)), s.t. x∈A.

According to Theorem 4.1, (x̄, ȳ) is a Benson proper minimizer efficient of (UVP)_T.

Remark 5.1

Theorem 5.1 extends Theorem 4.3 of [22] and Theorem 5.5 of [24]. In Theorem 5.1, the condition (iv) icr (D) ⊂ icr[cone (G(A)) + icr (D)] can be replaced by the condition aff (D) = aff[cone (G(A)) + icr (D)]. Actually, if aff (D) = aff[cone (G(A)) + icr (D)], icr (D) ⊂ icr[cone (G(A)) + icr (D)]. However, the example below indicates that the condition icr (D) ⊂ icr[cone (G(A)) + icr (D)] is weaker than aff (D) = aff[cone (G(A)) + icr (D)].

Example 5.1

Let X = Z = ℝ², D = {(y₁, 0) | y₁ ≥ 0} and A = {(x₁, x₂) | − 2 ≤ x₁ + x₂ ≤ 3}. The set-valued map F: A ⇉ Y is defined as follows:

F(x1,x2)={(x1,x2)}, ∀(x1,x2)∈A.

Clearly, icr (D) ⊂ icr(cone (F(A)) + icr (D)). However, aff (D) = {(y₁, 0) | y₁ ∈ ℝ} ≠ ℝ² = aff(cone (F(A)) + icr (D)).

Analogously to Theorem 5.2 in [14], we can obtain the following theorem.

Theorem 5.2

If there exists a pair (x̄, ȳ) with x̄ ∈ S and ȳ ∈ F(x̄) and a positive linear operator T̄ ∈ L⁺(Z, Y) such that:

0 ∈ T̄(G(x̄) ∩ − D);
(x̄, ȳ) is a Benson proper efficient element of (UVP)_T̄.

Then (x̄, ȳ) is a Benson proper efficient element of (CVP).

6 Benson proper saddle points

Definition 6.1

A pair (x̄, T̄) ∈ A × L⁺(Z, Y) is said to be a Benson proper saddle point of L(x, T) if

L(x¯,T¯)∩PMinL(A,T¯),C∩PMaxL(x¯,L+),C≠∅,

where L(A, T̄) = ⋃_x∈AL(x, T̄) and L(x̄, L⁺) = ⋃_{T∈L⁺(Z,Y)} L(x̄, T).

In the following, we will present an important equivalent characterization for Benson proper saddle point of the Lagrangian map L(x, T) by virtue of a strong separation theorem.

Theorem 6.1

Let D be v-closed. Then, (x̄, T̄) ∈ A × L⁺(Z, Y) is a Benson proper saddle point of L(x, T) if and only if there exist ȳ ∈ F(x̄) and z̄ ∈ G(x̄) such that

ȳ ∈ PMin (L(A, T̄), C);
G(x̄) ⊂ − D;
T̄(z̄) = 0;
ȳ ∈ PMax(F(x̄), C).

Proof

Necessity. Since (x̄, T̄) is a Benson proper saddle point of L(x, T), there exist ȳ ∈ F(x̄), z̄ ∈ G(x̄) such that

y¯+T¯(z¯)∈PMinL(A,T¯),C, (6.1)

and

y¯+T¯(z¯)∈PMaxL(x¯,L+),C. (6.2)

From (6.2), it follows that

vcl[cone(L(x¯,L+)−C−(y¯+T¯(z¯)))]∩C={0}. (6.3)

Now, for any T ∈ L⁺(Z, Y),

T(z¯)−T¯(z¯)=(y¯+T(z¯))−(y¯+T¯(z¯))∈F(x¯)+T(G(x¯))−(y¯+T¯(z¯))=L(x¯,T)−(y¯+T¯(z¯)).

As a result,

⋃T∈L+(Z,Y)T(z¯)−C−T¯(z¯)⊂L(x¯,L+)−C−(y¯+T¯(z¯)).

Hence, by (6.3), one has

vcl(cone(⋃T∈L+(Z,Y)T(z¯)−C−T¯(z¯)))∩C={0}. T∈L+(Z,Y). (6.4)

Define a map f : L⁺(Z, Y) → Y as

f(T)=−T(z¯).

Then, (6.4) is equivalent to

vcl(cone(f(L+)+C−f(T¯))∩(−C)={0}.

This implies that T̄ ∈ L⁺(Z, Y) is an Benson proper efficient solution of the following vector optimization problem

Min f(T)s.t. T∈L+(Z,Y).

Since f is a linear map, clearly, f is relatively solid generalized C-subconvexlike on L⁺(Z, Y). Thus, by Theorem 4.2, there exists φ ∈ C⁺ⁱ such that

φ(−T¯(z¯))=φ(f(T¯))≤φ(f(T))=φ(−T(z¯)),∀T∈L+(Z,Y),

that is

φ(T¯(z¯))≥φ(T(z¯)), ∀T∈L+(Z,Y). (6.5)

We affirm that − z̄ ∈ D. Otherwise, 0 ∉ D + z̄. Since D is v-closed, vcl(D + z̄) = D + z̄. Thus, 0 ∉ vcl(D + z̄). From Lemma 2.5, icr(D + z̄) = icr(D)+z̄ ≠ ∅. By Lemma 2.5, there exists z^∗ ∈ Z^∗ ∖ {0} such that

z∗(λd)>z∗(−z¯) ∀λ≥0, ∀d∈D. (6.6)

Taking λ = 0 in (6.6), we obtain

z∗(z¯)>0.

By (6.6), we have

z∗(d)>1λz∗(−z¯), ∀λ>0, ∀d∈D. (6.7)

Letting λ → +∞ gives

z∗(d)>0, ∀d∈D.

Thus, z^∗ ∈ D⁺ ∖ {0}. Choose c₀ ∈ icr(C) be fixed, and define the map T₀ : Z → Y as

T0(z)=z∗(z)z∗(z¯)c0+T(z¯), ∀z∈Z.

Obviously, T₀ ∈ L⁺(Z, Y) and

T0(z¯)−T(z¯)=c0, ∀z∈Z.

It follows from φ ∈ C⁺ⁱ and Remark 2.1 that

φ(T0(z¯))−φ(T(z¯))=φ(c0)>0,

which contradicts (6.5) and so − z̄ ∈ D. Thus, − T̄(z̄) ∈ C. If T̄(z̄) ≠ 0, then − T̄(z̄) ∈ C ∖ {0}, and so

φ(T¯(z¯))<0. (6.8)

Taking T = 0 ∈ L⁺(Z, Y) in (6.5) gives φ(T̄(z̄)) ≥ 0. Which conflicts with (6.8). So condition (iii) holds. Together with (6.1), condition (i) holds.

Now we show G(x̄) ⊂ − D. If not, there exist z₁ ∈ G(x̄) such that − z₁ ∉ D. Analogously to the above proof of −z̄ ∈ D, there exists z1∗ ∈ D⁺ ∖ {0} such that z1∗ (z₁) > 0. Choose c₁ ∈ icr(C) be fixed and define the map T₁ : Z → Y as follows

T1(z)=z1∗(z)c1.

Clearly, T₁ ∈ L⁺(Z, Y) and

T1(z1)=z1∗(z1)c1∈icr(C)⊂C∖{0}. (6.9)

On the other hand, from condition (iii) and (6.2), we have

y¯∈PMaxL(x¯,L+),C⊂MaxL(x¯,L+),C,

i.e.,

y−y¯∉C∖{0}, ∀y∈⋃T∈L+(Z,Y)L(x¯,T).

y¯+T1(z1)∈F(x¯)+T1[G(x¯)]=L(x¯,T1)⊂⋃T∈L+(Z,Y)L(x¯,T).

Therefore,

T1(z1)=y¯+T1(z1)−y¯∉C∖{0}.

Which contradicts (6.9). So, G(x̄) ⊂ − D holds, i.e., condition (ii) holds.

Finally, from (6.3),

vcl(cone(F(x¯)+T(G(x¯))−C−(y¯+T¯(z¯)))∩C={0}, ∀T∈L+(Z,Y). (6.10)

Taking T = 0 ∈ L⁺(Z, Y) in (6.10)

vcl(cone(F(x¯)−C−y¯))∩C={0},

which means that condition (iv) holds.

Sufficiency. Since condition (i) and condition (iii), we only prove ȳ ∈ L(x̄, T̄) ∩ PMax (L(x̄, L⁺), C). From ȳ ∈ F(x̄), z̄ ∈ G(x̄) and condition (iii), it follows that

y¯=y¯+T¯(z¯)∈F(x¯)+T¯(G(x¯))=L(x¯,T¯) (6.11)

By condition (ii),

T(G(x¯))⊂−C, ∀T∈L+(Z,Y).

Hence,

L(x¯,L+)−C−y¯=F(x¯)+⋃T∈L+(Z,Y)T(G(x¯))−C−y¯⊂F(x¯)−C−y¯. (6.12)

From condition (iv) and (6.12), it follows that

vcl(cone(L(x¯,L+)−C−y¯))∩C={0},

i.e., ȳ ∈ PMax (L(x̄, L⁺), C).

Now, the example below explains the sufficiency of Theorem 6.1.

Example 6.1

Let X = Y = Z = ℝ², C = D = R+2 and A = {0} × [−1, 0]. The set-valued maps F : X ⇉ Y and G : X ⇉ Z are, respectively, defined as

F(x1,x2)={(y1,y2)∈R2∣(y1,y2)∈{4}×[−1+x2,1+x2]}, ∀(x1,x2)∈A;G(x1,x2)={(z1,z2)∈R2∣(z1,z2)=λ(x2,x2), λ∈[0,1]}, ∀(x1,x2)∈A.

Take x̄ = (0, −1), ȳ = (4, 0), z̄ = (0, 0), T̄(z₁, z₂) = (0.5z₁, 0.5z₂) ∈ L⁺(Z, Y). Clearly, all conditions of Theorem 6.1 are satisfied. Consequently, (x̄, T̄) is a Benson proper saddle point of the Lagrangian set-valued map L(x, T).

Remark 6.1

Theorem 6.1 extends Proposition 6.1 in [14] which were done in the framework of topological linear spaces.

Theorem 6.2

Let D be v-closed. (x̄, T̄) ∈ A × L⁺(Z, Y) is a Benson proper saddle point of L(x, T). Then, there exist ȳ ∈ F(x̄) such that (x̄, ȳ) is a Benson proper efficient element of (VP).

Proof

It follows directly from the necessity of Theorem 6.1 and Theorem 5.2.

Theorem 6.3

Let C, D be v-closed and C⁺ is solid. Let x̄ ∈ S and ȳ ∈ F(x̄). Assume that the following conditions hold:

(x̄, ȳ) is a Benson proper efficient element of (CVP) and F − ȳ is relatively solid generalized C-subconvexlike on S;
(F − ȳ, G) is relatively solid generalized C × D-subconvexlike on A;
(CVP) satisfies the generalized Slater constraint condition;
icr (D) ⊂ icr[cone (G(A)) + icr (D)];
G(x̄) ⊂ −D, ȳ ∈ PMax(F(x̄), C).

Then, there exists T̄ ∈ L⁺(Z, Y) such that (x̄, T̄) ∈ A × L⁺(Z, Y) is a Benson proper saddle point of L. Furthermore, 0 ∈ T̄(G(x̄ ∩ − D)).

Proof

This conclusion follows by Theorem 5.1 and the sufficiency of Theorem 6.1.

7 Duality

A set-valued map Φ: L⁺(Z, Y) ⇉ Y defined by

Φ(T):=PMinL(A,T),C, T∈L+(Z,Y)

is called a proper dual map of (CVP).

The vector maximization problem with set-valued map Φ

(VD) Max ⋃T∈L+Φ(T)

is said to be a Lagrange dual problem of (CVP).

Definition 7.1

A point ȳ ∈ Y is called a feasible point of (VD) if ȳ ∈ ⋃_T∈L⁺ Φ(T). A feasible point ȳ is called an efficient point of (VD) if

y−y¯∉C∖{0}, ∀y∈⋃T∈L+Φ(T).

Analogously to Theorem 7.1 and Theorem 7.2 in [14], we can obtain the following two theorems.

Theorem 7.1

Weak Duality. Let x̄ ∈ S and ȳ be any feasible point of (VD). Then

y¯∉F(x¯)+C∖{0}.

Theorem 7.2

Let x̄ ∈ S. Then, ȳ ∈ F(x̄) ∩ ⋃_T∈L⁺Φ(T) if and only if x̄ is a Benson proper efficient solution of (CVP) and ȳ is an efficient point of (VD).

The following strong duality results show us a necessary and sufficient condition of proper efficiency.

Theorem 7.3

Strong duality. Let C is v-closed and C⁺ is solid. Let (CVP) satisfies the generalized Slater constraint condition such that

F is relatively solid generalized C-subconvexlike on S;
(F, G) is relatively solid generalized C × D-subconvexlike on A;
icr (D) ⊂ icr[cone (G(A)) + icr (D)].

Then, x̄ is a Benson proper efficient solution of (CVP) if and only if x̄ ∈ S and there exists ȳ ∈ F(x̄) such that ȳ is an efficient point of (VD).

Proof

Suppose that x̄ is a Benson proper efficient solution of (CVP). By Definition 5.1, x̄ ∈ S and there exists ȳ ∈ F(x̄) such that (x̄, ȳ) is a Benson proper efficient element of (CVP). By Theorem 5.1, there exists T̄ ∈ L⁺(Z, Y) with 0 ∈ T̄(G(x̄) ∩ − D) such that

y¯∈PMin(L(x,T¯),C)=Φ(T¯)⊂⋃T∈L+Φ(T).

By Theorem 7.2, ȳ is an efficient point of (VD).

Conversely, suppose that x̄ ∈ S, and there exists ȳ ∈ F(x̄) such that ȳ is an efficient point of (VD). Then, ȳ ∈ ⋃_T∈L⁺Φ(T). Again, by Theorem 7.2, we get x̄ is a Benson proper efficient solution of (CVP).

Remark 7.1

Theorem 7.3 extends Theorem 3.6 and Theorem 3.7 in [23] to vector optimization problems with set-valued maps.

8 Conclusions

In this paper, we introduce the concept of relatively solid generalized cone-subconvexlike set-valued maps. With the help of the concept, we establish scalarization theorems, Lagrangian type duality theorems, saddle-point theorems and duality theorems to characterize Benson proper efficient solutions.

Acknowledgments

This work was supported by the National Natural Science Foundation of China No.71471140 and No.11871383.

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Received: 2018-06-23

Accepted: 2019-09-23

Published Online: 2019-10-18

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0104

Keywords for this article

set-valued maps; relative algebraic interior; vector closure; Benson proper efficiency; generalized cone subconvexlikeness

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