Globally proper efficiency of set optimization problems based on the certainly set less order relation

Abstract

In this paper, we investigate the globally proper efficiency of set optimization problems. Firstly, we use the so-called certainly set less order relation to define a new kind of set order relation. Based on the new set order relation, we introduce the notion of the globally proper efficient solution of the set optimization problem. Secondly, we establish Lagrange multiplier rule of the set optimization problem. Finally, we obtain Lagrangian duality theorems and saddle point theorems. We also give some examples to illustrate our results.

Keywords:

• Set optimization
• globally proper efficient solution
• Lagrangian multiplier
• duality
• saddle point

AMS Subject Classifications:

• 90C26
• 90C29
• 90C30
• 90C46

1. Introduction

Vector optimization is an important topic in optimization theory and has been investigated by many scholars. The model of the vector optimization problem is formed as follows: $(\mathrm{VOP})\mathrm{Min}f(x),x\in X,$where $f:X\to Y$ is a vector-valued map, and X and Y are a linear space and a locally convex space, respectively. Many scholars studied different kinds of solutions of (VOP) (see [1–4] and the references therein), for example, properly minimal solutions, weakly minimal solutions, or approximately minimal solutions, see [2].

With the development of set-valued analysis, several scholars have been paying attention to the following set-valued optimization problem: $(\mathrm{SVOP})\mathrm{Min}F(x),x\in X,$where $F:X\rightrightarrows Y$ is a set-valued map. When $F(x)$ is a singleton set for any $x\in X$, (SVOP) reduces to (VOP). Among the introduced minimality notions of (SVOP) are Benson properly efficient solution [5], Henig properly efficient solution [6] and super efficient solution [7]. Not only the topological properties of these solutions such as the connectedness of the solution set have been investigated [8], but also optimality conditions of (SVOP) in the sense of proper efficiency have been established.

In order to define the solution $x_0$ of (VOP), we only need to compare $f(x_0)$ with every element in $f(X)$, where $x_0\in X$. However, when we define the solution $x_0$ of (SVOP), we need to compare $y_0\in F(x_0)$ with every element in $F(X)$. This described the so-called vector approach to a set-valued optimization problem. Note that when defining the solutions of (VOP) and (SVOP), we usually make some comparisons between two vectors. Recently, several scholars have been using the ordered convex cone to make comparisons between two sets and formed the following set optimization problem (SOP): $(\mathrm{SOP})\mathrm{Min}\{F(x)|x\in X\}.$When we define the solution $x_0$ of (SOP), we need to compare $F(x_0)$ with every element in $\{F(x)|x\in X\}$. This is the so-called set approach, see [9]. The set approach to a set-valued optimization problem has gained tremendous interest in the last years. Very recently, Ansari et al. [10] have studied the convergence of the solution set of set optimization problems. Moreover, it is interesting to mention that there exist several publications that deal with a vectorization approach of (SOP), see, for example, [11–13]. The extremal value functions used in vectorization have been intensely studied in Gerlach and Rocktäschel [14].

To generalize the cone convexity of set-valued maps, Kuroiwa et al. [15] introduced six kinds of set-relations based on cone ordering. Hernández and Rodríguez-Marín [16] introduced the efficient solution and the weakly efficient solution of (SOP) and investigated duality, Lagrangian multiplier rule and saddle points of (SOP). Dhingra and Lalitha [17] investigated the set optimization involving improvement sets [18]. Very recently, the existence of solutions [19,20], scalarizations [21,22] and the connectedness [23,24] of the solutions set for (SOP) have been studied.

In the theory of vector optimization, we always consider a partial order relation to compare two vectors or sets. However, the set order relation induced by the improvement set in [17] is not a partial order relation. On the other hand, it is well-known that the set of (weakly) efficient solutions for (VOP) or (SVOP) is usually too big, which poses difficulties for the decision maker. To overcome this defect, the notion of properly efficient solution of (VOP) and (SVOP) has been introduced. Therefore, using the partial order relation to introduce the notion of proper efficiency of (SOP) is an interesting point that we are tackling in this work. To the best of our knowledge, there are no publications introducing the notions of properly efficient solutions of (SOP). The purpose of this paper is to use the certainly set less order relation [25,26] to introduce a globally proper efficient solution of (SOP).

This paper is organized as follows. In Section 2, we give some notations and preliminaries. In Section 3, we obtain a Lagrange multiplier rule of (SOP). In Section 4, we establish duality theorems of (SOP) and finally, in Section 5, we give saddle point theorems of (SOP).

2. Notations and preliminaries

Let Y and Z be two real locally convex spaces. $Y^{\ast }$ and $Z^{\ast }$ are dual spaces of Y and Z, respectively. The zero element of every space is denoted by 0. Write $\mathcal{P}(Y):=\{A\subseteq Y|A\ne \mathrm{\varnothing }\}$. $C\in \mathcal{P}(Y)$ is called a convex set iff $\lambda x_1+(1-\lambda )x_2\in C,\mathrm{\forall }{x}_1,x_2\in C,\mathrm{\forall }\lambda \in [0,1].$$C\in \mathcal{P}(Y)$ is called a cone iff $\lambda x\in C,\mathrm{\forall }x\in C,\mathrm{\forall }\lambda \ge 0.$$C\in \mathcal{P}(Y)$ is pointed iff $C\cap (-C)=\{0\}$. $C\in \mathcal{P}(Y)$ is nontrivial iff $C\ne \{0\}$ and $C\ne Y$. The interior and the closure of C are denoted by $\mathrm{int}C$ and $\mathrm{cl}C$, respectively. Unless otherwise specified, we suppose that C and D are two nontrivial point convex cones of Y and Z with $\mathrm{int}C\ne \mathrm{\varnothing }$ and $\mathrm{int}D\ne \mathrm{\varnothing }$, respectively.

Definition 2.1

[27]

Let $A\in \mathcal{P}(Y)$. $y\in A$ is called a minimal element of A (denoted by $y\in \mathrm{Min}A$) iff $(A-y)\cap (-C\setminus \{0\})=\mathrm{\varnothing }$.

We denote the set ${\mathcal{H}}_C:=\{H\in \mathcal{P}(Y)|H\mathrm{isapointedconvexconewith}C\setminus \{0\}\subseteq \mathrm{int}H\}$.

Definition 2.2

[28]

Let $A\in \mathcal{P}(Y)$. $y\in A$ is called a globally minimal element of A (denoted by $y\in \mathrm{GMin}A$) iff there exists $H\in {\mathcal{H}}_C$ such that $(A-y)\cap (-H\setminus \{0\})=\mathrm{\varnothing }$.

Remark 2.1

It is clear that $\mathrm{GMin}A\subseteq \mathrm{Min}A$. However, the following example shows that $\mathrm{Min}A⊈\mathrm{GMin}A$. Hence, Definition 2.1 generalizes Definition 2.2.

Example 2.1

Let $C:=\{(x,y)\mid x\ge 0,y\ge 0,(x,y)\in {\mathbb{R}}^2\},$ $A:=\{(x,y)\mid -1<x<0,1<y<3,(x,y)\in {\mathbb{R}}^2\}\cup \{(0,1)\}.$ Let $a=(0,1)$. It is easy to check $a\in \mathrm{Min}A$ and $a\notin \mathrm{GMin}A$. Therefore, $\mathrm{Min}A⊈\mathrm{GMin}A$.

To compare two elements of $\mathcal{P}(Y)$, we establish the following the set order relation defined between two nonempty sets.

Definition 2.3

[25,26]

Let $A,B\in \mathcal{P}(Y)$. We define the following set order relation: $A{\le }_C^{c}B\iff A=B\mathrm{or}B-A\subseteq C\mathrm{whenever}A\ne B.$

Remark 2.2

According to Proposition 6.2 in Ansari [25], the set order relation ${\le }_C^c$ is a partial order relation since C is a nontrivial point convex cone.

Let C be replaced by $C+H\mathrm{\setminus }\{0\}$ in the set order relation ${\le }_C^c$, we can introduce a new set order relation ${\le }_{C+H\mathrm{\setminus }\{0\}}^c$ as follows: $A{\le }_{C+H\mathrm{\setminus }\{0\}}^{c}B\iff A=B\mathrm{or}B-A\subseteq C+H\mathrm{\setminus }\{0\}\mathrm{whenever}A\ne B.$

Remark 2.3

The set order relation ${\le }_{C+H\mathrm{\setminus }\{0\}}^c$ is transitive. Indeed, let $A_1{\le }_{C+H\mathrm{\setminus }\{0\}}^c{A}_2$ and $A_2{\le }_{C+H\mathrm{\setminus }\{0\}}^c{A}_3$. If $A_1\ne A_2$ we have $b-a\in C+H\mathrm{\setminus }\{0\},\mathrm{\forall }b\in A_2,\mathrm{\forall }a\in A_1$and $c-b\in C+H\mathrm{\setminus }\{0\},\mathrm{\forall }c\in A_3,\mathrm{\forall }b\in A_2.$Thus, we have $c-a=(c-b)+(b-a)\in C+H\mathrm{\setminus }\{0\}+C+H\mathrm{\setminus }\{0\}\subseteq C+H\mathrm{\setminus }\{0\},\mathrm{\forall }c\in A_3,a\in A_1,$which implies $A_3-A_1\subseteq C+H\mathrm{\setminus }\{0\}$, i.e. $A_1{\le }_{C+H\mathrm{\setminus }\{0\}}^c{A}_3$. Clearly, $A_1{\le }_{C+H\mathrm{\setminus }\{0\}}^c{A}_3$ holds when $A_1=A_2$. Hence, the set order relation ${\le }_{C+H\mathrm{\setminus }\{0\}}^c$ is transitive. Clearly, the set order relation ${\le }_{C+H\mathrm{\setminus }\{0\}}^c$ is reflexive. We claim that the set order relation ${\le }_{C+H\mathrm{\setminus }\{0\}}^c$ is antisymmetric. Let $B_1{\le }_{C+H\mathrm{\setminus }\{0\}}^c{B}_2$ and $B_2{\le }_{C+H\mathrm{\setminus }\{0\}}^c{B}_1$. If $B_1\ne B_2$, then we have $B_2-B_1\subseteq C+H\mathrm{\setminus }\{0\}$ and $B_1-B_2\subseteq C+H\mathrm{\setminus }\{0\}$. Thus, $B_2-B_1\subseteq (C+H\mathrm{\setminus }\{0\})\cap (-(C+H\mathrm{\setminus }\{0\}))=\mathrm{\varnothing }$, which is a contradiction. Therefore, $B_1=B_2$. Thus, the set order relation ${\le }_{C+H\mathrm{\setminus }\{0\}}^c$ is a partial order relation.

Next, based on the above set order relation, we define the globally proper minimal element of $\mathcal{S}\subseteq \mathcal{P}(Y)$.

Definition 2.4

Let $\mathcal{S}\subseteq \mathcal{P}(Y)$.

1. $A\in \mathcal{S}$ is a c-globally minimal element of $\mathcal{S}$ (denoted by $A\in c$-$\mathrm{GMin}\mathcal{S}$) iff there exists $H\in {\mathcal{H}}_C$ such that $B\in \mathcal{S}$ and $B{\le }_{C+H\mathrm{\setminus }\{0\}}^{c}A$ imply $A{\le }_{C+H\mathrm{\setminus }\{0\}}^{c}B$.

2. $A\in \mathcal{S}$ is a c-globally maximal element of $\mathcal{S}$ (denoted by $A\in c$-$\mathrm{GMax}\mathcal{S}$) iff there exists $H\in {\mathcal{H}}_C$ such that $B\in \mathcal{S}$ and $A{\le }_{C+H\mathrm{\setminus }\{0\}}^{c}B$ imply $B{\le }_{C+H\mathrm{\setminus }\{0\}}^{c}A$.

Remark 2.4

When $\mathcal{S}\subseteq \mathcal{P}(Y)$ becomes $\mathcal{S}\in \mathcal{P}(Y)$, Definition 2.4(i) reduces to Definition 2.2.

The following proposition shows a relationship between a c-globally minimal element of $\mathcal{S}$ and the set order relation $B{\nleqq }_{C+H\mathrm{\setminus }\{0\}}^{c}A$ for all $B\in \mathcal{S}$.

Proposition 2.1

Let $\mathcal{S}\subseteq \mathcal{P}(Y)$ and $A\in \mathcal{S}$. We have $A\in c$-$\mathrm{GMin}\mathcal{S}$ iff there exists $H\in {\mathcal{H}}_C$ such that $B{\nleqq }_{C+H\mathrm{\setminus }\{0\}}^{c}A,\mathrm{\forall }B\in \mathcal{S}.$

Proof.

The sufficiency is clear. Now, we prove the necessity. We suppose that, for any $H\in {\mathcal{H}}_C$, there exists $B\in \mathcal{S}$ such that (1) $B{\le }_{C+H\mathrm{\setminus }\{0\}}^{c}A.$(1) If A = B, then $0\in A-B\subseteq C+H\mathrm{\setminus }\{0\}$ which contradicts with $0\notin C+H\mathrm{\setminus }\{0\}$. If $A\ne B$, it follows from Definition 2.4 that $A{\le }_{C+H\mathrm{\setminus }\{0\}}^{c}B$. Thus, we obtain (2) $B-A\subseteq C+H\mathrm{\setminus }\{0\}.$(2) By (1(1) $B{\le }_{C+H\mathrm{\setminus }\{0\}}^{c}A.$(1) ), we have (3) $A-B\subseteq C+H\mathrm{\setminus }\{0\}.$(3) It follows from (2(2) $B-A\subseteq C+H\mathrm{\setminus }\{0\}.$(2) ) and (3(3) $A-B\subseteq C+H\mathrm{\setminus }\{0\}.$(3) ) that (4) $A-B\subseteq (C+H\mathrm{\setminus }\{0\})\cap (-(C+H\mathrm{\setminus }\{0\})).$(4) Since $(H\mathrm{\setminus }\{0\})\cap (-(H\mathrm{\setminus }\{0\}))=\mathrm{\varnothing }$, $(C+H\mathrm{\setminus }\{0\})\cap (-(C+H\mathrm{\setminus }\{0\}))=\mathrm{\varnothing }$, which contradicts (4(4) $A-B\subseteq (C+H\mathrm{\setminus }\{0\})\cap (-(C+H\mathrm{\setminus }\{0\})).$(4) ).

3. Lagrangian multiplier

In this section, we will establish a Lagrange multiplier rule of the set optimization problem. From now on, we suppose that X is a linear space and $\mathcal{A}$ is a nonempty subset of X.

Definition 3.1

[5]

The set-valued map $F:X\rightrightarrows Y$ is said to be C-convexlike on $\mathcal{A}$ iff $\lambda F(x_1)+(1-\lambda )F(x_2)\subseteq F(\mathcal{A})+C,\mathrm{\forall }{x}_1,x_2\in \mathcal{A},\mathrm{\forall }\lambda \in [0,1].$

Remark 3.1

[5]

F is C-convexlike on $\mathcal{A}$ iff $F(\mathcal{A})+C$ is a convex set of Y.

Definition 3.2

[29]

Let B be a nonempty subset of Y. B is said to be C-bounded iff, for any neighborhood U of zero in Y, there exists t>0 such that $B\subseteq tU+C$.

Lemma 3.1

Let $B\subseteq Y$ be an C-bounded set and $H\in {\mathcal{H}}_C$. Then, $\bigcap _{b\in B}(b-\mathrm{int}H)\ne \mathrm{\varnothing }$.

Proof.

Let $h\in \mathrm{int}H$. Write $U:=-h+\mathrm{int}H$. It is clear that U is a neighborhood of zero. Since $B\subseteq Y$ is C-bounded, there exists t>0 such that $B\subseteq tU+C=t(-h+\mathrm{int}H)+C\subseteq -th+C+\mathrm{int}H\subseteq -th+H\mathrm{\setminus }\{0\},$which shows that $-th\in \bigcap _{b\in B}(b-\mathrm{int}H)$. Hence, $\bigcap _{b\in B}(b-\mathrm{int}H)\ne \mathrm{\varnothing }$.

Let $F:X\rightrightarrows Y$ and $G:X\rightrightarrows Z$ be two set-valued maps on $\mathcal{A}$. Now, we consider the following set optimization problem: $(\mathrm{SOP})\mathrm{Min}\{F(x)|x\in \mathrm{\Omega }\},$where $\mathrm{\Omega }:=\{x\in \mathcal{A}|G(x)\cap (-D)\ne \mathrm{\varnothing }\}$.

Let $\mathcal{L}(Z,Y)$ denote the set of all continuous linear maps from Z to Y. Write ${\mathcal{L}}_{+}(Z,Y):=\{T\in \mathcal{L}(Z,Y)|T(D)\subseteq C\}$ and $\mathrm{\Lambda }(Z,Y):=\{h|h(\cdot )=T(\cdot )+m,T\in {\mathcal{L}}_{+}(Z,Y),m\in -C\}$.

Definition 3.3

[29]

(SOP) satisfies the generalized Slater constraint qualification iff there exists $x^{\prime }\in \mathcal{A}$ such that $G(x^{\prime })\cap (-\mathrm{int}D)\ne \mathrm{\varnothing }$.

Definition 3.4

$x_0\in \mathrm{\Omega }$ is called a c-globally minimal solution of (SOP) iff $F(x_0)\in c\text{-}\mathrm{GMin}\{F(x)|x\in \mathrm{\Omega }\}.$

Theorem 3.1

Let $H\in {\mathcal{H}}_C$ and $x_0\in \mathcal{A}$. Suppose that the following conditions are satisfied.

1. (SOP) satisfies generalized Slater constraint qualification;

2. $F(x_0)$ is C-bounded;

3. The set-valued map $(F,G):X\rightrightarrows Y\times Z$ is $(H\mathrm{\setminus }\{0\})\times D$-convexlike on $\mathcal{A}$;

4. Let $B:=\bigcap _{y_0\in F(x_0)}(y_0-H\mathrm{\setminus }\{0\})$ and $M:=\{(y^{\ast },z^{\ast })\in ((H\mathrm{\setminus }\{0\}{)}^{+}\mathrm{\setminus }\{0\})\times D^{+}|(y^{\ast },z^{\ast })$ separates $(F,G)(\mathcal{A})$ and $B\times (-D)\}$. If $M\ne \mathrm{\varnothing }$, there exist $(y_0^{\ast },z_0^{\ast })\in M$ and $(y_0,z_0)\in F(x_0)\times G(x_0)$ such that (5) $inf\{⟨y,y_0^{\ast }⟩+⟨z,z_0^{\ast }⟩|(y,z)\in F(\mathcal{A})\times G(\mathcal{A})\}=⟨y_0,y_0^{\ast }⟩+⟨z_0,z_0^{\ast }⟩.$(5)

5. There does not exist $x\in \mathcal{A}$ such that $(F(x_0)-F(x))\cap (H\mathrm{\setminus }\{0\})\ne \mathrm{\varnothing }$.Then, there exists $T\in {\mathcal{L}}_{+}(Z,Y)$ such that (6) $T(G(x_0)\cap (-D))\subseteq -\mathrm{int}C\cup \{0\}$(6) and $x_0$ is a c-globally minimal solution of the following unconstrained set optimization problem: $(\mathrm{USOP})\mathrm{Min}\{F(x)+(T\circ G)(x)|x\in \mathcal{A}\}.$

Proof.

Clearly, B is a convex set in Y. Since $F(x_0)$ is C-bounded, it follows from Lemma 3.1 that $\mathrm{int}B=\bigcap _{y_0\in F(x_0)}(y_0-\mathrm{int}H)\ne \mathrm{\varnothing }$. Thus, $B\times (-D)$ is a convex set with $\mathrm{int}(B\times (-D))\ne \mathrm{\varnothing }$ in $Y\times Z$. According to Condition (iii), $(F,G)(\mathcal{A})+(H\mathrm{\setminus }\{0\})\times D$ is a convex set in $Y\times Z$.

We assert that (7) $((F,G)(\mathcal{A})+(H\mathrm{\setminus }\{0\})\times D)\cap \mathrm{int}(B\times (-D))=\mathrm{\varnothing }.$(7) Otherwise, there exists $(\overline{y},\overline{z})\in (F,G)(\mathcal{A})+H\mathrm{\setminus }\{0\}\times D$ such that $(\overline{y},\overline{z})\in \mathrm{int}(B\times (-D))$. Thus, there exists $\overline{x}\in \mathcal{A}$ such that (8) $\overline{y}\in (F(\overline{x})+H\mathrm{\setminus }\{0\})\cap \mathrm{int}B$(8) and (9) $\overline{z}\in (G(\overline{x})+D)\cap \mathrm{int}(-D).$(9) (9(9) $\overline{z}\in (G(\overline{x})+D)\cap \mathrm{int}(-D).$(9) ) implies $\overline{x}\in \mathrm{\Omega }$. By (8(8) $\overline{y}\in (F(\overline{x})+H\mathrm{\setminus }\{0\})\cap \mathrm{int}B$(8) ), there exist $y^{\prime }\in F(\overline{x})$ and $h\in H\mathrm{\setminus }\{0\}$ such that (10) $\overline{y}=y^{\prime }+h\in \mathrm{int}B.$(10) Clearly, $B-H\mathrm{\setminus }\{0\}\subseteq B.$ Therefore, we have (11) $\mathrm{int}B-H\mathrm{\setminus }\{0\}\subseteq \mathrm{int}B.$(11) It follows from (10(10) $\overline{y}=y^{\prime }+h\in \mathrm{int}B.$(10) ) and (11(11) $\mathrm{int}B-H\mathrm{\setminus }\{0\}\subseteq \mathrm{int}B.$(11) ) that (12) $y^{\prime }=\overline{y}-h\in \mathrm{int}B=\bigcap _{y_0\in F(x_0)}(y_0-\mathrm{int}H)\subseteq \bigcap _{y_0\in F(x_0)}(y_0-H\mathrm{\setminus }\{0\}).$(12) (12(12) $y^{\prime }=\overline{y}-h\in \mathrm{int}B=\bigcap _{y_0\in F(x_0)}(y_0-\mathrm{int}H)\subseteq \bigcap _{y_0\in F(x_0)}(y_0-H\mathrm{\setminus }\{0\}).$(12) ) shows that $y^{\prime }\in y_0-H\mathrm{\setminus }\{0\},\mathrm{\forall }{y}_0\in F(x_0),$i.e. $F(x_0)-y^{\prime }\subseteq H\mathrm{\setminus }\{0\},$which implies that $(F(x_0)-F(\overline{x}))\cap (H\mathrm{\setminus }\{0\})\ne \mathrm{\varnothing },$which contradicts Condition (v). Therefore, (7(7) $((F,G)(\mathcal{A})+(H\mathrm{\setminus }\{0\})\times D)\cap \mathrm{int}(B\times (-D))=\mathrm{\varnothing }.$(7) ) holds.

By the separation theorem of convex sets, there exists $(y^{\ast },z^{\ast })\in (Y^{\ast }\times Z^{\ast })\mathrm{\setminus }\{(0,0)\}$ such that (13) $⟨y,y^{\ast }⟩+⟨z,z^{\ast }⟩\ge ⟨b,y^{\ast }⟩+⟨-d,z^{\ast }⟩,\mathrm{\forall }(y,z)\in (F,G)(\mathcal{A})+(H\mathrm{\setminus }\{0\})\times D,b\in B,d\in D.$(13) According to (13(13) $⟨y,y^{\ast }⟩+⟨z,z^{\ast }⟩\ge ⟨b,y^{\ast }⟩+⟨-d,z^{\ast }⟩,\mathrm{\forall }(y,z)\in (F,G)(\mathcal{A})+(H\mathrm{\setminus }\{0\})\times D,b\in B,d\in D.$(13) ), $(y^{\ast },z^{\ast })\in (H\mathrm{\setminus }\{0\}{)}^{+}\times D^{+}.$ We assert that $y^{\ast }\ne 0$. Otherwise, $z^{\ast }\ne 0$. By (13(13) $⟨y,y^{\ast }⟩+⟨z,z^{\ast }⟩\ge ⟨b,y^{\ast }⟩+⟨-d,z^{\ast }⟩,\mathrm{\forall }(y,z)\in (F,G)(\mathcal{A})+(H\mathrm{\setminus }\{0\})\times D,b\in B,d\in D.$(13) ), we have (14) $⟨z,z^{\ast }⟩\ge 0,\mathrm{\forall }z\in G(\mathcal{A}).$(14) On the other hand, it follows from Condition (i) that there exists $x^{\prime }\in \mathcal{A}$ such that $z^{\prime }\in G(x^{\prime })\cap (-\mathrm{int}D)$. Since $z^{\ast }\in D^{+}$, we have $⟨z^{\prime },z^{\ast }⟩<0,$which contradicts (14(14) $⟨z,z^{\ast }⟩\ge 0,\mathrm{\forall }z\in G(\mathcal{A}).$(14) ). Therefore, $y^{\ast }\ne 0$. According to (13(13) $⟨y,y^{\ast }⟩+⟨z,z^{\ast }⟩\ge ⟨b,y^{\ast }⟩+⟨-d,z^{\ast }⟩,\mathrm{\forall }(y,z)\in (F,G)(\mathcal{A})+(H\mathrm{\setminus }\{0\})\times D,b\in B,d\in D.$(13) ), we have (15) $⟨y,y^{\ast }⟩+⟨z,z^{\ast }⟩\ge ⟨b,y^{\ast }⟩+⟨-d,z^{\ast }⟩,\mathrm{\forall }(y,z)\in (F,G)(\mathcal{A}),b\in B,d\in D.$(15) Since $(y^{\ast },z^{\ast })\in ((H\mathrm{\setminus }\{0\}{)}^{+}\mathrm{\setminus }\{0\})\times D^{+}$, it follows from (15(15) $⟨y,y^{\ast }⟩+⟨z,z^{\ast }⟩\ge ⟨b,y^{\ast }⟩+⟨-d,z^{\ast }⟩,\mathrm{\forall }(y,z)\in (F,G)(\mathcal{A}),b\in B,d\in D.$(15) ) that $M\ne \mathrm{\varnothing }$. By Condition (iv), there exist $(y_0^{\ast },z_0^{\ast })\in M$ and $(y_0,z_0)\in F(x_0)\times G(x_0)$ such that (5(5) $inf\{⟨y,y_0^{\ast }⟩+⟨z,z_0^{\ast }⟩|(y,z)\in F(\mathcal{A})\times G(\mathcal{A})\}=⟨y_0,y_0^{\ast }⟩+⟨z_0,z_0^{\ast }⟩.$(5) ) holds. By (5(5) $inf\{⟨y,y_0^{\ast }⟩+⟨z,z_0^{\ast }⟩|(y,z)\in F(\mathcal{A})\times G(\mathcal{A})\}=⟨y_0,y_0^{\ast }⟩+⟨z_0,z_0^{\ast }⟩.$(5) ), we have (16) $⟨y,y_0^{\ast }⟩+⟨z,z_0^{\ast }⟩\ge ⟨y_0,y_0^{\ast }⟩+⟨z_0,z_0^{\ast }⟩,\mathrm{\forall }x\in \mathcal{A},(y,z)\in (F,G)(x).$(16) Let $c_0\in \mathrm{int}C$ be fixed. Since $\mathrm{int}C\subseteq \mathrm{int}H$ and $y_0^{\ast }\in (H\mathrm{\setminus }\{0\}{)}^{+}$, $⟨c_0,y_0^{\ast }⟩>0$. We define the map $T:Z\rightrightarrows Y$ as follows (17) $T(z)=\frac{⟨z,z_0^{\ast }⟩}{⟨c_0,y_0^{\ast }⟩}{c}_0,\mathrm{\forall }z\in Z.$(17) Clearly, $T\in {\mathcal{L}}_{+}(Z,Y)$. It follows from $z_0^{\ast }\in D^{+}$ that (18) $⟨z,z_0^{\ast }⟩\le 0,\mathrm{\forall }z\in G(x_0)\cap (-D).$(18) By (17(17) $T(z)=\frac{⟨z,z_0^{\ast }⟩}{⟨c_0,y_0^{\ast }⟩}{c}_0,\mathrm{\forall }z\in Z.$(17) ) and (18(18) $⟨z,z_0^{\ast }⟩\le 0,\mathrm{\forall }z\in G(x_0)\cap (-D).$(18) ), (6(6) $T(G(x_0)\cap (-D))\subseteq -\mathrm{int}C\cup \{0\}$(6) ) holds.

We assert (19) $F(x)+(T\circ G)(x){\nleqq }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_0)+(T\circ G)(x_0),\mathrm{\forall }x\in \mathcal{A}.$(19) Otherwise, there exists $\tilde{x}\in \mathcal{A}$ such that $F(\tilde{x})+(T\circ G)(\tilde{x}){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_0)+(T\circ G)(x_0).$Hence, we have $F(x_0)+(T\circ G)(x_0)-(F(\tilde{x})+(T\circ G)(\tilde{x}))\subseteq C+H\mathrm{\setminus }\{0\}.$Let $y_0\in F(x_0)$ and $z_0\in G(x_0)$. Then, there exist $\tilde{y}\in F(\tilde{x}),\tilde{z}\in G(\tilde{x}),\tilde{c}\in C$ and $\tilde{h}\in H\mathrm{\setminus }\{0\}$ such that (20) $y_0+\frac{⟨z_0,z_0^{\ast }⟩}{⟨c_0,y_0^{\ast }⟩}{c}_0=\tilde{y}+\frac{⟨\tilde{z},z_0^{\ast }⟩}{⟨c_0,y_0^{\ast }⟩}{c}_0+\tilde{c}+\tilde{h}.$(20) It follows from (20(20) $y_0+\frac{⟨z_0,z_0^{\ast }⟩}{⟨c_0,y_0^{\ast }⟩}{c}_0=\tilde{y}+\frac{⟨\tilde{z},z_0^{\ast }⟩}{⟨c_0,y_0^{\ast }⟩}{c}_0+\tilde{c}+\tilde{h}.$(20) ) that (21) $⟨y_0,y_0^{\ast }⟩+⟨z_0,z_0^{\ast }⟩-⟨\tilde{y},y_0^{\ast }⟩-⟨\tilde{z},z_0^{\ast }⟩=⟨\tilde{c}+\tilde{h},y_0^{\ast }⟩.$(21) Since $y_0^{\ast }\in (H\mathrm{\setminus }\{0\}{)}^{+}$, we have (22) $⟨\tilde{c}+\tilde{h},y_0^{\ast }⟩>0.$(22) Using (21(21) $⟨y_0,y_0^{\ast }⟩+⟨z_0,z_0^{\ast }⟩-⟨\tilde{y},y_0^{\ast }⟩-⟨\tilde{z},z_0^{\ast }⟩=⟨\tilde{c}+\tilde{h},y_0^{\ast }⟩.$(21) ) and (22(22) $⟨\tilde{c}+\tilde{h},y_0^{\ast }⟩>0.$(22) ), we obtain $⟨y_0,y_0^{\ast }⟩+⟨z_0,z_0^{\ast }⟩>⟨\tilde{y},y_0^{\ast }⟩+⟨\tilde{z},z_0^{\ast }⟩,$which contradicts (16(16) $⟨y,y_0^{\ast }⟩+⟨z,z_0^{\ast }⟩\ge ⟨y_0,y_0^{\ast }⟩+⟨z_0,z_0^{\ast }⟩,\mathrm{\forall }x\in \mathcal{A},(y,z)\in (F,G)(x).$(16) ). Therefore, (19(19) $F(x)+(T\circ G)(x){\nleqq }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_0)+(T\circ G)(x_0),\mathrm{\forall }x\in \mathcal{A}.$(19) ) holds. Thus, by Proposition 2.1, $x_0$ is a c-globally minimal solution of (USOP).

The following example is used to illustrate Theorem 3.1.

Example 3.1

Let $Y:={\mathbb{R}}^2$, $Z:=\mathbb{R}$, $C:=\{(x,y)\mid y\le 3x,y\ge \frac{1}{3}x,(x,y)\in {\mathbb{R}}^2\}$, $D:={\mathbb{R}}^{+}$, $\mathcal{A}:=[0,2]$ and $H:={\mathbb{R}}_{+}^2$. The set-valued maps $F:X\rightrightarrows Y$ and $G:X\rightrightarrows Z$ on $\mathcal{A}$ are defined as: $F(x)=\{\begin{array}{ll}[(0,x+1),(x+1,0)],& x\in [0,1)\\ [(0,1),(1,0)],& x=1\\ [(0,x),(x,0)],& x\in (1,2)\\ [(0,2),(2,0)],& x=2\end{array}$and $G(x)=\{\begin{array}{ll}[-x,0],& x\in [0,1)\\ [-1,0],& x=1\\ [-x+1,0],& x\in (1,2)\\ (0,1],& x=2\end{array}.$It is clear that $H\in {\mathcal{H}}_C$ and $\mathrm{\Omega }=[0,2)$. Let $x_0=1\in \mathcal{A}$. It is easy to check that Conditions (i)–(v) hold. Hence, there exists T defined as follows: $T(z)=\frac{⟨z,z_0^{\ast }⟩}{⟨c_0,y_0^{\ast }⟩}{c}_0,\mathrm{\forall }z\in Z,$where $(y_0^{\ast },z_0^{\ast })=((1,1),1)\in M$ and $c_0=(\frac{1}{2},\frac{1}{2})\in \mathrm{int}C$. For the above T, (6) holds and $x_0=1$ is a c-globally minimal solution of (USOP).

4. Duality

Let $h\in \mathrm{\Lambda }(Z,Y)$ with $h(.)=T(.)+m$ and $m\in -C$. The set-valued map $h\circ G:X\rightrightarrows Y$ is defined as follows $(h\circ G)(x)=\bigcup _{z\in G(x)}T(z)+m,\mathrm{\forall }x\in \mathcal{A}.$We define the Lagrangian set-valued map $L:X\times \mathrm{\Lambda }(Z,Y)\rightrightarrows Y$ as follows $L(x,h)=F(x)+(h\circ G)(x),\mathrm{\forall }(x,h)\in \mathcal{A}\times \mathrm{\Lambda }(Z,Y).$The dual set-valued map ${\mathrm{\Phi }}_G:\mathrm{\Lambda }(Z,Y)\rightrightarrows Y$ is write as ${\mathrm{\Phi }}_G(h)=c\text{-}\mathrm{GMin}\{L(x,h)|x\in \mathcal{A}\},\mathrm{\forall }h\in \mathrm{\Lambda }(Z,Y).$The dual problem associated to (SOP) is the following optimization problem: $(\mathrm{DSOP})\mathrm{Max}\{{\mathrm{\Phi }}_G(h)|h\in \mathrm{\Lambda }(Z,Y)\}.$

Definition 4.1

$h_0\in \mathrm{\Lambda }(Z,Y)$ is called a c-globally maximal solution of (DSOP) iff there exists $x_0\in \mathcal{A}$ such that $L(x_0,h_0)\in {\mathrm{\Phi }}_G(h_0)$ and $L(x_0,h_0)\in c$-$\mathrm{GMax}\{{\mathrm{\Phi }}_G(h)|h\in \mathrm{\Lambda }(Z,Y)\}$.

Definition 4.2

$(x_0,h_0)\in X\times \mathrm{\Lambda }(Z,Y)$ is called a feasible pair of (DSOP) iff $h_0\in \mathrm{\Lambda }(Z,Y)$ and $F(x_0)+(h_0\circ G)(x_0)\in {\mathrm{\Phi }}_G(h_0).$

Theorem 4.1

Weak Duality

Let $(h\circ G)(x_0)\subseteq -C$ and $(x^{\prime },h)$ be a feasible pair of (DSOP). If there exists $H\in {\mathcal{H}}_C$ such that (23) $F(x_0){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x^{\prime })+(h\circ G)(x^{\prime }),$(23) then $F(x^{\prime })+(h\circ G)(x^{\prime }){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_0)$.

Proof.

By (23(23) $F(x_0){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x^{\prime })+(h\circ G)(x^{\prime }),$(23) ), the conclusion holds when $F(x_0)=F(x^{\prime })+(h\circ G)(x^{\prime })$. When $F(x_0)\ne F(x^{\prime })+(h\circ G)(x^{\prime })$, we obtain (24) $F(x^{\prime })+(h\circ G)(x^{\prime })-F(x_0)\subseteq C+H\mathrm{\setminus }\{0\}.$(24) According to $(h\circ G)(x_0)\subseteq -C$ and (24(24) $F(x^{\prime })+(h\circ G)(x^{\prime })-F(x_0)\subseteq C+H\mathrm{\setminus }\{0\}.$(24) ), we obtain $F(x^{\prime })+(h\circ G)(x^{\prime })-(F(x_0)+(h\circ G)(x_0))\subseteq C+H\mathrm{\setminus }\{0\},$which implies (25) $F(x_0)+(h\circ G)(x_0){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x^{\prime })+(h\circ G)(x^{\prime }).$(25) Because $(x^{\prime },h)$ is a feasible pair of (DSOP), it follows from Definition 4.2 that (26) $F(x^{\prime })+(h\circ G)(x^{\prime })\in {\mathrm{\Phi }}_G(h).$(26) According to (25(25) $F(x_0)+(h\circ G)(x_0){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x^{\prime })+(h\circ G)(x^{\prime }).$(25) ), (26(26) $F(x^{\prime })+(h\circ G)(x^{\prime })\in {\mathrm{\Phi }}_G(h).$(26) ) and Definition 2.4, we have $F(x^{\prime })+(h\circ G)(x^{\prime }){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_0)+(h\circ G)(x_0),$which implies (27) $F(x_0)+(h\circ G)(x_0)-(F(x^{\prime })+(h\circ G)(x^{\prime }))\subseteq C+H\mathrm{\setminus }\{0\}.$(27) By (27(27) $F(x_0)+(h\circ G)(x_0)-(F(x^{\prime })+(h\circ G)(x^{\prime }))\subseteq C+H\mathrm{\setminus }\{0\}.$(27) ), we have (28) $F(x_0)-(F(x^{\prime })+(h\circ G)(x^{\prime }))\subseteq -a+C+H\mathrm{\setminus }\{0\},\mathrm{\forall }a\in (h\circ G)(x_0).$(28) Since $(h\circ G)(x_0)\subseteq -C$, it follows from (28(28) $F(x_0)-(F(x^{\prime })+(h\circ G)(x^{\prime }))\subseteq -a+C+H\mathrm{\setminus }\{0\},\mathrm{\forall }a\in (h\circ G)(x_0).$(28) ) that $F(x_0)-(F(x^{\prime })+(h\circ G)(x^{\prime }))\subseteq C+H\mathrm{\setminus }\{0\},$which implies $F(x^{\prime })+(h\circ G)(x^{\prime }){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_0)$.

Corollary 4.1

Let $(x_1,h_1)\in X\times \mathrm{\Lambda }(Z,Y)$ and $(h\circ G)(x)\subseteq -C$ with $x\in \mathrm{\Omega }$ and $h\in \mathrm{\Lambda }(Z,Y)$. If there exists $H\in {\mathcal{H}}_C$ such that $(x_1,h_1)$ is a feasible pair of (DSOP) and (29) $F(x_0){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_1)+(h_1\circ G)(x_1),$(29) then $x_0$ is a c-globally minimal solution of (SOP) and $h_1$ is a c-globally maximal solution of (DSOP).

Proof.

Let $x^{\prime }\in \mathrm{\Omega }$ and $F(x^{\prime }){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_0)$. $x_0$ is a c-globally minimal solution of (SOP) when $F(x^{\prime })=F(x_0)$. When $F(x^{\prime })\ne F(x_0)$, we have (30) $F(x_0)-F(x^{\prime })\subseteq C+H\mathrm{\setminus }\{0\}.$(30) We assert that $x_0$ is a c-globally minimal solution of (SOP).

Case 1. $F(x_0)=F(x_1)+(h_1\circ G)(x_1)$. Since $F(x^{\prime }){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_0)$, it follows from (29(29) $F(x_0){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_1)+(h_1\circ G)(x_1),$(29) ) that (31) $F(x^{\prime }){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_1)+(h_1\circ G)(x_1).$(31) By (31(31) $F(x^{\prime }){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_1)+(h_1\circ G)(x_1).$(31) ) and Theorem 4.1, we have $F(x_1)+(h_1\circ G)(x_1){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x^{\prime })$. Hence, $F(x_0){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x^{\prime })$, which shows $x_0$ is a c-globally minimal solution of (SOP).

Case 2. $F(x_0)\ne F(x_1)+(h_1\circ G)(x_1)$. By (29(29) $F(x_0){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_1)+(h_1\circ G)(x_1),$(29) ), there is (32) $F(x_1)+(h_1\circ G)(x_1)-F(x_0)\subseteq C+H\mathrm{\setminus }\{0\}$(32) It follows from (30(30) $F(x_0)-F(x^{\prime })\subseteq C+H\mathrm{\setminus }\{0\}.$(30) ) and (32(32) $F(x_1)+(h_1\circ G)(x_1)-F(x_0)\subseteq C+H\mathrm{\setminus }\{0\}$(32) ) that $\begin{array}{rl}F(x_1)+(h_1\circ G)(x_1)-F(x^{\prime })& \subseteq F(x_1)+(h_1\circ G)(x_1)-F(x_0)+C+H\mathrm{\setminus }\{0\}\\ & \subseteq C+H\mathrm{\setminus }\{0\}+C+H\mathrm{\setminus }\{0\}\\ & \subseteq C+H\mathrm{\setminus }\{0\},\end{array}$which implies $F(x^{\prime }){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_1)+(h_1\circ G)(x_1).$Combining with Theorem 4.1, we obtain (33) $F(x_1)+(h_1\circ G)(x_1){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x^{\prime }).$(33) Using (29(29) $F(x_0){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_1)+(h_1\circ G)(x_1),$(29) ) and (33(33) $F(x_1)+(h_1\circ G)(x_1){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x^{\prime }).$(33) ), we have $F(x_0){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x^{\prime }),$which shows that $x_0$ is a c-globally minimal solution of (SOP).

Next, we prove that $h_1$ is a c-globally maximal solution of (DSOP). Since $(x_1,h_1)$ is a feasible pair of (DSOP), $F(x_1)+(h_1\circ G)(x_1)\in {\mathrm{\Phi }}_G(h_1).$ Let $(x_2,h_2)\in X\times \mathrm{\Lambda }(Z,Y)$ be a feasible pair of (DSOP) such that (34) $F(x_1)+(h_1\circ G)(x_1){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_2)+(h_2\circ G)(x_2).$(34) It follows from (29(29) $F(x_0){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_1)+(h_1\circ G)(x_1),$(29) ) and (34(34) $F(x_1)+(h_1\circ G)(x_1){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_2)+(h_2\circ G)(x_2).$(34) ) that (35) $F(x_0){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_2)+(h_2\circ G)(x_2).$(35) Because $(x_2,h_2)$ is a feasible pair of (DSOP), it follows from (35(35) $F(x_0){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_2)+(h_2\circ G)(x_2).$(35) ) and Theorem 4.1 that (36) $F(x_2)+(h_2\circ G)(x_2){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_0).$(36) By (29(29) $F(x_0){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_1)+(h_1\circ G)(x_1),$(29) ) and (36(36) $F(x_2)+(h_2\circ G)(x_2){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_0).$(36) ), we have (37) $F(x_2)+(h_2\circ G)(x_2){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_1)+(h_1\circ G)(x_1).$(37) According to (34(34) $F(x_1)+(h_1\circ G)(x_1){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_2)+(h_2\circ G)(x_2).$(34) ), (37(37) $F(x_2)+(h_2\circ G)(x_2){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_1)+(h_1\circ G)(x_1).$(37) ) and Definition 2.4, we obtain $L(x_1,h_1)\in c$-$\mathrm{GMax}\{{\mathrm{\Phi }}_G(h)|h\in \mathrm{\Lambda }(Z,Y)\}$. Hence, $h_1$ is a c-globally maximal solution of (DSOP).

Theorem 4.2

Let $x_0\in \mathrm{\Omega }$ be a c-globally minimal solution of (SOP) and $H\in {\mathcal{H}}_C$. There exists $u_0\in {\mathcal{L}}_{+}(Z,Y)$ such that $(u_0\circ G)(x)\subseteq C$ with $x\in \mathrm{\Omega }$ and $x_0$ is a solution of c-$\mathrm{GMin}\{F(x)+(u_0\circ G)(x)|x\in \mathcal{A}\}.$ If there exists $x^{\prime }\in \mathrm{\Omega }$ such that (38) $F(x^{\prime })+(u_0\circ G)(x^{\prime }){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_0)+(u_0\circ G)(x_0),$(38) then $u_0$ is a c-globally maximal solution of (DSOP).

Proof.

Since $x_0$ is a solution of c-$\mathrm{GMin}\{F(x)+(u_0\circ G)(x)|x\in \mathcal{A}\},$ $(x_0,u_0)$ is a feasible pair of (DSOP). Therefore, (39) $F(x_0)+(u_0\circ G)(x_0)\in {\mathrm{\Phi }}_G(u_0).$(39) By (38(38) $F(x^{\prime })+(u_0\circ G)(x^{\prime }){\le }_{C+H\mathrm{\setminus }\{0\}}^{c}F(x_0)+(u_0\circ G)(x_0),$(38) ), $u_0$ is a c-globally maximal solution of (DSOP) when $F(x^{\prime })+(u_0\circ G)(x^{\prime })=F(x_0)+(u_0\circ G)(x_0)$. When $F(x^{\prime })+(u_0\circ G)(x^{\prime })\ne F(x_0)+(u_0\circ G)(x_0)$, we have (40) $F(x_0)+(u_0\circ G)(x_0)-(F(x^{\prime })+(u_0\circ G)(x^{\prime }))\subseteq C+H\setminus \{0\}.$(40) It follows from (40(40) $F(x_0)+(u_0\circ G)(x_0)-(F(x^{\prime })+(u_0\circ G)(x^{\prime }))\subseteq C+H\setminus \{0\}.$(40) ) that (41) $F(x_0)+(u_0\circ G)(x_0)-F(x^{\prime })\subseteq a+C+H\setminus \{0\},\mathrm{\forall }a\in (u_0\circ G)(x^{\prime }).$(41) According to $(u_0\circ G)(x^{\prime })\subseteq C$ and (41(41) $F(x_0)+(u_0\circ G)(x_0)-F(x^{\prime })\subseteq a+C+H\setminus \{0\},\mathrm{\forall }a\in (u_0\circ G)(x^{\prime }).$(41) ), we obtain $F(x_0)+(u_0\circ G)(x_0)-F(x^{\prime })\subseteq C+H\setminus \{0\}$which implies (42) $F(x^{\prime }){⪯}_{C+H\setminus \{0\}}^{c}F(x_0)+(u_0\circ G)(x_0).$(42) According to (39(39) $F(x_0)+(u_0\circ G)(x_0)\in {\mathrm{\Phi }}_G(u_0).$(39) ), (42(42) $F(x^{\prime }){⪯}_{C+H\setminus \{0\}}^{c}F(x_0)+(u_0\circ G)(x_0).$(42) ) and Corollary 4.1, $u_0$ is a c-globally maximal solution of (DSOP).

5. Proper saddle points

In this section, we introduce and study the notion of proper saddle points of the Lagrangian map L.

Definition 5.1

The pair $(x_0,h_0)\in X\times \mathrm{\Lambda }(Z,Y)$ is called proper saddle point of Lagrangian map L iff the following conditions hold:

1. $L(x_0,h_0)\in c$-$\mathrm{GMin}\{F(x)+(h_0\circ G)(x)|x\in \mathcal{A}\}$;

2. $L(x_0,h_0)\in c$-$\mathrm{GMax}\{F(x_0)+(h\circ G)(x_0)|h\in \mathrm{\Lambda }(Z,Y)\}$.

Let $A\in \mathcal{P}(Y)$ and $\mathrm{\Theta }\subseteq Z^{\ast }$. Write $\mathrm{WMax}A:=\{a\in A|(A-a)\cap \mathrm{int}C=\mathrm{\varnothing }\}$. From now on, we suppose that (43) $D:=\{z\in Z|⟨z,\phi ⟩\ge 0,\mathrm{\forall }\phi \in \mathrm{\Theta }\}.$(43) It is clear that D is a closed convex cone in Z.

Theorem 5.1

Let $(x_0,h_0)\in \mathcal{A}\times \mathrm{\Lambda }(Z,Y)$ and $H\in {\mathcal{H}}_C$. Suppose that the following conditions are satisfied:

1. $(x_0,h_0)$ is a proper saddle point of L;

2. $\mathrm{WMax}\bigcup _{h\in \mathrm{\Lambda }(Z,Y)}(F(x_0)+(h\circ G)(x_0))\ne \mathrm{\varnothing };$

3. There exists $x^{\prime }\in \mathrm{\Omega }$ such that (44) $(h_0\circ G)(x^{\prime })\subseteq C$(44) and (45) $F(x_0)+(h_0\circ G)(x_0)-(F(x^{\prime })+(h_0\circ G)(x^{\prime }))\subseteq C+H\mathrm{\setminus }\{0\}.$(45) Then, $G(x_0)\subseteq -D$. Moreover, $x^{\prime }$ is a c-globally minimal solution of (SOP) and $(x_0,h_0)$ is a feasible pair of (DSOP).

Proof.

Firstly, we prove $G(x_0)\subseteq -D$. Let $z_0\in G(x_0)$. Suppose that $z_0\notin -D$. According to (43(43) $D:=\{z\in Z|⟨z,\phi ⟩\ge 0,\mathrm{\forall }\phi \in \mathrm{\Theta }\}.$(43) ), there exists $\phi \in \mathrm{\Theta }$ such that (46) $⟨z_0,\phi ⟩>0.$(46) Let $e_0\in \mathrm{int}C$. For any $n\in \mathbb{N}$, we define the map $T_n:Z\to Y$ as follows (47) $T_n(z)=n{e}_0⟨z,\phi ⟩,\mathrm{\forall }z\in Z.$(47) It follows from (43(43) $D:=\{z\in Z|⟨z,\phi ⟩\ge 0,\mathrm{\forall }\phi \in \mathrm{\Theta }\}.$(43) ) that $T_n\in \mathrm{\Lambda }(Z,Y)$ for any $n\in \mathbb{N}$. By (46(46) $⟨z_0,\phi ⟩>0.$(46) ) and (47(47) $T_n(z)=n{e}_0⟨z,\phi ⟩,\mathrm{\forall }z\in Z.$(47) ), we have $T_n(z_0)\in \mathrm{int}C$ for any $n\in \mathbb{N}.$ Since $\mathrm{WMax}\bigcup _{h\in \mathrm{\Lambda }(Z,Y)}(F(x_0)+(h\circ G)(x_0))\ne \mathrm{\varnothing }$, we let (48) $y_0\in \mathrm{WMax}\bigcup _{h\in \mathrm{\Lambda }(Z,Y)}(F(x_0)+(h\circ G)(x_0)).$(48) Using (48(48) $y_0\in \mathrm{WMax}\bigcup _{h\in \mathrm{\Lambda }(Z,Y)}(F(x_0)+(h\circ G)(x_0)).$(48) ), we obtain (49) $y_0\in \bigcup _{h\in \mathrm{\Lambda }(Z,Y)}(F(x_0)+(h\circ G)(x_0))$(49) and (50) $(\bigcup _{h\in \mathrm{\Lambda }(Z,Y)}(F(x_0)+(h\circ G)(x_0))-y_0)\cap \mathrm{int}C=\mathrm{\varnothing }.$(50) By (49(49) $y_0\in \bigcup _{h\in \mathrm{\Lambda }(Z,Y)}(F(x_0)+(h\circ G)(x_0))$(49) ), there exists $h^{\prime }\in \mathrm{\Lambda }(Z,Y)$ such that $y_0\in F(x_0)+(h^{\prime }\circ G)(x_0).$ Therefore, there exist $y_0^{\prime }\in F(x_0),z^{\prime }\in G(x_0),T\in {\mathcal{L}}_{+}(Z,Y)$ and $m\in -C$ such that (51) $y_0=y_0^{\prime }+T(z^{\prime })+m.$(51) Let (52) $U:=e_0-\mathrm{int}C.$(52) Clearly, U is a neighborhood of 0 in Y. Because U is absorbent, there exists $n_0\in \mathbb{N}$ such that (53) $\frac{1}{n_0⟨z_0,\phi ⟩}(T(z^{\prime })-T(z_0))\in U.$(53) By (52(52) $U:=e_0-\mathrm{int}C.$(52) ) and (53(53) $\frac{1}{n_0⟨z_0,\phi ⟩}(T(z^{\prime })-T(z_0))\in U.$(53) ), we have (54) $T(z^{\prime })-T(z_0)\in n_0⟨z_0,\phi ⟩e_0-n_0⟨z_0,\phi ⟩\mathrm{int}C\subseteq n_0⟨z_0,\phi ⟩e_0-\mathrm{int}C.$(54) According to (54(54) $T(z^{\prime })-T(z_0)\in n_0⟨z_0,\phi ⟩e_0-n_0⟨z_0,\phi ⟩\mathrm{int}C\subseteq n_0⟨z_0,\phi ⟩e_0-\mathrm{int}C.$(54) ), there exists $m^{\prime }\in -\mathrm{int}C$ such that (55) $T(z^{\prime })-T(z_0)=n_0⟨z_0,\phi ⟩e_0+m^{\prime }.$(55) Let (56) $n^{\prime }:=n_0+1$(56) and (57) $\beta :=y_0^{\prime }+(T+T_{n^{\prime }})(z_0)+m+m^{\prime }.$(57) We define the map $h^{″}:Z\to Y$ as follows (58) $h^{″}(z)=(T+T_{n^{\prime }})(z)+m+m^{\prime },\mathrm{\forall }z\in Z.$(58) Clearly, $h^{″}\in \mathrm{\Lambda }(Z,Y)$. By (57(57) $\beta :=y_0^{\prime }+(T+T_{n^{\prime }})(z_0)+m+m^{\prime }.$(57) ) and (58(58) $h^{″}(z)=(T+T_{n^{\prime }})(z)+m+m^{\prime },\mathrm{\forall }z\in Z.$(58) ), we obtain (59) $\beta =y_0^{\prime }+h^{″}(z_0)\in F(x_0)+(h^{″}\circ G)(x_0)\subseteq \bigcup _{h\in \mathrm{\Lambda }(Z,Y)}(F(x_0)+(h\circ G)(x_0)).$(59) By (47(47) $T_n(z)=n{e}_0⟨z,\phi ⟩,\mathrm{\forall }z\in Z.$(47) ), (51(51) $y_0=y_0^{\prime }+T(z^{\prime })+m.$(51) ), (55(55) $T(z^{\prime })-T(z_0)=n_0⟨z_0,\phi ⟩e_0+m^{\prime }.$(55) ), (56(56) $n^{\prime }:=n_0+1$(56) ) and (57(57) $\beta :=y_0^{\prime }+(T+T_{n^{\prime }})(z_0)+m+m^{\prime }.$(57) ), we have (60) $\beta -y_0=⟨z_0,\phi ⟩e_0\in \mathrm{int}C.$(60) It follows from (59(59) $\beta =y_0^{\prime }+h^{″}(z_0)\in F(x_0)+(h^{″}\circ G)(x_0)\subseteq \bigcup _{h\in \mathrm{\Lambda }(Z,Y)}(F(x_0)+(h\circ G)(x_0)).$(59) ) and (60(60) $\beta -y_0=⟨z_0,\phi ⟩e_0\in \mathrm{int}C.$(60) ) that $(\bigcup _{h\in \mathrm{\Lambda }(Z,Y)}(F(x_0)+(h\circ G)(x_0))-y_0)\cap \mathrm{int}C\ne \mathrm{\varnothing },$which contradicts (50(50) $(\bigcup _{h\in \mathrm{\Lambda }(Z,Y)}(F(x_0)+(h\circ G)(x_0))-y_0)\cap \mathrm{int}C=\mathrm{\varnothing }.$(50) ). Hence, $G(x_0)\subseteq -D$.

By Condition (i) and Definition 4.2, $(x_0,h_0)$ is a feasible pair of (DSOP). It follows from (45(45) $F(x_0)+(h_0\circ G)(x_0)-(F(x^{\prime })+(h_0\circ G)(x^{\prime }))\subseteq C+H\mathrm{\setminus }\{0\}.$(45) ) that (61) $F(x_0)+(h_0\circ G)(x_0)-F(x^{\prime })\subseteq a+C+H\mathrm{\setminus }\{0\},\mathrm{\forall }a\in (h_0\circ G)(x^{\prime }).$(61) Using (61(61) $F(x_0)+(h_0\circ G)(x_0)-F(x^{\prime })\subseteq a+C+H\mathrm{\setminus }\{0\},\mathrm{\forall }a\in (h_0\circ G)(x^{\prime }).$(61) ) and (44(44) $(h_0\circ G)(x^{\prime })\subseteq C$(44) ), we have $F(x_0)+(h_0\circ G)(x_0)-F(x^{\prime })\subseteq C+H\mathrm{\setminus }\{0\}$which implies (62) $F(x^{\prime }){\le }_{C+H\setminus \{0\}}^{c}F(x_0)+(h_0\circ G)(x_0).$(62) According to (62(62) $F(x^{\prime }){\le }_{C+H\setminus \{0\}}^{c}F(x_0)+(h_0\circ G)(x_0).$(62) ) and Corollary 4.1, $x^{\prime }$ is a c-globally minimal solution of (SOP).

6. Conclusions

In this paper, we investigate set optimization problems. The notion of the globally proper efficient element of the power set is introduced. Some properties of the globally proper efficient element are characterized. Under the assumption of the cone convexlikeness of the set-valued map, a Lagrange multiplier rule of the set optimization problem is presented. Lagrangian duality theorems, including a weak duality theorem and a strong duality theorem, are established. Optimality conditions involving saddle points are obtained. It will be interesting to establish optimality conditions in the sense of new proper efficiency of set optimization problems, such as Benson proper efficiency, Henig proper efficiency and super efficiency.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by the National Nature Science Foundation of China (12171061 and 12071379), the Science and Technology Research Program of Chongqing Education Commission (KJZD-K202001104) and the Graduate Innovation Project of Chongqing (CYS21473).