The Continuity and Convexity of a Nonlinear Scalarization Function with Applications in Set Optimization Problems Involving a Partial Order Relation

Abstract

In this paper, we deal with the properties and applications of a nonlinear scalarization function for sets by using the Minkowski difference. Under some suitable assumptions, the continuity and convexity concerned with the nonlinear scalarization function for sets are presented. As applications, the path connectedness of the solution sets to set optimization problems and the continuity of the solution mappings of parametric set optimization problems are established. The results achieved do not impose the monotonicity of the set-valued objective mapping, which are obviously different from the related ones in the literature.

1. Introduction

In recent years, the extension of the vector-valued optimization problem to the set-valued optimization problem has received increasing attention due to its wide applications in many fields such as optimal control, differential inclusions, game theory, robust optimization, fuzzy optimization, welfare economics, and mathematical finance; see, for instance [1,2,3,4,5] and the references therein. In the literature, there are mainly two types of criteria of solutions for set-valued optimization problems. The classical one is the vector criterion: a solution of a set-valued map is defined via the minimal element of the image set of the map with respect to the usual ordering relation of vector optimization. Although it is of mathematical interest, it does not seem natural whenever one needs to consider preferences over sets, since only one element does not necessarily imply that the whole image set is in a certain sense minimal with respect to all image sets. In order to overcome this drawback, the set criterion is proposed by Kuroiwa [6]. The set criterion is that a solution is defined via the minimal set of the collection of all image sets with respect to set order relations. Regarding this criterion, there are some other set order relations considered in set-valued optimization problems. We refer the reader to [7,8] and the references therein for more details. A set-valued optimization problem with a set criterion is called a set optimization problem.

It is well known that nonlinear scalarization functions are the most essential tools in vector or set optimization. There are mainly two types of scalarization functions in vector optimization. They are the Gerstewitz function [1,9,10] and the oriented distance function [11,12]. Accordingly, in set optimization, there are also two kinds of scalarization functions have been used: the extensions of the Gerstewitz function [8,13,14,15,16,17] and the extensions of the oriented distance function of Hiriart-Urruty [13,18,19,20]. In terms of scalarization techniques, several theoretical aspects of set optimization were discussed, such as the characterization of several types of optimal solutions, alternative theorems, optimality conditions, and the well-posedness; see, for instance, [13,14,16,17,20,21,22,23,24,25,26] and the references therein.

As we know, the continuity and convexity of the extensions of the Gerstewitz function play a significant role in the existence and the stability of set optimization problems. For more details, we refer the reader to [14,27]. Therefore, it is necessary and interesting to investigate the continuity and convexity for the extensions of the Gerstewitz function. In this paper, we give the continuity and convexity of the nonlinear scalarization function proposed by Karaman et al. in [8], which is defined by a partial set order relation involving the Minkowski difference. As applications, we study the path connectedness and stability of set optimization problems.

The rest of the paper is organized as follows. In Section 2, we introduce some definitions and previous results required throughout the paper. In Section 3, by using some properties of nonlinear scalarization functions defined in [8] and well-known results for set-valued mappings, we prove the continuity and convexity of a nonlinear scalarization function for sets. In Section 4, we explore an application of properties of the nonlinear scalarization function to the path connectedness of the solution set of a set optimization problem. In Section 5, we provide the other application of the nonlinear scalarization function to the continuity of solution mappings of a parametric set optimization problem. In Section 6, we give the concluding remarks of the paper.

2. Preliminaries

In this section, we recall some basic definitions and properties which are necessary for this study. Throughout this paper, X and Y are two normed vector spaces. Given a subset A of Y, the closure, the complement, the topological interior, the boundary, and the convex hull of A are denoted, respectively, by cl A, $A^c$, int A, $\mathrm{bd}A$, and $\mathrm{conv}A$. We denote by ${\mathbb{B}}_Y$ the closed unit ball in Y. The family of nonempty proper subsets of Y, the family of nonempty bounded subsets of Y, and the family of nonempty compact subsets of Y are denoted by ${\mathcal{P}}_0\left(Y\right)$, ${\mathcal{B}}^{*}\left(Y\right)$, and ${\mathcal{B}}^{**}\left(Y\right)$, respectively. For every $A,B\in {\mathcal{P}}_0\left(Y\right)$ and $\lambda \in \mathbb{R}$, we denote, respectively,

$$A-B=\{y_1-y_2:y_1\in A,y_2\in B\},\lambda A=\{\lambda y:y\in A\}$$

by the algebraic difference of the sets A and B and the scalar multiplication of the set A.

A nonempty subset $C\subseteq {\mathbb{R}}^m$ is said to be a cone if $tC\subseteq C$ for all $t\ge 0$. A cone $C\subseteq {\mathbb{R}}^m$ is said to be convex (pointed) if and only if $C+C\subseteq C$($C\cap (-C)=\left\{0_{{\mathbb{R}}^m}\right\})$. Throughout the paper, we assume that C is a closed, convex, and pointed cone with a nonempty interior. Let $Y^{*}$ be the topological dual space of Y and the dual cone of C be denoted by $C^{*}$, which is defined by $C^{*}=\{f\in Y^{*}:f\left(c\right)\ge 0,\forall c\in C\}$.

For A, $B\in {\mathcal{P}}_0\left(Y\right)$, the Minkowski difference of A and B, considered in [28], is given by

$$B\dot{-}A:=\{z\in Z:z+A\subseteq B\}=\bigcap _{a\in A}(B-a).$$

It is worth mentioning that the Minkowski difference of a set and a vector coincide with the algebraic difference of them, that is, $A\dot{-}b=A-b$ for all $A\in {\mathcal{P}}_0\left(Y\right)$ and $b\in Y$. Also, $\left(A\dot{-}B\right)-b=A\dot{-}B-b=A\dot{-}(B+b)$ and $(A-b)\dot{-}B=\left(A\dot{-}B\right)-b$ for all $A,B\in {\mathcal{P}}_0\left(Y\right)$ and $b\in Y$.

We now recall some order relations on ${\mathcal{P}}_0\left(Y\right)$. The first one is the lower set order relation ${⪯}_C^l$ and the upper set order relation ${⪯}_C^u$ on ${\mathcal{P}}_0\left(Y\right)$, which are discussed by [6].

$$A{⪯}_C^{l}B\iff B\subseteq A+C;A{⪯}_C^{u}B\iff A\subseteq B-C.$$

What is noteworthy is that ${⪯}_C^l$ and ${⪯}_C^u$ are pre-order relations, i.e., reflexive and transitive relations, on ${\mathcal{P}}_0\left(Y\right)$. Recently, by using the Minkowski difference, Karaman et al. [8] introduced the following partial order relations on the family of nonempty bounded sets, which are reflexive, transitive, and antisymmetric on ${\mathcal{B}}^{*}\left(Y\right)$.

Definition 1 

([8]). Let $A,B\in {\mathcal{P}}_0\left(Y\right)$.

(i) 

$A{⪯}_C^{m_1}B⟺\left(B\dot{-}A\right)\cap C\ne Ø.$

(ii) 

$A{\prec }_C^{m_1}B⟺\left(B\dot{-}A\right)\cap \mathrm{int}C\ne Ø.$

(iii) 

$A{⪯}_C^{m_2}B⟺\left(A\dot{-}B\right)\cap -C\ne Ø.$

(iv) 

$A{\prec }_C^{m_2}B⟺\left(A\dot{-}B\right)\cap -\mathrm{int}C\ne Ø.$

In the rest of this paper, we only consider the order relation ${⪯}_C^{m_2}$ since we can obtain the corresponding results for the order relation ${⪯}_C^{m_1}$.

Let K be a nonempty subset of X. Let $F:K\rightrightarrows Y$ be a set-valued mapping. We consider the following set optimization problem with set order ${⪯}_C^{m_2}$ (for short, $m_2$-SOP):

$$\left(m_2\text{-}\mathrm{SOP}\right){\min }_{C}F\left(x\right)\mathrm{subject}\mathrm{to}x\in K.$$

When the set K and the mapping F are perturbed by a parameter λ which varies over a subset Λ in a normed space, we consider the following parametric set optimization problem with set order ${⪯}_C^{m_2}$ (for short, m₂-PSOP):

$$\left(m_2\text{-}\mathrm{PSOP}\right){\min }_{C}F(x,\lambda )\mathrm{subject}\mathrm{to}x\in K\left(\lambda \right).$$

Definition 2 

([8]). An element $x_0\in K$ is said to be

(i) 

a $m_2$-minimal solution of ($m_2$-SOP) if there does not exist any $x\in K$ with $F\left(x\right)\ne F\left(x_0\right)$ such that $F\left(x\right){⪯}_C^{m_2}F\left(x_0\right)$, that is, either $F\left(x\right){⋠}_C^{m_2}F\left(x_0\right)$ or $F\left(x\right)=F\left(x_0\right)$ for any $x\in K$;

(ii) 

a weak $m_2$-minimal solution of ($m_2$-SOP) if there does not exist any $x\in K$ such that $F\left(x\right){\prec }_C^{m_2}F\left(x_0\right)$.

Let ${\mathrm{argmin}}_{m_2}(K,F)$ and ${\mathrm{argmin}}_{w{m}_2}(K,F)$ denote the $m_2$-minimal solution set of ($m_2$-SOP) and the weak $m_2$-minimal solution set of ($m_2$-SOP), respectively. In addition, the solution concepts of the problem $(m_2$-PSOP) can be similarly defined. For each $\lambda \in \Lambda$, let $S\left(\lambda \right)$ and $S_w\left(\lambda \right)$ denote the $m_2$-minimal solution set of ($m_2$-PSOP) and the weak $m_2$-minimal solution set of ($m_2$-PSOP), respectively. Throughout the paper, we always assume that ${\mathrm{argmin}}_{m_2}(K,F)\ne Ø$ and $S\left(\lambda \right)\ne Ø$.

To give the relationship between the sets ${\mathrm{argmin}}_{m_2}$ (K, F) and ${\mathrm{argmin}}_{{wm}_2}$ (K, F), we need to recall a vital conclusion presented by Karaman et al. in [8].

Lemma 1. 

If $A\in {\mathcal{B}}^{*}\left(Y\right)$, then $A\dot{-}A=\left\{0_Y\right\}$.

Proposition 1. 

Assume that $F\left(x\right)$ is bounded for each $x\in K$. Then

$${\mathrm{argmin}}_{m_2}(K,F)\subseteq {\mathrm{argmin}}_{w{m}_2}(K,F).$$

Proof. 

Suppose, to the contrary, that there exists $x_0\in {\mathrm{argmin}}_{m_2}(K,F)$ such that $x_0\notin {\mathrm{argmin}}_{w{m}_2}(K,F)$. Then, there exists $y_0\in K$ such that

$$\left(F\left(y_0\right)\dot{-}F\left(x_0\right)\right)\cap (-\mathrm{int}C)\ne Ø.$$

(1)

This together with Lemma 1 gives us $F\left(x_0\right)\ne F\left(y_0\right)$. Moreover, (1) also implies that $F\left(y_0\right){⪯}_C^{m_2}F\left(x_0\right)$. Thus, it follows that $x_0\notin {\mathrm{argmin}}_{m_2}(K,F)$, which leads to a contradiction. □

The following example is given to show that the statement ${\mathrm{argmin}}_{m_2}(K,F)\subseteq {\mathrm{argmin}}_{w{m}_2}(K,F)$ may be not true if the values of F are unbounded.

Example 1. 

Let $K=[0,1]$, $Y={\mathbb{R}}^2$, $C={\mathbb{R}}_{+}^2$ and $e=(1,1)$. Let $F:K\rightrightarrows Y$ be defined by

$$F\left(x\right)=\left\{\begin{array}{cc}\mathbb{R}e,\hfill & if x\in \{0,1\},\hfill \\ \left\{0_{{\mathbb{R}}^2}\right\},\hfill & if x\in (0,1).\hfill \end{array}\right.$$

It is easy to verify that ${\mathrm{argmin}}_{m_2}(K,F)=\{0,1\}$ and ${\mathrm{argmin}}_{w{m}_2}(K,F)=Ø$.

Definition 3. 

Let $\Omega$ be a nonempty convex subset of X. A set-valued mapping $\Phi :X\rightrightarrows Y$ is said to be

(i) 

$m_2$-naturally C-quasiconvex on $\Omega$ if, for any $x_1,x_2\in \Omega$ and for any $\alpha \in [0,1]$, there exists $\beta \in [0,1]$ such that

$$\Phi \left(\alpha x_1+(1-\alpha )x_2\right){⪯}_C^{m_2}\beta \Phi \left(x_1\right)+(1-\beta )\Phi \left(x_2\right).$$

(ii) 

strictly $m_2$-naturally C-quasiconvex on $\Omega$ if, for any $x_1,x_2\in \Omega$ with $x_1\ne x_2$ and for any $\alpha \in (0,1)$, there exists $\beta \in [0,1]$ such that

$$\Phi \left(\alpha x_1+(1-\alpha )x_2\right){\prec }_C^{m_2}\beta \Phi \left(x_1\right)+(1-\beta )\Phi \left(x_2\right).$$

(iii) 

$m_2$-properly C-quasiconvex if, for any $x_1,x_2\in \Omega$ and for any $\alpha \in [0,1]$, one has

$$\mathrm{either}\Phi \left(\alpha x_1+(1-\alpha )x_2\right){⪯}_C^{m_2}\Phi \left(x_1\right)\mathrm{or}\Phi \left(\alpha x_1+(1-\alpha )x_2\right){⪯}_C^{m_2}\Phi \left(x_2\right)$$

(iv) 

strictly $m_2$-properly C-quasiconvex if, for any $x_1,x_2\in \Omega$ with $x_1\ne x_2$ and for any $\alpha \in (0,1)$, one has

$$\mathrm{either}\Phi \left(\alpha x_1+(1-\alpha )x_2\right){\prec }_C^{m_2}\Phi \left(x_1\right)\mathrm{or}\Phi \left(\alpha x_1+(1-\alpha )x_2\right){\prec }_C^{m_2}\Phi \left(x_2\right)$$

Remark 1. 

(i) 

If $\Phi$ is a single-valued mapping, then the above definitions of $m_2$-natural C-quasiconvexity (strictly $m_2$-natural C-quasiconvexity) and $m_2$-proper C-quasiconvexity (strictly $m_2$-proper C-quasiconvexity) reduce to classic definitions of natural C-quasiconvexity in [29] and proper C-quasiconvexity in [30], respectively.

(ii) 

It is easy to see that $m_2$-proper C-quasiconvexity (strictly $m_2$-proper C-quasiconvexity) leads to $m_2$-natural C-quasiconvexity (strictly $m_2$-natural C-quasiconvexity), respectively. However, the converse may be not valid. The following example is given to show the case.

Example 2. 

Let $X=\mathbb{R}$, $Y={\mathbb{R}}^2$, and $C={\mathbb{R}}_{+}^2$. Let $\Omega =[-1,1]$ and

$$\Phi \left(x\right)=(x,-x)+{\mathbb{B}}_Y.$$

It is not hard to check that Φ is $m_2$-naturally C-quasiconvex on Ω. Indeed, for any $x_1,x_2\in \Omega$ and for any $\alpha \in [0,1]$, there exists $\beta =\alpha$ such that

$$\Phi \left(\alpha x_1+(1-\alpha )x_2\right){⪯}_C^{m_2}\beta \Phi \left(x_1\right)+(1-\beta )\Phi \left(x_2\right).$$

However, it is not $m_2$-properly C-quasiconvex on Ω. Indeed, let $x_1=-1$, $x_2=1$, and $\alpha ={\displaystyle \frac{1}{2}}$. Let $A=\Phi \left(\alpha x_1+(1-\alpha )x_2\right)=\left\{(x,y)\in {\mathbb{R}}^2:x^2+y^2\le 1\right\}$, $B=\Phi \left(x_1\right)=\left\{(x,y)\in {\mathbb{R}}^2:{(x+1)}^2+{(y-1)}^2\le 1\right\}$, and $D=\Phi \left(x_2\right)=\left\{(x,y)\in {\mathbb{R}}^2:{(x-1)}^2+{(y+1)}^2\le 1\right\}$ (see Figure 1). It follows from a direct computation that $A\dot{-}B=\left\{e\right\}=\left\{(1,-1)\right\}$ (see Figure 2) and $A\dot{-}D=\left\{f\right\}=\left\{(-1,1)\right\}$ (see Figure 2). Hence, $\left(A\dot{-}B\right)\cap -C=Ø$ and $\left(A\dot{-}D\right)\cap -C=Ø$. Consequently, Φ is not $m_2$-properly C-quasiconvex on Ω.

Definition 4 

([31]). A topological space T is said to be path-connected (or arcwise-connected) if, for any $x,y\in T$, there exists a continuous mapping $\gamma :[0,1]\to T$ such that $\gamma \left(0\right)=x$ and $\gamma \left(1\right)=y$.

Definition 5 

([32]). Let $T_1$ and $T_2$ be two topological vector spaces. A set-valued mapping $G:T_1\rightrightarrows T_2$ is said to be

(i) 

lower semicontinuous (for short, l.s.c.) at $\overline{t}\in T_1$ if, for every open set $V\subseteq T_2$ with $G\left(\overline{t}\right)\cap V\ne Ø$, there is a neighborhood $N\left(\overline{t}\right)$ of $\overline{t}$ such that for any $t\in N\left(\overline{t}\right)$ with $G\left(t\right)\cap V\ne Ø$;

(ii) 

upper semicontinuous (for short, u.s.c.) at $\overline{t}\in T_1$ if, for every open set $V\subseteq T_2$ with $G\left(\overline{t}\right)\subseteq V$, there is a neighborhood $N\left(\overline{t}\right)$ of $\overline{t}$ such that for any $t\in N\left(\overline{t}\right)$ with $G\left(t\right)\subseteq V$;

(iii) 

Hausdorff upper semicontinuous (for short, H-u.s.c.) at $\overline{t}\in T_1$ if, for each neighborhood U of the zero point $0_{T_2}$, there is a neighborhood $N\left(\overline{t}\right)$ of $\overline{t}$ such that for any $t\in N\left(\overline{t}\right)$ with $G\left(t\right)\subseteq G\left(\overline{t}\right)+U$.

We say that G is l.s.c. (u.s.c.) on $T_1$ if it is l.s.c. (u.s.c.) at each $t\in T_1$. G is said to be continuous on $T_1$ if it is both l.s.c. and u.s.c. on $T_1$.

Proposition 2 

([33,34]). Assume that $T_1$ and $T_2$ be two normed vector spaces. Let $G:T_1\rightrightarrows T_2$ be a set-valued mapping. Then, the following statements are true.

(i) 

G is l.s.c. at $\overline{t}$ if and only if for any sequence $\left\{t_n\right\}\subset T_1$ with $t_n\to \overline{t}$ and for any $\overline{x}\in G\left(\overline{t}\right)$, there exists $x_n\in G\left(t_n\right)$ such that $x_n\to \overline{x}$.

(ii) 

If G has compact values at $\overline{t}$, then G is u.s.c. at $\overline{t}$ if and only if for any net $\left\{t_n\right\}\subset T_1$ with $t_n\to \overline{t}$ and any $x_n\in G\left(t_n\right)$, there exist $\overline{x}\in G\left(\overline{t}\right)$ and a subsequence $\left\{x_{n_k}\right\}$ of $\left\{x_n\right\}$ such that $x_{n_k}\to \overline{x}$.

Lemma 2 

(See Proposition 23 of Chapter 3 in [33]). Let $W:X\times Y\to \mathbb{R}$ be a real-valued function and $G:Y\rightrightarrows X$ be a set-valued mapping. The marginal set-valued mapping $M:Y\rightrightarrows X$ is defined by $M\left(y\right)=\{z\in G\left(y\right):W(z,y)={sup}_{x\in G\left(y\right)}W(x,y)\}$. If W is continuous on $X\times Y$ and G is continuous with compact values on Y, then M is u.s.c. on Y.

Definition 6 

([8]). For $e\in \mathrm{int}C$, the function $I_e^{m_2}(·,·):{\mathcal{P}}_0\left(Y\right)\times {\mathcal{P}}_0\left(Y\right)\to \mathbb{R}\cup \{\pm \infty \}$ is defined as

$$I_e^{m_2}(A,B):=inf\{t\in \mathbb{R}:A{⪯}_C^{m_2}te+B\}.$$

Proposition 3 

([8]).

(i) 

If $A\in {\mathcal{B}}^{*}\left(Y\right)$ and $B\in {\mathcal{P}}_0\left(Y\right)$, then $I_e^{m_2}(A,B)>-\infty$;

(ii) 

Let $A,B\in {\mathcal{P}}_0\left(Y\right)$. Then, $I_e^{m_2}(A,B)=+\infty$ if and only if $A\dot{-}B=Ø$.

Proposition 4 

([8]). If $A,B\in {\mathcal{P}}_0\left(Y\right)$ and $A\dot{-}B$ is compact, then $I_e^{m_2}(A,B)=min\{t\in \mathbb{R}:A{⪯}_C^{m_2}te+B\}$.

Proposition 5 

([8]). Let $A,B\in {\mathcal{P}}_0\left(Y\right)$ and $r\in \mathbb{R}$. Then, the following statements hold.

(i) 

$I_e^{m_2}(A,B)<r$ if and only if $A{\prec }_C^{m_2}re+B$.

(ii) 

Let $A\dot{-}B$ be compact, then $I_e^{m_2}(A,B)\le r$ if and only if $A{⪯}_C^{m_2}re+B$.

Proposition 6 

([8]). If $A\in {\mathcal{B}}^{*}\left(Y\right)$, then $I_e^{m_2}(A,A)=0$.

Proposition 7 

([8]).

(i) 

Let $B\in {\mathcal{P}}_0\left(Y\right)$. Then, $I_e^{m_2}(·,B)$ is $m_2$-increasing on ${\mathcal{P}}_0\left(Y\right)$, that is, if $D,E\in {\mathcal{P}}_0\left(Y\right)$ and $D{⪯}_C^{m_2}E$ implies that $I_e^{m_2}(D,B)\le I_e^{m_2}(E,B)$.

(ii) 

Let $B\in {\mathcal{P}}_0\left(Y\right)$. Then, $I_e^{m_2}(·,B)$ is strictly $m_2$-increasing on ${\mathcal{B}}^{**}\left(Y\right)$, that is, if $D,E\in {\mathcal{B}}^{**}\left(Y\right)$ and $D{\prec }_C^{m_2}E$, this implies that $I_e^{m_2}(D,B)<I_e^{m_2}(E,B)$.

Proposition 8 

([8]).

(i) 

Let $A\in {\mathcal{P}}_0\left(Y\right)$. Then, $I_e^{m_2}(A,·)$ is $m_2$-decreasing on ${\mathcal{P}}_0\left(Y\right)$, that is, if $D,E\in {\mathcal{P}}_0\left(Y\right)$ and $D{⪯}_C^{m_2}E$, this implies that $I_e^{m_2}(A,D)\ge I_e^{m_2}(A,E)$.

(ii) 

Let $A\in {\mathcal{B}}^{**}\left(Y\right)$. Then, $I_e^{m_2}(A,·)$ is strictly $m_2$-decreasing on ${\mathcal{P}}_0\left(Y\right)$, that is, if $D,E\in {\mathcal{P}}_0\left(Y\right)$ and $D{\prec }_C^{m_2}E$, this implies that $I_e^{m_2}(A,D)>I_e^{m_2}(A,E)$.

Proposition 9. 

If $A,B\in {\mathcal{P}}_0\left(Y\right)$, then $I_e^{m_2}(\delta A,\delta B)=\delta I_e^{m_2}(A,B)$ for all $\delta \ge 0$.

Proof. 

It follows from Proposition 6 that the conclusion obviously holds for $\delta =0$. Now, we show that it is valid for $\delta >0$. Indeed, for $A,B\in {\mathcal{P}}_0\left(Y\right)$ and $\delta >0$, we have

$$\begin{array}{ccc}\hfill I_e^{m_2}(\delta A,\delta B)& =& inf\{t\in \mathbb{R}:\delta A{⪯}_C^{m_2}te+\delta B\}\hfill \\ & =& inf\{t\in \mathbb{R}:A{⪯}_C^{m_2}{\displaystyle \frac{t}{\delta }}e+B\}\hfill \\ & =& inf\{\delta \gamma \in \mathbb{R}:A{⪯}_C^{m_2}\gamma e+B\}\hfill \\ & =& \delta inf\{\gamma \in \mathbb{R}:A{⪯}_C^{m_2}\gamma e+B\}\hfill \\ & =& \delta I_e^{m_2}(A,B).\hfill \end{array}$$

Therefore, the proof is complete. □

3. Continuity and Convexity of a Nonlinear Scalarization Function

In this section, we shall show the continuity and convexity of a nonlinear scalarization function based on the function $I_e^{m_2}(·,·)$.

Let ${\Lambda }_1$ and ${\Lambda }_2$ be two normed vector spaces. Assume that $A:{\Lambda }_1\rightrightarrows Y$ and $B:{\Lambda }_2\rightrightarrows Y$ are two set-valued mappings. We define $\omega :{\Lambda }_1\times {\Lambda }_2\to \mathbb{R}\cup \{\pm \infty \}$ as follows:

$$\omega (\mu ,\eta ):=I_e^{m_2}\left(A\left(\mu \right),B\left(\eta \right)\right)=inf\left\{t\in \mathbb{R}:A\left(\mu \right){⪯}_C^{m_2}te+B\left(\eta \right)\right\},(\mu ,\eta )\in {\Lambda }_1\times {\Lambda }_2.$$

In the sequel, we always assume that, for any given $\mu \in {\Lambda }_1$, $\eta \in {\Lambda }_2$, $A\left(\mu \right)\in {\mathcal{B}}^{*}\left(Y\right)$, $B\left(\eta \right)\in {\mathcal{P}}_0\left(Y\right)$, and $A\left(\mu \right)\dot{-}B\left(\eta \right)\ne Ø$. Then, by Proposition 3, we have

$$-\infty <\omega (\mu ,\eta )<+\infty ,\forall (\mu ,\eta )\in {\Lambda }_1\times {\Lambda }_2.$$

Proposition 10. 

Assume that $A\left(\mu \right)\dot{-}B\left(\eta \right)\ne Ø$ for any $(\mu ,\eta )\in {\Lambda }_1\times {\Lambda }_2$. Assume that $A(·)$ and $B(·)$ are continuous with nonempty compact values. Then, $\omega (·,·)$ is continuous on ${\Lambda }_1\times {\Lambda }_2$.

Proof. 

Firstly, we prove that $\omega (·,·)$ is lower semicontinuous on ${\Lambda }_1\times {\Lambda }_2$. For any $r\in \mathbb{R}$, let

$$M_l=\left\{(\mu ,\eta )\in {\Lambda }_1\times {\Lambda }_2:\omega (\mu ,\eta )\le r\right\}.$$

It suffices to show that $M_l$ is a closed subset of ${\Lambda }_1\times {\Lambda }_2$. Let $\left\{({\mu }_n,{\eta }_n)\right\}\subseteq M_l$ with $({\mu }_n,{\eta }_n)\to ({\mu }_0,{\eta }_0)$. Suppose that $({\mu }_0,{\eta }_0)\notin M_l$. Then, $A\left({\mu }_0\right){⋠}_C^{m_2}re+B\left({\eta }_0\right)$ by Proposition 5 (ii). This indicates that $\left(A\left({\mu }_0\right)\dot{-}B\left({\eta }_0\right)-re\right)\cap -C=Ø$. According to $A\left({\mu }_0\right)\dot{-}B\left({\eta }_0\right)\ne Ø$, we have

$$x-re\notin -C,\forall x\in A\left({\mu }_0\right)\dot{-}B\left({\eta }_0\right).$$

(2)

Now, we claim that there exists $N_0\in \mathbb{N}$ such that when $n\ge N_0$,

$$x_n-re\notin -C,\forall x_n\in A\left({\mu }_n\right)\dot{-}B\left({\eta }_n\right).$$

(3)

If not, then for any n, there exist $m_n\ge n$ and ${\overline{x}}_{m_n}\in A\left({\mu }_{m_n}\right)\dot{-}B\left({\eta }_{m_n}\right)$ such that ${\overline{x}}_{m_n}-re\in -C$. Without loss of generality, we assume that there exists ${\overline{x}}_n$ with

$${\overline{x}}_n+B\left({\eta }_n\right)\subseteq A\left({\mu }_n\right),$$

(4)

such that

$${\overline{x}}_n-re\in -C,\forall n.$$

(5)

Thanks to (4), we have ${\overline{x}}_n+y_n\in A\left({\mu }_n\right)$ for any $y_n\in B\left({\eta }_n\right)$. As $A(·)$ is u.s.c. with compact values at ${\mu }_0$ and $B(·)$ is u.s.c. with compact values at ${\eta }_0$, we see that the sequences $\left\{y_n\right\}$ and $\{{\overline{x}}_n+y_n\}$ have convergent subsequences by Proposition 2 (ii). Consequently, $\left\{{\overline{x}}_n\right\}$ has a convergent subsequence $\left\{{\overline{x}}_{n_k}\right\}$. Without loss of the generality, we assume that ${\overline{x}}_n\to x_0$. Now, we show that $x_0\in A\left({\mu }_0\right)\dot{-}B\left({\eta }_0\right)$. Indeed, for any $b\in B\left({\eta }_0\right)$, by the lower semicontinuity of the mapping $B(·)$, there exists $b_n\in B\left({\eta }_n\right)$ such that $b_n\to b$. It follows from (4) that there exists $a_n\in A\left({\mu }_n\right)$ such that

$${\overline{x}}_n+b_n=a_n.$$

(6)

Since $A(·)$ is u.s.c. with compact values at ${\mu }_0$, by Proposition 2 (ii), there exist $a_0\in A\left({\mu }_0\right)$ and a subsequence $\left\{a_{n_k}\right\}$ of $\left\{a_n\right\}$ such that $a_{n_k}\to a_0$. Combining this with ${\overline{x}}_n\to x_0$, $b_n\to b$ and (6), we arrive at $x_0+b=a_0\in A\left({\mu }_0\right)$. Moreover, by the arbitrariness of $b\in B\left({\eta }_0\right)$, we have $x_0+B\left({\eta }_0\right)\subseteq A\left({\mu }_0\right)$, i.e., $x_0\in A\left({\mu }_0\right)\dot{-}B\left({\eta }_0\right)$. With the help of (5) and ${\overline{x}}_n\to x_0$, and taking into account the closedness of C, we have

$$x_0-re\in -C,$$

which contradicts (2). Hence, (3) holds. This implies that $A\left({\mu }_n\right){⋠}_C^{m_2}re+B\left({\eta }_n\right)$ and so $\omega ({\mu }_n,{\eta }_n)>r$ by Proposition 5 (ii). This leads to a contradiction. Therefore, $M_l$ is a closed subset of ${\Lambda }_1\times {\Lambda }_2$.

Next, we prove that $\omega (·,·)$ is upper semicontinuous on ${\Lambda }_1\times {\Lambda }_2$. For any $r\in \mathbb{R}$, let

$$M_u=\{x\in X:\omega (\eta ,\mu )\ge r\}.$$

It suffices to show that $M_u$ is a closed subset of ${\Lambda }_1\times {\Lambda }_2$. Indeed, let $\left\{({\eta }_n,{\mu }_n)\right\}\subseteq M_u$ with $({\eta }_n,{\mu }_n)\to ({\eta }_0,{\mu }_0)$. Then, $A\left({\mu }_n\right){⋠}_C^{m_2}re+B\left({\eta }_n\right)$ by Proposition 5 (i). This implies that

$$\left(A\left({\mu }_n\right)\dot{-}B\left({\eta }_n\right)-re\right)\cap -\mathrm{int}C=Ø,\forall n\in \mathbb{N}.$$

(7)

Now, we claim that $\left(A\left({\mu }_0\right)\dot{-}B\left({\lambda }_0\right)-re\right)\cap -\mathrm{int}C=Ø$, that is,

$$x-re\notin -\mathrm{int}C,\forall x\in A\left({\mu }_0\right)\dot{-}B\left({\eta }_0\right).$$

(8)

If not, then there exists a point $x_0$ with

$$x_0+B\left({\eta }_0\right)\subseteq A\left({\mu }_0\right)$$

(9)

such that

$$x_0-re\in -\mathrm{int}C.$$

(10)

By (9), we have $x_0\in A\left({\mu }_0\right)-u$ for any $u\in B\left({\eta }_0\right)$. Since $A(·)$ is l.s.c. at ${\mu }_0$, there exists

$$\left\{x_n\right\}\subseteq A\left({\mu }_n\right)-u,\forall u\in B\left({\eta }_0\right)$$

(11)

such that $x_n\to x_0$.

For any $u_n\in B\left({\eta }_n\right)$, since $B(·)$ is u.s.c. with compact values at ${\eta }_0$, by Proposition 2 (ii), there exist $u_0\in B\left({\eta }_0\right)$ and a subsequence $\left\{u_{n_k}\right\}$ of $\left\{u_n\right\}$ such that $u_{n_k}\to u_0$. Without loss of generality, we assume that $u_n\to u_0$. As a result, for any $ϵ>0$, there exists $N\in \mathbb{N}$ such that $u_n-u_0\in ϵ\mathbb{B},\forall n\ge N$. Hence, it follows from (11) that

$$x_n+u_n\in A\left({\mu }_n\right)+u_n-u_0\subseteq A\left({\mu }_n\right)+ϵ\mathbb{B},\forall n\ge N.$$

This together with the arbitrariness of $u_n\in B\left({\eta }_n\right)$ gives rise to

$$x_n+B\left({\eta }_n\right)\subseteq A\left({\mu }_n\right)+ϵ\mathbb{B},\forall n\ge N,$$

i.e., $x_n\in \left(A\left({\mu }_n\right)\dot{-}B\left({\eta }_n\right)+ϵ\mathbb{B}\right)$, $\forall n\ge N$. Moreover, it follows from (7) that

$$\left(\left\{x_n\right\}-re\right)\cap -\mathrm{int}C\subseteq \epsilon \mathbb{B},\forall n\ge N.$$

(12)

We claim that

$$x_0-re\notin -\mathrm{int}C.$$

(13)

Granting to this, we get a contradiction with (10). Hence, (8) holds. This leads to $\left(A\left({\mu }_0\right)\dot{-}B\left({\eta }_0\right)-re\right)\cap -\mathrm{int}C=Ø$, and so $A\left({\mu }_0\right){⋠}_C^{m_1}re+B\left({\eta }_0\right)$. Hence, $\omega ({\eta }_0,{\mu }_0)\ge r$ by Proposition 5 (i). Consequently, $({\eta }_0,{\mu }_0)\in M_u$.

Now, we turn our attention to the claim (13). If not, according to $x_n\to x_0$, there exists $N_1\in \mathbb{N}$ with $N_1\ge N$ such that $x_n-re\in -\mathrm{int}C$, $\forall n\ge N_1$. Set $ϵ={\displaystyle \frac{\parallel x_{N_1}-re\parallel }{2}}$, then we have $(x_{N_1}-re)\cap -\mathrm{int}C=x_{N_1}-re\notin ϵ\mathbb{B}$, which contradicts (12). Therefore, (13) is valid. □

Proposition 11. 

Assume that $B\left({\eta }_0\right)\in {\mathcal{B}}^{*}\left(Y\right)$ for some ${\eta }_0\in {\Lambda }_2$.

(i) 

If $A(·)$ is $m_2$-naturally C-quasiconvex with bounded values on ${\Lambda }_1$, then $\omega (·,{\eta }_0)$ is naturally quasiconvex on ${\Lambda }_1$, that is, for any ${\mu }_1$, ${\mu }_2\in {\Lambda }_1$ and for any $\alpha \in [0,1]$, there exists $\beta \in [0,1]$ such that

$$\omega \left(\alpha {\mu }_1+(1-\alpha ){\mu }_2,{\eta }_0\right)\le \beta \omega ({\mu }_1,{\eta }_0)+(1-\beta )\omega ({\mu }_2,{\eta }_0).$$

(ii) 

If $A(·)$ is strictly $m_2$-naturally C-quasiconvex with nonempty compact values on ${\Lambda }_1$, then $\omega (·,{\eta }_0)$ is strictly naturally quasiconvex on ${\Lambda }_1$, that is, for any ${\mu }_1$, ${\mu }_2\in {\Lambda }_1$ with ${\mu }_1\ne {\mu }_2$ and for any $\alpha \in (0,1)$, there exists $\beta \in [0,1]$ such that

$$\omega \left(\alpha {\mu }_1+(1-\alpha ){\mu }_2,{\eta }_0\right)<\beta \omega ({\mu }_1,{\eta }_0)+(1-\beta )\omega ({\mu }_2,{\eta }_0).$$

Proof. 

We only prove statement (ii), since statement (i) is obtained by the same techniques given in the proof of (ii). As $A(·)$ is strictly $m_2$-naturally C-quasiconvex on ${\Lambda }_1$, for any ${\mu }_1$, ${\mu }_2\in {\Lambda }_1$ with ${\mu }_1\ne {\mu }_2$ and for any $\alpha \in (0,1)$, there exists $\beta \in [0,1]$ such that

$$A\left(\alpha {\mu }_1+(1-\alpha ){\mu }_2\right){\prec }_C^{m_2}\beta A\left({\mu }_1\right)+(1-\beta )A\left({\mu }_2\right).$$

In terms of Proposition 7 (ii) and Proposition 9, we have

$$\begin{array}{ccc}\hfill \omega \left(\alpha {\mu }_1+(1-\alpha ){\mu }_2,{\eta }_0\right)& =& I_e^{m_2}\left(A\left(\alpha {\mu }_1+(1-\alpha ){\mu }_2\right),B\left({\eta }_0\right)\right)\hfill \\ & <& I_e^{m_2}\left(\beta A\left({\mu }_1\right)+(1-\beta )A\left({\mu }_2\right),B\left({\eta }_0\right)\right)\hfill \\ & =& inf\left\{t\in \mathbb{R}:\beta A\left({\mu }_1\right)+(1-\beta )A\left({\mu }_2\right){⪯}_C^{m_2}te+B\left({\eta }_0\right)\right\}\hfill \\ & \le & inf[\left\{t\in \mathbb{R}:\beta A\left({\mu }_1\right){⪯}_C^{m_2}te+\beta B\left({\lambda }_0\right)\right\}\hfill \\ & & +\left\{t\in \mathbb{R}:(1-\beta )A\left({\mu }_2\right){⪯}_C^{m_2}te+(1-\beta )B\left({\lambda }_0\right)\right\}]\hfill \\ & =& inf\left\{t\in \mathbb{R}:\beta A\left({\mu }_1\right){⪯}_C^{m_2}te+\beta B\left({\eta }_0\right)\right\}\hfill \\ & & +inf\left\{t\in \mathbb{R}:(1-\beta )A\left({\mu }_2\right){⪯}_C^{m_2}te+(1-\beta )B\left({\eta }_0\right)\right\}\hfill \\ & =& \beta \omega ({\mu }_1,{\eta }_0)+(1-\beta )\omega ({\mu }_2,{\eta }_0).\hfill \end{array}$$

The above strict inequality is from Proposition 7 (ii). The inequality holds because of $\left\{t\in \mathbb{R}:\beta A\left({\mu }_1\right){⪯}_C^{m_2}te+\beta B\left({\lambda }_0\right)\right\}+\left\{t\in \mathbb{R}:(1-\beta )A\left({\mu }_2\right){⪯}_C^{m_2}te+(1-\beta )B\left({\lambda }_0\right)\right\}\subseteq \left\{t\in \mathbb{R}:\beta A\left({\mu }_1\right)+(1-\beta )A\left({\mu }_2\right){⪯}_C^{m_2}te+B\left({\lambda }_0\right)\right\}$. The third equality is valid due to Proposition 3 and the fact

$$inf(D+F)=infD+infF,$$

where $D,F\subseteq \mathbb{R}$, $infD>-\infty$, and $infF>-\infty$. The last equality holds by Proposition 9. Henceforth, $\omega (·,{\eta }_0)$ is strictly naturally quasiconvex on ${\Lambda }_1$. □

Proposition 12. 

Let ${\mu }_0\in {\Lambda }_1$.

(i) 

If $B(·)$ is $m_2$-properly C-quasiconvex on ${\Lambda }_2$, then $\omega ({\mu }_0,·)$ is properly quasiconcave on ${\Lambda }_2$, that is, for any ${\eta }_1$, ${\eta }_2\in {\Lambda }_2$ and for any $\alpha \in [0,1]$, one has

$$\mathrm{either}\omega \left({\mu }_0,\alpha {\eta }_1+(1-\alpha ){\eta }_2\right)\ge \omega ({\mu }_0,{\eta }_1)\mathrm{or}\omega \left({\mu }_0,\alpha {\eta }_1+(1-\alpha ){\eta }_2\right)\ge \omega ({\mu }_0,{\eta }_2).$$

(ii) 

If $A\left({\mu }_0\right)$ is compact and $B(·)$ is strictly $m_2$-properly C-quasiconvex on ${\Lambda }_2$, then $\omega ({\mu }_0,·)$ is strictly properly quasiconcave on ${\Lambda }_2$, that is, for any ${\eta }_1$, ${\eta }_2\in {\Lambda }_2$ with ${\eta }_1\ne {\eta }_2$ and for any $\alpha \in (0,1)$, one has

$$\mathrm{either}\omega \left({\mu }_0,\alpha {\eta }_1+(1-\alpha ){\eta }_2\right)>\omega ({\mu }_0,{\eta }_1)\mathrm{or}\omega \left({\mu }_0,\alpha {\eta }_1+(1-\alpha ){\eta }_2\right)>\omega ({\mu }_0,{\eta }_2).$$

Proof. 

We only prove statement (ii), since the proof of the statement (i) follows on the similar line. As $B(·)$ is strictly $m_2$-properly C-quasiconvex on ${\Lambda }_2$, for any ${\eta }_1$, ${\eta }_2\in {\Lambda }_2$ with ${\eta }_1\ne {\eta }_2$ and for any $\alpha \in (0,1)$, one has

$$\mathrm{either}B\left(\alpha {\eta }_1+(1-\alpha ){\eta }_2\right){\prec }_C^{m_2}B\left({\eta }_1\right)\mathrm{or}B\left(\alpha {\eta }_1+(1-\alpha ){\eta }_2\right){\prec }_C^{m_2}B\left({\eta }_2\right).$$

Due to Proposition 8 (ii), one has

$$\begin{array}{ccc}\hfill \mathrm{either}\omega \left({\mu }_0,\alpha {\eta }_1+(1-\alpha ){\eta }_2\right)& =& I_e^{m_2}\left(A\left({\mu }_0\right),B\left(\alpha {\eta }_1+(1-\alpha ){\eta }_2\right)\right)\hfill \\ & >& I_e^{m_2}\left(A\left({\mu }_0\right),B\left({\eta }_1\right)\right)\hfill \\ & =& \omega ({\mu }_0,{\eta }_1),\hfill \end{array}$$

or

$$\begin{array}{ccc}\hfill \omega \left({\mu }_0,\alpha {\eta }_1+(1-\alpha ){\eta }_2\right)& =& I_e^{m_2}\left(A\left({\mu }_0\right),B\left(\alpha {\eta }_1+(1-\alpha ){\eta }_2\right)\right)\hfill \\ & >& I_e^{m_2}\left(A\left({\mu }_0\right),B\left({\eta }_2\right)\right)\hfill \\ & =& \omega ({\mu }_0,{\eta }_2).\hfill \end{array}$$

Consequently, $\omega ({\mu }_0,·)$ is strictly properly quasiconcave on ${\Lambda }_2$. □

4. Path Connectedness of (${\mathit{m}}_{\mathbf{2}}$-SOP)

In this section, we implement an application of the continuity of the function $\omega (·,·)$ to the path connectedness of the $m_2$-minimal solution set and the weak $m_2$-minimal solution set of ($m_2$-SOP).

The following result shows that the sets ${\mathrm{argmin}}_{m_2}(K,F)$ and ${\mathrm{argmin}}_{w{m}_2}(K,F)$ are the same, when F is strictly $m_2$-naturally C-quasiconvex on K.

Lemma 3. 

Assume that the following conditions hold:

(i) 

Assume that K is a convex subset of X;

(ii) 

F has bounded values on K;

(iii) 

F is strictly $m_2$-naturally C-quasiconvex on K.

Then, ${\mathrm{argmin}}_{m_2}(K,F)={\mathrm{argmin}}_{w{m}_2}(K,F).$

Proof. 

By Proposition 1, it suffices to prove that ${\mathrm{argmin}}_{w{m}_2}(K,F)\subseteq {\mathrm{argmin}}_{m_2}(K,F)$. Assume that there exists $x_0\in {\mathrm{argmin}}_{w{m}_2}(K,F)$ such that $x_0\notin {\mathrm{argmin}}_{m_2}(K,F)$. Then, there exists $\overline{x}\in K$ such that

$$F\left(\overline{x}\right){⪯}_C^{m_2}F\left(x_0\right),$$

(14)

and

$$F\left(\overline{x}\right)\ne F\left(x_0\right).$$

(15)

Hence, it follows from (15) that $\overline{x}\ne x_0$. Since F is strictly $m_2$-naturally C-quasiconvex on K, for any $\alpha \in (0,1)$, there exists $\beta \in [0,1]$ such that

$$F\left(\alpha \overline{x}+(1-\alpha )x_0\right){\prec }_C^{m_2}\beta F\left(\overline{x}\right)+(1-\beta )F\left(x_0\right).$$

Combining this with (14), $F\left(x_0\right)\subseteq {\beta }_{0}F\left(x_0\right)+(1-{\beta }_0)F\left(x_0\right)$, and the transitivity of the set order relation ${⪯}_C^{m_2}$, we have $F\left(\alpha \overline{x}+(1-\alpha )x_0\right){\prec }_C^{m_2}F\left(x_0\right)$, $\forall \alpha \in (0,1)$. This together with the convexity of K shows that $x_0\notin {\mathrm{argmin}}_{w{m}_2}(K,F)$. This leads to a contradiction because of $x_0\in {\mathrm{argmin}}_{w{m}_2}(K,F)$. Thus, the proof is complete. □

Inspired by works in [35,36], we introduce the following definition of the level mapping with respect to the order relation ${⪯}_C^{m_2}$.

Definition 7. 

Let $F:X\to Y$ be a set-valued mapping and $K\subseteq X$. A set-valued mapping $le{v}_F:K\rightrightarrows K$ defined by

$$le{v}_F\left(x\right)=\left\{y\in K:F\left(y\right){⪯}_C^{m_2}F\left(x\right)\right\},x\in K,$$

is called a level mapping.

Remark 2. 

It is easy to see that ${\mathrm{argmin}}_{m_2}\left(le{v}_F\left(x\right),F\right)\subseteq {\mathrm{argmin}}_{m_2}(K,F)$ for any $x\in K$.

Lemma 4. 

Assume that K is a convex subset of X. Then, the following statements hold.

(i) 

If F is strictly $m_2$-naturally C-quasiconvex on K, then $le{v}_F\left(x_0\right)=\left\{x_0\right\}$ when $x_0\in {\mathrm{argmin}}_{w{m}_2}(K,F)$.

(ii) 

If F is $m_2$-naturally C-quasiconvex on K, then $le{v}_F\left(x\right)$ is convex for any $x\in K$.

Proof. 

(i) It is easy to see that $x_0\in le{v}_F\left(x_0\right)$. Suppose that there exists $\overline{x}\in le{v}_F\left(x_0\right)$ such that $\overline{x}\ne x_0$. Since F is strictly $m_2$-naturally C-quasiconvex on K, for any $\alpha \in (0,1)$, there exists ${\beta }_0\in [0,1]$ such that

$$F\left(\alpha x_0+(1-\alpha )\overline{x}\right){\prec }_C^{m_2}{\beta }_{0}F\left(x_0\right)+(1-{\beta }_0)F\left(\overline{x}\right).$$

(16)

It follows from $\overline{x}\in le{v}_F\left(x_0\right)$ that

$$F\left(\overline{x}\right){⪯}_C^{m_2}F\left(x_0\right).$$

Combining this with (16), the transitivity of the set order relation ${⪯}_C^{m_2}$, and $F\left(x_0\right)\subseteq {\beta }_{0}F\left(x_0\right)+(1-{\beta }_0)F\left(x_0\right)$, we arrive at

$$F\left(\alpha x_0+(1-\alpha )\overline{x}\right){\prec }_C^{m_2}F\left(x_0\right),\forall \alpha \in (0,1).$$

This contradicts $x_0\in {\mathrm{argmin}}_{w{m}_2}(K,F)$. Therefore, $le{v}_F\left(x_0\right)=\left\{x_0\right\}$. □

(ii) Let $y_1,y_2\in le{v}_F\left(x\right)$, that is,

$$F\left(y_i\right){⪯}_C^{m_2}F\left(x\right),i=1,2.$$

(17)

Since F is $m_2$-naturally C-quasiconvex on K, for any $\alpha \in [0,1]$, there exists ${\beta }_0\in [0,1]$ such that

$$F\left(\alpha y_1+(1-\alpha )y_2\right){⪯}_C^{m_2}{\beta }_{0}F\left(y_1\right)+(1-{\beta }_0)F\left(y_2\right).$$

(18)

By (17) and (18), and taking into account $F\left(x_0\right)\subseteq {\beta }_{0}F\left(x_0\right)+(1-{\beta }_0)F\left(x_0\right)$ and the transitivity of the set order relation ${⪯}_C^{m_2}$, we obtain $F\left(\alpha y_1+(1-\alpha )y_2\right){⪯}_C^{m_2}F\left(x\right)$ for any $\alpha \in [0,1]$. This implies that $\alpha y_1+(1-\alpha )y_2\in le{v}_F\left(x\right)$ for any $\alpha \in [0,1]$ and the proof is complete. □

Lemma 5. 

Assume that the following conditions hold:

(i) 

K is closed and convex;

(ii) 

F is continuous with nonempty compact values on K;

(iii) 

$F\left(x\right)\dot{-}F\left(y\right)\ne Ø$ for each $x,y\in K$.

Then, $le{v}_F\left(x\right)$ is closed for any $x\in K$.

Proof. 

Let $\left\{y_n\right\}\subseteq le{v}_F\left(x\right)$ with $y_n\to y_0$. It suffices to prove that $y_0\in le{v}_F\left(x\right)$. It is clear that $y_0\in K$. It follows from $\left\{y_n\right\}\subseteq le{v}_F\left(x\right)$ that

$$F\left(y_n\right){⪯}_C^{m_2}F\left(x\right),\forall n\in \mathbb{N}.$$

(19)

Then, we deduce $I_e^{m_2}\left(F\left(y_n\right),F\left(x\right)\right)\le 0$ from Proposition 5 (ii). By the similar proof for the closedness of $M_l$ in Proposition 10, we can obtain $I_e^{m_2}\left(F\left(y_0\right),F\left(x\right)\right)\le 0$ and so $F\left(y_0\right){⪯}_C^{m_2}F\left(x\right)$. Therefore, $le{v}_F\left(x\right)$ is closed for any $x\in K$. □

Lemma 6. 

Assume that K is a nonempty compact and F is continuous on K with nonempty compact values. Then, $le{v}_F(·)$ is u.s.c. on K.

Proof. 

Suppose to the contrary that there exists $x_0\in K$ such that $le{v}_F(·)$ is not u.s.c. at $x_0$. Then, there exist an open set $W_0$ with $le{v}_F\left(x_0\right)\subseteq W_0$ and a sequence $\left\{x_n\right\}$ with $x_n\to x_0$ such that $le{v}_F\left(x_n\right)⊈W_0,\forall n\in \mathbb{N}$. This implies that there is $y_n\in le{v}_F\left(x_n\right)$ such that

$$y_n\notin W_0,\forall n\in \mathbb{N}.$$

(20)

Since K is compact and $\left\{y_n\right\}\subseteq K$, without loss of generality, we can assume that $y_n\to y_0\in K$. It follows from $y_n\in le{v}_F\left(x_n\right)$ that there exists $c_n\in C$ such that

$$F\left(x_n\right)-c_n\subseteq F\left(y_n\right),\forall n\in \mathbb{N}.$$

(21)

Consequently, for any $v_n\in F\left(x_n\right)$, we have $v_n-c_n\in F\left(y_n\right)$. Since F is u.s.c. with compact values on K, sequences $\left\{v_n\right\}$ and $\{v_n-c_n\}$ have convergent subsequences by Proposition 2 (ii). This implies that $\left\{c_n\right\}$ has a convergent subsequence. Without loss of generality, we assume that $c_n\to c_0$. Thanks to $\left\{c_n\right\}\subseteq C$ and the closedness of C, we have $c_0\in C$.

We claim that $F\left(x_0\right)-c_0\subseteq F\left(y_0\right)$, i.e., $y_0\in le{v}_F\left(x_0\right)$. Indeed, for any $z_0\in F\left(x_0\right)$, by the lower semicontinuity of F, there exists $z_n\in F\left(x_n\right)$ such that $z_n\to z_0$. It follows from (21) that there exists $u_n\in F\left(y_n\right)$ such that

$$u_n=z_n-c_n.$$

(22)

By the compactness and the upper semicontinuity of F and by Proposition 2 (ii), we see that there exist $u_0\in F\left(y_0\right)$ and a subsequence $\left\{u_{n_k}\right\}$ of $\left\{u_n\right\}$ such that $u_{n_k}\to u_0$. Combining this with $z_n\to z_0$, $c_n\to c_0$, and (22), we have

$$z_0-c_0=u_0\in F\left(y_0\right).$$

In view of the arbitrariness of $z_0\in F\left(x_0\right)$, we have $F\left(y_0\right){⪯}_C^{m_2}F\left(x_0\right)$. This states that $y_0\in le{v}_F\left(x_0\right)$. With the help of $y_n\to y_0$ and $le{v}_F\left(x_0\right)\subseteq W_0$, we see that $y_n\in W_0$ for n is large enough, which contradicts (20). Therefore, $le{v}_F(·)$ is u.s.c. on K. □

Lemma 7. 

Suppose that the following conditions hold:

(i) 

K is a nonempty and convex subset of X;

(ii) 

F is strictly $m_2$-naturally C-quasiconvex on K;

(iii) 

$F\left(x\right)\dot{-}F\left(y\right)\ne Ø$ for each $x,y\in K$;

(iv) 

F is l.s.c. on K.

Then, $le{v}_F(·)$ is l.s.c. on K.

Proof. 

Suppose to the contrary that there exists $x_0\in K$ such that $le{v}_F(·)$ is not l.s.c. at $x_0$. Then, there exist $y_0\in le{v}_F\left(x_0\right)$, a neighborhood $W_0$ of $0_X$ and a sequence $\left\{x_n\right\}$ with $x_n\to x_0$ such that

$$(y_0+W_0)\cap le{v}_F\left(x_n\right)=Ø,\forall n\in N.$$

(23)

It is clear that $x_0\in le{v}_F\left(x_0\right)$. Now, we consider the following two cases:

$\mathbf{Case}\mathbf{one}$. $y_0=x_0$. Then, by $x_n\in le{v}_F\left(x_n\right)$ and $x_n\to x_0=y_0$, we have $x_n\in (y_0+W_0)\cap le{v}_F\left(x_n\right)$ for n, which is large enough. This contradicts (23).

$\mathbf{Case}\mathbf{two}$. $y_0\ne x_0$. Since F is strictly $m_2$-naturally C-quasiconvex on K, for any $\alpha \in (0,1)$, there exists $\beta \in [0,1]$ such that

$$F\left(\alpha x_0+(1-\alpha )y_0\right){\prec }_C^{m_2}\beta F\left(x_0\right)+(1-\beta )F\left(y_0\right).$$

This together with $y_0\in le{v}_F\left(x_0\right)$ and the transitivity of ${⪯}_C^{m_2}$ result in

$$F\left(\alpha x_0+(1-\alpha )y_0\right){\prec }_C^{m_2}\beta F\left(x_0\right)+(1-\beta )F\left(x_0\right).$$

According to $F\left(x_0\right)\subseteq \beta F\left(x_0\right)+(1-\beta )F\left(y_0\right)$, we have

$$F\left(\alpha x_0+(1-\alpha )y_0\right){\prec }_C^{m_2}F\left(x_0\right),\forall \alpha \in (0,1).$$

(24)

Let $y\left(\alpha \right)=\alpha x_0+(1-\alpha )y_0$, $\alpha \in (0,1)$. It is clear that there exists ${\alpha }_0\in (0,1)$ such that

$$y\left({\alpha }_0\right)\in y_0+W_0.$$

(25)

Now, we claim that there exists $N_0\in \mathbb{N}$ such that, when $n\ge N_0$,

$$F\left(y\left({\alpha }_0\right)\right){\prec }_C^{m_2}F\left(x_n\right).$$

(26)

Indeed, if not, without loss of generality, we can assume that $F\left(y\left({\alpha }_0\right)\right){⋠}_C^{m_2}F\left(x_n\right)$ for all $n\in \mathbb{N}$. This implies that

$$\left(F\left(y\left({\alpha }_0\right)\right)\dot{-}F\left(x_n\right)\right)\cap -\mathrm{int}C=Ø,\forall n\in \mathbb{N}.$$

Similar to the proof of Proposition 10 (see (7) and (8)), we can prove that

$$\left(F\left(y\left({\alpha }_0\right)\right)\dot{-}F\left(x_0\right)\right)\cap -\mathrm{int}C=Ø.$$

This contradicts (24). Hence, (26) holds and so $y\left({\alpha }_0\right)\in le{v}_F\left(x_n\right)$. With the help of (25), one has $y\left({\alpha }_0\right)\in (y_0+W_0)\cap le{v}_F\left(x_n\right)$, which contradicts (23). Consequently, $le{v}_F\left(x\right)$ is l.s.c. on K. □

Let $B\in {\mathcal{P}}_0\left(Y\right)$. We define the function $\varphi :X\to \mathbb{R}\cup \{\pm \infty \}$ by $\varphi \left(x\right)=I_e^{m_2}\left(F\left(x\right),B\right)$ and define a set-valued mapping $H_m:K\rightrightarrows K$ as follows:

$$H_m\left(x\right)=\left\{z\in le{v}_F\left(x\right):\varphi \left(z\right)=\underset{y\in le{v}_F\left(x\right)}{inf}\varphi \left(y\right)\right\}.$$

Lemma 8. 

Suppose that the following conditions hold:

(i) 

K is a nonempty, compact, and convex subset of X;

(ii) 

F is continuous with nonempty compact values on K;

(iii) 

$F\left(x\right)\dot{-}F\left(y\right)\ne Ø$ for each $x,y\in K$;

(iv) 

F is strictly $m_2$-naturally C-quasiconvex.

Then, the following two statements are true:

(a) 

$H_m\left(x\right)$ is nonempty for any $x\in K$.

(b) 

$H_m\left(x\right)\subseteq {\mathrm{argmin}}_{w{m}_2}\left(le{v}_F\left(x\right),F\right)={\mathrm{argmin}}_{m_2}\left(le{v}_F\left(x\right),F\right)\subseteq {\mathrm{argmin}}_{m_2}(K,F),\forall x\in K$.

Proof. 

(a) It follows from Lemma 5 that $le{v}_F\left(x\right)$ is closed. Since K is compact and $le{v}_F\left(x\right)\subseteq K$, we see that $le{v}_F\left(x\right)$ is compact. It is clear that $x\in le{v}_F\left(x\right)$ and so it is nonempty. By Proposition 10, we can see that $\varphi$ is continuous on K. Therefore, $H_m\left(x\right)$ is nonempty for any $x\in K$.

(b) For any $x\in K$, let $z_0\in H_m\left(x\right)$. Suppose that $z_0\notin {\mathrm{argmin}}_{w{m}_2}\left(le{v}_F\left(x\right),F\right)$. Then there exists $\overline{z}\in le{v}_F\left(x\right)$ such that $F\left(\overline{z}\right){\prec }_C^{m_2}F\left(z_0\right)$. It follows from Proposition 7 (ii) that $\varphi \left(\overline{z}\right)<\varphi \left(z_0\right)$, which contradicts $z_0\in H_m\left(x\right)$. As a result, $H_m\left(x\right)\subseteq {\mathrm{argmin}}_{w{m}_2}(le{v}_F\left(x\right),F)$. By Lemma 3, we have ${\mathrm{argmin}}_{w{m}_2}(le{v}_F\left(x\right),F)={\mathrm{argmin}}_{m_2}(le{v}_F\left(x\right),F)$. Moreover, it follows from Remark 3 that ${\mathrm{argmin}}_{m_2}\left(le{v}_F\left(x\right),F\right)\subseteq {\mathrm{argmin}}_{m_2}(K,F)$. Therefore, the proof is complete. □

Lemma 9. 

Suppose that the following conditions hold:

(i) 

K is a nonempty, compact, and convex subset of X;

(ii) 

F is continuous with nonempty compact values on K;

(iii) 

$F\left(x\right)\dot{-}F\left(y\right)\ne Ø$ for each $x,y\in K$;

(iv) 

F is strictly $m_2$-naturally C-quasiconvex.

Then, $H_m(·)$ is continuous on K.

Proof. 

First, we show that $H_m\left(x\right)$ is a singleton for each $x\in K$. Suppose that there exists $x_0\in K$ such that $H_m\left(x_0\right)$ is not a singleton. Then, there exist $z_1,z_2\in H_m\left(x_0\right)$ with $z_1\ne z_2$, that is,

$$z_1,z_2\in le{v}_F\left(x_0\right)and\varphi \left(z_1\right)=\varphi \left(z_2\right)=\underset{y\in le{v}_F\left(x_0\right)}{inf}\varphi \left(y\right).$$

(27)

It follows from Lemma 4 (ii) that $le{v}_F\left(x_0\right)$ is convex. Hence, for any $\alpha \in (0,1)$, one has $\alpha z_1+(1-\alpha )z_2\in le{v}_F\left(x_0\right)$. Moreover, by Proposition 11 (ii), we can see that $\varphi$ is strictly naturally quasiconvex on K. Then, for any $\alpha \in (0,1)$, there exists $\beta \in [0,1]$ such that

$$\varphi \left(\alpha z_1+(1-\alpha )z_2\right)<\beta \varphi \left(z_1\right)+(1-\beta )\varphi \left(z_2\right).$$

Due to (27), we have

$$\varphi \left(\alpha z_1+(1-\alpha )z_2\right)<\underset{y\in le{v}_F\left(x_0\right)}{inf}\varphi \left(y\right).$$

This leads to a contradiction because of $\alpha z_1+(1-\alpha )z_2\in le{v}_F\left(x_0\right)$. Henceforth, $H_m\left(x\right)$ is a singleton for each $x\in K$.

Next, we prove that $H_m(·)$ is continuous on K. Indeed, it follows from Lemma 6 and Lemma 7 that $le{v}_F(·)$ is continuous on K. Combining with the compactness of K, $le{v}_F\left(x\right)\subseteq K$ and Lemma 5, we see that $Le{v}_F\left(x\right)$ is compact for each $x\in K$. It is clear that $x\in Le{v}_F\left(x\right)$ and so it is nonempty. By Proposition 10, we can see that $\varphi$ is continuous on K. Let $\psi \left(x\right)=-\varphi \left(x\right)$ for $x\in X$. Then, we have

$$H_m\left(x\right)=\{z\in le{v}_F\left(x\right):\psi \left(z\right)=\underset{y\in le{v}_F\left(x\right)}{sup}\psi \left(y\right)\}.$$

By Lemma 2, it can be seen that $H_m(·)$ is u.s.c. on K. Therefore, $H_m(·)$ is continuous on K by the single-valuedness of $H_m\left(x\right)$ for each $x\in K$. □

Theorem 1. 

Assume that the following assumptions are valid:

(i) 

K is a nonempty, compact, and convex subset of X;

(ii) 

F is continuous with nonempty compact values on K;

(iii) 

$F\left(x\right)\dot{-}F\left(y\right)\ne Ø$ for each $x,y\in K$;

(iv) 

F is strictly $m_2$-naturally C-quasiconvex on K.

Then, ${\mathrm{argmin}}_{m_2}(K,F)$ is path-connected. Moreover, ${\mathrm{argmin}}_{w{m}_2}(K,F)$ is path-connected.

Proof. 

For any $x\in {\mathrm{argmin}}_{m_2}(K,F)$, by Lemma 4 (i), one has $le{v}_F\left(x\right)=\left\{x\right\}$ and so $H_m\left(x\right)=\left\{x\right\}$. For any $x_1,x_2\in {\mathrm{argmin}}_{m_2}(K,F)$, let

$$g\left(\lambda \right)=H_m\left(\lambda x_1+(1-\lambda )x_2\right),\lambda \in [0,1].$$

It follows from Lemma 8 that $g\left(\lambda \right)\in {\mathrm{argmin}}_{m_2}(K,F)$ for any $\lambda \in [0,1]$. By Lemma 9, $H_m(·)$ is continuous on K and so $g(·)$ is continuous on $[0,1]$. In addition, we can see that $g\left(0\right)=H_m\left(x_2\right)=\left\{x_2\right\}$ and $g\left(1\right)=H_m\left(x_1\right)=\left\{x_1\right\}$. Therefore, ${\mathrm{argmin}}_{m_2}(K,F)$ is path-connected. Furthermore, ${\mathrm{argmin}}_{w{m}_2}(K,F)$ is path-connected because of ${\mathrm{argmin}}_{w{m}_2}(K,F)={\mathrm{argmin}}_{m_2}(K,F)$ by Lemma 3. □

Remark 3. 

It is worth pointing out that our main results in this section are different from that in [36]. In fact, Han et al. [36] established the path connectedness of the solution sets for set optimization problems involving the lower set order relation ${⪯}_C^l$ by linear scalarization function $f\in C^{*}\setminus \left\{0\right\}$, while, in this paper, with the help of the nonlinear scalarization function $I_e^{m_2}(·,·)$, we present the path connectedness of the solution sets for set optimization problems with the order relation ${⪯}_C^{m_2}$.

Now, we give an example to illustrate Theorem 1.

Example 3. 

Let $X=\mathbb{R}$, $Y={\mathbb{R}}^2$, and $C={\mathbb{R}}_{+}^2$. Let $K=[-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}]$ and

$$F\left(x\right)=(x,sinx)+{\mathbb{B}}_Y.$$

It is not hard to check that the assumptions (i)–(iii) are satisfied. Now, we check that the assumption (iv) is valid. Indeed, for any $x_1,x_2\in K$ with $x_1\ne x_2$ (without loss of the generality, we assume that $x_1<x_2$) and for any $\alpha \in (0,1)$, there exists ${\beta }_0=0$ such that

$$F\left(\alpha x_1+(1-\alpha )x_2\right){\prec }_C^m{\beta }_{0}F\left(x_1\right)+(1-{\beta }_0)F\left(x_2\right)=F\left(x_2\right)=(x_2,sin{x}_2)+{\mathbb{B}}_Y,$$

since $\left(F\left(\alpha x_1+(1-\alpha )x_2\right)\dot{-}F\left(x_2\right)\right)\cap -\mathrm{int}C=\left(\alpha (x_1-x_2),sin\left(\alpha x_1+(1-\alpha )x_2\right)-sin{x}_2\right)$. Hence, ${\mathrm{argmin}}_{m_2}(K,F)$ and ${\mathrm{argmin}}_{w{m}_2}(K,F)$ are path-connected by Theorem 1.

5. Continuity of the Solution Set Mappings of (${\mathit{m}}_{\mathbf{2}}$-PSOP)

In the section, we explore an application of the nonlinear scalarization function for sets to the continuity of the weak $m_2$-minimal solution set mapping $S_w(·)$ and the $m_2$-minimal solution set mapping $S(·)$ of ($m_2$-PSOP).

We define $\phi :X\times X\times \Lambda \to \mathbb{R}\cup \{\pm \infty \}$ by

$$\phi (x,y,\lambda )=I_e^{m_2}\left(F(y,\lambda ),F(x,\lambda )\right),\forall (x,y)\in X\times X.$$

Lemma 10. 

For each $\lambda \in \Lambda$, one has

$$S_w\left(\lambda \right)=\left\{x\in K\left(\lambda \right):\phi (x,y,\lambda )\ge 0,\forall y\in K\left(\lambda \right)\right\}.$$

Proof. 

By the definition of the $m_2$-minimal solution set of ($m_2$-PSOP), for each $\lambda \in \Lambda$, we have $x\in S_w\left(\lambda \right)$ if and only if

$$F(y,\lambda ){⋠}_C^{m_2}F(x,\lambda ),\forall y\in K\left(\lambda \right).$$

This indicates that $\phi (x,y,\lambda )\ge 0$, for any $y\in K\left(\lambda \right)$ by Proposition 5 (i). Henceforth, the proof is complete. □

Theorem 2. 

Let ${\lambda }_0\in \Lambda$. Assume that

(i) 

$K(·)$ is continuous with nonempty compact values at ${\lambda }_0$;

(ii) 

$F(·,·)$ is continuous with nonempty compact values on $K\left({\lambda }_0\right)\times \left\{{\lambda }_0\right\}$;

(iii) 

$F(y,\lambda )\dot{-}F(x,\lambda )\ne Ø$, for any $x,y\in K\left({\lambda }_0\right)$ and for any $\lambda \in \Lambda$.

Then, $S_w(·)$ is u.s.c. with compact values at ${\lambda }_0$.

Proof. 

Suppose to the contrary that $S_w(·)$ is not u.s.c. at ${\lambda }_0$. Then, there exist an open set $W_0$ with $S_w\left({\lambda }_0\right)\subseteq W_0$ and a sequence $\left\{{\lambda }_n\right\}$ with ${\lambda }_n\to {\lambda }_0$ such that $S_w\left({\lambda }_n\right)⊈W_0,\forall n\in \mathbb{N}$. This implies that there is $x_n\in S_w\left({\lambda }_n\right)$ such that

$$x_n\notin W_0,\forall n\in \mathbb{N}.$$

(28)

As $K(·)$ is u.s.c. with compact values at ${\lambda }_0$ and $x_n\in K\left({\lambda }_n\right)$, by Proposition 2 (ii), there exist $x_0\in K\left({\lambda }_0\right)$ and a subsequence $\left\{x_{n_k}\right\}$ of $x_n$ such that $x_{n_k}\to x_0$. Without loss of generality, we assume that $x_n\to x_0$.

Now, we claim that $x_0\in S_w\left({\lambda }_0\right)$. Indeed, suppose that $x_0\notin S_w\left({\lambda }_0\right)$. Then, it follows from Lemma 10 that there exists $y_0\in K\left({\lambda }_0\right)$ such that

$$\phi (x_0,y_0,{\lambda }_0)<0.$$

(29)

Since $K(·)$ is l.s.c. at ${\lambda }_0$ and $y_0\in K\left({\lambda }_0\right)$, it follows that there exists $y_n\in K\left({\lambda }_n\right)$ such that $y_n\to y_0$ by Proposition 2 (i). In terms of Proposition 10, we see that $\phi (·,·,·)$ is continuous on $K\left({\lambda }_0\right)\times K\left({\lambda }_0\right)\times \left\{{\lambda }_0\right\}$. As a result, it follows from (29) that

$$\phi (x_n,y_n,{\lambda }_n)<0,\mathrm{for}\mathrm{n}\mathrm{large}\mathrm{enough},$$

which contradicts $x_n\in S_w\left({\lambda }_n\right)$ by Lemma 10. Thus, $x_0\in S_w\left({\lambda }_0\right)$. This together with $x_n\to x_0$ and $S_w\left({\lambda }_0\right)\subseteq W_0$ shows that $x_n\in W_0$ for n that is large enough, which contradicts (28). Therefore, $S_w(·)$ is u.s.c. at ${\lambda }_0$.

Next, we show that $S_w\left({\lambda }_0\right)$ is compact. Since $K\left({\lambda }_0\right)$ is compact and $S_w\left({\lambda }_0\right)\subseteq K\left({\lambda }_0\right)$, it suffices to prove the closedness of $S_w\left({\lambda }_0\right)$. Indeed, let $\left\{x_n\right\}\subseteq S_w\left({\lambda }_0\right)$ with $x_n\to x_0$. Then, by Lemma 10, we have

$$\phi (x_n,y,{\lambda }_0)\ge 0,\forall y\in K\left({\lambda }_0\right).$$

This together with the continuity of $\phi$ indicates that $x_0\in S_W\left({\lambda }_0\right)$. As a consequence, the proof is complete. □

Next, we prove the lower semicontinuity of $S_w(·)$ and $S(·)$. Let $\widehat{S}:\Lambda \rightrightarrows X$ be defined by

$$\widehat{S}\left(\lambda \right):=\left\{x\in K\left(\lambda \right):\phi (x,y,\lambda )>0,\forall y\in K\left(\lambda \right)\right\}.$$

In the sequel, we assume that $\widehat{S}\left(\lambda \right)\ne Ø$ for each $\lambda \in \Lambda$. In order to establish the main results, we need the following several lemmas.

Lemma 11 

(See Theorem 1.1.2 in [37]). If $A\subseteq Y$ is convex and $\mathrm{int}A\ne Ø$, then $\mathrm{cl}\left(\mathrm{int}A\right)=\mathrm{cl}A$.

Lemma 12. 

Let ${\lambda }_0\in \Lambda$. Assume that

(i) 

$K(·)$ is continuous with nonempty compact values at ${\lambda }_0$;

(ii) 

$F(·,·)$ is continuous with nonempty compact values on $K\left({\lambda }_0\right)\times \left\{{\lambda }_0\right\}$;

(iii) 

$F(y,\lambda )\dot{-}F(x,\lambda )\ne Ø$ for any $x,y\in K\left({\lambda }_0\right)$ and for any $\lambda \in \Lambda$.

Then, $\widehat{S}(·)$ is l.s.c. at ${\lambda }_0$.

Proof. 

To prove the result by contradiction, suppose that $\widehat{S}(·)$ is not l.s.c. at ${\lambda }_0$. Then, by Proposition 2 (i), there exist a sequence $\left\{{\lambda }_n\right\}$ with ${\lambda }_n\to {\lambda }_0$ and $x_0\in \widehat{S}\left({\lambda }_0\right)$ such that for any $x_n\in \widehat{S}\left({\lambda }_n\right)$, we have $x_n\nrightarrow x_0$.

From $x_0\in \widehat{S}\left({\lambda }_0\right)$, we have $x_0\in K\left({\lambda }_0\right)$. As $K(·)$ is l.s.c. at ${\lambda }_0$, there exists ${\overline{x}}_n\in K\left({\lambda }_n\right)$ such that ${\overline{x}}_n\to x_0$. By the above contradiction assumption, there exists a subsequence $\left\{{\overline{x}}_{n_k}\right\}$ of $\left\{{\overline{x}}_n\right\}$ such that ${\overline{x}}_{n_k}\notin \widehat{S}\left({\lambda }_{n_k}\right)$ for any $k\in \mathbb{N}$. Without loss of the generality, we assume that ${\overline{x}}_n\notin \widehat{S}\left({\lambda }_n\right)$ for any $n\in \mathbb{N}$. Then, there exists $y_n\in K\left({\lambda }_n\right)$ such that

$$\phi ({\overline{x}}_n,y_n,{\lambda }_n)\le 0,\forall n\in \mathbb{N}.$$

(30)

Since $K(·)$ is u.s.c. with nonempty compact values at ${\lambda }_0$, by Proposition 2 (ii), there exist $y_0\in K\left({\lambda }_0\right)$ and a subsequence $\left\{y_{n_k}\right\}$ of $\left\{y_n\right\}$ such that $y_{n_k}\to y_0$. It follows from Proposition 10 that $\phi (·,·,·)$ is continuous on $K\left({\lambda }_0\right)\times K\left({\lambda }_0\right)\times \left\{{\lambda }_0\right\}$. This together with (30) states that

$$\phi (x_0,y_0,{\lambda }_0)\le 0,$$

which contradicts $x_0\in \widehat{S}\left({\lambda }_0\right)$. Thus, $\widehat{S}(·)$ is l.s.c. at ${\lambda }_0$. □

Lemma 13. 

Let ${\lambda }_0\in \Lambda$. Assume that the following conditions hold:

(i) 

$K(·)$ is continuous with nonempty compact values at ${\lambda }_0$;

(ii) 

$F(·,·)$ is continuous with nonempty compact values on $K\left({\lambda }_0\right)\times \left\{{\lambda }_0\right\}$;

(iii) 

$F(·,{\lambda }_0)$ is $m_2$-properly C-quasiconvex on $K\left({\lambda }_0\right)$;

(iv) 

$F(y,{\lambda }_0)\dot{-}F(x,{\lambda }_0)\ne Ø$ for any $x,y\in K\left({\lambda }_0\right)$.

Then, we have

$$\widehat{S}\left({\lambda }_0\right)\subseteq S\left({\lambda }_0\right)\subseteq S_w\left({\lambda }_0\right)=\mathrm{cl}\widehat{S}\left({\lambda }_0\right).$$

(31)

Proof. 

The proof is divided into the following two steps.

$\mathbf{Step}\mathbf{1}.$ We show that

$$\widehat{S}\left({\lambda }_0\right)\subseteq S\left({\lambda }_0\right)\subseteq S_w\left({\lambda }_0\right).$$

(32)

Indeed, for any $x\in \widehat{S}\left({\lambda }_0\right)$, by Proposition 5 (ii), we have

$$F(y,{\lambda }_0){⋠}_C^{m_2}F(x,{\lambda }_0),\forall y\in K\left({\lambda }_0\right).$$

This implies that $x\in S\left({\lambda }_0\right)$ and so $\widehat{S}\left({\lambda }_0\right)\subseteq S\left({\lambda }_0\right)$. Simultaneously, taking into account $S\left({\lambda }_0\right)\subseteq S_w\left({\lambda }_0\right)$, we see that (32) is valid.

$\mathbf{Step}\mathbf{2}.$ We claim that $S_w\left({\lambda }_0\right)=\mathrm{cl}\widehat{S}\left({\lambda }_0\right)$. To this end, by Lemma 11, we need to prove the set $S_w\left({\lambda }_0\right)$ is closed and convex. Firstly, by Lemma 10, we see that $S_w\left({\lambda }_0\right)=\{x\in K\left({\lambda }_0\right):\phi (x,y,{\lambda }_0)\ge 0,\forall y\in K\left({\lambda }_0\right)\}$. It follows from the assumption (ii) and Proposition 10 that $\phi (·,·,{\lambda }_0)$ is continuous on $K\left({\lambda }_0\right)\times K\left({\lambda }_0\right)$. Hence, it is easy to see that the set $S_w\left({\lambda }_0\right)$ is closed.

Secondly, by the assumption (iii) and Proposition 12 (i), we see that $\phi (·,y,{\lambda }_0)$ is properly quasiconcave on $K\left(\lambda \right)$ for each $y\in K\left({\lambda }_0\right)$. Hence, for any $x_1,x_2\in S_w\left({\lambda }_0\right)$ and for any $\alpha \in [0,1]$, we have

$$\mathrm{either}\phi \left(\alpha x_1+(1-\alpha )x_2,y,{\lambda }_0\right)\ge \phi (x_1,y,{\lambda }_0)\ge 0,\forall y\in K\left({\lambda }_0\right)$$

or

$$\phi \left(\alpha x_1+(1-\alpha )x_2,y,{\lambda }_0\right)\ge \phi \left(x_2,y,{\lambda }_0\right)\ge 0,\forall y\in K\left({\lambda }_0\right).$$

This implies that $\alpha x_1+(1-\alpha )x_2\in S_w\left({\lambda }_0\right)$ for any $\alpha \in [0,1]$ and so $S_w\left({\lambda }_0\right)$ is convex. Taking into account $\mathrm{int}{S}_w\left({\lambda }_0\right)=\widehat{S}\left({\lambda }_0\right)\ne Ø$ and with the help of Lemma 11, we have

$$S_w\left({\lambda }_0\right)=\mathrm{cl}\widehat{S}\left({\lambda }_0\right).$$

This together with (32) indicates that (31) is valid. □

Theorem 3. 

Let ${\lambda }_0\in \Lambda$. Assume that the following conditions hold:

(i) 

$K(·)$ is continuous with nonempty compact values at ${\lambda }_0$;

(ii) 

$F(·,·)$ is continuous with nonempty compact values on $K\left({\lambda }_0\right)\times \left\{{\lambda }_0\right\}$;

(iii) 

$F(y,\lambda )\dot{-}F(x,\lambda )\ne Ø$ for any $x,y\in K\left({\lambda }_0\right)$ and for any $\lambda \in \Lambda$;

(iv) 

$F(·,{\lambda }_0)$ is $m_2$-properly C-quasiconvex on $K\left({\lambda }_0\right)$.

Then, $S(·)$ is H-u.s.c. at ${\lambda }_0$.

Proof. 

Suppose to the contrary that $S(·)$ is not H-u.s.c. at ${\lambda }_0$. Then, there exists an open set $V_0$ of $0_X$ such that for any neighborhood $U_0$ of ${\lambda }_0$ and there exists ${\lambda }^{\prime }\in U_0$ with $S\left({\lambda }^{\prime }\right)⊈S\left({\lambda }_0\right)+V_0$. Hence, there exists a sequence $\left\{{\lambda }_n\right\}$ with ${\lambda }_n\to {\lambda }_0$ such that $S\left({\lambda }_n\right)⊈S\left({\lambda }_0\right)+V_0,\forall n\in \mathbb{N}$. This implies that there is $x_n\in S\left({\lambda }_n\right)$ such that

$$x_n\notin S\left({\lambda }_0\right)+V_0,\forall n\in \mathbb{N}.$$

(33)

Taking into account $S\left({\lambda }_n\right)\subseteq S_w\left({\lambda }_n\right)$, we have $x_n\in S_w\left({\lambda }_n\right)$, $\forall n\in \mathbb{N}$. Since $S_w(·)$ is u.s.c. with compact values at ${\lambda }_0$ by Theorem 2, it follows from Proposition 2 (ii) that there exist a subsequence $\left\{x_{n_k}\right\}$ of $\left\{x_n\right\}$ and $x_0\in S_w\left({\lambda }_0\right)$ such that $x_{n_k}\to x_0$. It follows from the closedness of $S_w\left({\lambda }_0\right)$ and Lemma 13 that

$$\mathrm{cl}S\left({\lambda }_0\right)=S_w\left({\lambda }_0\right).$$

Consequently,

$$x_0\in \mathrm{cl}S\left({\lambda }_0\right)\subseteq S\left({\lambda }_0\right)+V_0.$$

This together with $x_{n_k}\to x_0$ gives us $x_{n_k}\in S\left({\lambda }_0\right)+V_0$ for k if it is large enough, which contradicts (33). Therefore, $S(·)$ is H-u.s.c. at ${\lambda }_0$. □

Theorem 4. 

Let ${\lambda }_0\in \Lambda$. Assume that the following conditions hold:

(i) 

$K(·)$ is continuous with nonempty compact values at ${\lambda }_0$;

(ii) 

$F(·,·)$ is continuous with nonempty compact values on $K\left({\lambda }_0\right)\times \left\{{\lambda }_0\right\}$;

(iii) 

$F(y,\lambda )\dot{-}F(x,\lambda )\ne Ø$ for any $x,y\in K\left({\lambda }_0\right)$ and for any $\lambda \in \Lambda$;

(iv) 

$F(·,{\lambda }_0)$ is $m_2$-properly C-quasiconvex on $K\left({\lambda }_0\right)$.

Then, $S(·)$ is l.s.c. at ${\lambda }_0$. Moreover, $S_w(·)$ is l.s.c. at ${\lambda }_0$.

Proof. 

Since the proof is similar to one for the mapping $S_w(·)$, we only prove that $S(·)$ is l.s.c. at ${\lambda }_0$. Indeed, for any $x\in S\left({\lambda }_0\right)$ and for any neighborhood $U\left(x\right)$ of x, and noting that $S\left({\lambda }_0\right)\subseteq \mathrm{cl}\widehat{S}\left({\lambda }_0\right)$ obtained in Lemma 13, we have

$$U\left(x\right)\cap \widehat{S}\left({\lambda }_0\right)\ne Ø.$$

By Lemma 12, we hold that $\widehat{S}(·)$ is l.s.c. at ${\lambda }_0$. Thus, there exists a neighborhood $U\left({\lambda }_0\right)$ of ${\lambda }_0$ such that

$$\widehat{S}\left(\lambda \right)\cap U\left(x\right)\ne Ø,\forall \lambda \in U\left({\lambda }_0\right).$$

Since $\widehat{S}\left(\lambda \right)\subset S\left(\lambda \right)$ for each $\lambda \in \Lambda$, we have

$$S\left(\lambda \right)\cap U\left(x\right)\ne Ø,\forall \lambda \in U\left({\lambda }_0\right).$$

This means that $S(·)$ is l.s.c. at ${\lambda }_0$. □

Remark 4. 

We would like to mention that our main results in this section are different from those in [35,38,39,40]. In fact, we study the semicontinuity of the minimal solution mapping $S(·)$ and the weak minimal solution mapping $S_w(·)$ for parametric set optimization problems with set order relation ${⪯}_C^{m_2}$, while [35,38,39,40] discuss the semicontinuity of the minimal solution mapping $S(·)$ and the weak minimal solution mapping $S_w(·)$ for parametric set optimization problems involving the lower set less relation ${⪯}^l$ or upper set less relation ${⪯}^u$.

In addition, in this section, we used the nonlinear scalarization method to establish the density result and then give the sufficient conditions for the semicontinuity of the minimal solution mapping $S(·)$ and the weak minimal solution mapping $S_w(·)$ to ($m_2$-PSOP). Our method improves the level set mappings or monotonicity approaches proposed in [35,38,39,40] since they use the monotonicity and strictly C-quasiconvexity assumptions.

Remark 5. 

Recently, Preechasilp and Wangkeeree [41] obtained the upper semicontinuity of the $m_1$-minimal solution mapping under the assumption that the objective mapping F has converse $m_1$-property. It is worth noting that the property can lead to the following conclusion when $x_n\to x_0$, $y_n\to y_0$, and ${\lambda }_n\to {\lambda }_0$,

$$F(y_0,{\lambda }_0){⪯}_C^{m_1}F(x_0,{\lambda }_0)\Rightarrow F(y_{n_0},{\lambda }_{n_0}){⪯}_C^{m_1}F(x_{n_0},{\lambda }_{n_0})\mathrm{for}\mathrm{some}{n}_0\in \mathbb{N}.$$

This may be not suitable from the point of view of the locally sign-preserving property of the limit.

Now, we give an example to illustrate that Theorem 3 and Theorem 4 are applicable, but Theorem 3.6 in [41] is not applicable.

Example 4. 

Let $X=\mathbb{R}$, $Y={\mathbb{R}}^2$, and $C={\mathbb{R}}_{+}^2$. Let $\Lambda =[0,1]$ and $K:\Lambda \rightrightarrows X$ be defined by $K\left(\lambda \right)=\{x\in \mathbb{R}:[-\lambda ,\lambda \left]\right\}$ for each $\lambda \in \Lambda$. Let $F:X\times \Lambda \rightrightarrows Y$ be defined by

$$F(x,\lambda )=\left(x+\lambda ,sinx\right)+{\mathbb{B}}_Y.$$

It is not hard to check that the assumptions (i)–(iii) are satisfied in Theorem 3 and Theorem 4. Now, we check that the assumption (iv) holds. Indeed, for any $\lambda \in \Lambda$, for any $x_1,x_2\in K\left(\lambda \right)$ (without loss of generality, we assume that $x_1\le x_2$), and for any $\alpha \in [0,1]$, one has

$$F\left(\alpha x_1+(1-\alpha )x_2,\lambda \right){⪯}_C^{m_2}F(x_2,\lambda ).$$

Hence, $S(·)$ is H-u.s.c and l.s.c. on Λ by Theorems 3 and 4. However, F does not have converse $m_1$-property. Indeed, let $x_0=y_0={\lambda }_0=0$. Obviously, $F(y_0,{\lambda }_0){⪯}_C^{m_1}F(x_0,{\lambda }_0)$, and there exist the sequences $\left\{x_n\right\}=\left\{{\displaystyle \frac{1}{2n}}\right\}$, $\left\{y_n\right\}=\left\{{\displaystyle \frac{1}{n}}\right\}$, and $\left\{{\lambda }_n\right\}=\left\{{\displaystyle \frac{1}{n}}\right\}$ such that for all $n\in N$ with $F(y_n,{\lambda }_n){⋠}_C^{m_1}F(x_n,{\lambda }_n)$. Hence F does not have the converse $m_1$-property and so Theorem 3.6 in [41] is not applicable in this example.

6. Conclusions

This paper mainly investigates the continuity and the convexity of a nonlinear scalarization function for sets, which is an extension of the Gerstewitz function. The function is defined by the partial set order relation ${⪯}_C^{m_2}$, which involves the Minkowski difference. We have shown that our results have wide applications in the path connectedness of the set optimization problems and the continuity of the solution mappings of parametric set optimization problems. In the future, it would be interesting to explore applications of the nonlinear scalarization function to the characterizations of robust counterparts of uncertain multiobjective optimization problems and the computation for robust traffic network equilibrium problems.