Stability of Set-Valued Optimization Problems with Naturally Quasi-Functions

Abstract

In this paper, we discuss the stability of three kinds of minimal point sets and three kinds of minimizer sets of naturally quasi-functional set-valued optimization problems when the data of the approximate problems converges to the data of the original problems in the sense of Painlevé–Kuratowski. Our main results improve and extend the results of the recent papers.

1 Introduction

Stability is widely studied in optimization theory and methodology. In recent years, some papers have appeared, which are devoted to this topic for vector-valued optimization problems; see, e.g., [1–12] and references therein. Attouch and Riahi [1] first studied this topic. Lucchetti and Miglierina [4] discussed stability for a convex vector-valued optimization problem based on the concept of the continuous convergence of the vector-valued mappings. Oppezzi and Rossi [6, 7] extended the definition of gamma-convergence from scalar-valued to vector-valued and applied it to this topic for a convex vector-valued problem. Lalitha and Chatterjee [9, 10] further discussed this topic for a properly quasi-convex vector-valued problem and improved the main results of [6, 7].

Also, set-valued optimization problems have been intensively investigated and applied to various problems, such as game theory, duality principles, robust optimization, and fuzzy optimization; see, e.g., [13–15] and the references therein. However, the studies dealing with the stability of set-valued optimization problems have so far been very limited [2, 3]. The line of proof in [2, 3] is similar to ones used for vector-valued optimization problems. To the best of our knowledge, the studies of stability based on the other ways for set-valued optimization problems are not available. Maybe, the main reason is that the notion of set-valued mappings is more complicated than vector-valued ones. Therefore, we must employ new analysis tools and techniques, which are different from the ones in [2, 3], to discuss the stability of set-valued optimization problems.

The aim of this paper is to use some techniques similar to the ones of [4, 6–11] to study the stability for set-valued optimization problems. Following the idea of [4, 6–11], we first establish the Painlevé–Kuratowski convergence of three kinds of minimal point sets of naturally quasi-functional set-valued optimization problems. Furthermore, we investigate the Painlevé–Kuratowski set-convergence of three kinds of minimizer sets for set-valued optimization problems. Our results are extensions and improvements in the corresponding ones for vector-valued optimization problems in [9–11].

The paper is organized as follows. In Sect. 2, we recall some concepts and establish fundament results. In Sect. 3, we establish the Painlevé–Kuratowski set-convergence results of three kinds of minimal point sets of naturally quasi-functional set-valued optimization problems. In Sect. 4, we furthermore establish the Painlevé–Kuratowski set-convergence results of three kinds of minimizer sets. In Sect. 5, we provide a conclusion of the work presented.

2 Preliminaries

Throughout this paper, let P be a pointed (i.e., $P\cap (-P)=\{0\}$), closed and convex cone in $\mathbb {R}^l$ with nonempty interior. For a nonempty set D of $\mathbb {R}^l$, let $\text{ int }\,D$, $\partial \,D$ and $D^c$ denote the interior, boundary and complement of D, respectively. Let $\mathbb {B}(0, r)$ denote the closed ball centered at 0 and with radius r. Some fundamental terminologies are presented as follows.

Definition 2.1

Let D be a nonempty set of $\mathbb {R}^l$. A point $y\in D$ is said to be

• (i) a minimal (resp. weak minimal) point of D iff $(D-y)\cap (-P)=\{0\}$ (resp. $(D-y)\cap (-\text {int}\,P)=\emptyset$), and $\text {Min}_P\, D$ (resp. $\text {Min}_{\mathrm{int}\,P}\, D$) denotes the set of all minimal (resp. weak minimal) points of D;

• (ii) a Henig proper minimal point of D iff there exists a convex cone $P'$ with $\text {int}\,P'\ne \emptyset$ such that $P\backslash \{0\}\subseteq \text {int}\,P'$ and $(D-y)\cap (-P')=\{0\},$ and $\text {Min}^H_P\, D$ denotes the set of all Henig properly minimal points of D.

It is easy to check that $\text {Min}^H_P\, D\subseteq \text {Min}_P\, D\subseteq \text {Min}_{\mathrm{int}\,P}\, D$.

We now consider the following constrained set-valued optimization problem

where $F:\mathbb {R}^k\rightrightarrows \mathbb {R}^l$ is a set-valued mapping and $A\subseteq \mathbb {R}^k$ is a nonempty set.

Based on the above notions of minimality for a set, we denote by $\text {Min}_PF(A)$, $\text {Min}_{\mathrm{int}\, P}F(A)$ and $\text {Min}^H_PF(A)$ the sets of all minimal, weak minimal and Henig proper minimal points, respectively.

Definition 2.2

A pair $(x_0, y_0)$ with $x_0\in A$ and $y_0\in F(x_0)$ is said to be a minimizer (resp. weak minimizer, Henig minimizer) of the problem (A, F) iff $y_0\in \text {Min}_P\, F(A)$ (resp. $y_0\in \text {Min}_{\mathrm{int}\,P}\, F(A)$, $y_0\in \text {Min}^H_P\, F(A)$). We denote by $\text {E}_P\, (A, F)$, $\text {E}_{\mathrm{int}\,P}\, (A, F)$ and $\text {E}^H_P\, (A, F)$ the set of all minimizers, weak minimizers and Henig minimizers of the problem (A, F), respectively.

Definition 2.3

(see [5]) A sequence of sets $\{D_n\}$ of $\mathbb {R}^k$ converges to a set D in the sense of Painlevé–Kuratowski (for short, P.K.) convergence, denoted by $D_n\overset{\mathrm{P.K.}}{\longrightarrow }D$, iff $\lim \sup _n D_n\subseteq D\subseteq \lim \inf _n D_n,$ where

$$\begin{aligned}&\lim \inf \nolimits _n D_n:=\{x\in \mathbb {R}^m:x=\lim \nolimits _{n\rightarrow \infty }x_n, x_n\in D_n,\quad \forall n\in \mathbb {N}\}\hbox { and}\\&\lim \sup \nolimits _n D_n:=\{x\in \mathbb {R}^m:x=\lim \nolimits _{k\rightarrow \infty }x_k, x_k\in D_{n_k},\quad \forall k, \{n_k\}\subseteq \mathbb {N}\}. \end{aligned}$$

Definition 2.4

(see [3, 16]) A sequence of set-valued mappings $F_n:\mathbb {R}^k\rightrightarrows \mathbb {R}^l$ $(n\in \mathbb {N})$ converges to $F:\mathbb {R}^k\rightrightarrows \mathbb {R}^l$ in the sense of P.K. convergence, denoted by $F_n\overset{\mathrm{P.K.}}{\longrightarrow }F$, iff $\text {epi}F_n \overset{\mathrm{P.K.}}{\longrightarrow }\text {epi}F,$ where $\text {epi}F:=\{(x, z)\in \mathbb {R}^k\times \mathbb {R}^l:z\in F(x)+P\}$.

We introduce a virtual element $+\infty$ in $\mathbb {R}^l$ meaning that for any $y\in \mathbb {R}^l$, $+\infty \in y+P$. Then, we recall the following concept.

Definition 2.5

Let $F, F_n:\mathbb {R}^k\rightarrow \mathbb {R}^l(n\in \mathbb {N})$ be set-valued mappings, $A, A_n$ $(n\in \mathbb {N})$ be sets in $\mathbb {R}^k$ and $\{(A_n, F_n): n\in \mathbb {N}\}$ be the corresponding sequence pair. We say that $(A_n, F_n)$ converges to (A, F) in the sense of P.K. convergence, denoted by $(A_n, F_n)\overset{\mathrm{P.K.}}{\longrightarrow }(A, F)$, iff $\bar{F}_n\overset{\mathrm{P.K.}}{\longrightarrow }\bar{F}$, where

$$\begin{aligned} \bar{F}_n(x)=\left\{ \begin{array}{ll} F_n(x), &{} x\in A_n; \\ \{+\infty \}, &{} x\in \mathbb {R}^k\setminus A_n \end{array} \right. \quad \text {and}\quad \bar{F}(x)=\left\{ \begin{array}{ll} F(x), &{} x\in A; \\ \{ +\infty \}, &{} x\in \mathbb {R}^k\setminus A. \end{array} \right. \end{aligned}$$

Definition 2.6

(see [17, 18]) Let A be a nonempty and convex subset of $\mathbb {R}^k$. A set-valued mapping $F:\mathbb {R}^k\rightrightarrows \mathbb {R}^l$ is said to be:

• (i) P-function (for short, P-F) on A iff for every $x_1,$ $x_2\in A$ and $\lambda \in \, [0,1]\,$, $\lambda F(x_1)+(1-\lambda ) F(x_2)\subseteq F(x_\lambda )+P,$ where $x_\lambda :=\lambda x_1+(1-\lambda )x_2$;

• (ii) P-like-function   (for short, P-LF)  on A iff for every $x_1, x_2\in A$ and $\lambda \in \, [0,1]\,$, there exists $z\in A$ such that $\lambda F(x_1)+(1-\lambda )F(x_2)\subseteq F(z)+P;$

• (iii) properly quasi-P-function (for short, P-PQF) on A iff for every $x_1, x_2\in A$ and $\lambda \in \, [0,1]\,$, $F(x_1)\subseteq F(x_\lambda )+P\quad \text{ or }\quad F(x_2)\subseteq F(x_\lambda )+P;$

• (iv) naturally quasi-P-function (for short, P-NQF) on A iff for every $x_1, x_2\in A$, $y_1\in F(x_1), y_2\in F(x_2)$ and $\lambda \in \, [0,1]\,$, there exists $\eta \in [0,1]$ such that $\eta y_1+(1-\eta )y_2\in F(x_\lambda )+P.$

We say F is strictly P-function ( for short, P-SF) on A iff for every $x_1,$ $x_2\in A,$ $x_1\ne x_2$ and $\lambda \in \, ]0,1[\,$ such that the inequality of (i) holds whenever P is replaced by $\text {int}\,P$. Similarly, F is P-SLF, P-SPQF and P-SNQF on A.

Proposition 2.1

(see [17]) The following statements hold

• (1) P-SF $\Rightarrow$ P-F $\Rightarrow$ P-LF and P-SF $\Rightarrow$ P-SLF $\Rightarrow$ P-LF;

• (2) P-SPQF $\Rightarrow$ P-PQF $\Rightarrow$ P-NQF and P-SPQF $\Rightarrow$ P-SNQF $\Rightarrow$ P-NQF.

Proposition 2.2

(see [17]) A set-valued mapping $F:\mathbb {R}^k\rightrightarrows \mathbb {R}^l$ is P-LF on a nonempty and convex subset A of $\mathbb {R}^k$ if and only if $F(A)+P$ is a convex set.

Definition 2.7

(see [19]) A set-valued mapping $F:\mathbb {R}^k\rightrightarrows \mathbb {R}^l$ is said to be compact-valued (resp. convex-valued) on a set A in $\mathbb {R}^k$ if for any $x\in A$, F(x) is a compact (resp. convex) subset of $\mathbb {R}^l$.

Lemma 2.1

(see [19]) Let $F:\mathbb {R}^k \rightrightarrows \mathbb {R}^l$ be a set-valued map and $x_0\in A$ of $\mathbb {R}^k$ be a given point. If F be compact-valued on A, then F is upper semicontinuous (for short, u.s.c) at $x_0\in A$ if and only if for any sequence $\{x_n\}\subseteq A$ with $x_n\rightarrow x_0$ and for every $y_n\in F(x_n),$ there exist $y_0\in F(x_0)$ and a subsequence $\{y_{n_k}\}$ of $\{y_n\}$ such that $y_{n_k}\rightarrow y_0$.

Lemma 2.2

(see [17]) Let $F:\mathbb {R}^k \rightrightarrows \mathbb {R}^l$ be u.s.c. and convex-valued on a convex set A in $\mathbb {R}^k$. If F is P-NQF on A, then F is P-FL on A, i.e., $F(A)+P$ is a convex subset.

Definition 2.8

(see [20]) A nonempty and convex subset A of $\mathbb {R}^k$ is said to be rotund if the boundary of A does not contain line segments, i.e., for any x, $x'\in A:x\ne x'$, $]x,x'[\,\cap (\partial \, A)^c\ne \emptyset ,$ where $]x,x'[\,:=\{\lambda x+(1-\lambda )x':\lambda \in ]0, 1[\,\}.$

For the set-valued mapping, we introduce a similar concept as following.

Definition 2.9

A convex-valued mapping $F:\mathbb {R}^k\rightrightarrows \mathbb {R}^l$ is said to be rtound-valued on $A\subseteq \mathbb {R}^k$ iff for any $x\in A$, F(x) is a rotund subset of $\mathbb {R}^l$.

Remark 2.1

If F is a single-valued mapping, then F is both rotund-valued and compact-valued on $A\subseteq \mathbb {R}^k$.

Motivated by the idea of Theorem 4.3 in [21], we get the following result.

Lemma 2.3

If A is a convex set, F is P-SNQF and F is convex-valued and rotund-valued on A, then $Min _{\mathrm{int}\,P} F(A)=Min _P F(A).$

Proof

It suffices to show that $\text{ Min }_{\mathrm{int}\,P} F(A)\subseteq \text{ Min }_P F(A)$. Let $y\in \text{ Min }_{\mathrm{int}\,P} F(A)$, hence there exists $x\in A$ such that

$$\begin{aligned} y\in F(x)\quad \text{ and }\quad (F(A)-y)\cap (-\text{ int }\,P)=\emptyset . \end{aligned}$$

(1)

By the contradiction, assume $y\not \in \text{ Min }_P F(A)$, i.e., $(F(A)-y)\cap (-P)\ne \{0\}$. Then there exist $\bar{x}\in A$ and $\bar{y}\in F(\bar{x})$ such that $y\in \bar{y}+P\backslash \{0\}.$ The following two cases would be considered: Case 1. If $x= \bar{x}$, then it follows from $y\in \bar{y}+P\backslash \{0\}$ and the rotundity of F(x) that $]y, \bar{y}[\,\cap \, (\partial \,F(x))^c\ne \emptyset ,$ i.e., $\,]y, \bar{y}[\,\cap \, \text{ int }\,F(x)\ne \emptyset .$ Hence there exists $y'\in \,]y, \bar{y}[\,\cap \, \text{ int }\,F(x).$ We can choose $\epsilon \in \text{ int }\,P$ such that $y'-\epsilon \in F(x)\cap (y-\text{ int }\,P),$ which contradicts (1).

Case 2. If $x\ne \bar{x}$, by the P-SNQF property of F, for every $\lambda \in ]0, 1[$, there exist $\eta \in [0, 1]$ and $y'\in F(\lambda x+(1-\lambda )\bar{x})$ such that $\eta y+(1-\eta )\bar{y}\in y'+\text{ int }\,P,$ which with $y\in \bar{y}+P\backslash \{0\}$ yields that $y\in y'+\text{ int }\,P,$ which contradicts (1). $\square$

Corollary 2.1

Assume the conditions of Lemma 2.3 are satisfied. Then we have $\text{ E }_{\mathrm{int}\,P}(F, A)=\text{ E }(F, A).$

The following example is given to illustrate Lemma 2.3 and Corollary 2.1.

Example 2.1

Let $P=\mathbb {R}^2_+$, $A=\mathbb {R}$ and $F:\mathbb {R}\rightrightarrows \mathbb {R}^2$ be defined as

$$\begin{aligned} F(x)=\{(r, s): x^2\le r\le x^2+1, x^2\le s\le x^2+1\},\quad \forall x\in \mathbb {R}. \end{aligned}$$

Clearly, we have: (i) F is P-SNQF on A; (ii) $\text{ Min }_P\,F(A)=\{(0,0)\}$ and $\text{ Min }_{\mathrm{int}\,P}\,F(A)=\{(0, s): 0\le s\le 1\}\cup \{(r, 0): 0\le r\le 1\}$. So, Lemma 2.3 and Corollary 2.1 do not hold in the absence of rotundity of F.

Definition 2.10

For $\alpha \in \mathbb {R}^l$, the sublevel set of $F:\mathbb {R}^k\rightrightarrows \mathbb {R}^l$ on $A\subseteq \mathbb {R}^k$ at height $\alpha$ is $F^\alpha :=\{x\in A:\alpha \in F(x)+P\}.$

For any $\alpha \in \mathbb {R}^l$, we have: (i) $F^\alpha =\bar{F}^\alpha$, where $\bar{F}:\mathbb {R}^k\rightrightarrows \mathbb {R}^l$ is defined as in Definition 2.5; (ii) $F^\alpha$ is a convex set whenever F is a P-NQF mapping.

Definition 2.11

(see [22]) For a convex set A of $\mathbb {R}^k$, the recession cone of A is the set $0^+(A):=\{d\in \mathbb {R}^m:a+td\in A,\quad \forall \, a\in A,\quad \forall \, t\ge 0\}.$

It is known that if A is a closed and convex set in $\mathbb {R}^k$, then $0^+(A)=\{0\}$ iff A is a bounded set. By Theorem 1.1.17 of [5], for any convex and closed set A in $\mathbb {R}^k$, $0^+(A)=\{d\in \mathbb {R}^k:\exists \, a\in A,\,a+td\in A,\quad \forall \, t\ge 0\}.$

Lemma 2.4

Let A be a nonempty, closed and convex set in $\mathbb {R}^k$ and F be an u.s.c. P-NQF and compact-valued mapping on A. If $A\cap F^\alpha \ne \emptyset$ for $\alpha \in \mathbb {R}^l$, then $0^+(A\cap F^\alpha )=\{0\}$ if and only if $A\cap F^\alpha$ is bounded.

Proof

If $A\cap F^\alpha \ne \emptyset$ for $\alpha \in \mathbb {R}^l$, then $A\cap F^\alpha$ is obviously a convex set. To apply the remark above, we only need to verify that $A\cap F^\alpha$ is a closed set. Indeed, let $x_n\in A\cap F^\alpha$ with $x_n\rightarrow x\in A$ as A is closed. Then there exists $y_n\in F(x_n)$ such that $\alpha \in y_n+P$. By Lemma 2.1, there exists a subsequence $\{y_{n_k}\}$ of $\{y_n\}$ such that $y_{n_k}\rightarrow y\in F(x)$, and so $\alpha \in F(x)+P$. Therefore, $x\in A\cap F^\alpha$, which means that $A\cap F^\alpha$ is closed.

[ $\Leftarrow$ ]. This implication follows from the remark above.

[ $\Rightarrow$ ]. Assume to the contrary that there exists a sequence $\{x_n\}$ in $A\cap F^\alpha$ such that $\Vert x_n\Vert \rightarrow \infty$. By passing to a subsequence if necessary, we can assume that $\frac{x_n}{\Vert x_n\Vert }\rightarrow d$ and $\Vert d\Vert =1$. Next we show that in such case $d\in 0^+(A\cap F^\alpha )$. Indeed, let $x\in A\cap F^\alpha$ be arbitrarily given. From the convexity of A, it follows that for any real number $t\ge 0$, $z_n:=(1-\frac{t}{\Vert x_n\Vert })x+\frac{t}{\Vert x_n\Vert }x_n\in A$ for sufficiently large n. It is easy to see that $z_n\rightarrow z:=x+td\in A$ as A is closed. Since $x_n, x\in A\cap F^\alpha$, there exist $y_n\in F(x_n)$ and $y\in F(x)$ such that

$$\begin{aligned} \alpha \in y_n+P\quad \hbox {and}\quad \alpha \in y+P. \end{aligned}$$

(2)

As F is P-NQF on A, there is $\eta \in [0, 1]$ such that $\eta y_n+(1-\eta ) y\in F(z_n)+P,$ which together with (2) yields that $\alpha \in F(z_n)+P$, i.e., $z_n\in A\cap F^\alpha$. Hence, there exists $w_n\in F(z_n)$ such that $\alpha -w_n\in P$. By Lemma 2.1, there exists a subsequence $\{w_{n_k}\}$ of $\{w_n\}$ such that $w_{n_k}\rightarrow w\in F(z)$. So, we have $\alpha -w\in P$. This implies that $\alpha \in F(z)+P$, i.e., $z\in F^\alpha$. Thus $z=x+td\in A\cap F^\alpha$ and $0\ne d\in 0^+(A\cap F^\alpha )=\{0\}$, which is a contradiction. The proof is complete. $\square$

Lemma 2.5

Let A be a nonempty, closed and convex subset of $\mathbb {R}^k$ and F be an u.s.c., P-NQF and compact-valued mapping on A. If $0^+(A\cap F^\alpha )=\{0\}$ whenever $A\cap F^\alpha \ne \emptyset$ for $\alpha \in \mathbb {R}^l$, then $F(A)+P$ is a closed set.

Proof

Take a sequence $\{y_n\}$ in $F(A)+P$ such that $y_n\rightarrow y$. There exists a sequence $\{x_n\}$ in A such that $y_n\in F(x_n)+P$. By $y_n\rightarrow y$, for any $\epsilon \in \text{ int }\,P$ sufficiently large n, we have $y+\epsilon \in y_n+P$. Hence, $y+\epsilon \in F(x_n)+P$ for sufficiently large n. So, we have $x_n\in A\cap F^{y+\epsilon }$ and there exists $w_n\in F(x_n)$ such that $y+\epsilon -w_n\in P$. As $0^+(A\cap F^{y+\epsilon })=\{0\}$, it follows from Lemma 2.4 that $\{x_n\}$ is bounded. By passing to a subsequence if necessary, we can assume that $x_n\rightarrow x\in A$ as A is closed. Therefore, from Lemma 2.1 it follows that there exists a subsequence $\{w_{n_k}\}$ of $\{w_{n}\}$ such that $w_{n_k}\rightarrow w\in F(x)$. So, by the arbitrariness of $\epsilon$ and the closedness of P, we have $y-w\in P$. This implies that $y\in F(x)+P$ and the proof is complete. $\square$

3 Stability of the Minimal Point Sets

In the sequel, unless otherwise specified, let $F, F_n:\mathbb {R}^k\rightarrow \mathbb {R}^l(n\in \mathbb {N})$ be set-valued mappings and $A, A_n(n\in \mathbb {N})$ be nonempty, closed and convex sets in $\mathbb {R}^k$. In this section, we mainly investigate the P.K. convergence of three kinds of minimal point sets of set-valued optimization problems.

Proposition 3.1

Let $A, A_n(n\in \mathbb {N})$ be nonempty, closed and convex subsets of $\mathbb {R}^k$. Let $F, F_n:\mathbb {R}^k\rightarrow \mathbb {R}^l(n\in \mathbb {N})$ be P-NQF mappings on $A, A_n$, respectively. Assume that: (i) $A_n\overset{\mathrm{P.K.}}{\longrightarrow }A$; (ii) $(A_n, F_{n})\overset{\mathrm{P.K.}}{\longrightarrow }(A, F)$; (iii) if $A\cap F^\alpha \ne \emptyset$ for some $\alpha \in \mathbb {R}^l$, then $0^+(A\cap F^\alpha )=\{0\}$.

Then $0^+(A_n\cap F^\alpha _{n})=\{0\}$ for sufficiently large n whenever $A_n\cap F^\alpha _{n}\ne \emptyset$.

Proof

Assume to the contrary that for some $\alpha \in \mathbb {R}^l$, we have $A_n\cap F^\alpha _n\ne \emptyset$ and $0^+(A_n\cap F^\alpha _n)\ne \{0\}$ for infinite number of indices n. Then, there exists a subsequence $\{d_k\}$ such that $d_k\in 0^+(A_{n_k}\cap F^\alpha _{n_k}),\, d_k \rightarrow d, \Vert d_k\Vert =\Vert d\Vert =1.$ Let $x\in A$ be fixed, and let $\beta \in F(x)$. Clearly, $A\cap F^\beta \ne \emptyset .$ As $\text {int}\,P\ne \emptyset$, there exit $\epsilon \in \text {int}\,P$ and $\lambda >0$ such that $\lambda \epsilon \in \alpha +P$ and $\lambda \epsilon \in \beta +P$. Letting $\gamma =\lambda \epsilon$, we have $\gamma \in \alpha +P\quad \text{ and }\quad \gamma \in \beta +P.$ Clearly, $x\in A\cap F^\gamma$. Since $(A_n, F_n)\overset{\mathrm{P.K.}}{\longrightarrow }(A, F)$, there exists a subsequence $\{(x'_k, \gamma _k)\}$ such that

$$\begin{aligned}&(x'_k, \gamma _k)\rightarrow (x, \gamma ), \end{aligned}$$

(3)

$$\begin{aligned}&x'_k\in A_{n_k}\quad \text{ and }\quad \gamma _k\in F_{n_k}(x'_k)+P. \end{aligned}$$

(4)

By (3), we have $\gamma _k\rightarrow \gamma$. Then, for any $e\in \text{ int }\,P$ and for sufficiently large k,

$$\begin{aligned} \gamma +e\in \gamma _k+P. \end{aligned}$$

(5)

Then it follows from (4) and (5) that for sufficiently large k,$\gamma +e\in F_{n_k}(x'_k)+P.$ This means that

$$\begin{aligned} x'_k\in A_{n_k}\cap F^{\gamma +e}_{n_k}. \end{aligned}$$

(6)

Now, since $d_k\in 0^+(A_{n_k}\cap F^\alpha _{n_k})\subseteq 0^+(A_{n_k}\cap F^{\gamma +e}_{n_k}),$ which together with (6) yields that for any $\mu \ge 0$, $x'_k+\mu d_k\in A_{n_k}\cap F^{\gamma +e}_{n_k}.$ By $(A_{n_k}, F_{n_k})\overset{\mathrm{P.K.}}{\longrightarrow }(A, F)$, we have that $x'_k+\mu d_k\rightarrow x+\mu d\in A\cap F^{\gamma +e}$ for any $\mu \ge 0$. That implies that $0\ne d\in 0^+(A\cap F^{\gamma +e})=\{0\}$, which is a contradiction. $\square$

Proposition 3.2

Assume that all conditions of Proposition 3.1 are satisfied. Then for every $\alpha \in \mathbb {R}^l$ with $A\cap F^\alpha \ne \emptyset$ and for every $r>0$, there exists $k_r\in \mathbb {N}$ such that, $A_n\cap F^\alpha _{n}\subseteq A\cap F^\alpha +\mathbb {B}(0, r),\quad \forall n\ge k_r.$

Proof

Suppose to the contrary that there exist $\alpha \in \mathbb {R}^l$, $r>0$ with $A\cap F^\alpha \ne \emptyset$ and the conclusion is not true. Then there exists a subsequence $\{x_k\}$ such that

$$\begin{aligned} x_k\in A_{n_k}\cap F^\alpha _{n_k}\quad \text{ and }\quad d(x_k, A\cap F^\alpha )>r, \end{aligned}$$

(7)

where $d(x, A):=\inf _{a\in A}\Vert x-a\Vert$. If $\{x_k\}$ is bounded, by passing eventually to a subsequence, we have $x_k\rightarrow x_0\in A$. Then, by $x_k\in A_{n_k}\cap F^\alpha _{n_k}$, we have $(x_k, \alpha )\in \text {epi}\bar{F}_{n_k}$ and $(x_k, \alpha )\rightarrow (x_0, \alpha )$. From $(A_n,F_n)\overset{\mathrm{P.K.}}{\longrightarrow }(A, F)$ it follows that $(x_0, \alpha )\in \text {epi}\bar{F}$. This implies that $x_0\in A\cap F^\alpha$, which contradicts (7).

When $\{x_k\}$ is unbounded, we can assume that $\Vert x_k\Vert \rightarrow \infty .$ Let $x'\in A\cap F^\alpha$ be given. By $(A_n,F_n)\overset{\mathrm{P.K.}}{\longrightarrow }(A, F)$, there exists a subsequence $\{(x'_k, \gamma _k)\}$ in $\text {epi}\bar{F}_{n_k}$ such that $(x'_k, \gamma _k)\rightarrow (x',\alpha )$. Clearly, for any k,

$$\begin{aligned} \gamma _k\in F_{n_k}(x'_k)+P. \end{aligned}$$

(8)

By passing to a subsequence if necessary, we can assume that for any $t\ge 0$,

$$\begin{aligned} \frac{t}{\Vert x_k\Vert }x_k\rightarrow td\quad \text{ and }\quad \Vert d\Vert =1. \end{aligned}$$

(9)

From the convexity of $A_{n_k}$ it follows that

$$\begin{aligned} z_k:=\left( 1-\frac{t}{\Vert x_k\Vert }\right) x'_k+\frac{t}{\Vert x_k\Vert }x_k\in A_{n_k}\,\text {for }k \text { sufficiently large}. \end{aligned}$$

(10)

By (9), (10) and $A_n\overset{\mathrm{P.K.}}{\longrightarrow }A$, we have

$$\begin{aligned} z_k\rightarrow z:=x'+td\in A. \end{aligned}$$

(11)

Since $\gamma _k\rightarrow \alpha$, for any $\epsilon \in \text{ int }\,C$, there exists $k_\epsilon \in \mathbb {N}$ such that,

$$\begin{aligned} \alpha +\epsilon \in \gamma _k+P, \quad \forall k\ge k_\epsilon . \end{aligned}$$

(12)

By (7), (8) and (12), there exist $y_k\in F_{n_k}(x_k)$ and $y'_k\in F_{n_k}(x'_k)$ such that

$$\begin{aligned} \alpha \in y_k+P\quad \text{ and }\quad \alpha +\epsilon \in y'_k+P. \end{aligned}$$

(13)

As $F_{n_k}$ is P-NQF on $A_{n_k}$, there exists $\eta \in [0, 1]$ such that

$$\begin{aligned} \eta y_k+(1-\eta )y'_k\subseteq F_{n_k}(z_k)+P. \end{aligned}$$

(14)

Combining (13) and (14), we have $\alpha +(1-\eta )\epsilon \in F_{n_k}(z_k)+P,$ which implies that $z_k\in A_{n_k}\cap F^{\alpha +(1-\eta )\epsilon }_{n_k}$. Then from $(A_n,F_n)\overset{\mathrm{P.K.}}{\longrightarrow }(A, F)$ and (11) it follows that $z\in A\cap F^{\alpha +(1-\eta )\epsilon }$. That is $0\ne d\in 0^+(A\cap F^{\alpha +(1-\eta )\epsilon })=\{0\},$ which is a contradiction. Thus, the conclusion holds and the proof is complete. $\square$

Proposition 3.3

Assume that all conditions of Proposition 3.1 hold and F is an u.s.c. and compact-valued map on A. Then $F_n(A_n)+P\overset{\mathrm{P.K.}}{\longrightarrow }F(A)+P.$

Proof

Firstly we prove that $F(A)+P\subseteq \lim \inf _n (F_n(A_n)+P)$. Let $y\in F(A)+P$ be arbitrarily given. Then there exists $x\in A$ such that $y\in F(x)+P$. Since $(A_n, F_{n})\overset{\mathrm{P.K.}}{\longrightarrow }(A, F)$, there exists a sequence $\{(x_n, y_n)\}$ in $\text {epi}\,\bar{F}_{n}$ such that $(x_n, y_n)\rightarrow (x, y)$. This means that $y\in \lim \inf _n(F_n(A_n)+P)$.

Secondly we prove that $\lim \sup _n (F_n(A_n)+P)\subseteq F(A)+P.$ Take y in $\lim \sup _n(F_n(A_n)+P)$, hence there exists a subsequence $\{y_k\}$ in $F_{n_k}(A_{n_k})+P$ such that $y_k\rightarrow y$. Then we can choose $x_k\in A_{n_k}$ such that $y_k\in F_{n_k}(x_k)+P$. Noting that $y_k\rightarrow y$, then for any $\epsilon \in \text{ int }\,P$, there exists $k_\epsilon \in \mathbb {N}$ such that $y+\epsilon \in y_k+P$, $\forall k>k_\epsilon$. So, $y+\epsilon \in F_{n_k}(x_k)+P$, $\forall k>k_\epsilon$. This means that $x_k\in A_{n_k}\cap F^{y+\epsilon }_{n_k}$. In virtue of Lemma 2.4 and Proposition 3.2, we have that $\{x_k\}$ is bounded. By passing to a subsequence if necessary, we assume that $x_k\rightarrow x\in A$. From $(A_n, F_n)\overset{\mathrm{P.K.}}{\longrightarrow }(A, F)$, we get $(x_k, y_k)\rightarrow (x, y)\in \text {epi}\,F$, that is, $y\in F(x)+P\subseteq F(A)+P$. The proof is completed. $\square$

Lemma 3.1

(see [4]) Let $D_n(n\in \mathbb {N})$ and D be nonempty, closed and convex subsets of $\mathbb {R}^l$. Assume that $D_n\overset{\mathrm{P.K.}}{\longrightarrow }D$. Then

$$\begin{aligned} (i)\quad \mathrm{Min}_P D\subseteq \lim \inf _n \mathrm{Min}_P D_n; (ii)\quad \mathrm{Min}^H_P D\subseteq \lim \inf _n \mathrm{Min}^H_P D_n. \end{aligned}$$

Theorem 3.1

Assume all conditions of Proposition 3.1 are satisfied and $F, F_n\,(n\in \mathbb {N})$ are u.s.c. and compact-valued mappings on $A, A_n (n\in \mathbb {N})$, respectively. Then

$$\begin{aligned} (i) Min _P F(A)\subseteq \lim \inf _n Min _P F_n(A_n); (ii) Min ^H_P F(A)\subseteq \lim \inf _n Min ^H_P F_n(A_n). \end{aligned}$$

Proof

(i) From Proposition 3.3, we have $F_n( A_n)+P\overset{\mathrm{P.K.}}{\longrightarrow }F(A)+P.$ Since $F_n$ is an u.s.c., P-NQF and compact-valued map on $A_n$, by Lemmas 2.2 and 2.5, $F_n(A_n)+P$ is closed and convex for sufficiently large n. So, by Lemma 3.1 (i) and Proposition 3.3, we have $\text{ Min }_P (F(A)+P)\subseteq \lim \inf _n\text{ Min }_P(F_n(A_n)+P).$ As $\text{ Min }_P(A+P)=\text{ Min }_P A$ for any $A\subseteq \mathbb {R}^k$, $\text{ Min }_P F(A)\subseteq \lim \inf _n\text{ Min }_P F_n(A_n).$

(ii) The proof follows on similar lines by using Lemma 3.1 (ii). $\square$

Corollary 3.1

Assume that all conditions of Theorem 3.1 hold and F is P-SNQF and rotund-valued on A. Then $Min _{\mathrm{int}\,P} F(A)\subseteq \lim \inf _{n} Min _{\mathrm{int}\,P} F_n(A_n).$

Proof

By Lemma 2.3, we have $\text {Min}_{\mathrm{int}\mathbf \,P}\,F(A)=\text {Min}_P\,F(A),$ which together with Theorem 3.1 yields that $\text {Min}_{\mathrm{int}\,P}\,F(A)\subseteq {\lim \inf }_n\text {Min}_P\,F_n(A_n).$ For any set $A_n\subseteq \mathbb {R}^k$, we have ${\lim \inf }_n\text {Min}_P\,F_n(A_n)\subseteq {\lim \inf }_n\text {Min}_{\mathrm{int}\,P}\,F_n(A_n).$ So, the conclusion is true and the proof is complete. $\square$

Theorem 3.2

If $(A_n, F_n)\overset{\mathrm{P.K.}}{\longrightarrow }(A, F)$, $\lim \sup _{n} Min _{\mathrm{int}\,P} F_n(A_n)\subseteq Min _{\mathrm{int}\,P} F(A).$

Proof

For any $y\in \lim \sup _{n} \text {Min}_{\mathrm{int}\,P} F_n(A_n)$, there exists a subsequence $\{(x_{k}, y_k)\}$ in $\text{ E }_{\mathrm{int}\,P}\,(A_{n_k}, F_{n_k})$ such that $y_k\rightarrow y$. Suppose $y\not \in \text{ Min }_{\mathrm{int}\,P}\,F(A)$. Then there exist $x_0\in A$ and $y_0\in F(x_0)$ such that $y_0-y\in -\text{ int }\,P.$ Let $\epsilon :=y-y_0\in \text{ int }\,P$. Since $y_k\rightarrow y$, there exists $k_\epsilon \in \mathbb {N}$ such that $k\ge k_\epsilon$,

$$\begin{aligned} y_k\in y-\frac{\epsilon }{4}+\text{ int }\,P. \end{aligned}$$

(15)

As $(x_0, y_0)\in \text{ epi }\bar{F}$ and $(A_{n}, F_{n})\overset{\mathrm{P.K.}}{\longrightarrow }(A, F)$, there exists a sequence $\{(u_n, v_n)\}$ in $\text{ epi }\, \bar{F}_n$ such that $(u_n, v_n)\rightarrow (x_0, y_0).$ Then there exists $k'_\epsilon \ge k_\epsilon$ such that for any $n\ge k'_\epsilon$,

$$\begin{aligned} v_n\in y_0+\frac{\epsilon }{2}-\text{ int }\,P=y-\frac{\epsilon }{2}-\text{ int }\,P. \end{aligned}$$

(16)

Combining (15) and (16), we have for any $k\ge k'_\epsilon$,

$$\begin{aligned} v_k\in y_k-\frac{\epsilon }{4}-\text{ int }\,P\in y_k-\text{ int }\,P. \end{aligned}$$

(17)

As $(u_k, v_k)\in \text{ epi }\, F_{n_k}$, there exists $v'_k\in F_{n_k}(u_k)$ such that $v'_k\in v_k-P$, which together with (17) yields that $v'_k\in y_k-\text{ int }\,P.$ This contradicts the fact $(x_k, y_k)\in \text{ E }_{\mathrm{int}\,P}(A_{n_k}, F_{n_k})$. Therefore, $y\in \text{ Min }_{\mathrm{int}\,P}\,F(A)$ and the proof is complete. $\square$

Corollary 3.2

Suppose that $(A_n, F_n)\overset{\mathrm{P.K.}}{\longrightarrow }(A, F)$ and F is P-SNQF and rotund-valued on A. Then, we have $\lim \sup _{n} Min _P\,F_n(A_n)\subseteq Min _P\,F(A).$

Proof

By Theorem 3.2 and Lemma 2.3, $\lim \sup _{n} \text{ Min }_{\mathrm{int}\,P}\,F_n(A_n)\subseteq \text{ Min }_P\,F(A).$ Then, we have $\lim \sup _{n} \text{ Min }_{\mathrm{int}\,P}\,F_n(A_n)\subseteq \text{ Min }_P\,F(A),$ which together with the fact ${\lim \sup }_n\text{ Min }_P\,F_n(A_n)\subseteq {\lim \sup }_n\text{ Min }_{\mathrm{int}\,P}\,F_n(A_n),\quad \forall A_n\subseteq \mathbb {R}^k$ yields that $\lim \sup _{n} \text{ Min }_P\,F_n(A_n)\subseteq \text{ Min }_P\,F(A).$ So, the proof is complete. $\square$

By strengthening the conditions of Theorem 3.2, we can establish the upper part of convergence of Henig minimal point sets for perturbed problems.

Theorem 3.3

Assume that all conditions of Proposition 3.1 are satisfied. Let $F, F_n$ $(n\in \mathbb {N})$ be u.s.c., P-SPQF and compact-valued maps on $A, A_n(n\in \mathbb {N})$, respectively. Furthermore, assume that $epi F$ is a closed set. Then, we have

$$\begin{aligned} \lim \sup _n Min ^H_P\,F_n(A_n)\subseteq Min ^H_P\,F(A). \end{aligned}$$

Proof

Let $y\in \lim \sup _n\text{ Min }^H_P F_n(A_n)$ be any given. By Theorem 3.2, we have $y\in \lim \sup _n\text{ Min }^H_P F_n(A_n)\subseteq \text{ Min }_{\mathrm{int}\,P} F(A)\subseteq F(A).$ Hence there exists $x\in A$ such that $y\in F(x)$. We claim that $y\in \text{ Min }^H_P F(A)$. By the contradiction, we suppose that for any pointed convex cone $P_1$ with $P\backslash \{0\}\subseteq \text{ int }\,P_1$, there exist $x_0\in A$ and $y_0\in F(x_0)$ such that

$$\begin{aligned} 0\ne z:=y_0-y\in -P_1. \end{aligned}$$

(18)

As $y\in \lim \sup _n\text{ Min }^H_P F_n(A_n)$, there exists a sequence $\{y_k\}$ in $\text{ Min }^H_P F_{n_k}(A_{n_k})$ such that $y_k\rightarrow y$. Then there exits $x_k\in A_{n_k}$ such that $y_k\in F_{n_k}(x_k)$. Let $\epsilon \in \text{ int }\,P$ be an arbitrary element. As $y_k\rightarrow y$, it follows that $y+\epsilon \in F_{n_k}(x_k)+P$ for sufficiently large k. This means that $x_k\in A_{n_k}\cap F^{y+\epsilon }_{n_k}$. By Lemma 2.4 and Proposition 3.2, the sequence $\{x_k\}$ is bounded and has a convergent subsequence. Without loss of generality, we assume that $x_k\rightarrow \hat{x}\in A$. Since $(x_0, y_0)\in \text{ epi }\bar{F}$ and $(A_n, F_{n})\overset{\mathrm{P.K.}}{\longrightarrow }(A, F)$, there exists a sequence $\{(u_k, v_k)\}$ in $\text{ epi }\,\bar{F}_{n_k}$ such that $(u_k, v_k)\rightarrow (x_0, y_0)$. Then for any $\epsilon \in \text{ int }\,P$, there exists $k_\epsilon \in \mathbb {N}$ such that

$$\begin{aligned} y_0+\epsilon \in F_{n_k}(u_k)+P,\,\hbox {for any }\, k\ge k_\epsilon . \end{aligned}$$

(19)

For any $k>1$, set $s_k:=\frac{1}{k}u_k+(1-\frac{1}{k})x_k$, which together with the convexity of $A_{n_k}$ yields that $s_k\in A_{n_k}$. Noting that $A_{n}\overset{\mathrm{P.K.}}{\longrightarrow }A$, we have that $s_k\rightarrow \hat{x}\in A$. If $x_k=u_k$ for any k, then

$$\begin{aligned} \hat{x}=x_0 \text{ and } F(\hat{x})=F(x_0)\ni y_0. \end{aligned}$$

(20)

If $x_k\ne u_k$ for some k, then the P-SPQF property of $F_{n_k}$ implies that, either $F_{n_k}(x_k)\subseteq F_{n_k}(s_k)+\text{ int }\,P\quad \text{ or }\quad F_{n_k}(u_k)\subseteq F_{n_k}(s_k)+\text{ int }\,P.$ Since $y_k\in F_{n_k}(x_k)$ and $y_k\in \text{ Min }^H_P F_{n_k}(A_{n_k})$, $F_{n_k}(x_k)\subseteq F_{n_k}(s_k)+\text{ int }\,P$ does not hold. Therefore, we have $F_{n_k}(u_k)\subseteq F_{n_k}(s_k)+\text{ int }\,P$ for any k. Then from (19) it follows that $y_0+\epsilon \in F_{n_k}(s_k)+\text{ int }\,P, \,\hbox {for any }\, k\ge k_\epsilon .$ Then, $(s_k, y_0+\epsilon )\in \text{ epi }F_{n_k}$ and $(s_k, y_0+\epsilon )\rightarrow (\hat{x}, y_0+\epsilon )$. By the fact $(F_n, A_{n})\overset{\mathrm{P.K.}}{\longrightarrow }(F, A)$, we have that $(\hat{x}, y_0+\epsilon )\in \text{ epi }F$. This implies that

$$\begin{aligned} y_0+\epsilon \in F(\hat{x})+P. \end{aligned}$$

(21)

So, from (20) and (21) we always have $y_0+\epsilon \in F(\hat{x})+P$. By the arbitrariness of $\epsilon$ and the closedness of $\text{ epi }\,F$, we have $y_0\in F(\hat{x})+P$, that is, there exists $p_0\in P$ such that $y_0-p_0\in F(\hat{x}).$

The following two cases would be considered:

Case 1. If $x=\hat{x}$, then $y\in F(x)=F(\hat{x})$. We claim that $y\ne y_0-p_0$. Otherwise, we have

$$\begin{aligned} z=y_0-y=p_0\in P\subseteq P_1. \end{aligned}$$

(22)

Hence, from (18) and (22) it follows that $0\ne z\in P_1\cap (-P_1)$. As $P_1$ is a pointed cone, $0\ne z\in P_1\cap (-P_1)=\{0\}$, which is a contradiction. Thus $y\ne y_0-p_0$, which with the rotundity of $F(\hat{x})$ yields that $]y, y_0-p_0[\,\cap \, (\partial \,F(\hat{x}))^c\ne \emptyset$, that is, $]y, y_0-p_0[\,\cap \, \text{ int }\,F(\hat{x})\ne \emptyset .$ Hence there exists $y'\in \,]y, y_0-p_0[\,\cap \, \text{ int }\,F(\hat{x})$. Now we can choose $\epsilon \in \text{ int }\,P$ such that $y'-\epsilon \in F(\hat{x})\cap (y-\text{ int }\,P),$ which contradicts the fact $y\in \text{ Min }_{\mathrm{int}\,P} F(A)$.

Case 2. If $x\ne \hat{x}$, then P-SPQF property of F implies that for any $\lambda \in \,]0,1[$, either $F(x)\subseteq F(\lambda \hat{x}+(1-\lambda )x)+\text{ int }\,P$ or $F(\hat{x})\subseteq F(\lambda \hat{x}+(1-\lambda )x)+\text{ int }\,P.$ As $y\in F(x)$ and $y\in \text{ Min }_{\mathrm{int}\,P} F(A)$, $F(x)\subseteq F(\lambda \hat{x}+(1-\lambda )x)+\text{ int }\,P$ does not hold. Hence, for any $\lambda \in \,]0, 1[$, $F(\hat{x})\subseteq F(\lambda \hat{x}+(1-\lambda )x)+\text{ int }\,P$, which together with $y_0\in F(\hat{x})+P$ yields that $y_0\in F(\lambda \hat{x}+(1-\lambda )x)+\text{ int }\,P.$ Since $\text{ epi }\,F$ is closed, by taking limit as $\lambda \rightarrow 0_+$, we have $(x,y_0)\in \text{ epi }\,F$. Then there exists $p'_0\in P$ such that $y_0-p'_0\in F(x)$. Similar to the Case 1, we can show that $y\ne y_0-p'_0$. By the rotundity of F(x), we have $]y, y_0-p'_0[\,\cap \, \text{ int }\,F( x)\ne \emptyset .$ Hence there exists $y'\in \,]y, y_0-p'_0[\,\cap \, \text{ int }\,F( x)$. Now we can choose $\epsilon \in \text{ int }\,P$ such that $y'-\epsilon \in F( x)\cap (y-\text{ int }\,P),$ which contradicts the fact $y\in \text{ Min }_{\mathrm{int}\,P} F(A)$.

So, from Cases 1 and 2 we can see that $y\in \text{ Min }^H_P F(A)$. $\square$

Summarizing Theorems 3.1 (i), 3.2 and Corollaries 3.1, 3.2, we have established the P.K. convergence of the (weak) minimal point sets.

Theorem 3.4

Let $A, A_n(n\in \mathbb {N})$ be nonempty, closed and convex subsets of $\mathbb {R}^k$. Let $F :\mathbb {R}^k\rightarrow \mathbb {R}^l$ be an u.s.c., P-SNQF, compact-valued and rotund-valued mapping on A and $F_n:\mathbb {R}^k\rightarrow \mathbb {R}^l(n\in \mathbb {N})$ be u.s.c., P-NQF and compact-valued mappings on $A_n(n\in \mathbb {N})$. Assume that: (i) $A_n\overset{\mathrm{P.K.}}{\longrightarrow }A$; (ii) $(A_n, F_{n})\overset{\mathrm{P.K.}}{\longrightarrow }(A, F)$; (iii) if $A\cap F^\alpha \ne \emptyset$ for some $\alpha \in \mathbb {R}^l$, then $0^+(A\cap F^\alpha )=\{0\}$. Then, we have

$$\begin{aligned} (a)\quad Min _P F_n(A_n)\overset{\mathrm{P.K.}}{\longrightarrow }Min _P F(A); \quad (b)\quad Min _{\mathrm{int}\,P} F_n(A_n)\overset{\mathrm{P.K.}}{\longrightarrow }Min _{\mathrm{int}\,P} F(A). \end{aligned}$$

Remark 3.1

Theorem 3.4 extends the corresponding ones of [4, 6–11] in the following aspects: (i) Theorem 3.4 extends the corresponding ones of [4, 6–11] from the vector-valued optimization problem to the set-valued case; (ii) The convexity of objective mappings $F, F_n(n\in \mathbb {N})$ in [4, 6–9, 11] are weakened to P-NQF; (iii) The convergence of objective mappings $F_n (n\in \mathbb {N})$ in [4, 6, 7, 9, 10] is weakened to P.K. convergence (see, Examples 3.2 and 3.3 in [11]).

The following example is given to illustrate Remark 3.1 (i) and (ii).

Example 3.1

Let $A_n=A=[0,1]$ and $P=\mathbb {R}^2_+$. Let $F_n, F:\mathbb {R}\rightrightarrows \mathbb {R}^2$ be respectively defined as $F_n(x)=(x^3, 1-x^2+\frac{1}{n})\quad \text{ and }\quad F(x)=(x^3, 1-x^2).$ Indeed, we have $\text{ Min }_{\mathrm{int}\,P} F(A)=\text{ Min }_P F(A)=\{(x^3, 1-x^2):x\in [0,1]\}$ and $\text{ Min }_{\mathrm{int}\,P} F_n(A_n)=\text{ Min }_P F_n(A_n)=\{(x^3, 1-x^2+\frac{1}{n}):x\in [0,1]\}$. Therefore, $\text{ Min }_{\mathrm{int}\,P} F_n(A_n)=\text{ Min }_P F_n(A_n)\overset{\mathrm{P.K.}}{\longrightarrow } \text{ WMin }_{\mathrm{int}\,P} F(A)=\text{ Min }_P F(A)$ and Theorem 3.4 is applicable.

However, the corresponding ones in [4, 6–11] are not applicable. The main reasons are that $F_n$ and F are neither P-PQF nor P-F. So, Theorems 3.4 is an improvement of the corresponding ones in [4, 6–11].

Combining Proposition 2.1 (2), Theorems 3.1 (ii) and 3.3, we have established the P.K. convergence of the Henig minimal sets.

Theorem 3.5

Let $A, A_n$ $(n\in \mathbb {N})$ be nonempty, closed and convex subsets of $\mathbb {R}^k$. Let F be an u.s.c., P-SPQF and compact-valued mapping on A and $F_n$ $(n\in \mathbb {N})$ be P-SPQF and compact-valued maps on $A_n (n\in \mathbb {N})$. Assume that: (i) $A_n\overset{\mathrm{P.K.}}{\longrightarrow }A$; (ii) $(A_n, F_{n})\overset{\mathrm{P.K.}}{\longrightarrow }(A, F)$; (iii) if $A\cap F^\alpha \ne \emptyset$ for some $\alpha \in \mathbb {R}^l$, then $0^+(A\cap F^\alpha )=\{0\}$; (iv) $\text{ epi }\,F$ is closed. Then, we have

$$\begin{aligned} Min ^H_P F_n(A_n)\overset{\mathrm{P.K.}}{\longrightarrow }Min ^H_P F(A). \end{aligned}$$

4 Stability of the Minimizer Sets

In this section, we investigate the P.K. convergence of three kinds of minimizer sets in the given space.

Theorem 4.1

Suppose that all conditions of Theorem 3.4 hold. Then we have

$$\begin{aligned} (a)\quad E _{\mathrm{int}\,P}\,(A_n, F_n)\overset{\mathrm{P.K.}}{\longrightarrow }E _{\mathrm{int}\,P}\,(A, F); \quad (b)\,E _P\,(A_n, F_n)\overset{\mathrm{P.K.}}{\longrightarrow }E _P\,(A, F). \end{aligned}$$

Proof

(a) Firstly, we show $\lim \sup _n\text{ E }_{\mathrm{int}\,P}\,(A_n, F_n)\subseteq \text{ E }_{\mathrm{int}\,P}\,(A, F).$ Let (x, y) in $\lim \sup _n\text{ E }_{\mathrm{int}\,P}\,(A_n, F_n)$, hence there exists a subsequence $\{(x_k, y_k)\}$ in $\text{ E }_{\mathrm{int}\,P}\,(A_{n_k}, F_{n_k})$ such that $(x_k, y_k)\rightarrow (x, y)$. We claim that $(x, y)\in \text{ E }_{\mathrm{int}\,P}\,(A, F)$. Otherwise, there exist $x'\in A$ and $y'\in F(x')$ such that $0\ne y'-y\in -\text{ int }\,P$. Then we can choose $\epsilon \in \text{ int }\,P$ such that

$$\begin{aligned} y'-y+2\epsilon \in -\text{ int }\,P. \end{aligned}$$

(23)

By the fact $y'\in F(x')$ and $(A_n, F_n)\overset{\mathrm{P.K.}}{\longrightarrow }(A, F)$, there exists a sequence $(x'_k, y'_k)\in \,\text{ epi }\, F_{n_k}$ such that $(x'_k, y'_k)\rightarrow (x', y')$. As $y_k\rightarrow y$ and $y'_k\rightarrow y'$, then for sufficiently large k and the above $\epsilon$, we have $y_k\in y-\epsilon +\text{ int }\,P$ and $y'_k\in y'+\epsilon -\text{ int }\,P,$ which together with (23) yields that $y'_k-y_k\in -\text{ int }\,P$. As $(x'_k, y'_k)\in \,\text{ epi }\, F_{n_k}$ for any k, there exists $w_k\in F_{n_k}(x'_k)$ such that $w_k\in y'_k-P$. So, $w_k-y_k\in -\text{ int }\,P$, which contradicts $(x_k, y_k)\in \text{ E }_{\mathrm{int}\,P}\,(A_{n_k}, F_{n_k})$.

Secondly, we prove $\text{ E }_{\mathrm{int}\,P}\,(A, F)\subseteq \lim \inf _n\text{ E }_{\mathrm{int}\,P}\,(A_n, F_n).$ Taking (x, y) in $\text{ E }_{\mathrm{int}\,P}\,(A, F)$, we have $y\in F(x)$ and $y\in \text{ Min }_{\mathrm{int}\,P}F(A)$. By Theorem 3.2, there exists a sequence $\{y_n\}$ in $\text{ Min }_{\mathrm{int}\,P} F_n(A_n)$ such that $y_n\rightarrow y$. We can choose a sequence $\{x_n\}$ in $A_n$ such that $y_n\in F_n(x_n)$ and $(x_n, y_n)\in \text{ E }_{\mathrm{int}\,P}\,(A_n, F_n)$. Let $\epsilon \in \text{ int }\,P$ be any element. From the fact $F_n(x_n)\ni y_n\rightarrow y$, it follows that $y+\epsilon \in y_n+P\subseteq F_n(x_n)+P$ for sufficiently large n. This means that $x_n\in A_n\cap F^{y+\epsilon }_n$. By Lemma 2.4 and Proposition 3.2, the sequence $\{x_n\}$ is bounded and has a convergent subsequence. Without loss of generality, we assume that $x_n\rightarrow \hat{x}\in A$. Since $(A_n, F_n)\overset{\mathrm{P.K.}}{\longrightarrow }(A, F)$, we have that $\lim \sup _n \text{ epi }\bar{F}_n\subseteq \text{ epi }\bar{F}$. According to $(x_n, y_n)\in \text{ epi }\bar{F}_n$ and $(x_n, y_n)\rightarrow (\hat{x}, y)$, we get that $(\hat{x}, y)\in \text{ epi }\bar{F}$. That is, there exists $\hat{y}\in F(\hat{x})$ such that $y\in \hat{y}+P$.

Now we show that $x=\hat{x}$. Suppose to the contrary that $x\ne \hat{x}$. As F is P-SNQF on A and $y\in F(x)$ and $\hat{y}\in F(\hat{x})$, for any $\lambda \in ]0,1[$, there exists $\eta \in [0, 1]$ such that $\eta \hat{y}+(1-\eta )y\in F(\lambda x+(1-\lambda )\hat{x})+\text{ int }\,P$ which together with the fact $y\in \hat{y}+P$ yields $y\in F(\lambda x+(1-\lambda )\hat{x})+\text{ int }\,P.$ This contradicts the fact $(x, y)\in \text{ E }_{\mathrm{int}\,P}\,(A, F)$. Thus, every possible convergent subsequence of $\{x_n\}$ converges to x and hence the entire sequence $\{x_n\}$ converges to x. So, $(x_n, y_n)\in \text{ E }_{\mathrm{int}\,P}\,(A_n, F_n)$, $(x_n, y_n)\rightarrow (x, y)$ and $\text{ E }_{\mathrm{int}\,P}\,(A, F)\subseteq \lim \inf _n \text{ E }_{\mathrm{int}\,P}\,(A_n, F_n).$

(b) The proof follows on similar lines of (a). $\square$

Theorem 4.2

Suppose that all conditions of Theorem 3.5 hold. Then we have

$$\begin{aligned} E ^H_P\,(A_n, F_n)\overset{\mathrm{P.K.}}{\longrightarrow }E ^H_P\,(A, F). \end{aligned}$$

Proof

As the proof is similar to the one of Theorem 4.1, we omit it. $\square$

5 Conclusions

This paper considered the stability of a P-NQF set-valued optimization problem based on the concept of P.K. convergence of the feasible sets and objective set-valued mappings. The results improved and extended the corresponding results of the recent papers. The generalization is threefold: The objective mappings are extended from vector-valued mappings to set-valued ones; the convexity of objective mappings is weakened to P-NQF; the convergence of the sequences of objective mappings is weakened to P.K. convergence.