Levitin–Polyak well-posedness for set optimization problems involving set order relations

Abstract

In this paper set optimization problems with three types of set order relations are concerned. We introduce various types of Levitin–Polyak (LP) well-posedness for set optimization problems and survey their relationships. After that, sufficient and necessary conditions for the reference problems to be LP well-posed are given. Furthermore, using the Kuratowski measure of noncompactness, we study characterizations of well-posedness for set optimization problems. Moreover, the links between stability and LP well-posedness of such problems are established via the study on approximating solution mappings. Tools and techniques used in this study and our results are different from existing ones in the literature.

1 Introduction

The concept of well-posedness was firstly introduced by Tikhonov [36]. Beside the requirement about the uniqueness of the solution, Tikhonov well-posedness requested the convergence of each minimizing sequence to the unique solution. Therefore, well-posedness plays a vital role to make connections between stability properties and solution methods for problems related to optimization. On this topic, many research results have been devoted to a lot of important problems such as variational inequalities [7], equilibrium problems [1], inclusion problems [37] and the references therein. Several generalizations of Tikhonov well-posedness were introduced and investigated for various kinds of optimization problems [4, 10, 30,31,32]. Levitin–Polyak (LP) well-posedness is an extension of Tikhonov well-posedness and was originally proposed in [28]. Every minimizing sequence must belong to the feasible set in Tikhonov well-posedness, whereas it can be outside of the feasible region but the distance between it and this set has to approach zero in LP well-posedness. There have been many studies of LP well-posedness (see, e.g., [18, 27] and the references therein).

Kuroiwa et al. [26] proposed set order relations including lower set less relation, upper set less relation and set less relation (combination of the lower and the upper set less relation). This gave a new way to formulate the solution of set-valued optimization problems which is called solutions concept based on the set approach [25], and hence the optimization problems in this approach are called set optimization problems involving set order relations [21, 24]. As pointed out in [19], the set less relation is generalized and more appropriate in practical problems than both the lower and upper set less relations. Furthermore, the set less relation plays a center role in relationships with other new order relations for sets proposed in [5, 19] which are more useful in set optimization. Although set optimization is a new direction in the field of optimization, it has attracted a great deal of attention of researchers with many important and interesting results [11, 13, 17, 20]. Useful applications of set optimization in practical problems were reported, for example, the application in socio-economic [34] (to manage noise disturbance in the region surrounding the Frankfurt Airport in Germany), the application in finance [14] (to evaluate the risk of a multivariate random outcome). Moreover, relationships between set optimizations and other important problems such as variational inequalities [8], Ky Fan inequality problems (so-called equilibrium problems) [35] were investigated. For further reading and references, we refer to books [15, 21].

The first introduction of well-posedness for set optimization problems was presented by Zhang et al. [38]. The authors established both sufficient and necessary conditions for set optimization problems involving the lower set less relation to be well-posed and obtained criteria as well as characterizations of well-posedness for these problems by the scalarization method. This research was generalized by Gutiérrez et al. [12] under assumptions of cone properness. After that, Dhingra and Lalitha [9] introduced a concept of well-setness for such problems and proved that it is an extension of the generalized well-posedness which was considered in [38]. Recently, well-posedness of set optimization problems involving not only the lower but also the upper set less relation have been discussed in [16, 22, 29]. For LP well-posedness, to the best of our knowledge there is only the paper of Khoshkhabar-amiranloo and Khorram [22] which studied set optimization problem involving the lower set less relation. Of course such an important topic as LP well-posedness for set optimization problems must be the aim of many works. Moreover, the scalarization method which is the main tool used in papers mentioned above investigates difficultly set optimization problems involving different set order relations, some other approaches to study well-posedness for such problems should be considered.

Motivated and inspired by works mentioned above, in this paper, without using the scalarization method, we investigate different types of LP well-posedness for set optimization problems involving several kinds of set order relations. More precisely, we concern set optimization problems involving three types of set order relations. Then, we introduce concepts of LP well-posedness for such problems and discuss relationships among them. Moreover, necessary and/or sufficient conditions for these concepts of well-posedness are investigated. Applying Kuratowski measure of noncompactness, we study characterizations of such concepts. Finally, approximating solution mappings and their stability are studied to build the connection between stability of approximating problem and LP well-posedness of the set optimization problem.

The outline of this paper is given as follows. In Sect. 2, we recall some definitions and results needed in what follows. Sect. 3 introduces various kinds of LP well-posedness for set optimization problems and investigates their relationships. Furthermore, sufficient and/or necessary conditions of pointwise LP well-posedness for such problems are also obtained. In this section, characterizations of these types of pointwise LP well-posedness are surveyed by using measure of noncompactness. In the last section, Sect. 4, we study sufficient conditions for such problem to be metrically LP well-posed and their relationships.

2 Preliminaries

Let X be a normed space and Y be a real Hausdorff topological linear space. Let K be a closed convex pointed cone in Y with $\text {int}K \not = \emptyset$, where intK denotes the interior of K. The space Y is endowed with an order relation induced by cone K in the following way

$$\begin{aligned} x \le _K y\Leftrightarrow & {} y-x \in K,\\ x <_K y\Leftrightarrow & {} y-x \in \text {int}K. \end{aligned}$$

The cone K induces various set orderings in Y. These such orderings as the following were presented in [19, 21, 25]. Let $\mathscr {P}(Y )$ be the family of all nonempty subsets of Y. For $A, B \in \mathscr {P}(Y )$, lower set less relation, upper set less relation and set less relation, respectively, are defined by

$$\begin{aligned}&A \le ^l B \, \text {if and only if} \, B \subset A + K,\\&A \le ^u B \, \text {if and only if} \, A \subset B- K,\\&A \le ^s B \, \text {if and only if} \, A \subset B- K \,\, \text {and} \, \, B \subset A + K. \end{aligned}$$

Remark 2.1

The relationship between the lower set less relation $\le ^l$ and the upper set less relation $\le ^u$ was given by Remark 2.6.10 in [21] as the following

$$\begin{aligned} A \le ^l B \Leftrightarrow -B\le ^u -A. \end{aligned}$$

Definition 2.1

[19] The binary relation $\le$ is said to be

• (i) compatible with the addition if and only if $A \le B$ and $D \le E$ imply $A + D \le B + E$ for all $A,B,D,E \in \mathscr {P}(Y)$;

• (ii) compatible with the multiplication with a nonnegative real number if and only if $A \le B$ implies $\lambda A \le \lambda B$ for all scalars $\lambda \ge 0$ and all $A,B \in \mathscr {P}(Y)$;

• (iii) compatible with the conlinear structure of $\mathscr {P}(Y)$ if and only if it is compatible with both the addition and the multiplication with a nonnegative real number.

Proposition 2.1

[19]

• (i) The order relations $\le ^l$, $\le ^u$ and $\le ^s$ are pre-order (i.e., these relations are reflexive and transitive).

• (ii) The order relations $\le ^l$, $\le ^u$ and $\le ^s$ are compatible with the conlinear structure of $\mathscr {P}(Y)$.

• (iii) In general, the order relations $\le ^l$, $\le ^u$ and $\le ^s$ are not antisymmetric; more precisely, for arbitrary sets $A,B \in \mathscr {P}(Y)$ we have

  $$\begin{aligned}&(A \le ^l B \quad \text{ and } \quad B \le ^l A) \Leftrightarrow A +K = B +K,\\&(A \le ^u B \quad \text{ and } \quad B \le ^u A) \Leftrightarrow A-K = B-K,\\&(A \le ^s B \quad \text{ and } \quad B \le ^s A) \Leftrightarrow (A +K = B +K \quad \text{ and } \quad A-K = B-K). \end{aligned}$$

For $\alpha \in \{l, u, s\}$, we say that

$$\begin{aligned} A \sim ^\alpha B \, \text {if and only if} \, A \le ^\alpha B \, \text {and} \, B \le ^\alpha A. \end{aligned}$$

Let $F : X \rightrightarrows Y$ be a set-valued mapping with nonempty values on X. For each $\alpha \in ~\{l, u, s\}$, we consider the following set optimization problem

$$\begin{aligned}&(\mathrm{P}_{\alpha }) \qquad {\alpha }{\text {-Min}} \textit{F(x)}\\&\qquad \qquad \text {subject to} \quad x \in M, \end{aligned}$$

where M is a nonempty closed subset of X. A point $\bar{x} \in M$ is said to be an $\alpha \text {-minimal}$ solution of $(\mathrm{P}_\alpha )$ if and only if for any $x \in M$ such that $F(x) \le ^{\alpha } F( \bar{x})$ then $F( \bar{x}) \le ^{\alpha } F(x)$. The set of all $\alpha \text {-minimal}$ solutions of $(\mathrm{P}_\alpha )$ is denoted by $S_{\alpha {{\text {-Min}}}F}$.

Remark 2.2

It can be seen that if $\bar{x} \in$$S_{\alpha {{\text {-Min}}}F}$ and $F(\bar{x}) \sim ^{\alpha }F(\hat{x})$ for some $\hat{x} \in M$, then $\hat{x} \in S_{\alpha {{\text {-Min}}}F}$.

We recall the following definitions of semicontinuity for a set-valued mapping and their properties used in the sequel.

Definition 2.2

([3], p. 38, 39) Let $F: X \rightrightarrows Y$ be a set-valued mapping.

• (i) F is said to be upper semicontinuous at $x_0 \in X$ if and only if for any open subset U of Y with $F(x_0) \subset U$, there is a neighborhood N of $x_0$ such that $F(x) \subset U$ for every $x \in N$.

• (ii) F is said to be lower semicontinuous at $x_0 \in X$ if and only if for any open subset U of Y with $F(x_0) \cap U \not = \emptyset$, there is a neighborhood N of $x_0$ such that $F(x) \cap U \not = \emptyset$ for all $x \in N$.

• (iii) F is said to be lower (upper) semicontinuous on a subset S of X if and only if it is lower (upper) semicontinuous at every $x \in S$.

Lemma 2.1

Let $F: X \rightrightarrows Y$ be a set-valued mapping.

• (i) ([3], p. 39) F is lower semicontinuous at $x_0 \in X$ if and only if for any net $\{x_\alpha \} \subset X$ converging to $x_0$ and for any $y \in F(x_0)$, there exist $y_\alpha \in F(x_\alpha )$ such that $\{y_\alpha \}$ converges to y.

• (ii) ([2]) If $F(x_0)$ is compact, then F is upper semicontinuous at $x_0\in X$ if and only if for any net $\{x_\alpha \}$ converging to $x_0$ and for any $y_\alpha \in F(x_\alpha )$, there exist $y_0\in F(x_0)$ and a subnet $\{y_\beta \}$ of $\{y_\alpha \}$ such that $\{y_\beta \}$ converges to $y_0$. If, in addition, $F(x_0) = \{y_0\}$ is a singleton, then for the above nets, $\{y_\beta \}$ converges to $y_0$.

Now we recall the concepts of Hausdorff distance and Hausdorff convergence of sequence of sets. If S is a nonempty subset of X and $x \in X$, then the distance d between x and S is defined by

$$\begin{aligned} d(x, S):= \inf _{u\in S}\left\| x-u\right\| . \end{aligned}$$

If $S_1$ and $S_2$ are two nonempty subsets of X, then Hausdorff distance between $S_1$ and $S_2$, denoted by $H(S_1, S_2)$, is defined by

$$\begin{aligned} H(S_1, S_2):=\text {max}\{H^*(S_1, S_2), H^*(S_2, S_1) \}, \end{aligned}$$

where $H^*(S_1, S_2) := \sup _{x\in S_1}d(x, S_2)$.

Definition 2.3

([23], p. 359) Let $\{A_n\}$ be a sequence of subsets of X. We say that $A_n$ converge to $A \subset X$ in the sense of the Hausdorff metric, denoted by $A_n\rightarrow A$, if and only if $H(A_n, A) \rightarrow 0$ as $n \rightarrow \infty$.

Next, we recall the concept of Kuratowski measure of noncompactness and it’s properties used in the sequel.

Definition 2.4

([33], Definition 2.1) Let M be a nonempty subset of X. The Kuratowski measure of noncompactness $\mu$ of the set M is defined by

$$\begin{aligned} \mu (M):=\inf \left\{ \varepsilon >0 \mid M\subset \bigcup _{i=1}^n M_i,\text {diam}{M_i} <\varepsilon , i=1,\cdots ,n \text { for some } n \in \mathbb {N} \right\} , \end{aligned}$$

where $\text {diam} M_i:=\sup \{d(x,y) \mid x,y \in M_i\}$ is the diameter of $M_i$.

Lemma 2.2

([33], Proposition 2.3) The following assertions are true:

• (i) $\mu (M)=0$ if M is compact ; 

• (ii) $\mu (M)\le \mu (N)$ whenever $M\subset N;$

• (iii) if $\{M_n\}$ is a sequence of closed subsets in X satisfying $M_{n+1}\subset M_n$ for every $n\in \mathbb {N}$ and $\lim _{n\rightarrow \infty }\mu (M_n)=0$, then $K:=\bigcap _{n \in \mathbb {N}}M_n$ is nonempty compact and $\lim _{n\rightarrow \infty }H(M_n,K)$$=0$.

Lemma 2.3

Let X be a normed space and A, B be subsets of X. If A is compact and B is closed, then $A+B$ is closed.

Proof

Assume that $\{a_n+b_n\}$, $a_n \in A, b_n \in B$, converges to c for some $c\in X$. We show that $c \in A+B$. In fact, since A is compact, there exist a subsequence $\{a_{n_k }\}$ of sequence $\{a_n\}$ and $a \in A$ such that $\{a_{n_k }\}$ converges to a. We have

$$\begin{aligned} \left\| b_{n_k}-c+a\right\| = \left\| (b_{n_k }+a_{n_k})-c+(a-a_{n_k })\right\| \le \left\| b_{n_k }+a_{n_k}-c\right\| +\left\| a-a_{n_k }\right\| . \end{aligned}$$

We obtain that $\{b_{n_k }\}$ converges to $c-a$. Since B is closed, we get $c-a\in B$. Hence, there exists $b \in B$ such that $b=c-a$. Then, $c=a+b\in A+B$. So, $A+B$ is closed. $\square$

Lemma 2.4

Let M be a nonempty subset of a normed space X. Then, for every $x,y \in X$, |d(x, M) $-d(y,M)|\le \left\| x-y\right\|$.

Proof

Let $x,y \in X$, we have $\left\| x-y\right\| +d(y,M)=\left\| x-y\right\| +\inf _{z\in M}\left\| y-z\right\| =\inf _{z\in M}\{\left\| x-y\right\| +\left\| y-z\right\| \} \ge \inf _{z\in M}\left\| x-z\right\| =d(x,M)$. Hence, $\left\| x-y\right\| \ge d(x,M)-d(y,M)$. Similarly, we also get $\left\| x-y\right\| \ge d(y,M)-d(x,M)$. We conclude that $|d(x,M)-d(y,M)|\le \left\| x-y\right\|$. $\square$

3 Pointwise LP well-posedness and generalized pointwise LP well-posedness

Motivated by the study [22] on the pointwise $\textit{LP}$ well-posedness for $(\mathrm{P}_l)$, we are going to establish characterizations of this type of well-posedness for $(\mathrm{P}_\alpha )$ without using the scalarization method. Consider the problem $(\mathrm{P}_\alpha )$, for a given $\bar{x} \in S_{ \alpha {{\text {-Min}}}F}$, the LP approximating solution mapping at $\bar{x}$, $S_{ \alpha {{\text {-Min}}}F} (\bar{x}, \cdot ): \{\bar{x}\} \times \mathbb {R}_+ \rightrightarrows M$ is defined by

$$\begin{aligned} S_{\alpha {{\text {-Min}}}F}(\bar{x},\varepsilon ):=\{ x \in X \mid d(x,M)\le \varepsilon , F(x)\le ^\alpha F(\bar{x})+\varepsilon e \}, \end{aligned}$$

for each $\varepsilon \in \mathbb {R}_+$.

Inspired by ideas in [22] (Definition 5.1), we extend some notions for the problem $(\mathrm{P}_l)$ in [22] to the problem $(\mathrm{P}_\alpha )$ and propose some new concepts for the problem $(\mathrm{P}_\alpha )$. Let $e \in \mathrm{int}K.$

Definition 3.1

Let $\bar{x} \in S_{ \alpha {{\text {-Min}}}F}$ be given. A sequence $\{x_n\} \subset X$ is said to be a $\textit{LP}$-minimizing sequence for the problem $(\mathrm{P}_\alpha )$ at $\bar{x}$ if and only if there exists a sequence $\{\varepsilon _n\} \subset \mathbb {R}_+ {\setminus } \{0\}$ converging to 0 such that

$$\begin{aligned} d(x_n, M) \le \varepsilon _n, \quad F(x_n) \le ^\alpha F(\bar{x}) + \varepsilon _ne. \end{aligned}$$

The equivalence of this concept is given by the following result.

Proposition 3.1

$\{x_n\}\subset X$ is a $\textit{LP}$-minimizing sequence for the problem $(\mathrm{P}_\alpha )$ at $\bar{x}\in S_{ \alpha {{\text {-Min}}}F}$ if and only if there exists a sequence $\{d_n\} \subset K {\setminus } \{0\}$ converging to 0 such that

$$\begin{aligned} d(x_n, M) \rightarrow 0, \quad F(x_n) \le ^\alpha F(\bar{x}) + d_n. \end{aligned}$$

Proof

We only prove the assertion for the case $\alpha =s$; the proofs of the assertion for the cases $\alpha =l$ and $\alpha =u$ are similar. Let $\{x_n\}\subset X$ and $\{d_n\} \subset K {\setminus } \{0\}$ converging to 0 such that $d(x_n, M) \rightarrow 0$ and $F(x_n) \le ^s F(\bar{x}) + d_n$, i.e.,

$$\begin{aligned} F(\bar{x}) + d_n \subset F(x_n)+K , \quad F(x_n) \subset F(\bar{x}) + d_n -K. \end{aligned}$$

(1)

Since $e-K$ is a neighborhood of the origin 0 in Y, there exists $\varepsilon > 0$ such that $\varepsilon B(0,1)$$\subset e-K$ where B(x, r) is the closed ball centered x with radius r. For a given $n\in \mathbb {N}$, we have $d_n \in \left\| d_n\right\| B(0,1)\subset \left\| d_n\right\| \varepsilon ^{-1} (e-K)=\left\| d_n\right\| \varepsilon ^{-1}e-K$. For each $n \in \mathbb {N}$, taking $\varepsilon _n=\left\| d_n\right\| \varepsilon ^{-1}$, then $\{\varepsilon _n\} \subset \mathbb {R}_+ {\setminus } \{0\}$ converges to 0 and $\varepsilon _n e-d_n \in K$. It follows from (1) that

$$\begin{aligned} F(\bar{x}) + \varepsilon _n e \subset F(x_n)+K , \quad F(x_n) \subset F(\bar{x}) + \varepsilon _n e -K, \end{aligned}$$

i.e., $F(x_n) \le ^s F(\bar{x}) + \varepsilon _n e$. So, $\{x_n\}$ is a $\textit{LP}$-minimizing sequence for $(\mathrm{P}_s)$ at $\bar{x}$. Conversely, it is clear that if $\{x_n\}\subset X$ is a $\textit{LP}$-minimizing sequence for $(\mathrm{P}_s)$ at $\bar{x}\in S_{s{\text {-Min}}F}$, then the assertion is satisfied by setting $d_n=\varepsilon _ne$. $\square$

Definition 3.2

The problem $(\mathrm{P}_\alpha )$ is said to be

• (i) $\textit{LP}$ well-posed at $\bar{x} \in S_{ \alpha {{\text {-Min}}}F}$ if and only if any $\textit{LP}$-minimizing sequence for $(\mathrm{P}_\alpha )$ at $\bar{x}$ converges to $\bar{x}$;

• (ii) generalized $\textit{LP}$ well-posed at $\bar{x} \in S_{ \alpha {{\text {-Min}}}F}$ if and only if any $\textit{LP}$-minimizing sequence for $(\mathrm{P}_\alpha )$ at $\bar{x}$ has a subsequence converging to an element $\hat{x} \in S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)$.

Remark 3.1

When $\alpha =l$, the concept of pointwise well-posedness becomes the corresponding concepts studied in [22] (Definitions 5.1 and 5.2, respectively), even for this special case, the concept of generalized well-posedness is a new one.

The following examples illustrate the above-introduced concepts.

Example 3.1

Let $X=Y=\mathbb {R}$, $M=\mathbb {R}$, $K={\mathbb {R}}_+$. Let $F : X \rightrightarrows Y$ be defined by $F(x)=[0,1]$ for all $x \in X$. Let $e=1\in \text {int}K$ and $\bar{x}=0$. We have $S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)=S_{ \alpha {{\text {-Min}}}F}=\mathbb {R}$. Setting $x_n=n$, $\{x_n\}$ is a $\textit{LP}$-minimizing sequence for $(\mathrm{P}_\alpha )$ at $\bar{x}=0$. Since $\{x_n\}$ admits no convergent subsequence, $(\mathrm{P}_\alpha )$ is not both $\textit{LP}$ well-posed and generalized $\textit{LP}$ well-posed at 0.

Example 3.2

Let $X=Y=\mathbb {R}$, $M=\mathbb {R}$, $K={\mathbb {R}}_+$. Let $F : X \rightrightarrows Y$ be defined by

$$\begin{aligned} F(x)= \left\{ \begin{array}{ll} {[}0,1], &{}\text{ if }\quad -1 \le x \le 1,\\ {[}1,1+x^2], &{} \text{ otherwise }. \end{array}\right. \end{aligned}$$

Let $e=1\in \text {int}K$ and $\bar{x}=0$. By direct computations, we have $S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)=[-1,1]$. Setting $x_n=1-\frac{1}{n}$, $\{x_n\}$ is a $\textit{LP}$-minimizing sequence for $(\mathrm{P}_\alpha )$ at $\bar{x}=0$ and converges to 1. Hence, $(\mathrm{P}_\alpha )$ is not $\textit{LP}$ well-posed at 0, but it is generalized $\textit{LP}$ well-posed at 0. Indeed, if $\{{{\hat{x}}}_n\}$ is a $\textit{LP}$-minimizing sequence for $(\mathrm{P}_\alpha )$ at $\bar{x}=0$, then there is a sequence $\{\varepsilon _n\} \subset \mathbb {R}_+ {\setminus } \{0\}$ converging to 0 such that $d({{\hat{x}}}_n, M) \le \varepsilon _n$ and $F({{\hat{x}}}_n) \le ^\alpha F(0) + \varepsilon _ne=[\varepsilon _n,1+\varepsilon _n]$. This implies that $-1 \le {{\hat{x}}}_n \le 1$ for n sufficiently large, and hence there exists a subsequence of $\{{{\hat{x}}}_n\}$ converging to some point of $S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)$.

Example 3.3

Let $X=\mathbb {R}$, $M=\mathbb {R}$, $K={\mathbb {R}}_+$. Let $F : X \rightrightarrows Y$ be defined by $F(x)=[x^2,2x^2]$ for all $x \in X$. Let $e=1\in \text {int}K$ and $\bar{x}=0$. Direct cacullations give us $S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)=\{0\}$. Let $\{x_n\}$ be a $\textit{LP}$-minimizing sequence for $(\mathrm{P}_\alpha )$ at $\bar{x}=0$. Then, there exists a sequence $\{\varepsilon _n\} \subset \mathbb {R}_+ {\setminus } \{0\}$ converging to 0 such that $d(x_n, M) \le \varepsilon _n$ and $F(x_n) \le ^\alpha F(0) + \varepsilon _ne=\{\varepsilon _n\}$. It leads to $x_n^2 \le \varepsilon _n$, so $\{x_n\}$ converges to 0. Therefore, $(\mathrm{P}_\alpha )$ is $\textit{LP}$ well-posed at 0.

Lemma 3.1

If $(P_\alpha )$ is generalized LP well-posed at $\bar{x} \in S_{\alpha {{\text {-Min}}}F}$, then $S_{\alpha {{\text {-Min}}}F}(\bar{x},0)$ is compact.

Proof

For every sequence $\{x_n\}\subset S_{\alpha {{\text {-Min}}}F}(\bar{x},0)$, we always have $d(x_n,M)=0$ and

$$\begin{aligned} F(x_n)\le ^\alpha F(\bar{x})+\varepsilon _n e \end{aligned}$$

for any $\{\varepsilon _n\} \subset \mathbb {R}_+ {\setminus } \{0\}$ converging to 0. This means that $\{x_n\}$ is a LP-minimizing sequence for $(\text {P}_\alpha )$ at $\bar{x}$. By the generalized LP well-posedness of $(P_\alpha )$ at $\bar{x}$, there exists a subsequence $\{x_{n_k }\}$ of $\{x_n\}$ such that $\{x_{n_k }\}$ converges to an element $\hat{x} \in S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)$. This leads to the compactness of $S_{\alpha {{\text {-Min}}}F}(\bar{x},0)$. $\square$

The next results give some properties of the mapping $S_{ \alpha {{\text {-Min}}}F} (\bar{x},\cdot )$ which are useful in the sequel.

Proposition 3.2

Let $\bar{x} \in S_{ \alpha {{\text {-Min}}}F}$ be given. Then, the following statements are true:

• (i) $S_{\alpha {{\text {-Min}}}F}= \bigcup _{z \in S_{\alpha {{\text {-Min}}}F}} S_{\alpha {{\text {-Min}}}F}(z,0) ;$

• (ii) if $\varepsilon _1 \le \varepsilon _2$, then $S_{\alpha {{\text {-Min}}}F}(\bar{x},\varepsilon _1) \subset S_{\alpha {{\text {-Min}}}F}(\bar{x},\varepsilon _2);$

• (iii) $\bigcap _{\varepsilon >0} S_{\alpha {{\text {-Min}}}F}(\bar{x},\varepsilon ) =S_{\alpha {{\text {-Min}}}F}(\bar{x},0)$ if F is compact-valued on M.

Proof

We only demonstrate the proof of the assertions (i)-(iii) for the case $\alpha =s$; the proofs of these assertions for the cases $\alpha =l$ and $\alpha =u$ are similar.

(i) Let $z \in S_{s{\text {-Min}}F}$ be given. Since $z \in S_{s{\text {-Min}}F}(z,0)$, $S_{s{\text {-Min}}F} \subset \bigcup _{z \in S_{s{\text {-Min}}F}} S_{s{\text {-Min}}F}(z,0)$. Moreover, let $x \in \bigcup _{z\in S_{s{\text {-Min}}F}} S_{s{\text {-Min}}F}(z,0)$, there exists $z \in S_{s{\text {-Min}}F}$ such that $x \in S_{s{\text {-Min}}F}(z,0)$. Therefore, $d(x,M)=0$ and $F(x) \le ^s F(z)$. Since $z \in S_{s{\text {-Min}}F}$, $x \in S_{s{\text {-Min}}F}$. It implies that $\bigcup _{z \in S_{s{\text {-Min}}F}} S_{s{\text {-Min}}F}(z,0) \subset$$S_{s{\text {-Min}}F}$.

(ii) Assume $\varepsilon _1 \le \varepsilon _2$. Let $x \in S_{s{\text {-Min}}F}(\bar{x},\varepsilon _1)$, then $d(x,M)\le \varepsilon _1$ and $F(x) \le ^s F(\bar{x})+\varepsilon _1 e$. It follows from the definition of set less relation $\le ^s$ that

$$\begin{aligned} F(x) \le ^l F(\bar{x})+\varepsilon _1 e \quad \text {and} \quad F(x) \le ^u F(\bar{x})+\varepsilon _1 e, \end{aligned}$$

i.e.,

$$\begin{aligned} F(\bar{x})+\varepsilon _1 e \subset F(x)+K \quad \text {and} \quad F(x)\subset F(\bar{x})+\varepsilon _1 e -K. \end{aligned}$$

We observe that

$$\begin{aligned} F(\bar{x})+\varepsilon _2 e -K= F(\bar{x})+\varepsilon _1 e -K+(\varepsilon _2 -\varepsilon _1)e . \end{aligned}$$

Combining the convexity of K with Proposition 2.1, we obtain that

$$\begin{aligned} F(\bar{x})+\varepsilon _2 e \subset F(x)+K, \end{aligned}$$

and

$$\begin{aligned} F(x) \subset F(\bar{x})+\varepsilon _1 e -K \subset F(\bar{x})+\varepsilon _2 e -K. \end{aligned}$$

Thus, $F(x) \le ^l F(\bar{x})+\varepsilon _2 e$ and $F(x) \le ^u F(\bar{x})+\varepsilon _2 e.$ Since $\bar{x} \in S_{s{\text {-Min}}F}$, $F(x) \le ^s F(\bar{x})+\varepsilon _2 e$. Moreover, $d(x,M)\le \varepsilon _2$ as $d(x,M)\le \varepsilon _1$. Therefore, $x \in s{\text {-Min}} F(\bar{x},\varepsilon _2)$. We conclude that $S_{s{\text {-Min}}F}(\bar{x},\varepsilon _1) \subset S_{s{\text {-Min}}F}(\bar{x},\varepsilon _2)$.

(iii) Let $x \in S_{s{\text {-Min}}F}(\bar{x},0)$. It is clear that $x \in S_{s{\text {-Min}}F}(\bar{x},\varepsilon )$ for any $\varepsilon >0$. Therefore, $x \in \bigcap _{\varepsilon >0} S_{s{\text {-Min}}F}(\bar{x},\varepsilon )$. For the converse, let $x \in \bigcap _{\varepsilon >0}S_{s{\text {-Min}}F}(\bar{x},\varepsilon )$, we have $x \in S_{s{\text {-Min}}F}(\bar{x},\varepsilon )$ for any $\varepsilon >0$. It follows from definition of $S_{s{\text {-Min}}F}(\bar{x},\varepsilon )$ that $d(x,M)\le \varepsilon$ and $F(x) \le ^s F(\bar{x})+\varepsilon e$, i.e.,

$$\begin{aligned} d(x,M)\le \varepsilon ,\, F(\bar{x})+\varepsilon e\subset F(x)+K\quad \text {and} \quad F(x)\subset F(\bar{x})+\varepsilon e-K. \end{aligned}$$

(2)

By the compact-valuedness of F and Lemma 2.3, $F(x)-K$ and $F(\bar{x})-K$ are closed. From (2), let $\varepsilon \rightarrow 0$, we obtain that

$$\begin{aligned} d(x,M)=0,\, F(\bar{x})\subset F(x)+K \quad \text {and} \quad F(x)\subset F(\bar{x})-K, \end{aligned}$$

i.e.,

$$\begin{aligned} d(x,M)=0,\, F(x)\le ^s F(\bar{x}). \end{aligned}$$

Hence, $x \in S_{s{\hbox {-}}{\text {Min}}F}(\bar{x},0)$. We get $\bigcap _{\varepsilon >0} S_{s{\hbox {-}}{\text {Min}}F}(\bar{x},\varepsilon ) \subset S_{s{\hbox {-}}{\text {Min}}F}(\bar{x},0)$. $\square$

Next, using the Kuratowski measure of noncompactness of LP approximating solution sets, we establish metric characterizations of two types of pointwise LP well-posedness for $(\text {P}_\alpha )$.

Theorem 3.1

• (i) If $(\text {P}_\alpha )$ is generalized LP well-posed at $\bar{x} \in S_{\alpha {{\text {-Min}}}F}$, then $\mu (S_{\alpha {{\text {-Min}}}F}(\bar{x}, \varepsilon ))$$\rightarrow 0$ as $\varepsilon \rightarrow 0$.

• (ii) If $(\text {P}_\alpha )$ is LP well-posed at $\bar{x} \in S_{\alpha {{\text {-Min}}}F}$, then diam$(S_{\alpha {{\text {-Min}}}F}(\bar{x}, \varepsilon )) \rightarrow 0$ as $\varepsilon \rightarrow 0$.

Proof

(i) Suppose that $(\text {P}_\alpha )$ is generalized LP well-posed at $\bar{x} \in S_{ \alpha {{\text {-Min}}}F}$. First of all, we show that $H(S_{\alpha {{\text {-Min}}}F}(\bar{x},\varepsilon ), S_{\alpha {{\text {-Min}}}F}(\bar{x},0)) \rightarrow 0$ as $\varepsilon \rightarrow 0$. Indeed, we observe that, for each $\varepsilon >0$, $S_{\alpha {{\text {-Min}}}F}(\bar{x},0) \subset S_{\alpha {{\text {-Min}}}F}(\bar{x},\varepsilon )$, and hence

$$\begin{aligned} H^*(S_{\alpha {{\text {-Min}}}F}(\bar{x},0),S_{\alpha {{\text {-Min}}}F}(\bar{x},\varepsilon ))=0. \end{aligned}$$

It is sufficient to show that $H^*(S_{\alpha {{\text {-Min}}}F}(\bar{x},\varepsilon ),S_{\alpha {{\text {-Min}}}F}(\bar{x},0))\rightarrow 0$ as $\varepsilon \rightarrow 0$. Suppose by contrary that there exist a real number $r>0$ and a sequence $\{\varepsilon _n \} \subset \mathbb {R}_+ {\setminus } \{0\}$ converging to 0, and for each $n \in \mathbb {N}$ there exists $x_n\in S_{\alpha {{\text {-Min}}}F}(\bar{x},\varepsilon _n)$ such that $d(x_n,S_{\alpha {{\text {-Min}}}F}(\bar{x},0))\ge r$. We have $d(x_n,M)\le \varepsilon _n$ and $F(x_n)\le ^\alpha F(\bar{x})+\varepsilon _n e$. This means that $\{x_n\}$ is a LP-minimizing sequence for $(\text {P}_\alpha )$ at $\bar{x}$, and hence $\{x_n\}$ has a subsequence $\{x_{n_k}\}$ converging to some point $\hat{x} \in S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)$. Therefore, for $n_k$ sufficiently large, we have $\left\| x_{n_k}-\hat{x}\right\| <r$ which is a contradiction.

Next, we prove that $\mu (S_{\alpha {{\text {-Min}}}F}(\bar{x}, \varepsilon )) \rightarrow 0$ as $\varepsilon \rightarrow 0$. By Lemma 3.1, $S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)$ is compact. Now, for any $\varepsilon >0$, there are sets $M_1, M_2, \ldots M_n$ for some $n \in \mathbb {N}$ such that $S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)$$\subset \cup _{i=1}^{n}M_i$ with $\text {diam} M_i\le \varepsilon$ for all $i=1,\ldots ,n$. For each $i\in \{1,\ldots ,n \}$, denote

$$\begin{aligned} N_i:=\{x\in X\mid d(x,M_i)\le H(S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon ),S_{ \alpha {{\text {-Min}}}F}(\bar{x},0))\}. \end{aligned}$$

We claim that $S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon )\subset \cup _{i=1}^{n}N_i$. Indeed, let $x\in S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon )$ be arbitrary, we have

$$\begin{aligned} d(x,S_{ \alpha {{\text {-Min}}}F}(\bar{x},0))\le H(S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon ),S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)) . \end{aligned}$$

Since $S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)\subset \cup _{i=1}^{n}M_i$, we conclude that

$$\begin{aligned} d(x,\cup _{i=1}^{n}M_i)\le H(S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon ), S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)) . \end{aligned}$$

So, there is $k_0\in \{1,2,\ldots ,n\}$ such that

$$\begin{aligned} d(x,M_{k_0}) \le H(S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon ), S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)). \end{aligned}$$

It means that $x\in N_{k_0}$. Therefore, $S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon )\subset \cup _{i=1}^{n}N_i$. Notice further that

$$\begin{aligned} \text {diam} N_i= & {} \text {diam} M_i+2H(S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon ), S_{ \alpha {{\text {-Min}}}F}(\bar{x},0))\\\le & {} \varepsilon + 2H(S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon ), S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)). \end{aligned}$$

Hence, we get

$$\begin{aligned} \mu (S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon ))\le \mu (S_{ \alpha {{\text {-Min}}}F}(\bar{x},0))+2H(S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon ), S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)). \end{aligned}$$

Since $S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)$ is compact, we have $\mu (S_{ \alpha {{\text {-Min}}}F}(\bar{x},0))=0$. Therefore,

$$\begin{aligned} \mu (S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon ))\le 2H(S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon ), S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)). \end{aligned}$$

It follows that $\mu (S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon ))\rightarrow 0$ as $\varepsilon \rightarrow 0$.

(ii) Suppose, to the contrary, that there exist a sequence $\{\varepsilon _n\}\subset \mathbb {R}_+ {\setminus } \{0\}$ and a positive real number r such that $\mathrm{diam}(S_{\alpha {{\text {-Min}}}F}(\bar{x}, \varepsilon _n)) > r$. Because $\bar{x} \in S_{\alpha {{\text {-Min}}}F}(\bar{x}, \varepsilon _n)$, for each n, there exists $x_n \in S_{\alpha {{\text {-Min}}}F}(\bar{x}, \varepsilon _n)$ such that $\left\| x_n-\bar{x}\right\| > \frac{r}{2}$. However, since $\{x_n\}$ is a LP-minimizing sequence for $(\text {P}_\alpha )$ at $\bar{x}$, $\{x_n\}$ converges to $\bar{x}$ which is a contradiction. $\square$

The next result gives sufficient conditions for the closedness of approximating solution set.

Proposition 3.3

$S_{ \alpha {{\text {-Min}}}F}(\bar{x},\varepsilon )$ is closed for each $\varepsilon \ge 0$ if F is continuous and compact-valued on M.

Proof

We only prove the assertion for the case $\alpha =s$. Taking $\varepsilon \ge 0$, let $\{x_n\}\subset S_{ s{\text {-Min}}F}(\bar{x},\varepsilon )$ converge to x, we need to prove that $x \in S_{ s{\text {-Min}}F}(\bar{x},\varepsilon )$. Since $x_n \in S_{ s {\text {-Min}}F}(\bar{x},\varepsilon )$, $d(x_n,M)\le \varepsilon$ and

$$\begin{aligned} F(x_n)\le ^s F(\bar{x})+\varepsilon e. \end{aligned}$$

(3)

By the continuity of the function d(., M), $d(x,M)\le \varepsilon$. Next, we show that $F(x)\le ^s F(\bar{x})+\varepsilon e.$ Indeed, from (3), we have

$$\begin{aligned} F(\bar{x})+\varepsilon e\subset F(x_n) +K, \end{aligned}$$

(4)

and

$$\begin{aligned} F(x_n) \subset F(\bar{x})+\varepsilon e-K. \end{aligned}$$

(5)

Let $y \in F(x)$ be arbitrary. Since F is lower semicontinuous and $\{x_n\}$ converges to x, there exist $y_n \in F(x_n)$ such that $\{y_n\}$ converges to y. Combining this with (5), there exist $w_n \in F(\bar{x})$ such that

$$\begin{aligned} y_n \in w_n+\varepsilon e-K. \end{aligned}$$

(6)

Since $F(\bar{x})$ is compact, we can assume that $\{w_n\}$ converges to some $w \in F(\bar{x})$. By (6), there exist $k_n \in K$ such that $y_n = w_n+\varepsilon e-k_n$ . This leads to $\lim _{n \rightarrow \infty }k_n=w+\varepsilon e-y$. Moreover, we get $w+\varepsilon e-y \in K$ as K is closed. Therefore, there exists $k \in K$ such that $w+\varepsilon e-y=k$. We have $y=w+\varepsilon e-k \in w+\varepsilon e-K$. It yields that $y \in F(\bar{x})+\varepsilon e-K$ as $w \in F(\bar{x})$. We arrive at the fact that $F(x) \subset F(\bar{x})+\varepsilon e-K,$ i.e., $F(x) \le ^u F(\bar{x})+\varepsilon e.$

Similarly, let $t \in F(\bar{x})$ be arbitrary, it follows from (4) that, for each $n \in \mathbb {N}$, there exists $v_n \in F(x_n)$ such that

$$\begin{aligned} t \in v_n-\varepsilon e+K. \end{aligned}$$

(7)

Since F is upper semicontinuous and compact-valued at x, we can assume that $\{v_n\}$ converges to some element $v\in F(x)$. It implies from (7) that there exist $k_n \in K$ such that $t= v_n- \varepsilon e+k_n.$ Hence, $k_n =t+ \varepsilon e-v_n.$ This leads to $\lim _{n \rightarrow \infty }k_n=t+\varepsilon e-v$. Since K is closed, there exists $k \in K$ such that $t+\varepsilon e-v=k$. We get $t=v-\varepsilon e+k \in v-\varepsilon e+K$. It yields that $t \in F(x)-\varepsilon e+K$ as $v \in F(x)$. We have $F(\bar{x}) \subset F(x)-\varepsilon e+K.$ It means that $F(x) \le ^l F(\bar{x})+\varepsilon e.$ So, we obtain $F(x) \le ^s F(\bar{x})+\varepsilon e.$ The proof is complete. $\square$

Theorem 3.2

Suppose that F is continuous and compact-valued on M. Then,

• (i) $(\mathrm{P}_\alpha )$ is generalized LP well-posed at $\bar{x}\in S_{\alpha {{\text {-Min}}}F}$ if $\mu (S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon )) \rightarrow 0$ as $\varepsilon \rightarrow 0$.

• (ii) $(\mathrm{P}_\alpha )$ is LP well-posed at $\bar{x}\in S_{\alpha {{\text {-Min}}}F}$ if $\mathrm{diam} (S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon )) \rightarrow 0$ as $\varepsilon \rightarrow 0$.

Proof

(i) Suppose that $\mu (S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon )) \rightarrow 0$ as $\varepsilon \rightarrow 0$. Let $\{x_n\}$ be a LP-minimizing sequence for $(\mathrm{P}_\alpha )$ at $\bar{x}$. Therefore, there exists a sequence $\{\varepsilon _n\} \subset \mathbb {R}_+ {\setminus } \{0\}$ converging to 0 such that $d(x_n,M)\le \varepsilon _n$ and $F(x_n) \le ^{\alpha } F(\bar{x})+\varepsilon _ne$. This means that $x_n \in S_{ \alpha {{\text {-Min}}}F}(\bar{x},\varepsilon _n)$. It is clear that $\mu (S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon _n)) \rightarrow 0$ as $n \rightarrow \infty$, and hence by Lemma  2.2 (iii), we have $\cap _{n \in \mathbb {N}}S_{ \alpha {{\text {-Min}}}F}(\bar{x},\varepsilon _n)$ is a nonempty compact set and

$$\begin{aligned} H(S_{ \alpha {{\text {-Min}}}F}(\bar{x},\varepsilon _n), \cap _{n \in \mathbb {N}}S_{ \alpha {{\text {-Min}}}F}(\bar{x},\varepsilon _n))\rightarrow 0 \end{aligned}$$

as $n \rightarrow \infty$. Note further from Proposition 3.2 (iii) that

$$\begin{aligned} S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)=\cap _{n \in \mathbb {N}}S_{ \alpha {{\text {-Min}}}F}(\bar{x},\varepsilon _n). \end{aligned}$$

Hence, we conclude that $S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)$ is compact and

$$\begin{aligned} H(S_{ \alpha {{\text {-Min}}}F}(\bar{x},\varepsilon _n),S_{ \alpha {{\text {-Min}}}F}(\bar{x},0))\rightarrow 0 \end{aligned}$$

as $n \rightarrow \infty$. Thus,

$$\begin{aligned} d(x_n,S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)) \rightarrow 0. \end{aligned}$$

(8)

Therefore, there exists a sequence $\{{\hat{x}}_n\}\subset S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)$ such that $d(x_n,{\hat{x}}_n) \rightarrow 0$ as $n\rightarrow \infty$. Since $S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)$ is compact, there is a subsequence $\{{\hat{x}}_{n_k}\}$ of $\{{\hat{x}}_n\}$ converging to some ${\hat{x}} \in S_{ \alpha {{\text {-Min}}}F}$. This implies that $\{x_n\}$ has a corresponding subsequence $\{x_{n_k}\}$ converging to ${\hat{x}}$. Hence, $(\mathrm{P}_\alpha )$ is generalized LP well-posed at $\bar{x}$.

(ii) Assume that $\mathrm{diam} (S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon )) \rightarrow 0$ as $\varepsilon \rightarrow 0$. Then, $\mu (S_{ \alpha {{\text {-Min}}}F}(\bar{x},\varepsilon )) \rightarrow ~0$ as $\varepsilon \rightarrow 0$, and hence $(\mathrm{P}_\alpha )$ is generalized LP well-posed at $\bar{x}$ . By Proposition 3.2, $S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)$ is a singleton. By Lemma 2.1 (ii), $(\mathrm{P}_\alpha )$ is LP well-posed at $\bar{x}$. $\square$

The below example shows that Theorem 3.2 is applicable.

Example 3.4

Let $X =\mathbb {R}$, $Y =\mathbb {R}^2$, $M = [0, 1]$, $K={\mathbb {R}}_+^2$. Let $F : X \rightrightarrows Y$ be defined by

$$\begin{aligned} F(x)= [x, x+1] \times [x, x+1], \forall x \in X. \end{aligned}$$

Let $e=(1,1)\in \text {int}K$, $\bar{x}=0$. Clearly, all assumptions of Theroem 3.2 hold. By direct cacullations, we get $S_{ \alpha {{\text {-Min}}}F}(0,\varepsilon )=[0,\varepsilon ]$ and $S_{ \alpha {{\text {-Min}}}F}=\{0\}$. So, $\mathrm{diam} (S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon ))\rightarrow ~ 0$ as $\varepsilon \rightarrow 0$. Applying Theorem 3.2, the problem $(\mathrm{P}_\alpha )$ is LP well-posed at $\bar{x}=0$. In fact, if $\{x_n\}$ is a LP minimizing sequence for $(\mathrm{P}_\alpha )$ at $\bar{x}$, then there is a sequence $\{\varepsilon _n\} \subset \mathbb {R}_+ {\setminus } \{0\}$ converging to 0 such that $d(x_n, M) \le \varepsilon _n$ and $F(x_n) \le ^\alpha F(0) + \varepsilon _ne=[\varepsilon _n, \varepsilon _n+1] \times [\varepsilon _n, \varepsilon _n+1].$ We get $0\le x_n \le \varepsilon _n$, and hence $\{x_n\}$ converges to 0. So, $(\mathrm{P}_\alpha )$ is LP well-posed at 0.

The following example shows that the continuity of F in Theorem 3.2 is crucial.

Example 3.5

Let $X =\mathbb {R}$, $Y =\mathbb {R}^2$, $M = [-1, 1]$, $K={\mathbb {R}}_+^2$. Let $F : X \rightrightarrows Y$ be defined by

$$\begin{aligned} F(x)= \left\{ \begin{array}{ll} {[}0,1] \times [0,1], &{}\qquad \text{ if }\quad x < 0,\\ {[}1,2] \times [1,2], &{}\qquad \text{ if } \quad x \ge 0.\\ \end{array}\right. \end{aligned}$$

Let $e=(1,1)\in \text {int}K$, $\bar{x}=-\frac{1}{2}$. Then, F is compact-valued on M. Direct computations give us that $S_{ \alpha {{\text {-Min}}}F}(\bar{x},\varepsilon )=[-1, 0)$, and hence $\mu (S_{ \alpha {{\text {-Min}}}F}(\bar{x} ,\varepsilon )) \rightarrow 0$ as $\varepsilon \rightarrow 0$. However, the problem $(\mathrm{P}_\alpha )$ is not generalized LP well-posed at $\bar{x}$. Indeed, setting $x_n=-\frac{1}{n}$, we have $\{x_n\}$ is a LP minimizing sequence for $(\mathrm{P}_\alpha )$ at $\bar{x}$ but $\{x_n\}$ converges to $0 \not \in S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)$. The reason here is that F is not continuous.

Next, employing properties of the approximating solution mapping of ($\hbox {P}_\alpha$), the connection between LP well-posedness of ($\hbox {P}_\alpha$) and stability of approximating problem is established.

Theorem 3.3

Let $\bar{x}\in S_{\alpha {{\text {-Min}}}F}$.

• (i) Problem $(\mathrm{P}_\alpha )$ is generalized LP well-posed at $\bar{x}$ if and only if $S_{\alpha {{\text {-Min}}}F}(\bar{x},\cdot )$ is upper semicontinuous and compact-valued at 0.

• (ii) Problem $(\mathrm{P}_\alpha )$ is LP well-posed at $\bar{x}$ if and only if $S_{\alpha {{\text {-Min}}}F}(\bar{x},\cdot )$ is upper semicontinuous at 0 and $S_{\alpha {{\text {-Min}}}F}(\bar{x},0)=\{\bar{x}\}$.

Proof

(i) Suppose that $(\mathrm{P}_\alpha )$ is generalized LP well-posed at $\bar{x}$. By Lemma 3.1, $S_{\alpha {{\text {-Min}}}F}(\bar{x},0)$ is compact. Suppose by contrary that $S_{\alpha {{\text {-Min}}}F}(\bar{x},\cdot )$ is not upper semicontinuous at 0. Then, there exists an open set $N \supset S_{\alpha {{\text {-Min}}}F}(\bar{x},0)$ such that for any $\delta >0$, there exists $\varepsilon \in [0,\delta )$, $S_{\alpha {{\text {-Min}}}F}(\bar{x},\varepsilon ) \not \subset N$. It means that there exists a sequence $\{\varepsilon _n \}$ converging to 0 such that for each $n\in \mathbb {N}$, we have $S_{\alpha {{\text {-Min}}}F}(\bar{x},\varepsilon _n)\not \subset N.$ Thus, for each $n \in \mathbb {N}$, there is $x_n \in S_{\alpha {{\text {-Min}}}F}(\bar{x},\varepsilon _n)$, $x_n \not \in N$. Then, $d(x_n,M) \le \varepsilon _n$ and $F(x_n)\le ^\alpha F(\bar{x})+\varepsilon _n e,$ which imply that $\{x_n\}$ is a LP-minimizing sequence for $(\mathrm{P}_\alpha )$ at $\bar{x}$. Because $(\mathrm{P}_\alpha )$ is generalized LP well-posed at $\bar{x}$, there is a subsequence of $\{x_n\}$, denoted by $\{x_{n_k }\}$, converging to an element $\hat{x} \in S_{ \alpha {{\text {-Min}}}F}(\bar{x},0) \subset N$. This is impossible as $x_{n_k} \not \in N$ for all k.

Conversely, let $\{x_n\}\subset X$ be a LP-minimizing sequence for $(\mathrm{P}_\alpha )$ at $\bar{x}$. Then, there exists a sequence $\{\varepsilon _n\} \subset \mathbb {R}_+ {\setminus } \{0\}$ converging to 0 such that

$$\begin{aligned} d(x_n,M) \le \varepsilon _n, \quad F(x_n)\le ^\alpha F(\bar{x})+\varepsilon _n e. \end{aligned}$$

So, $x_n \in S_{ \alpha {{\text {-Min}}}F}(\bar{x},\varepsilon _n).$ It follows from the upper semicontinuity and compact-valuedness of $S_{\alpha {{\text {-Min}}}F}(\bar{x},\cdot )$ at 0, Lemma 2.1 (ii) implies that there exist an element $\hat{x} \in S_{ \alpha {{\text {-Min}}}F}(\bar{x},0)$ and a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that $\{x_{n_k}\}$ converges to $\hat{x}$. So, $(\mathrm{P}_\alpha )$ is generalized LP well-posed at $\bar{x}$.

(ii) Let $\{x_n\}\subset X$ be a LP-minimizing sequence for $(\mathrm{P}_\alpha )$ at $\bar{x}$, then there exists a sequence $\{\varepsilon _n\} \subset \mathbb {R}_+ {\setminus } \{0\}$ converging to 0 such that $d(x_n,M) \le \varepsilon _n$ and $F(x_n)\le ^\alpha F(\bar{x})+\varepsilon _n e.$ This means that, for each $n\in \mathbb {N}$,

$$\begin{aligned} x_n \in S_{\alpha {{\text {-Min}}}F}(\bar{x},\varepsilon _n). \end{aligned}$$

(9)

Since $S_{\alpha {{\text {-Min}}}F}(\bar{x},\cdot )$ is upper semicontinuous at 0, for any open set N, $S_{\alpha {{\text {-Min}}}F}(\bar{x},0) \subset N$, there is a neighborhood U of 0 such that for all $t\in U, t \ge 0$, we have $S_{\alpha {{\text {-Min}}}F}(\bar{x},t) \subset N$. Since $\{ \varepsilon _n\}$ converges to 0, there exists $n_0 \in \mathbb {N}$ such that $\varepsilon _n \in B(0, \frac{1}{n_0})$ for all $n \ge n_0$. Combining this with (9), we obtain $x_n \in S_{\alpha {{\text {-Min}}}F}(\bar{x},\varepsilon _n) \subset N$ for all $n \ge n_0$. Therefore, for every neighborhood W of 0, $x_n \in S_{\alpha {{\text {-Min}}}F}(\bar{x},0)+W$ for all $n \ge n_0$. Since $S_{\alpha {{\text {-Min}}}F}(\bar{x},0)=\{\bar{x}\}$, $\{x_n\}$ converges to $\bar{x}$. So, $(\mathrm{P}_\alpha )$ is LP well-posed at $\bar{x}$.

For the converse, suppose that $(\mathrm{P}_\alpha )$ is LP well-posed at $\bar{x}$. Using (i), $S_{\alpha {{\text {-Min}}}F}(\bar{x},\cdot )$ is upper semicontinuous and compact-valued at 0. We show that $S_{\alpha {{\text {-Min}}}F}(\bar{x},0)$ is a singleton. Suppose by the contrary that there exist $x_1, x_2\in S_{\alpha {{\text {-Min}}}F}(\bar{x},0)$ with $x_1 \not = x_2$. Putting $x_{2n+k}= x_k$ where $k=1$ or $k=2$. Clearly, $\{x_n\}$ is a LP-minimizing sequence for $(\mathrm{P}_\alpha )$ at $\bar{x}$. However, $\{x_n\}$ is not convergent. This is a contradiction. Therefore, $S_{\alpha {{\text {-Min}}}F}(\bar{x},0)$ is a singleton. Moreover, it is obvious that $\bar{x} \in S_{\alpha {{\text {-Min}}}F}(\bar{x},0)$. So, $S_{\alpha {{\text {-Min}}}F}(\bar{x},0)=\{\bar{x}\}$. $\square$

The assumption about the upper semicontinuity of approximating solution mapping of ($\hbox {P}_\alpha$) is used in Theorem 3.3. Next, we give the sufficient conditions for this assumption.

Proposition 3.4

Suppose that the following conditions hold:

• (i) M is compact; 

• (ii) F is continuous and compact-valued on M.

Then, $S_{\alpha {{\text {-Min}}}F}(\bar{x},\cdot )$ is upper semicontinuous at 0.

Proof

By the similarity, we only focus on the proof of the assertion for the case $\alpha =u$. By contradiction, suppose that $S_{u{\text {-Min}}F}(\bar{x},\cdot )$ is not upper semicontinuous at 0. Then, there exist an open set $N \supset S_{u{\text {-Min}}F}(\bar{x},0)$ and a sequence $\{\varepsilon _n\} \subset \mathbb {R}^+{\setminus }\{0\}$ converging to 0 such that for each n, there exists $x_n$ satisfying

$$\begin{aligned} x_n \in S_{u{\text {-Min}}F}(\bar{x},\varepsilon _n) {\setminus } W_0. \end{aligned}$$

(10)

Since $x_n \in S_{u{\text {-Min}}F}(\bar{x},\varepsilon _n)$,

$$\begin{aligned} d(x_n,M)\le \varepsilon _n \end{aligned}$$

(11)

and

$$\begin{aligned} F(x_n) \subset F(\bar{x})+\varepsilon _n e -K. \end{aligned}$$

(12)

It implies from (11) that there exist $\hat{x}_n \in M$ such that $d(x_n,\hat{x}_n)\le \varepsilon _n$. By the compactness of M, we can assume that $\{\hat{x}_n\}$ converges to an element $x_0 \in M$. Hence, $\{x_n\}$ converges to $x_0$. Next, we prove that

$$\begin{aligned} F(x_0) \subset F(\bar{x})-K. \end{aligned}$$

(13)

Indeed, by the compact-valuedness of F, the closedness of K and Lemma 2.3, $F(\bar{x})-K$ is closed. From (12), taking $n \rightarrow \infty$, we obtain (13). It means that $x_0 \in S_{u{\text {-Min}}F}(\bar{x},0)$. Combining this, (10) and the convergence to $x_0$ of $\{x_n\}$, we get a contradiction. Therefore, $S_{u{\text {-Min}}F}(\bar{x},\cdot )$ is upper semicontinuous at 0. $\square$

Corollary 3.1

Suppose that the following conditions hold:

• (i) M is compact; 

• (ii) F is continuous and compact-valued on M.

  Then,

  • (a) $(\mathrm{P}_\alpha )$ is generalized LP well-posed at $\bar{x}\in S_{\alpha {{\text {-Min}}}F}$ if $S_{\alpha {{\text {-Min}}}F}(\bar{x},0)$ is closed.

  • (b) $(\mathrm{P}_\alpha )$ is LP well-posed at $\bar{x}\in S_{\alpha {{\text {-Min}}}F}$ if $S_{\alpha {{\text {-Min}}}F}(\bar{x},0) =\{\bar{x}\}$.

4 Metrically LP well-posed set optimization problems

Picking up the ideas in [22], we introduce the following new concepts of LP well-posedness related to metrically approach for the problem $(\mathrm{P}_\alpha )$.

Definition 4.1

A sequence $\{x_n\} \subset X$ is said to be a

• (i) metrically LP-minimizing sequence for problem $(\text {P}_l)$ at $\bar{x} \in S_{ l{\text {-Min}}F}$ if and only if $H^*(F(\bar{x}), F(x_n)) \rightarrow 0$ and $d(x_n, M) \rightarrow 0$ as $n \rightarrow \infty$.

• (ii) metrically LP-minimizing sequence for problem $(\text {P}_u)$ at $\bar{x} \in S_{ u{\text {-Min}}F}$ if and only if $H^*( F(x_n),F(\bar{x})) \rightarrow 0$ and $d(x_n, M) \rightarrow 0$ as $n \rightarrow \infty$.

• (iii) metrically LP-minimizing sequence for problem $(\mathrm{P}_s)$ at $\bar{x} \in S_{ s{\text {-Min}}F}$ if and only if $H(F(\bar{x}), F(x_n)) \rightarrow 0$ and $d(x_n, M) \rightarrow 0$ as $n \rightarrow \infty$.

Definition 4.2

The problem $(\mathrm{P}_\alpha )$ is said to be metrically LP well-posed if and only if $S_{ \alpha {{\text {-Min}}}F} \not =\emptyset$ and for any metrically LP-minimizing sequence $\{x_n\}$ for problem $(\mathrm{P}_\alpha )$ at some $\bar{x} \in S_{ \alpha {{\text {-Min}}}F}$, we have $d(x_n, S_{ \alpha {{\text {-Min}}}F}) \rightarrow 0$ as $n \rightarrow \infty$.

Remark 4.1

When $\alpha =l$, concepts introduced in Definitions 4.1 (i) and 4.2 are similar to the corresponding ones studied in Definitions 4.5 (ii) and 4.6 (ii) in [22].

Example 4.1

• (a) Let $X=Y=\mathbb {R}$, $M=[0,1]$, $K={\mathbb {R}}_+$, and let $F : X \rightrightarrows Y$ be defined by $F(x)=[1,2]$ for all $x \in X$. Obviously, $S_{ \alpha {{\text {-Min}}}F}=[0,1]=M$, and the problem $(\mathrm{P}_\alpha )$ is metrically LP well-posed.

• (b) Let $X=Y=\mathbb {R}$, $M=K={\mathbb {R}}_+$. Let $F : X \rightrightarrows Y$ be defined by $F(x)=[x^2,3x^2]$ for all $x \in X$. Direct cacullations give us $S_{ \alpha {{\text {-Min}}}F}=\{0\}$ and the problem $(\mathrm{P}_\alpha )$ is metrically LP well-posed. Indeed, let $\{x_n\}$ be a metrically $\textit{LP}$-minimizing sequence for $(\mathrm{P}_\alpha )$ at $\bar{x}=0$, it implies from definition of metrically $\textit{LP}$-minimizing sequence for $(\mathrm{P}_\alpha )$ at $\bar{x}=0$ that $\{x_n\}$ converges to $\bar{x}$. Therefore, $d(x_n, S_{ \alpha {{\text {-Min}}}F}) \rightarrow 0$.

Example 4.2

Let $X=Y=\mathbb {R}$, $M=[-1,1]$, $K={\mathbb {R}}_+$, and $F : X \rightrightarrows Y$ is defined by

$$\begin{aligned} F(x)= \left\{ \begin{array}{ll} [0,1) , &{}\qquad \text{ if }\quad x \le 0,\\ (0,1], &{}\qquad \text{ if } \quad x > 0. \end{array}\right. \end{aligned}$$

By direct computations, we get $S_{ \alpha {{\text {-Min}}}F}=[-1,0]$. Taking $x_n=1+\frac{1}{n}$, then $\{x_n\}$ is a metrically LP-minimizing sequence for the problem $(\mathrm{P}_\alpha )$ at $\bar{x}=0 \in S_{ \alpha {{\text {-Min}}}F}$, but $d(x_n, S_{ \alpha {{\text {-Min}}}F})\rightarrow 1$. Therefore, the problem $(\mathrm{P}_\alpha )$ is not metrically LP well-posed.

Next, we introduce a generalized form of the above concept.

Definition 4.3

The problem $(\mathrm{P}_\alpha )$ is said to be generalized metrically LP well-posed if and only if $S_{ \alpha {{\text {-Min}}}F} \not =\emptyset$ and for any metrically LP-minimizing sequence $\{x_n\}$ for $(\mathrm{P}_\alpha )$ at some $\bar{x} \in S_{ \alpha {{\text {-Min}}}F}$, $\{x_n\}$ has a subsequence, denoted by $\{x_{n_k}\}$, such that $d(x_{n_k}, S_{ \alpha {{\text {-Min}}}F}) \rightarrow 0$ as $k \rightarrow \infty$.

It is clear that if $(\mathrm{P}_\alpha )$ is metrically LP well-posed, then it is generalized metrically LP well-posed.

These following results give the relationships between these kinds of LP well-posedness considered in this study.

Theorem 4.1

• (i) If $(\mathrm{P}_\alpha )$ is LP well-posed at all $\bar{x} \in S_{ \alpha {{\text {-Min}}}F}$, then $(\mathrm{P}_\alpha )$ is metrically LP well-posed.

• (ii) If $(\mathrm{P}_\alpha )$ is generalized LP well-posed at all $\bar{x} \in S_{ \alpha {{\text {-Min}}}F}$, then $(\mathrm{P}_\alpha )$ is generalized metrically LP well-posed.

Proof

(i) By the similarity we verify the assertions (i), (ii) for the case $\alpha =s$ as an example. Let $\{x_n\}$ be a metrically LP-minimizing sequence for problem $(\mathrm{P}_s)$ at some $\bar{x} \in S_{ s{\text {-Min}}F}$. We need to prove that $d(x_n, S_{ s{\text {-Min}}F}) \rightarrow 0$. In fact, since $\{x_n\}$ is a metrically LP-minimizing sequence for problem $(\mathrm{P}_s)$ at some $\bar{x} \in S_{ s{\text {-Min}}F}$, $d(x_n, M) \rightarrow 0$ and

$$\begin{aligned} H(F(\bar{x}), F(x_n)) \rightarrow 0. \end{aligned}$$

(14)

Observe that we can choose a sequence $\{\varepsilon _n\} \subset \mathbb {R}_+ {\setminus } \{0\}$ converging to 0 satisfying $d(x_n, M)$$\le \varepsilon _n$, both $-\varepsilon _ne+K$ and $\varepsilon _ne-K$ are neighborhoods of the origin in Y. By (14), there exists $n_0 \in \mathbb {N}$ such that for all $n>n_0$ we have

$$\begin{aligned} F(\bar{x}) \subset F(x_n)-\varepsilon _ne+K \,\, \text {and} \, \, F(x_n) \subset F(\bar{x})+\varepsilon _ne-K. \end{aligned}$$

This implies that $F(x_n) \le ^s F(\bar{x})+\varepsilon _ne$. Hence, $\{x_n\}$ is a LP-minimizing sequence for $(\mathrm{P}_s)$ at $\bar{x}$. By the LP well-posedness of $(\mathrm{P}_s)$ at $\bar{x}$, $\{x_n\}$ converges to $\bar{x}$. Moreover, since $\bar{x} \in S_{ s{\text {-Min}}F}$, $d(x_n, S_{ s{\text {-Min}}F}) \le \left\| x_n- \bar{x}\right\| \rightarrow 0$. So, $(\mathrm{P}_s)$ is metrically LP well-posed.

(ii) Using a similar argument with one above, we can prove that the statement (ii) is satisfied. $\square$

Remark 4.2

When $\alpha = l$, $(\mathrm{P}_\alpha )$ reduces to $(\text {P}_l)$ studied in [22]. To obtain the metrically LP well-posedness for $(\text {P}_l)$, the authors used an important assumption about the K-closed values of F on M, i.e., $F(x)+K$ is closed for all $x \in M$. Using another approach, as in Theorem  4.1, we can remove this assumption but also obtain the metrically LP well-posedness for $(\mathrm{P}_\alpha )$.

Combining Theorem 4.1 and Corollary 3.1, we obtain the following results.

Theorem 4.2

Suppose that the following conditions are satisfied:

• (i) M is compact; 

• (ii) F is continuous and compact-valued on M.

  Then,

  • (a) $(\mathrm{P}_\alpha )$ is metrically LP well-posed if $S_{\alpha {{\text {-Min}}}F}(\bar{x},0)=\{\bar{x}\}$ for every $\bar{x} \in S_{ \alpha {{\text {-Min}}}F}$.

  • (b) $(\mathrm{P}_\alpha )$ is generalized metrically LP well-posed if $S_{\alpha {{\text {-Min}}}F}(\bar{x},0)$ is closed for every $\bar{x} \in S_{ \alpha {{\text {-Min}}}F}$.

Remark 4.3

Very recently, in [6], the authors studied several kinds of well-posedness for set optimization problems via the lower set less relation, including B-well-posedness, L-well-posedness, DH-well-posedness, and they obtained many interesting results related to this topic. In this paper, we consider the Levitin–Polyak well-posedness and the generalized Levitin–Polyak well-posedness for set optimization problems involving various kinds of set less relations, and hence the concepts of well-posedness investigated in this paper are different from those in [6]. Therefore, it could not compare our results with theirs.