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¹¹institutetext: Nguyen Duy Cuong ²²institutetext: Department of Mathematics, College of Natural Sciences, Can Tho University, Can Tho, Vietnam
²²email: [email protected], ORCID: 0000-0003-2579-3601 ³³institutetext: A. Y. Kruger (corresponding author) ⁴⁴institutetext: Optimization Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
⁴⁴email: [email protected], ORCID: 0000-0002-7861-7380 ⁵⁵institutetext: Nguyen Hieu Thao ⁶⁶institutetext: School of Science, Engineering and Technology, RMIT University Vietnam, Ho Chi Minh City, Vietnam
⁶⁶email: [email protected], ORCID: 0000-0002-1455-9169

Extremality of families of sets and set-valued optimization

Nguyen Duy Cuong Alexander Y. Kruger Nguyen Hieu Thao

Abstract

The paper explores a new extremality model involving collections of arbitrary families of sets. We demonstrate its applicability to set-valued optimization problems with general preferences, weakening the assumptions of the known results and streamlining their proofs.

Keywords:

extremal principle separation optimality conditions set-valued optimization

MSC:

49J52 49J53 49K40 90C30 90C46

1 Introduction

The paper explores another extension of the extremal principle introduced recently in CuoKruTha24 . The conventional extremal principle KruMor80 ; MorSha96 ; Mor06.1 covers a wide range of problems in optimization and variational analysis as demonstrated, e.g., in the books BorZhu05 ; Mor06.1 ; Mor06.2 . At the same time, there exist problems, mainly in multiobjective and set-valued optimization described via closed preference relations, that cannot be covered within the framework of linear translations (as in the conventional principle). The first example of this kind was identified in Zhu Zhu00 . Fortunately, such problems can be handled with the help of a more flexible extended version of the extremal principle using nonlinear perturbations (deformations) of the sets defined by set-valued mappings. Such an extension was developed in Mordukhovich et al. MorTreZhu03 (see also BorZhu05 ; Mor06.2 ) and applied by other authors to various multiobjective problems ZheNg06 ; LiNgZhe07 ; Bao14.2 .

The model in MorTreZhu03 exploits non-intersection of perturbations of given sets $\Omega_{1},\ldots,\Omega_{n}$ . The perturbations are chosen from the respective families of sets $\Xi_{i}:=\{S_{i}(s)\mid s\in M_{i}\}$ $(i=1,\ldots,n)$ determined by given set-valued mappings $S_{i}:M_{i}\rightrightarrows X$ $(i=1,\ldots,n)$ on metric spaces. In the particular case of linear translations, i.e., when, for all $i=1,\ldots,n$ , $(M_{i},d_{i})=(X,d)$ and $S_{i}(a)=\Omega_{i}-a$ $(a\in X)$ , the model reduces to the conventional extremal principle. It was shown by examples in MorTreZhu03 ; Mor06.2 that the framework of set-valued perturbations is richer than that of linear translations. With minor modifications in the proof, the conventional extremal principle was extended to the set-valued setting producing a more advanced model.

This model has been refined in CuoKruTha24 , making it more flexible and, at the same time, simpler. The refined model studies extremality and stationarity of nonempty families of arbitrary sets $\Xi_{i}$ $(i=1,\ldots,n)$ and is applicable to a wider range of variational problems. The next definition and theorem are simplified versions of (CuoKruTha24, , Definition 3.2 and Theorem 3.4), respectively.

Definition 1 (Extremality and stationarity: families of sets).

Let $\Xi_{1},\ldots,\Xi_{n}$ be families of subsets of a normed space $X$ , and $\bar{x}\in X$ . The collection $\{\Xi_{1},\ldots,\Xi_{n}\}$ is

(i)

extremal at $\bar{x}$ if there is a $\rho\in(0,+\infty]$ such that, for any $\varepsilon>0$ , there exist $A_{i}\in\Xi_{i}$ $(i=1,\ldots,n)$ such that $\max_{1\leq i\leq n}d(\bar{x},A_{i})<\varepsilon$ and $\bigcap_{i=1}^{n}A_{i}\cap B_{\rho}(\bar{x})=\emptyset$ .
(ii)

stationary at $\bar{x}$ if, for any $\varepsilon>0$ , there exist a $\rho\in(0,\varepsilon)$ and $A_{i}\in\Xi_{i}$ $(i=1,\ldots,n)$ such that $\max_{1\leq i\leq n}d(\bar{x},A_{i})<\varepsilon\rho$ and $\bigcap_{i=1}^{n}A_{i}\cap B_{\rho}(\bar{x})=\emptyset$ .
(iii)

approximately stationary at $\bar{x}$ if, for any $\varepsilon>0$ , there exist a $\rho\in(0,\varepsilon)$ , $A_{i}\in\Xi_{i}$ and $x_{i}\in B_{\varepsilon}(\bar{x})$ $(i=1,\ldots,n)$ such that $\max_{1\leq i\leq n}d(x_{i},A_{i})<\varepsilon\rho$ and $\bigcap_{i=1}^{n}(A_{i}-x_{i})\cap(\rho\mathbb{B})=\emptyset$ .

Theorem 1.1.

Let $\Xi_{1},\ldots,\Xi_{n}$ be families of closed subsets of a Banach space $X$ , and $\bar{x}\in X$ . If $\{\Xi_{1},\ldots,\Xi_{n}\}$ is approximately stationary at $\bar{x}$ , then, for any $\varepsilon>0$ , there exist $A_{i}\in\Xi_{i}$ , $x_{i}\in A_{i}\cap B_{\varepsilon}(\bar{x})$ , and $x_{i}^{*}\in N^{C}_{A_{i}}(x_{i})$ $(i=1,\ldots,n)$ such that

\displaystyle\Big{\|}\sum_{i=1}^{n}x_{i}^{*}\Big{\|}<\varepsilon\quad\mbox{and}\quad\sum_{i=1}^{n}\|x_{i}^{*}\|=1.

If $X$ is Asplund, then $N^{C}$ in the above assertion can be replaced by $N^{F}$ .

The symbols $N^{C}$ and $N^{F}$ in the above theorem denote, respectively, the Clarke and Fréchet normal cones. Recall that a Banach space is Asplund if every continuous convex function on an open convex set is Fréchet differentiable on a dense subset Phe93 , or equivalently, if the dual of each its separable subspace is separable. We refer the reader to Phe93 ; Mor06.1 for discussions about and characterizations of Asplund spaces. All reflexive, particularly, all finite dimensional Banach spaces are Asplund. Most assertions involving Fréchet normals, subdifferentials and coderivatives are only valid in Asplund spaces; see MorSha96 .

Remark 1.

(i)

Part ii of Definition 1 is the explicit form of (CuoKruTha24, , Definition 3.2 (iii)), while part iii is a particular case of (CuoKruTha24, , Definition 3.2 (ii)) with $\Omega_{1}=\ldots=\Omega_{n}:=X$ , thus, representing the weakest version of the property in (CuoKruTha24, , Definition 3.2 (ii)).
(ii)

It is easy to see that i $\Rightarrow\$ ii $\Rightarrow\$ iii in Definition 1. Hence, the necessary conditions in Theorem 1.1 are also valid for the stationarity and extremality.
(iii)

Theorem 1.1 shows that approximate stationarity of a given collection of families of closed sets implies its fuzzy (up to $\varepsilon$ ) separation. Note that, unlike the general model discussed in BuiKru19 , not only the points $x_{i}$ and $x_{i}^{*}$ $(i=1,\ldots,n)$ usually involved in fuzzy separation statements depend on $\varepsilon$ , but also the sets $A_{1},\ldots,A_{n}$ .
(iv)

For each $i=1,\ldots,n$ , the sets $A_{i}$ making the family $\Xi_{i}$ in Definition 1 can be considered as perturbations of some given set $\Omega_{i}$ . With this interpretation in mind, Definition 1 i covers the definition of extremality with respect to set-valued perturbations in MorTreZhu03 ; Mor06.2 . Note that the “perturbation” sets in MorTreZhu03 ; Mor06.2 are rather loosely connected with the given sets.

Example 1.

Let $\Xi_{1}$ consist of a single one-point set $\{0\}\subset\mathbb{R}$ , and $\Xi_{2}$ be a family of singletons $\{1/n\}$ for $n\in\mathbb{N}$ . It is easy to see that $\{\Xi_{1},\Xi_{2}\}$ is extremal at $0$ in the sense of Definition 1 i (even with $\rho=+\infty$ ). The subsets of $\Xi_{1},\Xi_{2}$ may be considered as “perturbations” of the sets $\Omega_{1}=\Omega_{2}:=\mathbb{R}$ in the sense of MorTreZhu03 ; Mor06.2 . The pair $\{\Omega_{1},\Omega_{2}\}$ is clearly not extremal at $0$ in the conventional sense of KruMor80 ; Mor06.1 .

The more general (and simpler) model in Definition 1 and Theorem 1.1 is capable of treating a wider range of applications. In this paper, we demonstrate the applicability of Theorem 1.1 to set-valued optimization problems with general preference relations. This allows us to expand the range of set-valued optimization models studied in earlier publications, weaken their assumptions and streamline the proofs.

We study extremality/stationarity properties of the triple $\{F,\Omega,\Xi\}$ , where $F:X\rightrightarrows Y$ is a set-valued mapping between normed spaces, $\Omega$ is a subset of $X$ , and $\Xi$ is a nonempty family of subsets of $Y$ . The latter family may, in particular be determined by an abstract level-set mapping defining a preference relation on $Y$ .

Members of $\Xi$ do not have to be simply translations (deformations) of a fixed set (ordering cone). Extremality/stationarity properties of the triple $\{F,\Omega,\Xi\}$ reduce to the corresponding properties of the two special families of subsets of $X\times Y$ which involve ${\rm gph}\,F$ , and products of $\Omega$ and members of $\Xi$ . The properties are illustrated by examples. Application of Theorem 1.1 yields necessary conditions for approximate stationarity and, hence, also stationarity and extremality. Natural qualification conditions in terms of Clarke or Fréchet coderivatives and normal cones are provided, which allow one to write down the necessary conditions in the form of an abstract multiplier rule. The statements cover the corresponding results in MorTreZhu03 ; ZheNg05.2 ; ZheNg06 .

Requirements on preference relations defined by level-set mappings, making them meaningful in optimization and applications, are discussed. A certain subset of properties, which are satisfied by most conventional and many other preference relations, is established. The properties are shown to be in general weaker than those used in Mor06.2 ; Bao14.2 ; KhaTamZal15 , but still sufficient for the corresponding set-valued optimization problems to fall within the theory developed in the current paper. Several multiplier rules for problems with a single set-valued mapping, and then with multiple set-valued mappings are formulated.

The structure of the paper is as follows. Section 2 recalls some definitions and facts used throughout the paper. The applicability of Theorem 1.1 is illustrated in Sections 3–5 considering set-valued optimization problems with general preference relations. A model with a single set-valued mapping is studied in Section 3. A particular case of this model when the family $\Xi$ is determined by an abstract level-set mapping is considered in Section 4. A more general model with multiple set-valued mappings is briefly discussed in Section 5.

2 Preliminaries

Our basic notation is standard, see, e.g., Mor06.1 ; RocWet98 ; DonRoc14 ; Iof17 . Throughout the paper, if not explicitly stated otherwise, $X$ and $Y$ are normed spaces. Products of normed spaces are assumed to be equipped with the maximum norm. The topological dual of a normed space $X$ is denoted by $X^{*}$ , while $\langle\cdot,\cdot\rangle$ denotes the bilinear form defining the pairing between the two spaces. The open ball with center $x$ and radius $\delta>0$ is denoted by $B_{\delta}(x)$ . If $(x,y)\in X\times Y$ , we write $B_{\varepsilon}(x,y)$ instead of $B_{\varepsilon}((x,y))$ . The open unit ball is denoted by $\mathbb{B}$ with a subscript indicating the space, e.g., $\mathbb{B}_{X}$ and $\mathbb{B}_{X^{*}}$ . Symbols $\mathbb{R}$ and $\mathbb{N}$ stand for the real line and the set of all positive integers, respectively.

The interior and closure of a set $\Omega$ are denoted by ${\rm int}\,\Omega$ and ${\rm cl}\,\Omega$ , respectively. The distance from a point $x\in X$ to a subset $\Omega\subset X$ is defined by $d(x,\Omega):=\inf_{u\in\Omega}\|u-x\|$ , and we use the convention $d(x,\emptyset)=+\infty$ .

Given a subset $\Omega$ of a normed space $X$ and a point $\bar{x}\in\Omega$ , the sets (cf. Kru03 ; Cla83 )

	$\displaystyle N_{\Omega}^{F}(\bar{x}):=\Big{\{}x^{\ast}\in X^{\ast}\mid\limsup_{\Omega\ni x{\rightarrow}\bar{x},\;x\neq\bar{x}}\frac{\langle x^{\ast},x-\bar{x}}{\\|x-\bar{x}\\|}\leq 0\Big{\}},$		(1)
	$\displaystyle N_{\Omega}^{C}(\bar{x}):=\left\{x^{\ast}\in X^{\ast}\mid\left\langle x^{\ast},z\right\rangle\leq 0\quad\mbox{for all}\quad z\in T_{\Omega}^{C}(\bar{x})\right\}$		(2)

are the Fréchet and Clarke normal cones to $\Omega$ at $\bar{x}$ , where $T_{\Omega}^{C}(\bar{x})$ stands for the Clarke tangent cone to $\Omega$ at $\bar{x}$ :

T_{\Omega}^{C}(\bar{x}):=\big{\{}z\in X\mid\forall x_{k}{\rightarrow}\bar{x},\;x_{k}\in\Omega,\;\forall t_{k}\downarrow 0,\;\exists z_{k}\to z\\ \mbox{such that}\quad x_{k}+t_{k}z_{k}\in\Omega\quad\mbox{for all}\quad k\in\mathbb{N}\big{\}}.

The sets (1) and (2) are nonempty closed convex cones satisfying $N_{\Omega}^{F}(\bar{x})\subset N_{\Omega}^{C}(\bar{x})$ . If $\Omega$ is a convex set, they reduce to the normal cone in the sense of convex analysis:

\displaystyle N_{\Omega}(\bar{x}):=\left\{x^{*}\in X^{*}\mid\langle x^{*},x-\bar{x}\rangle\leq 0\quad\mbox{for all}\quad x\in\Omega\right\}.

By convention, we set $N_{\Omega}^{F}(\bar{x})=N_{\Omega}^{C}(\bar{x}):=\emptyset$ if $\bar{x}\notin\Omega$ .

A set-valued mapping $F:X\rightrightarrows Y$ between two sets $X$ and $Y$ is a mapping, which assigns to every $x\in X$ a (possibly empty) subset $F(x)$ of $Y$ . We use the notations ${\rm gph}\,F:=\{(x,y)\in X\times Y\mid y\in F(x)\}$ and ${\rm dom}\,\>F:=\{x\in X\mid F(x)\neq\emptyset\}$ for the graph and the domain of $F$ , respectively, and $F^{-1}:Y\rightrightarrows X$ for the inverse of $F$ . This inverse (which always exists with possibly empty values at some $y$ ) is defined by $F^{-1}(y):=\{x\in X\mid y\in F(x)\}$ , $y\in Y$ . Obviously ${\rm dom}\,F^{-1}=F(X)$ .

If $X$ and $Y$ are normed spaces, the Clarke coderivative $D^{*C}F(x,y)$ of $F$ at $(x,y)\in{\rm gph}\,F$ is a set-valued mapping defined by

\displaystyle D^{*C}F(x,y)(y^{*}):=\{x^{*}\in X^{*}\mid(x^{*},-y^{*})\in N^{C}_{{\rm gph}\,F}(x,y)\},\quad y^{*}\in Y^{*}.

(3)

Replacing the Clarke normal cone in (3) by the Fréchet one, we obtain the definition of the Fréchet coderivative.

Definition 2 (Aubin property).

A mapping $F:X\rightrightarrows Y$ between metric spaces has the Aubin property at $(\bar{x},\bar{y})\in{\rm gph}\,F$ if there exist $\tau>0$ and $\delta>0$ such that

\displaystyle d(y,F(x))\leq\tau d(x,x^{\prime})\quad\mbox{for all}\quad x,x^{\prime}\in B_{\delta}(\bar{x}),\;y\in F(x^{\prime})\cap B_{\delta}(\bar{y}).

(4)

The number $\tau$ is called the modulus of the Aubin property at $(\bar{x},\bar{y})$ .

Aubin property is among the most widely used properties of set-valued mappings in variational analysis (see, e.g., AubFra90 ; RocWet98 ; Mor06.1 ; DonRoc14 ; Iof17 ). It is known, in particular, to be equivalent to the metric regularity of the inverse mapping. It also yields estimates for the normals to the graph of the (given) mapping.

Lemma 1.

Let $X$ and $Y$ be normed spaces, $F:X\rightrightarrows Y$ , and $(\bar{x},\bar{y})\in{\rm gph}\,F$ .

(i)

If $F$ has the Aubin property at $(\bar{x},\bar{y})$ with modulus $\tau>0$ , then there is a $\delta>0$ such that

\displaystyle\|x^{*}\|\leq\tau\|y^{*}\|\quad\mbox{for all}\quad(x,y)\in{\rm gph}\,F\cap B_{\delta}(\bar{x},\bar{y}),\;(x^{*},y^{*})\in N^{F}_{{\rm gph}\,F}(x,y).

(5)

(ii)

If $\dim Y<+\infty$ , then $N^{F}$ in the above assertion can be replaced by $N^{C}$ .

Proof.

(i)

is well known; see, e.g., (Mor06.1, , Theorem 1.43(i)). (The latter theorem is formulated in Mor06.1 in the Banach space setting, but the proof is valid in arbitrary normed spaces.)
(ii)

Suppose $\dim Y<+\infty$ , and $F$ has the Aubin property at $(\bar{x},\bar{y})$ with modulus $\tau>0$ , i.e., condition (4) is satisfied for some $\delta>0$ . Let $(x,y)\in B_{\delta}(\bar{x},\bar{y})\cap{\rm gph}\,F$ and $(x^{*},y^{*})\in N_{{\rm gph}\,F}^{C}(x,y)$ . Take any sequences $(x_{k},y_{k})\in{\rm gph}\,F$ and $t_{k}>0$ such that $(x_{k},y_{k})\to(x,y)$ and $t_{k}\downarrow 0$ as $k\to+\infty$ . Fix an arbitrary $u\in X$ . Without lost generality, we can assume that $x_{k},x_{k}+t_{k}u\in B_{\delta}(\bar{x})$ and $y_{k}\in B_{\delta}(\bar{y})$ for all $k\in\mathbb{N}$ . By (4), for each $k\in\mathbb{N}$ , there exists a point $y_{k}^{\prime}\in F(x_{k}+t_{k}u)$ such that $\|y_{k}^{\prime}-y_{k}\|\leq\tau t_{k}\|u\|$ . Set $v_{k}:=(y_{k}^{\prime}-y_{k})/t_{k}$ . Then $\|v_{k}\|\leq\tau\|u\|$ . Passing to subsequences, we can suppose that $v_{k}\to v\in Y$ . Observe that $(u,v_{k})\to(u,v)$ as $k\to+\infty$ , and $(x_{k},y_{k})+t_{k}(u,v_{k})\in{\rm gph}\,F$ for each $k\in\mathbb{N}$ . Thus, $(u,v)\in T_{{\rm gph}\,F}^{C}(x,y)$ , and $\|v\|\leq\tau\|u\|$ . By the definition of the Clarke normal cone, we have $\left\langle x^{*},u\right\rangle\leq-\left\langle y^{*},v\right\rangle\leq\tau\|y^{*}\|\|u\|$ . Since vector $u$ is arbitrary, it follows that $\|x^{*}\|\leq\tau\|y^{*}\|$ .

3 Set-valued optimization: a single mapping

Let $X$ and $Y$ be normed spaces, $\Omega\subset X$ , $F:X\rightrightarrows Y$ , $\bar{x}\in\Omega$ and $\bar{y}\in F(\bar{x})$ . To model the setting of Definition 1, we consider a nonempty family $\Xi$ of subsets of $Y$ , and define two families of subsets of $X\times Y$ :

\displaystyle\Xi_{1}:=\{{\rm gph}\,F\}\quad\text{and}\quad\Xi_{2}:=\{\Omega\times A\mid A\in\Xi\}.

(6)

The first family consists of the single set ${\rm gph}\,F$ , and the first component of each member of the second family is always the given set $\Omega$ ; only the second component varies.

To emphasize the structure of the pair (6), when referring to the corresponding properties in Definition 1, we will talk about extremality/stationarity of the triple $\{F,\Omega,\Xi\}$ .

Definition 3.

The triple $\{F,\Omega,\Xi\}$ is extremal (resp., stationary, approximately stationary) at $(\bar{x},\bar{y})$ if the pair (6) is extremal (resp., stationary, approximately stationary) at $(\bar{x},\bar{y})$ .

The next proposition is a direct consequence of Definitions 1 and 3.

Proposition 1.

The triple $\{F,\Omega,\Xi\}$ is

(i)

extremal at $(\bar{x},\bar{y})$ if and only if there is a $\rho\in(0,+\infty]$ such that, for any $\varepsilon>0$ , there exists an $A\in\Xi$ such that $d(\bar{y},A)<\varepsilon$ , and

$\displaystyle F(\Omega\cap B_{\rho}(\bar{x}))\cap A\cap B_{\rho}(\bar{y})=\emptyset;$ (7)
(ii)

stationary at $(\bar{x},\bar{y})$ if and only if for any $\varepsilon>0$ , there exist a $\rho\in(0,\varepsilon)$ and an $A\in\Xi$ such that $d(\bar{y},A)<\varepsilon\rho$ , and condition (7) is satisfied;
(iii)

approximately stationary at $(\bar{x},\bar{y})$ if and only if, for any $\varepsilon>0$ , there exist a $\rho\in(0,\varepsilon)$ , an $A\in\Xi$ , and $(x_{1},y_{1}),(x_{2},y_{2})\in B_{\varepsilon}(\bar{x},\bar{y})$ such that $d((x_{1},y_{1}),{\rm gph}\,F)<\varepsilon\rho$ , $d(x_{2},\Omega)<\varepsilon\rho$ , $d(y_{2},A)<\varepsilon\rho$ , and

$\displaystyle F(x_{1}+(\Omega-x_{2})\cap(\rho\mathbb{B}_{X}))\cap(y_{1}+(A-y_{2})\cap(\rho\mathbb{B}_{Y}))=\emptyset.$

The next example illustrates relations between the properties in Proposition 1.

Example 2.

Let $X=Y=\Omega:=\mathbb{R}$ , $\Xi:=\{(-\infty,t]\mid t\in\mathbb{R}\}$ , and $F_{1},F_{2},F_{3},F_{4}:\mathbb{R}\rightrightarrows\mathbb{R}$ be given by

	$\displaystyle F_{1}(x)$	$\displaystyle:=[0,+\infty)\quad\mbox{for all}\;x\in\mathbb{R},\quad\;F_{2}(x):=\begin{cases}[x+1,+\infty)&\mbox{if }x<-1,\\ [0,+\infty)&\mbox{if }x\geq-1,\end{cases}$
	$\displaystyle F_{3}(x)$	$\displaystyle:=[-x^{2},+\infty)\;\mbox{for all}\;x\in\mathbb{R},\quad F_{4}(x):=\begin{cases}[x,+\infty)&\mbox{if }x<0,\\ [-x^{2},+\infty)&\mbox{if }x\geq 0.\end{cases}$

Then $0\in F_{i}(0)$ for all $i=1,2,3,4$ . The following assertions hold true:

(i)

$\{F_{1},\mathbb{R},\Xi\}$ is extremal at $(0,0)$ with $\rho=+\infty$ ;
(ii)

$\{F_{2},\mathbb{R},\Xi\}$ is extremal at $(0,0)$ with some $\rho\in(0,+\infty)$ but not with $\rho=+\infty$ ;
(iii)

$\{F_{3},\mathbb{R},\Xi\}$ is stationary but not extremal at $(0,0)$ ;
(iv)

$\{F_{4},\mathbb{R},\Xi\}$ is approximately stationary at $(0,0)$ but not stationary at $(0,0)$ .

The assertions are straightforward. We only prove assertion (iv). Let $\varepsilon\in(0,1)$ . Choose any $\rho\in(0,\varepsilon)$ and $t\in(\varepsilon\rho,\rho)$ . Then $-t\in\rho\mathbb{B}_{\mathbb{R}}$ , $-t\in F_{4}(-t)$ and $-t\in A$ for any $A:=(-\infty,-\eta]\in\Xi$ with $d(0,A)<\varepsilon\rho$ (i.e., for any $\eta<\varepsilon\rho$ ). By Proposition 1 (ii), $\{F_{4},\mathbb{R},\Xi\}$ is not stationary at $(0,0)$ .

Let $\varepsilon>0$ . Choose a $\rho\in(0,\min\{\varepsilon,1\}/3)$ and points $(x_{1},y_{1}):=(\rho,-\rho^{2})\in{\rm gph}\,F_{4}\cap(\varepsilon\mathbb{B}_{\mathbb{R}^{2}})$ , $x_{2}:=0\in\varepsilon\mathbb{B}_{\mathbb{R}}$ , $y_{2}:=0\in\varepsilon\mathbb{B}_{\mathbb{R}}$ . Observe that $A:=(-\infty,-3\rho^{2}]\in\Xi$ satisfies $d(0,A)<\varepsilon\rho$ . Then

	$\displaystyle F_{4}(x_{1}+(\rho\mathbb{B}_{\mathbb{R}}))=F_{4}(0,2\rho)=(-4\rho^{2},+\infty),$
	$\displaystyle y_{1}+(A-y_{2})\cap(\rho\mathbb{B}_{\mathbb{R}})=-\rho^{2}+(-\rho,-3\rho^{2}]=(-\rho^{2}-\rho,-4\rho^{2}],$

and consequently, $F_{4}(x_{1}+(\rho\mathbb{B}_{\mathbb{R}}))\cap(y_{1}+(A-y_{2})\cap(\rho\mathbb{B}_{\mathbb{R}}))=\emptyset$ . By Proposition 1 (iii), $\{F_{4},\mathbb{R},\Xi\}$ is approximately stationary at $(0,0)$ .

The following example shows that the family of sets $\Xi$ plays an important role in determining the properties.

Example 3.

Let $X=Y=\Omega:=\mathbb{R}$ and $F:\mathbb{R}\rightrightarrows\mathbb{R}$ be given by

\displaystyle F(x):=\begin{cases}\{x\sqrt{2}\}&\mbox{if }x\text{ is rational},\\ \{x\}&\mbox{otherwise}.\end{cases}

Then $0\in F(0)$ .

Let $\Xi:=\{(-\infty,t]\mid t\in\mathbb{R}\}$ and $\varepsilon\in(0,1)$ . Choose any $\rho\in(0,\varepsilon)$ , $(x_{1},y_{1})\in{\rm gph}\,F$ , $x_{2}\in\mathbb{R}$ , $y_{2}\in(-\varepsilon,0]$ and $A:=(-\infty,-t]\in\Xi$ with $d(y_{2},A)<\varepsilon\rho$ (i.e., $t<\varepsilon\rho-y_{2}$ ). Then $x_{1}+(\Omega-x_{2})\cap(\rho\mathbb{B}_{X})=B_{\rho}(x_{1})$ , $A-y_{2}=(-\infty,-\tau]$ , where $\tau:=y_{2}+t<\varepsilon\rho$ , and consequently, $y_{1}+(A-y_{2})\cap(\rho\mathbb{B}_{Y})\supset(y_{1}-\rho,y_{1}-\varepsilon\rho)$ . We next show that $F(B_{\rho}(x_{1}))\cap(y_{1}-\rho,y_{1}-\varepsilon\rho)\neq\emptyset$ . If $x_{1}$ is rational, then $y_{1}=x_{1}\sqrt{2}$ , and choosing a rational number $\hat{x}\in(x_{1}-\rho/\sqrt{2},x_{1}-\varepsilon\rho/\sqrt{2})\subset B_{\rho}(x_{1})$ , we get $\hat{y}:=\hat{x}\sqrt{2}\in F(\hat{x})\cap(y_{1}-\rho,y_{1}-\varepsilon\rho)$ . If $x_{1}$ is irrational, then $y_{1}=x_{1}$ , and choosing an irrational number $\hat{x}\in(x_{1}-\rho,x_{1}-\varepsilon\rho)\subset B_{\rho}(x_{1})$ , we get $\hat{y}:=\hat{x}\in F(\hat{x})\cap(y_{1}-\rho,y_{1}-\varepsilon\rho)$ . By Proposition 1 (iii), $\{F,\mathbb{R},\Xi\}$ is not approximately stationary at $(0,0)$ .

Let $\Xi:=\{\{-1/n\}\mid n\in\mathbb{N}\}$ . Since $F(\mathbb{R})$ only contains irrational numbers, we have $F(\mathbb{R})\cap A=\emptyset$ for all $A\in\Xi$ . By Proposition 1 (i), $\{F,\mathbb{R},\Xi\}$ is extremal at $(0,0)$ (with $\rho=+\infty$ ).

Application of Theorem 1.1 yields necessary conditions for approximate stationarity and, hence, also stationarity and extremality.

Theorem 3.1.

Let $X$ and $Y$ be Banach spaces, the sets $\Omega$ , ${\rm gph}\,F$ and all members of $\Xi$ be closed. If the triple $\{F,\Omega,\Xi\}$ is approximately stationary at $(\bar{x},\bar{y})$ , then, for any $\varepsilon>0$ , there exist $(x_{1},y_{1})\in{\rm gph}\,F\cap B_{\varepsilon}(\bar{x},\bar{y})$ , $x_{2}\in\Omega\cap B_{\varepsilon}(\bar{x})$ , $A\in\Xi$ , $y_{2}\in A\cap B_{\varepsilon}(\bar{y})$ , $(x_{1}^{*},y_{1}^{*})\in N^{C}_{{\rm gph}\,F}(x_{1},y_{1})$ , $x_{2}^{*}\in N^{C}_{\Omega}(x_{2})$ and $y_{2}^{*}\in N^{C}_{A}(y_{2})$ such that

\displaystyle\|(x_{1}^{*},y_{1}^{*})+(x_{2}^{*},y_{2}^{*})\|<\varepsilon\quad\mbox{and}\quad\|(x_{1}^{*},y_{1}^{*})\|+\|(x_{2}^{*},y_{2}^{*})\|=1.

If $X$ is Asplund, then $N^{C}$ in the above assertion can be replaced by $N^{F}$ .

The normalization condition $\|(x_{1}^{*},y_{1}^{*})\|+\|(x_{2}^{*},y_{2}^{*})\|=1$ in Theorem 3.1 ensures that normal vectors $(x_{1}^{*},y_{1}^{*})$ to ${\rm gph}\,F$ remain sufficiently large when $\varepsilon\downarrow 0$ , i.e., $x_{1}^{*}$ and $y_{1}^{*}$ cannot go to $0$ simultaneously. The case when vectors $y_{1}^{*}$ are bounded away from $0$ (hence, one can assume $\|y_{1}^{*}\|=1$ ) is of special interest as it leads to a proper multiplier rule. A closer look at the alternative: either $y_{1}^{*}$ are bounded away from $0$ as $\varepsilon\downarrow 0$ , or they are not (hence, vectors $x_{1}^{*}$ remain large), allows one to formulate the following consequence of Theorem 3.1.

Corollary 1.

(i)

there is an $M>0$ such that, for any $\varepsilon>0$ , there exist $(x_{1},y_{1})\in{\rm gph}\,F\cap B_{\varepsilon}(\bar{x},\bar{y})$ , $x_{2}\in\Omega\cap B_{\varepsilon}(\bar{x})$ , $A\in\Xi$ , $y_{2}\in A\cap B_{\varepsilon}(\bar{y})$ , and $y^{*}\in N^{C}_{A}(y_{2})+\varepsilon\mathbb{B}_{Y^{*}}$ such that $\|y^{*}\|=1$ and

\displaystyle 0\in D^{*C}F(x_{1},y_{1})(y^{*})+N^{C}_{\Omega}(x_{2})\cap(M\mathbb{B}_{X^{*}})+\varepsilon\mathbb{B}_{X^{*}};

(8)

(ii)

for any $\varepsilon>0$ , there exist $(x_{1},y_{1})\in{\rm gph}\,F\cap B_{\varepsilon}(\bar{x},\bar{y})$ , $x_{2}\in\Omega\cap B_{\varepsilon}(\bar{x})$ , $x_{1}^{*}\in D^{*C}F(x_{1},y_{1})(\varepsilon\mathbb{B}_{Y^{*}})$ and $x_{2}^{*}\in N^{C}_{\Omega}(x_{2})$ such that

$\displaystyle\|x_{1}^{*}+x_{2}^{*}\|<\varepsilon\quad\mbox{and}\quad\|x_{1}^{*}\|+\|x_{2}^{*}\|=1.$

If $X$ is Asplund, then $N^{C}$ and $D^{*C}$ in the above assertions can be replaced by $N^{F}$ and $D^{*F}$ , respectively.

Proof.

Let the triple $\{F,\Omega,\Xi\}$ be approximately stationary at $(\bar{x},\bar{y})$ . By Theorem 3.1, for any $j\in\mathbb{N}$ , there exist $(x_{1j},y_{1j})\in{\rm gph}\,F\cap B_{1/j}(\bar{x},\bar{y})$ , $x_{2j}\in\Omega\cap B_{1/j}(\bar{x})$ , ${A_{j}}\in\Xi$ , $y_{2j}\in A_{j}\cap B_{1/j}(\bar{y})$ , $(x_{1j}^{*},y_{1j}^{*})\in N^{C}_{{\rm gph}\,F}(x_{1j},y_{1j})$ , $x_{2j}^{*}\in N^{C}_{\Omega}(x_{2j})$ and $y^{*}_{2j}\in N^{C}_{A_{j}}(y_{2j})$ such that $\|(x_{1j}^{*},y_{1j}^{*})\|+\|(x_{2j}^{*},y_{2j}^{*})\|=1$ and $\|(x_{1j}^{*},y_{1j}^{*})+(x_{2j}^{*},y_{2j}^{*})\|<1/j$ . We consider two cases.

Case 1. $\limsup_{j\to+\infty}\|y_{1j}^{*}\|>\alpha>0$ . Note that $\alpha<1$ . Set $M:=1/\alpha$ . Let $\varepsilon>0$ . Choose a number $j\in\mathbb{N}$ so that $j^{-1}<\alpha\varepsilon$ and $\|y_{1j}^{*}\|>\alpha$ . Set $x_{1}:=x_{1j}$ , $y_{1}:=y_{1j}$ , $x_{2}:=x_{2j}$ , $A:=A_{j}$ , $y_{2}:=y_{2j}$ , $y^{*}:=-y_{1j}^{*}/\|y_{1j}^{*}\|$ , $x_{1}^{*}:=x_{1j}^{*}/\|y_{1j}^{*}\|$ , $x_{2}^{*}:=x_{2j}^{*}/\|y_{1j}^{*}\|$ and $y_{2}^{*}:=y_{2j}^{*}/\|y_{1j}^{*}\|$ . Then $(x_{1},y_{1})\in{\rm gph}\,F\cap B_{\varepsilon}(\bar{x},\bar{y})$ , $x_{2}\in\Omega\cap B_{\varepsilon}(\bar{x})$ , $y_{2}\in A\cap B_{\varepsilon}(\bar{y})$ , $x_{1}^{*}\in D^{*C}F(x_{1},y_{1})(y^{*})$ , $x_{2}^{*}\in N^{C}_{\Omega}(x_{2})$ , ${\|y^{*}\|=1}$ , $y^{*}_{2}\in N^{C}_{A}(y_{2})$ , $\|x_{2}^{*}\|<1/\alpha=M$ . Furthermore, $\|y^{*}-y_{2}^{*}\|=\|y_{1j}^{*}+y_{2j}^{*}\|/\|y_{1j}^{*}\|<1/(\alpha j)<\varepsilon$ , hence, $y^{*}\in N^{C}_{A}(y_{2})+\varepsilon\mathbb{B}_{Y^{*}}$ ; and $\|x_{1}^{*}+x_{2}^{*}\|=\|x_{1j}^{*}+x_{2j}^{*}\|/\|y_{1j}^{*}\|<1/(\alpha j)<\varepsilon$ , hence, condition (8) is satisfied. Thus, assertion (i) holds true.

Case 2. $\lim_{j\to+\infty}\|y_{1j}^{*}\|=0$ . Then $y_{2j}^{*}\to 0$ , $x_{1j}^{*}+x_{2j}^{*}\to 0$ and $1\geq\|x_{1j}^{*}\|+\|x_{2j}^{*}\|\to 1$ as $j\to+\infty$ . Let $\varepsilon>0$ . Choose a number $j\in\mathbb{N}$ so that $\|x_{1j}^{*}\|+\|x_{2j}^{*}\|>0$ and $\max\{j^{-1},\|y_{1j}^{*}\|,\|x_{1j}^{*}+x_{2j}^{*}\|/(\|x_{1j}^{*}\|+\|x_{2j}^{*}\|)\}<\varepsilon$ . Set $x_{1}:=x_{1j}$ , $y_{1}:=y_{1j}$ , $x_{2}:=x_{2j}$ , $x_{1}^{*}:=x_{1j}^{*}/(\|x_{1j}^{*}\|+\|x_{2j}^{*}\|)$ and $x_{2}^{*}:=x_{2j}^{*}/(\|x_{1j}^{*}\|+\|x_{2j}^{*}\|)$ . Then $(x_{1},y_{1})\in{\rm gph}\,F\cap B_{\varepsilon}(\bar{x},\bar{y})$ , $x_{2}\in\Omega\cap B_{\varepsilon}(\bar{x})$ , $x_{1}^{*}\in D^{*C}F(x_{1},y_{1})(\varepsilon\mathbb{B}_{Y^{*}})$ , $x_{2}^{*}\in N^{C}_{\Omega}(x_{2})$ , $\|x_{1}^{*}\|+\|x_{2}^{*}\|=1$ , and $\|x_{1}^{*}+x_{2}^{*}\|<\varepsilon$ . Thus, assertion (ii) holds true.

If $X$ is Asplund, then $N^{C}$ and $D^{*C}$ in the above arguments can be replaced by $N^{F}$ and $D^{*F}$ , respectively.

Remark 2.

(i)

Part (i) of Corollary 1 gives a kind of multiplier rule with $y^{*}$ playing the role of the vector of multipliers.
(ii)

Part (ii) corresponds to ‘singular’ behaviour of $F$ on $\Omega$ . It involves ‘horizontal’ normals to the graph of $F$ ; the $y^{*}$ component vanishes, and consequently, $\Xi$ plays no role.

The following condition is the negation of the condition in Corollary 1 (ii).

1. $(QC)_{C}$
  
  there is an $\varepsilon>0$ such that $\|x_{1}^{*}+x_{2}^{*}\|\geq\varepsilon$ for all $(x_{1},y_{1})\in{\rm gph}\,F\cap B_{\varepsilon}(\bar{x},\bar{y})$ , $x_{2}\in\Omega\cap B_{\varepsilon}(\bar{x})$ , $x_{1}^{*}\in D^{*C}F(x_{1},y_{1})(\varepsilon\mathbb{B}_{Y^{*}})$ and $x_{2}^{*}\in N^{C}_{\Omega}(x_{2}))$ such that $\|x_{1}^{*}\|+\|x_{2}^{*}\|=1$ .

It excludes the singular behavior mentioned in Remark 2 (ii) and serves as a qualification condition ensuring that only the condition in part (i) of Corollary 1 is possible. We denote by $(QC)_{F}$ the analogue of $(QC)_{C}$ with $N^{F}$ and $D^{*F}$ in place of $N^{C}$ and $D^{*C}$ , respectively.

Corollary 2.

Let $X$ and $Y$ be Banach spaces, $\Omega$ , ${\rm gph}\,F$ and all members of $\Xi$ be closed. Suppose that the triple $\{F,\Omega,\Xi\}$ is approximately stationary at $(\bar{x},\bar{y})$ . If condition $(QC)_{C}$ is satisfied, then assertion (i) in Corollary 1 holds true.

If $X$ is Asplund and condition $(QC)_{F}$ is satisfied, then assertion (i) in Corollary 1 holds true with $N^{F}$ and $D^{*F}$ in place of $N^{C}$ and $D^{*C}$ , respectively.

The next proposition provides two typical sufficient conditions for the fulfillment of conditions $(QC)_{C}$ and $(QC)_{F}$ .

Proposition 2.

Let $X$ and $Y$ be normed spaces.

(i)

If $F$ has the Aubin property at $(\bar{x},\bar{y})$ , then $(QC)_{F}$ is satisfied. If, additionally, $\dim Y<+\infty$ , then $(QC)_{C}$ is satisfied too.
(ii)

If $\bar{x}\in{\rm int}\,\Omega$ , then both $(QC)_{C}$ and $(QC)_{F}$ are satisfied.

Proof.

(i)

If $F$ has the Aubin property at $(\bar{x},\bar{y})$ , then, by Lemma 1, condition (5) is satisfied with some $\tau>0$ and $\delta>0$ , and, if $\dim Y<+\infty$ , then the latter condition is also satisfied with $N^{C}$ in place of $N^{F}$ . Hence, $(QC)_{F}$ is satisfied with $\varepsilon:=1/(2\tau+1)$ , as well as $(QC)_{C}$ if $\dim Y<+\infty$ . Indeed, if $(x_{1},y_{1})\in{\rm gph}\,F\cap B_{\delta}(\bar{x},\bar{y})$ and $x_{1}^{*}\in D^{*}F(x_{1},y_{1})(y_{1}^{*})$ (where $D^{*}$ stands for either $D^{*C}$ or $D^{*F}$ ), $x_{2}^{*}\in X^{*}$ , $\|x_{1}^{*}\|+\|x_{2}^{*}\|=1$ and $\|y_{1}^{*}\|<\varepsilon$ , then $\|x_{1}^{*}+x_{2}^{*}\|\geq\|x_{2}^{*}\|-\|x_{1}^{*}\|=1-2\|x_{1}^{*}\|>1-2\tau\varepsilon=\varepsilon.$
(ii)

If $\bar{x}\in{\rm int}\,\Omega$ , then $N^{C}_{\Omega}(x_{2})=N^{F}_{\Omega}(x_{2})=\{0\}$ for all $x_{2}$ near $\bar{x}$ , and consequently, for any normal vector $x_{2}^{*}$ to $\Omega$ at $x_{2}$ and any $x_{1}^{*}\in X^{*}$ , condition $\|x_{1}^{*}\|+\|x_{2}^{*}\|=1$ yields $\|x_{1}^{*}+x_{2}^{*}\|=1$ . Hence, both $(QC)_{C}$ and $(QC)_{F}$ are satisfied with any sufficiently small $\varepsilon$ .

Remark 3.

In view of Proposition 2, each of the Corollaries 1 and 2 covers (ZheNg05.2, , Theorems 3.1 and 4.1) and (ZheNg06, , Theorem 3.1 and Corollary 3.1), which give dual necessary conditions for Pareto optimality, as well as (MorTreZhu03, , Proposition 5.1) for a vector optimization problem with a general preference relation.

The next example illustrates the verification of the necessary conditions for approximate stationarity in Corollaries 1 and 2 for the triple $\{F,\Omega,\Xi\}$ , where $F_{4}$ is defined in Example 2.

Example 4.

Let $X=Y=\Omega:=\mathbb{R}$ , and $F_{4}$ and $\Xi$ be as in Example 2. Thus, the triple $\{F,\Omega,\Xi\}$ is approximately stationary at $(\bar{x},\bar{y})$ , and the conclusions of Corollary 1 must hold true. Moreover, the assumptions in both parts of Proposition 2 are satisfied, and consequently, condition $(QC)_{F}$ holds true. By Corollary 2, assertion (i) in Corollary 1 holds true with $N^{F}$ and $D^{*F}$ in place of $N^{C}$ and $D^{*C}$ , respectively. We now verify this assertion.

Let $M>0$ and $\varepsilon>0$ . Choose a $t\in(0,\min\{\varepsilon,1\})$ . Set $(x_{1},y_{1}):=(t/2,-t^{2}/4)\in{\rm gph}\,F_{4}\cap(\varepsilon\mathbb{B}_{\mathbb{R}^{2}})$ , $x_{2}:=0\in\Omega\cap{(\varepsilon\mathbb{B}_{\mathbb{R}})}$ , $A:=(-\infty,-t]\in\Xi$ , $y_{2}:=-t\in A\cap{(\varepsilon\mathbb{B}_{\mathbb{R}})}$ and $y^{*}:=1$ . Thus, $N^{F}_{\Omega}(x_{2})=\{0\}$ , $y^{*}\in N^{F}_{A}(y_{2})$ and $D^{*F}F_{4}(x_{1},y_{1})(y^{*})=\{-t\}$ . Hence,

\displaystyle D^{*F}F_{4}(x_{1},y_{1})(y^{*})+N^{F}_{\Omega}(x_{2})\cap(M\mathbb{B}_{\mathbb{R}})=\{-t\}\in\varepsilon\mathbb{B}_{\mathbb{R}},

i.e., assertion (i) in Corollary 1 holds true (in terms of Fréchet normals and coderivatives).

4 Abstract level-set mapping

We now consider a particular case of the model in Sections 3 when the family $\Xi$ is determined by an abstract level-set mapping $L:Y\rightrightarrows Y$ . The latter mapping defines a preference relation $\prec$ on $Y$ : $v\prec y$ if and only if $v\in L(y)$ ; see, e.g., (KhaTamZal15, , p. 67).

Given a point $y\in Y$ , we employ below the following notation:

\displaystyle L^{\circ}(y):=L(y)\setminus\{y\},\quad L^{-}(y):=L(y)\cup\{y\}.

(9)

Certain requirements are usually imposed on $L$ in order to make the corresponding preference relation meaningful in optimization and applications; see, e.g., Zhu00 ; MorTreZhu03 ; Mor06.2 ; KhaTamZal15 . In this section, we discuss the following properties of $L$ at or near the reference point $\bar{y}$ :

(O1)

$\liminf\limits_{L^{\circ}(\bar{y})\ni y\to\bar{y}}d(\bar{y},L(y))=0$ ;
(O2)

$\bar{y}\in{\rm cl}\,L^{\circ}(\bar{y})$ ;
(O3)

$\bar{y}\notin L(\bar{y})$ ;
(O4)

$y\in{\rm cl}\,L(y)$ for all $y$ near $\bar{y}$ ;
(O5)

if $y\in L^{\circ}(\bar{y})$ and $v\in{\rm cl}\,L(y)$ , then $v\in L^{\circ}(\bar{y})$ ;
(O6)

if $y\in L(\bar{y})$ and $v\in{\rm cl}\,L(y)$ , then $v\in L(\bar{y})$ .

Some characterizations of the properties and relations between them are collected in the next proposition.

Proposition 3.

Let $L:Y\rightrightarrows Y$ , and $L^{\circ}$ be given by (9). The following assertions hold true.

(i)

(O1) $\Leftrightarrow\$ $\{y\in Y\mid d(\bar{y},L(y))<\varepsilon\}\cap L^{\circ}(\bar{y})\cap B_{\varepsilon}(\bar{y})\neq\emptyset$ for all $\varepsilon>0$ .
(ii)

(O1) $\Rightarrow\$ (O2).
(iii)

(O3) $\Leftrightarrow\$ $[L(\bar{y})=L^{\circ}(\bar{y})]$ .
(iv)

(O2) & (O4) $\Rightarrow\$ (O1).
(v)

(O3) & (O4) $\Rightarrow\$ (O2).
(vi)

(O3) $\Rightarrow\$ $[\eqref{O5}\Leftrightarrow\eqref{O6}]$ .

Proof.

(i)

(O1) $\Leftrightarrow\$ $\inf_{y\in L^{\circ}(\bar{y})\cap B_{\varepsilon}(\bar{y})}d(\bar{y},L(y))=0$ for any $\varepsilon>0$ $\Rightarrow\$ for any $\varepsilon>0$ , there is a $y\in L^{\circ}(\bar{y})\cap B_{\varepsilon}(\bar{y})$ such that $d(\bar{y},L(y))<\varepsilon$ . This proves the ‘ $\Rightarrow$ ’ implication. Conversely, let $\delta:=\inf_{y\in L^{\circ}(\bar{y})\cap B_{\varepsilon}(\bar{y})}d(\bar{y},L(y))>0$ for some $\varepsilon>0$ . Then $\{y\in Y\mid d(\bar{y},L(y))<\delta\}\cap L^{\circ}(\bar{y})\cap B_{\varepsilon}(\bar{y})=\emptyset$ , and consequently, $\{y\in Y\mid d(\bar{y},L(y))<\varepsilon^{\prime}\}\cap L^{\circ}(\bar{y})\cap B_{\varepsilon^{\prime}}(\bar{y})=\emptyset$ , where $\varepsilon^{\prime}:=\min\{\varepsilon,\delta\}$ . The implication ‘ $\Leftarrow$ ’ follows.
(ii)

(O1) $\Rightarrow\$ there exists a sequence $\{y_{k}\}\subset L^{\circ}(\bar{y})$ with $y_{k}\to\bar{y}$ $\Leftrightarrow\$ (O2).
(iii)

The assertion is a consequence of the definition of $L^{\circ}$ in (9).

(iv)

Suppose conditions (O2) and (O4) are satisfied. Let $\varepsilon>0$ . Thanks to (O4), we can choose a $\xi\in(0,\varepsilon)$ such that $y\in{\rm cl}\,L(y)$ for all $y\in B_{\xi}(\bar{y})$ . If $y\in B_{\xi}(\bar{y})$ , then $d(\bar{y},L(y))=d(\bar{y},{\rm cl}\,L(y))\leq\|y-\bar{y}\|<\xi$ . Thus, $B_{\xi}(\bar{y})\subset\{y\in Y\mid d(\bar{y},L(y))<\xi\}$ . Thanks to (O2), we have

\{y\in Y\mid d(\bar{y},L(y))<\varepsilon\}\cap L^{\circ}(\bar{y})\cap B_{\varepsilon}(\bar{y})\supset L^{\circ}(\bar{y})\cap B_{\xi}(\bar{y})\neq\emptyset.

Since $\varepsilon$ is an arbitrary positive number, in view of i, this proves (O1).

(v)

(O4) $\Rightarrow\$ $\bar{y}\in{\rm cl}\,L(\bar{y})$ . The conclusion follows thanks to (iii).
(vi)

The assertion is a consequence of (iii).

Remark 4.

Properties (O4) and (O6) are components of the definition of closed preference relation (see (Mor06.2, , Definition 5.55), (Bao14.2, , p. 583), (KhaTamZal15, , p. 68)) widely used in vector and set-valued optimization. They are called, respectively, local satiation (around $\bar{y}$ ) and almost transitivity. Note that the latter property is actually stronger than the conventional transitivity. It is not satisfied for the preference defined by the lexicographical order (see (Mor06.2, , Example 5.57)) and some other natural preference relations important in vector optimization and its applications including those to welfare economics (see (KhaTamZal15, , Sect. 15.3)). Closed preference relations are additionally assumed in Mor06.2 ; Bao14.2 ; KhaTamZal15 to be nonreflexive, thus, satisfying, in particular, property (O3). In view of Proposition 3, if a preference relation satisfies properties (O3), (O4) and (O6), it also satisfies properties (O1), (O2) and (O5). In this section, we employ the weaker properties (O1) and (O5), which are satisfied by most conventional and many other preference relations. This makes our model applicable to a wider range of multiobjective and set-valued optimization problems compared to those studied in Mor06.2 ; Bao14.2 ; KhaTamZal15 .

The next two examples illustrate some characterizations of the level-set mapping.

Example 5.

Let $L(y):=\{y\}$ for all $y\in Y$ . Then $L^{\circ}(y)=\emptyset$ . Thus, properties (O4) and (O5) are obviously satisfied, while properties (O1) and (O2) are violated.

Example 6.

Let $L:\mathbb{R}^{2}\rightrightarrows\mathbb{R}^{2}$ be defined by

\displaystyle L(y_{1},y_{2}):=\begin{cases}\{(v_{1},v_{2})\in\mathbb{R}^{2}\mid v_{1}<y_{1},\;v_{2}<y_{2}\}&\text{if }(y_{1},y_{2})\neq(0,0),\\ \{(0,0)\}&\text{otherwise}.\end{cases}

Let $\bar{y}:=(0,0)$ . Then $L^{\circ}(y_{1},y_{2})=L(y_{1},y_{2})$ if $(y_{1},y_{2})\neq\bar{y}$ and $L^{\circ}(\bar{y})=\emptyset$ . As in Example 5, properties (O4) and (O5) are satisfied, while properties (O1) and (O2) are violated.

We are going to employ in our model the ‘localized’ family of sets

\displaystyle\Xi^{\delta}:=\{{\rm cl}\,L(y)\mid y\in L^{-}(\bar{y})\cap B_{\delta}(\bar{y})\}.

(10)

Note that members of $\Xi^{\delta}$ are not simply translations (deformations) of the fixed set $L(\bar{y})$ (or $L^{\circ}(\bar{y})$ ); they are defined by sets $L(y)$ where $y$ does not have to be equal to $\bar{y}$ .

Remark 5.

Given a set $K$ containing $\bar{y}$ , one can naturally define the level-set mapping by $L(y)=y-\bar{y}+K$ for all $y\in Y$ . Then (10) defines the family of perturbations as the traditional collection of translations of ${\rm cl}\,K$ , i.e., $\Xi^{\delta}=\{y-\bar{y}+{\rm cl}\,K\mid y\in K\cap B_{\delta}(\bar{y})\}$ .

In the current setting, the properties in Proposition 1 take the following form.

Proposition 4.

Let $\delta>0$ , and $\Xi^{\delta}$ be given by (10). The triple $\{F,\Omega,\Xi^{\delta}\}$ is

(i)

extremal at $(\bar{x},\bar{y})$ if and only if there is a $\rho\in(0,+\infty]$ such that, for any $\varepsilon>0$ , there exists a $y\in L^{-}(\bar{y})\cap B_{\delta}(\bar{y})$ such that $d(\bar{y},L(y))<\varepsilon$ , and

$\displaystyle F(\Omega\cap B_{\rho}(\bar{x}))\cap{\rm cl}\,L(y)\cap B_{\rho}(\bar{y})=\emptyset;$ (11)
(ii)

stationary at $(\bar{x},\bar{y})$ if and only if, for any $\varepsilon>0$ , there exist a $\rho\in(0,\varepsilon)$ and a $y\in L^{-}(\bar{y})\cap B_{\delta}(\bar{y})$ such that $d(\bar{y},L(y))<\varepsilon\rho$ , and condition (11) is satisfied;

(iii)

approximately stationary at $(\bar{x},\bar{y})$ if and only if, for any $\varepsilon>0$ , there exist a $\rho\in(0,\varepsilon)$ , a $y\in L^{-}(\bar{y})\cap B_{\delta}(\bar{y})$ , and $(x_{1},y_{1}),(x_{2},y_{2})\in B_{\varepsilon}(\bar{x},\bar{y})$ such that $d((x_{1},y_{1}),{\rm gph}\,F)<\varepsilon\rho$ , $d(x_{2},\Omega)<\varepsilon\rho$ , $d(y_{2},L(y))<\varepsilon\rho$ , and

\displaystyle F(x_{1}+(\Omega-x_{2})\cap(\rho\mathbb{B}_{X}))\cap(y_{1}+({\rm cl}\,L(y)-y_{2})\cap(\rho\mathbb{B}_{Y}))=\emptyset.

The statements of Theorem 3.1 and its corollaries can be easily adjusted to the current setting. For instance, Corollary 2 can be reformulated as follows.

Corollary 3.

Let $X$ and $Y$ be Banach spaces, $\Omega$ and ${\rm gph}\,F$ be closed, $\delta>0$ , and $\Xi^{\delta}$ be given by (10). Suppose that the triple $\{F,\Omega,\Xi^{\delta}\}$ is approximately stationary at $(\bar{x},\bar{y})$ . If condition $(QC)_{C}$ is satisfied, then there is an $M>0$ such that, for any $\varepsilon>0$ , there exist $(x_{1},y_{1})\in{\rm gph}\,F\cap B_{\varepsilon}(\bar{x},\bar{y})$ , $x_{2}\in\Omega\cap B_{\varepsilon}(\bar{x})$ , $y\in L^{-}(\bar{y})\cap B_{\delta}(\bar{y})$ , $y_{2}\in{\rm cl}\,L(y)\cap B_{\varepsilon}(\bar{y})$ , and $y^{*}\in N^{C}_{{\rm cl}\,L(y)}(y_{2})+\varepsilon\mathbb{B}_{Y^{*}}$ such that $\|y^{*}\|=1$ , and condition (8) holds true.

If $X$ is Asplund and condition $(QC)_{F}$ is satisfied, then the above assertion holds true with $N^{F}$ and $D^{*F}$ in place of $N^{C}$ and $D^{*C}$ , respectively.

The properties in Definition 3 are rather general. They cover various optimality and stationarity concepts in vector and set-valued optimization. With $\Omega$ , $F$ and $L$ as above, and points $\bar{x}\in\Omega$ and $\bar{y}\in F(\bar{x})$ , the next definition seems reasonable.

Definition 4.

The point $(\bar{x},\bar{y})$ is extremal for $F$ on $\Omega$ if there is a $\rho\in(0,+\infty]$ such that

\displaystyle F(\Omega\cap B_{\rho}(\bar{x}))\cap L^{\circ}(\bar{y})\cap B_{\rho}(\bar{y})=\emptyset.

(12)

Definition 4 covers both local ( $\rho<+\infty$ ) and global ( $\rho=+\infty$ ) extremality. The above concept is applicable, in particular, to solutions of the following set-valued minimization problem with respect to the preference determined by $L$ :

\displaystyle\text{minimize }\;F(x)\quad\text{subject to }\;x\in\Omega.

(

P

)

Remark 6.

(i)

The concept in Definition 4 is broader than just (local) minimality as $F$ is not assumed to be an objective mapping of an optimization problem. It can, for instance, be involved in modeling constraints.
(ii)

The property in Definition 4 is similar to the one in the definition of fully localized minimizer in (BaoMor10, , Definition 3.1) (see also (KhaTamZal15, , p. 68)). The latter definition uses the larger set ${\rm cl}\,L(\bar{y})\setminus\{\bar{y}\}$ in place of $L^{\circ}(\bar{y})$ in (12). It is not difficult to check that the two properties are equivalent (when $\rho<+\infty$ ). Unlike many solution concepts in vector optimization, the above definition involves “image localization” (hence, is in general weaker). It has proved to be useful when studying locally optimal allocations of welfare economics; cf. BaoMor10 ; KhaTamZal15 .

We next show that, under some mild assumptions on the level-set mapping $L$ , the extremality in the sense of Definition 4 can be treated in the framework of the extremality in the sense of Definition 3 (or its characterization in Proposition 4 (i)).

Proposition 5.

Let $\bar{x}\in\Omega$ , $\bar{y}\in F(\bar{x})$ , $\delta>0$ , and $\Xi^{\delta}$ be given by (10). Suppose $L$ satisfies conditions (O1) and (O5). If $(\bar{x},\bar{y})$ is extremal for $F$ on $\Omega$ , then the triple $\{F,\Omega,\Xi^{\delta}\}$ is extremal at $(\bar{x},\bar{y})$ .

Proof.

In view of (O1), it follows from Proposition 3 (i) that

\displaystyle\{y\in Y\mid d(\bar{y},L(y))<\varepsilon\}\cap L^{\circ}(\bar{y})\cap B_{\varepsilon}(\bar{y})\neq\emptyset\quad\mbox{for all}\quad\varepsilon>0.

(13)

Suppose $\{F,\Omega,\Xi^{\delta}\}$ is not extremal at $(\bar{x},\bar{y})$ . Let $\rho\in(0,+\infty]$ . By Proposition 4 (i), there exists an $\varepsilon>0$ such that, for any $y\in L^{-}(\bar{y})\cap B_{\delta}(\bar{y})$ with $d(\bar{y},L(y))<\varepsilon$ , it holds

\displaystyle F(\Omega\cap B_{\rho}(\bar{x}))\cap{\rm cl}\,L(y)\cap B_{\rho}(\bar{y})\neq\emptyset.

(14)

In view of (13), there is a point $y\in L^{\circ}(\bar{y})\cap B_{\delta}(\bar{y})\subset L^{-}(\bar{y})\cap B_{\delta}(\bar{y})$ with $d(\bar{y},L(y))<\varepsilon$ , and we can choose a point $\hat{y}$ belonging to the set in (14). Thus, $y\in L^{\circ}(\bar{y})$ and $\hat{y}\in{\rm cl}\,L(y)$ . Thanks to (O5), we have $\hat{y}\in L^{\circ}(\bar{y})$ , and consequently, $\hat{y}\in F(\Omega\cap B_{\rho}(\bar{x}))\cap L^{\circ}(\bar{y})\cap B_{\rho}(\bar{y})$ . Since $\rho\in(0,+\infty]$ is arbitrary, $(\bar{x},\bar{y})$ is not extremal for $F$ on $\Omega$ .

Thanks to Proposition 5, if the level-set mapping $L$ satisfies conditions (O1) and (O5), then extremal points of problem ( $P$ ) satisfy the necessary conditions in Theorem 3.1 and its corollaries.

5 Set-valued optimization: multiple mappings

It is not difficult to upgrade the model used in Definition 3 and the subsequent statements to make it directly applicable to constraint optimization problems: instead of a single mapping $F:X\rightrightarrows Y$ with $\bar{y}\in F(\bar{x})$ for some $\bar{x}\in\Omega\subset X$ and a single family $\Xi$ of subsets of $Y$ , one can consider finite collections of mappings $F_{i}:X\rightrightarrows Y_{i}$ between normed spaces together with points $\bar{y}_{i}\in F_{i}(\bar{x})$ , and nonempty families $\Xi_{i}$ of subsets of $Y_{i}$ $(i=1,\ldots,n)$ .

This more general setting can be viewed as a structured particular case of the set-valued optimization model considered in Section 3 if one sets

\displaystyle Y:=Y_{1}\times\ldots\times Y_{n},\quad F:=(F_{1},\ldots,F_{n}),\quad\bar{y}:=(\bar{y}_{1},\ldots,\bar{y}_{n})\and\Xi:=\Xi_{1}\times\ldots\times\Xi_{n}.

Thus, $\bar{y}\in F(\bar{x})$ , and $A\in\Xi$ means that $A=A_{1}\times\ldots\times A_{n}$ and $A_{i}\in\Xi_{i}$ $(i=1,\ldots,n)$ . To shorten the notation, we keep talking in this section about extremality/stationarity of the triple $\{F,\Omega,\Xi\}$ at $(\bar{x},\bar{y})$ .

Definition 5.

The triple $\{F,\Omega,\Xi\}$ is extremal (resp., stationary, approximately stationary) at $(\bar{x},\bar{y})$ if the collection of $n+1$ families of sets:

\displaystyle\widehat{\Xi}_{i}:=\{\Omega_{i}\}\;\;(i=1,\ldots,n)\and\widehat{\Xi}_{n+1}:=\{\Omega\times A\mid A\in\Xi\}.

is extremal (resp., stationary, approximately stationary) at $(\bar{x},\bar{y})$ , where $\Omega_{i}:=\{(x,y_{1},\ldots,y_{n})\in X\times Y_{1}\times\ldots\times Y_{n}\mid y_{i}\in F_{i}(x)\}$ $(i=1,\ldots,n)$ .

With the notation introduced above, Definitions 1 and 5 lead to characterizations of the extremality and stationarity of the triple $\{F,\Omega,\Xi\}$ given in parts (i) and (ii) of Proposition 1. The corresponding characterization of the approximate stationarity is a little different. It is formulated in the next proposition.

Proposition 6.

The triple $\{F,\Omega,\Xi\}$ is approximately stationary at $(\bar{x},\bar{y})$ if and only if, for any $\varepsilon>0$ , there exist a $\rho\in(0,\varepsilon)$ , $A_{i}\in\Xi_{i}$ $(i=1,\ldots,n)$ , $x_{i}\in B_{\varepsilon}(\bar{x})$ $(i=1,\ldots,n+1)$ , and $y_{i},v_{i}\in B_{\varepsilon}(\bar{y}_{i})$ $(i=1,\ldots,n)$ such that $d((x_{i},y_{i}),{\rm gph}\,F_{i})<\varepsilon\rho$ , $d(v_{i},A_{i})<\varepsilon\rho$ $(i=1,\ldots,n)$ , $d(x_{n+1},\Omega)<\varepsilon\rho$ and, for each $x\in\Omega\cap B_{\rho}(x_{n+1})$ , there is an $i\in\{1,\ldots,n\}$ such that

\displaystyle F_{i}(x_{i}+x-x_{n+1})\cap\big{(}y_{i}+(A_{i}-v_{i})\cap(\rho\mathbb{B}_{Y})\big{)}=\emptyset.

Application of Theorem 1.1 in the current setting produces necessary conditions for approximate stationarity and, hence, also stationarity and extremality extending Theorem 3.1 and its corollaries. Condition $({QC})_{C}$ can be extended as follows:

1. $(\widehat{QC})_{C}$
  
  there is an $\varepsilon>0$ such that $\big{\|}\sum_{i=1}^{n+1}x_{i}^{*}\big{\|}\geq\varepsilon$ for all $(x_{i},y_{i})\in{\rm gph}\,F_{i}\cap B_{\varepsilon}(\bar{x},\bar{y}_{i})$ , $x_{i}^{*}\in D^{*C}F_{i}(x_{i},y_{i})(\varepsilon\mathbb{B}_{Y_{i}^{*}})$ $(i=1,\ldots,n)$ , $x_{n+1}\in\Omega\cap B_{\varepsilon}(\bar{x})$ and $x_{n+1}^{*}\in N^{C}_{\Omega}(x_{n+1})$ such that $\sum_{i=1}^{n+1}\|x_{i}^{*}\|=1$ ,

while the corresponding extension $(\widehat{QC})_{F}$ of condition $({QC})_{F}$ is obtained by replacing $N^{C}$ and $D^{*C}$ in $(\widehat{QC})_{C}$ by $N^{F}$ and $D^{*F}$ , respectively. An extension of Corollary 3 takes the following form.

Theorem 5.1.

Let $X$ , $Y_{1},\ldots,Y_{n}$ be Banach spaces, $\Omega$ and, for each $i=1,\ldots,n$ , the graph ${\rm gph}\,F_{i}$ and all members of $\Xi_{i}$ be closed. Suppose $\{F,\Omega,\Xi\}$ is approximately stationary at $(\bar{x},\bar{y})$ . If condition $(\widehat{QC})_{C}$ is satisfied, then there is an $M>0$ such that, for any $\varepsilon>0$ , there exist $(x_{i},y_{i})\in{\rm gph}\,F_{i}\cap B_{\varepsilon}(\bar{x},\bar{y}_{i})$ , $A_{i}\in\Xi_{i}$ , $v_{i}\in A_{i}\cap B_{\varepsilon}(\bar{y}_{i})$ , $y^{*}_{i}\in N^{C}_{A_{i}}(v_{i})+\varepsilon\mathbb{B}_{Y_{i}^{*}}$ $(i=1,\ldots,n)$ , and $x_{n+1}\in\Omega\cap B_{\varepsilon}(\bar{x})$ such that $\sum_{i=1}^{n}\|y_{i}^{*}\|=1$ and

\displaystyle 0\in\sum_{i=1}^{n}D^{*C}F_{i}(x_{i},y_{i})(y_{i}^{*})+N^{C}_{\Omega}(x_{n+1})\cap(M\mathbb{B}_{X^{*}})+\varepsilon\mathbb{B}_{X^{*}}.

If $X$ is Asplund and condition $(\widehat{QC})_{F}$ is satisfied, then the above assertion holds true with $N^{F}$ and $D^{*F}$ in place of $N^{C}$ and $D^{*C}$ , respectively.

Remark 7.

(i)

Proposition 2 (with $F=(F_{1},\ldots,F_{n})$ in part (i)) gives two typical sufficient conditions for the fulfillment of conditions $(\widehat{QC})_{C}$ and $(\widehat{QC})_{F}$ .
(ii)

Theorem 5.1 covers (ZheNg06, , Theorems 3.1 and 3.2). In view of the previous item, it also covers (ZheNg06, , Corollary 3.2).
(iii)

Theorem 5.1 is a consequence of the dual necessary conditions for approximate stationarity of a collection of sets in Theorem 1.1. The latter theorem can be extended to cover a more general quantitative notion of approximate $\alpha$ -stationarity (with a fixed $\alpha>0$ ), leading to corresponding extensions of Theorem 5.1 and its corollaries covering, in particular, dual conditions for $\varepsilon$ -Pareto optimality in (ZheNg11, , Theorems 4.3 and 4.5).

Employing the multiple-mapping model studied in this section, one can consider a more general than ( $P$ ) optimization problem with set-valued constraints:

\displaystyle\text{minimize }\;F_{0}(x)\quad\text{subject to }\;F_{i}(x)\cap K_{i}\neq\emptyset\;(i=1,\ldots,n),\;\;x\in\Omega,

(

\mathcal{P}

)

where $F_{i}:X\rightrightarrows Y_{i}$ $(i=0,\ldots,n)$ are mappings between normed spaces, $\Omega\subset X$ , $K_{i}\subset Y_{i}$ $(i=1,\ldots,n)$ , and $Y_{0}$ is equipped with a level-set mapping $L$ . The “functional” constraints in ( $\mathcal{P}$ ) can model a system of equalities and inequalities as well as more general operator-type constraints.

Using the set of admissible solutions

$\widehat{\Omega}:=\{x\in\Omega\mid F_{i}(x)\cap K_{i}\neq\emptyset,\;i=1,\ldots,n\}$ ,

we say that a point $(\bar{x},\bar{y}_{0})\in X\times Y_{0}$ is extremal in problem ( $\mathcal{P}$ ) if it is extremal for $F_{0}$ on $\widehat{\Omega}$ . This means, in particular, that $\bar{x}\in\Omega$ , $\bar{y}_{0}\in F_{0}(\bar{x})$ , and there exist $\bar{y}_{i}\in F_{i}(\bar{x})\cap K_{i}$ $(i=1,\ldots,n)$ .

We are going to employ the model studied in the first part of this section with $n+1$ objects in place of $n$ . There are $n+1$ mappings $F_{0},\ldots,F_{n}$ and $n$ sets $K_{1},\ldots,K_{n}$ in ( $\mathcal{P}$ ). As in (10), we define $\Xi_{0}^{\delta}:=\{{\rm cl}\,L(y)\mid y\in L_{\delta}(\bar{y}_{0})\}$ $(\delta>0)$ , where $L_{\delta}(\bar{y}_{0})=(L(\bar{y}_{0})\cap B_{\delta}(\bar{y}))\cup\{\bar{y}\}$ . Now, set

\displaystyle Y:=Y_{0}\times\ldots\times Y_{n},\;\;F:=(F_{0},\ldots,F_{n}),\;\;\bar{y}:=(\bar{y}_{0},\ldots,\bar{y}_{n})\and\Xi^{\delta}:=\Xi_{0}^{\delta}\times K_{1}\times\ldots\times K_{n}.

Using the same arguments, one can prove the next extension of Proposition 5.

Proposition 7.

Let $\bar{x}\in\Omega$ , $\bar{y}_{0}\in F_{0}(\bar{x})$ , $\bar{y}_{i}\in F_{i}(\bar{x})\cap K_{i}$ $(i=1,\ldots,n)$ , $\delta>0$ , and $F$ , $\bar{y}$ and $\Xi^{\delta}$ be defined as above. Suppose $L$ satisfies conditions (O1) and (O5). If $(\bar{x},\bar{y}_{0})$ is extremal in problem ( $\mathcal{P}$ ), then $\{F,\Omega,\Xi^{\delta}\}$ is extremal at $(\bar{x},\bar{y})$ .

Condition $(\widehat{QC})_{C}$ in the current setting is reformulated as follows:

1. $(\widehat{QC})_{C}^{\prime}$
  
  there is an $\varepsilon>0$ such that $\big{\|}\sum_{i=0}^{n+1}x_{i}^{*}\big{\|}\geq\varepsilon$ for all $(x_{i},y_{i})\in{\rm gph}\,F_{i}\cap B_{\varepsilon}(\bar{x},\bar{y}_{i})$ , $x_{i}^{*}\in D^{*C}F_{i}(x_{i},y_{i})(\varepsilon\mathbb{B}_{Y_{i}^{*}})$ $(i=0,\ldots,n)$ , $x_{n+1}\in\Omega\cap B_{\varepsilon}(\bar{x})$ and $x_{n+1}^{*}\in N^{C}_{\Omega}(x_{n+1})$ such that $\sum_{i=0}^{n+1}\|x_{i}^{*}\|=1$ ,

while the corresponding reformulation $(\widehat{QC})_{F}^{\prime}$ of condition $(\widehat{QC})_{F}$ is obtained by replacing $N^{C}$ and $D^{*C}$ in $(\widehat{QC})_{C}^{\prime}$ by $N^{F}$ and $D^{*F}$ , respectively. In view of Proposition 7, Theorem 5.1 yields the following statement.

Corollary 4.

Let $X$ , $Y_{0},\ldots,Y_{n}$ be Banach spaces, the sets $\Omega$ , ${\rm gph}\,F_{i}$ $(i=0,\ldots,n)$ and $K_{i}$ $(i=1,\ldots,n)$ be closed, and $\delta>0$ . Suppose $L$ satisfies conditions (O1) and (O5). If $(\bar{x},\bar{y}_{0})$ is extremal in problem ( $\mathcal{P}$ ) and condition $(\widehat{QC})_{C}^{\prime}$ is satisfied, then there is an $M>0$ such that, for any $\varepsilon>0$ , there exist $(x_{i},y_{i})\in{\rm gph}\,F_{i}\cap B_{\varepsilon}(\bar{x},\bar{y}_{i})$ $(i=0,\ldots,n)$ , $x_{n+1}\in\Omega\cap B_{\varepsilon}(\bar{x})$ , $y\in B_{\delta}(\bar{y}_{0})$ , $v_{0}\in{\rm cl}\,L(y)\cap B_{\varepsilon}(\bar{y}_{0})$ , $y_{0}^{*}\in N^{C}_{{\rm cl}\,L(y)}(v_{0})+\varepsilon\mathbb{B}_{Y_{0}^{*}}$ , $v_{i}\in K_{i}\cap B_{\varepsilon}(\bar{y}_{i})$ and $y_{i}^{*}\in N^{C}_{K_{i}}(v_{i})+\varepsilon\mathbb{B}_{Y_{i}^{*}}$ $(i=1,\ldots,n)$ such that $\sum_{i=0}^{n}\|y_{i}^{*}\|=1$ and

\displaystyle 0\in\sum_{i=0}^{n}D^{*C}F(x_{i},y_{i})(y_{i}^{*})+N^{C}_{\Omega}(x_{n+1})\cap(M\mathbb{B}_{X^{*}})+\varepsilon\mathbb{B}_{X^{*}}.

If $X$ is Asplund and condition $(\widehat{QC})_{F}^{\prime}$ is satisfied, then the above assertion holds true with $N^{F}$ and $D^{*F}$ in place of $N^{C}$ and $D^{*C}$ , respectively.

Declarations

Funding. Nguyen Duy Cuong is supported by Vietnam National Program for the Development of Mathematics 2021-2030 under grant number B2023-CTT-09.

Conflict of interest. The authors have no competing interests to declare that are relevant to the content of this article.

Data availability. Data sharing is not applicable to this article as no datasets have been generated or analysed during the current study.

Acknowledgments

A part of the work was done during Alexander Kruger’s stay at the Vietnam Institute for Advanced Study in Mathematics in Hanoi. He is grateful to the Institute for its hospitality and supportive environment.

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