# First-order approximation of strong vector equilibria with application to
nondifferentiable constrained optimization

Amos Uderzo  Dept. of Mathematics and Applications, University of Milano -
Bicocca, Milano, Italy<EMAIL_ADDRESS>
###### Abstract.

Vector equilibrium problems are a natural generalization to the context of
partially ordered spaces of the Ky Fan inequality, where scalar bifunctions
are replaced with vector bifunctions. In the present paper, the local geometry
of the strong solution set to these problems is investigated through its
inner/outer conical approximations. Formulae for approximating the contingent
cone to the set of strong vector equilibria are established, which are
expressed via Bouligand derivatives of the bifunctions. These results are
subsequently employed for deriving both necessary and sufficient optimality
conditions for problems, whose feasible region is the strong solution set to a
vector equilibrium problem, so they can be cast in mathematical programming
with equilibrium constraints.

###### Key words and phrases:

Strong vector equilibrium, contingent cone, nondifferentiable optimization,
generalized differentiation, subdifferential, mathematical programming with
equilibrium constraint

###### 2010 Mathematics Subject Classification:

49J53, 49J52, 90C33

##  1\. Introduction

Given a mapping (vector-valued bifunction)
$f:\mathbb{R}^{n}\times\mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$, with
$\mathbb{R}^{m}$ being partially ordered by a (nontrivial) closed, convex and
pointed cone $C\subset\mathbb{R}^{m}$, and a nonempty, closed set
$K\subseteq\mathbb{R}^{n}$, by strong vector equilibrium problem the problem
is meant

$None$ $\hbox{ find $x\in K$ such that }f(x,z)\in C,\quad\forall z\in K.$

The set of all solutions (if any) to problem $({\rm VEP})$ will be denoted
throughout the paper by ${\mathcal{S}}{\mathcal{E}}$, namely

(1.1) ${\mathcal{S}}{\mathcal{E}}=\bigcap_{z\in K}f^{-1}(\cdot,z)(C)\cap K,$

and referred to as the set of strong vector equilibria. Clearly, strong vector
equilibrium problems are a natural generalization of the well-known Ky Fan
inequality to the more general context of partially ordered vector spaces.
Similarly as their scalar counterpart, they provide a convenient format to
treat in an unifying framework several different classes of problems, ranging
from multicriteria optimization problems, vector Nash equilibrium problems, to
vector variational inequalities and complementarity problems (see, for
instance, [1, 2, 3, 5, 9, 10, 16]).

As for many problems formalized by traditional or generalized equations, for
several purposes the mere knowledge of a single solution to $({\rm VEP})$ is
not enough. Very often, once a strong vector equilibrium
$\bar{x}\in{\mathcal{S}}{\mathcal{E}}$ has been found (or shown to exist), one
would need/aspire to glean insights into the behaviour of the set
${\mathcal{S}}{\mathcal{E}}$ around $\bar{x}$. The fact that $\bar{x}$ may be
an isolated element of ${\mathcal{S}}{\mathcal{E}}$ or lie in the boundary or,
instead, be an interior element of this set, might change dramatically the
outcome of a further analysis, where the local geometry of
${\mathcal{S}}{\mathcal{E}}$ around $\bar{x}$ does matter. On the other hand,
finding all the solutions of $({\rm VEP})$ around $\bar{x}$ could be a task
that one can hardly accomplish in many concrete cases. What is reasonably
achievable sometimes is only a local approximation of
${\mathcal{S}}{\mathcal{E}}$ near $\bar{x}$, yet suitable in specific
circumstances. To mention one of them, with connection with the subject of the
present paper, consider the successful approach to optimality conditions for
constrained problems, where at a certain step an approximated representation
of the feasible region already does the trick.

It is well known that in nonsmooth analysis tangent cones, working as a
surrogate of derivative for sets, are the main tools for formalizing first-
order (and beyond, if needed) approximations of sets. So the main aim of the
present paper is to provide elements for a conical approximation of strong
vector equilibria. It should be remarked that a difficulty in undertaking such
a task comes from the fact that the set ${\mathcal{S}}{\mathcal{E}}$ is not
explicitly defined. Besides, if addressing this question through the
reformulation of ${\mathcal{S}}{\mathcal{E}}$ as in $(\ref{eq:interEquiref})$,
classical results on the tangent cone representation of such sets as
$f^{-1}(\cdot,z)(C)\cap K$, now at disposal in nonsmooth analysis as a modern
development of the Lyusternik theorem (see [13, 15, 20]), seem not be readily
exploitable because of the intersection over $K$ appearing in
$(\ref{eq:interEquiref})$.

In this context, the findings exposed in what follows are focussed on
representing the contingent cone to ${\mathcal{S}}{\mathcal{E}}$ at a given
strong vector equilibrium $\bar{x}$, which is one of the most employed conical
approximations in the literature devoted to variational analysis and
optimization. The representation of such a cone will be performed by means of
first-order approximations of the problem data, namely generalized derivatives
of the bifunction $f$ and tangent cones of the set $K$ defining $({\rm VEP})$.
In other words, following a principle deep-rooted in many contexts of
nonlinear analysis, approximations of the solution set to a given problem are
obtained by means of exact solutions to approximated problems.

The paper is structured as follows. Section 2 aims at recalling preliminary
notions of nonsmooth analysis, which play a role in formulating and
establishing the achievements of the paper. Section 3 contains the main
results concerning the first-order approximation of the contingent cone to
${\mathcal{S}}{\mathcal{E}}$. In Section 4, these results are applied to
derive both necessary and sufficient optimality conditions for
nondifferentiable optimization problems, whose constraint systems are
formalized as a strong vector equilibrium problem.

Below, the basic notations employed in the paper are listed. The acronyms
l.s.c., u.s.c and p.h. stand for lower semicontinuous, upper semicontinuous
and positively homogeneous, respectively. $\mathbb{R}^{d}$ denotes the finite-
dimensional Euclidean space, with dimension $d\in\mathbb{N}$. The closed ball
centered at an element $x\in\mathbb{R}^{d}$, with radius $r\geq 0$, is denoted
by ${\rm B}\left(x;r\right)$. In particular, ${\mathbb{B}}={\rm
B}\left(\mathbf{0};1\right)$ stands for the unit ball, whereas ${\mathbb{S}}$
stands for the unit sphere, $\mathbf{0}$ denoting the null vector of an
Euclidean space. Given a subset $S\subseteq\mathbb{R}^{d}$, the distance of a
point $x$ from a set $S$ is denoted by ${\rm dist}\left(x;S\right)$, with the
convention that ${\rm dist}\left(x;\varnothing\right)=+\infty$. The prefix
${\rm int}\,S$ denotes the interior of $S$, ${\rm cl}\,S$ denotes its closure,
whereas ${\rm cone}\,S$ its conical hull, respectively. Given two subsets $A$
and $B$ of the same space, the excess of $A$ over $B$ is indicated by ${\rm
exc}(A;B)=\sup_{a\in A}{\rm dist}\left(a;B\right)$. By
$\mathscr{P}\hskip-2.84544pt\mathscr{H}(\mathbb{R}^{n},\mathbb{R}^{m})$ the
space of all continuous p.h. mappings acting between $\mathbb{R}^{n}$ and
$\mathbb{R}^{m}$ is denoted, equipped with the norm
$\|h\|_{\mathscr{P}\hskip-2.84544pt\mathscr{H}}=\sup_{u\in{\mathbb{S}}}\|h(u)\|$,
$h\in\mathscr{P}\hskip-2.84544pt\mathscr{H}(\mathbb{R}^{n},\mathbb{R}^{m})$,
while $\mathscr{L}(\mathbb{R}^{n},\mathbb{R}^{m})$ denotes its subspace of all
linear operators. The inner product of an Euclidean space will be denoted by
$\langle\cdot,\cdot\rangle$. Whenever $C$ is a cone in $\mathbb{R}^{n}$, by
${C}^{{}^{\ominus}}=\\{v\in\mathbb{R}^{n}:\ \langle v,c\rangle\leq
0,\quad\forall c\in C\\}$ the negative dual (a.k.a. polar) cone to $C$ is
denoted. Given a function
$\varphi:\mathbb{X}\longrightarrow\mathbb{R}\cup\\{\pm\infty\\}$, the symbol
$\partial\varphi(x)$ denotes the subdifferential of $\varphi$ at $x$ in the
sense of convex analysis (a.k.a. Fenchel subdifferential). The normal cone to
a set $S\subseteq\mathbb{R}^{q}$ at $x\in S$ in the sense of convex analysis
is denoted by ${\rm N}(x;S)=\\{v\in\mathbb{R}^{n}:\ \langle v,s-x\rangle,\
\forall s\in S\\}$.

##  2\. Preliminaries

###  2.1. Approximation of sets

Given a nonempty set $K\subseteq\mathbb{R}^{n}$ and $\bar{x}\in K$, in the
sequel the following different notions of tangent cone will be mainly
employed:

  * (i)

the contingent (a.k.a. Bouligand tangent) cone to $K$ at $\bar{x}$, which is
defined by

${\rm T}(\bar{x};K)=\\{v\in\mathbb{R}^{n}:\ \exists(v_{n})_{n},\ v_{n}\to v,\
\exists(t_{n})_{n},\ t_{n}\downarrow 0:\ \bar{x}+t_{n}v_{n}\in K,\ \forall
n\in\mathbb{N}\\};$

  * (ii)

the cone of radial (a.k.a. weak feasible) directions to $K$ at $\bar{x}$,
which is defined by

${\rm T}_{\rm r}(\bar{x};K)=\\{v\in\mathbb{R}^{n}:\ \forall\epsilon>0\ \exists
t_{\epsilon}\in(0,\epsilon):\ \bar{x}+t_{\epsilon}v\in K\\}.$

Clearly, for every $K\subseteq\mathbb{R}^{n}$ and $\bar{x}\in K$, it is ${\rm
T}_{\rm r}(\bar{x};K)\subseteq{\rm T}(\bar{x};K)$. Moreover ${\rm
T}(\bar{x};K)$ is always closed. If, in particular, $K$ is convex, then the
following representations hold

(2.1) ${\rm T}_{\rm r}(\bar{x};K)={\rm cone}\,(K-\bar{x})\quad\hbox{ and
}\quad{\rm T}(\bar{x};K)={\rm cl}\,({\rm cone}\,(K-\bar{x}))={\rm cl}\,{\rm
T}_{\rm r}(\bar{x};K)$

(see [20, Proposition 11.1.2(d)]). Thus, in such an event, both ${\rm T}_{\rm
r}(\bar{x};K)$ and ${\rm T}(\bar{x};K)$ are convex. It is well known that an
equivalent (variational) reformulation of the notion of contingent cone is
provided by the equality

(2.2) ${\rm T}(\bar{x};K)=\left\\{v\in\mathbb{R}^{n}:\ \liminf_{t\downarrow
0}{{\rm dist}\left(\bar{x}+tv;K\right)\over t}=0\right\\}.$

######  Remark 2.1.

Whenever a convex set $K\subseteq\mathbb{R}^{n}$ is, in particular,
polyhedral, one has ${\rm T}_{\rm r}(\bar{x};K)={\rm T}(\bar{x};K)$. To see
this, it suffices to exploit the formulae in $(\ref{eq:convexWTangcone})$ and
to observe that, in the present circumstance, ${\rm T}_{\rm r}(\bar{x};K)$
happens to be closed. The latter follows from the fact that, if $S$ is a
closed affine half-space in $\mathbb{R}^{n}$, then ${\rm T}_{\rm
r}(\bar{x};S)={\rm cone}\,(S-\bar{x})=S-\bar{x}$ is a closed set and from the
fact that, if $K_{1}$ and $K_{2}$ are convex sets with $\bar{x}\in\ K_{1}\cap
K_{2}$, then it holds ${\rm T}_{\rm r}(\bar{x};K_{1}\cap K_{2})={\rm T}_{\rm
r}(\bar{x};K_{1})\cap{\rm T}_{\rm r}(\bar{x};K_{2})$.

Along with the above cones, in the context of optimization problems some
further notions of first-order conical approximation will be needed:

  * (iii)

the cone of radial inner (a.k.a. feasible) directions to $K$ at $\bar{x}$,
which is defined by

${\rm T}_{\rm f}(\bar{x};K)=\\{v\in\mathbb{R}^{n}:\ \exists\epsilon>0:\
\forall t\in(0,\epsilon),\ \bar{x}+tv\in K\\};$

  * (vi)

the cone of inner directions (a.k.a. interior displacements) to $K$ at
$\bar{x}$, which is defined by

${\rm I}(\bar{x};K)=\\{v\in\mathbb{R}^{n}:\ \exists\epsilon>0:\ \forall
u\in{\rm B}\left(v;\epsilon\right),\ \forall t\in(0,\epsilon),\ \bar{x}+tu\in
K\\}.$

For a systematic discussion about properties of the above tangent cones and
their relationships, the reader is referred for instance to [4, Chapter 4],
[7, Chapter I.1], [8], [18, Chapter 2], and [20, Chapter 11].

###  2.2. Approximation of scalar functions

Given a function
$\varphi:\mathbb{R}^{n}\longrightarrow\mathbb{R}\cup\\{\pm\infty\\}$, let
$\bar{x}\in\varphi^{-1}(\mathbb{R})$. The set

$\widehat{\partial}^{+}\varphi(\bar{x})=\left\\{v\in\mathbb{R}^{n}:\
\limsup_{x\to\bar{x}}{\varphi(x)-\varphi(\bar{x})-\langle
v,x-\bar{x}\rangle\over\|x-\bar{x}\|}\leq 0\right\\}$

is called (Fréchet) upper subdifferential of $\varphi$ at $\bar{x}$. Any
element $v\in\widehat{\partial}^{+}\varphi(\bar{x})$ can be characterized by
the existence of a function $\psi:\mathbb{R}^{n}\longrightarrow\mathbb{R}$
such that $\varphi(\bar{x})=\psi(\bar{x})$, $\varphi(x)\leq\psi(x)$, for every
$x\in\mathbb{R}^{n}$, $\psi$ is (Fréchet) differentiable at $\bar{x}$ and
$v=\nabla\psi(\bar{x})$. If $\varphi:\mathbb{R}^{n}\longrightarrow\mathbb{R}$
is concave, then $\widehat{\partial}^{+}\varphi(\bar{x})$ coincides with the
superdifferential (a.k.a. upper subdifferential) in the sense of convex
analysis, i.e. $-\partial(-\varphi)(\bar{x})$.

Whenever $\varphi$ is an u.s.c. function, the upper subdifferential admits
another characterization in terms of Dini-Hadamard directional derivative, in
fact being equivalent to the Dini-Hadamard upper subdifferential (in finite-
dimensional spaces, the Fréchet bornology is equivalent to the Hadamard
bornology). More precisely, it holds

(2.3) $\widehat{\partial}^{+}\varphi(\bar{x})=\\{v\in\mathbb{R}^{n}:\ \langle
v,w\rangle\geq{\rm D}^{+}_{H}\varphi(\bar{x};w),\quad\forall
w\in\mathbb{R}^{n}\\},$

where

${\rm D}^{+}_{H}\varphi(\bar{x};w)=\limsup_{u\to w\atop t\downarrow
0}{\varphi(\bar{x}+tu)-\varphi(\bar{x})\over t}$

denotes the Dini-Hadamard upper directional derivative of $\varphi$ at
$\bar{x}$, in the direction $w\in\mathbb{R}^{n}$ (see [15, Chapter 1.3], [19,
Chapter 8.B]). Let us recall that, whenever $\varphi$ is locally Lipschitz
around $\bar{x}$, its Dini-Hadamard directional derivative at $\bar{x}$ takes
the following simpler form

${\rm D}^{+}_{D}\varphi(\bar{x};w)=\limsup_{t\downarrow
0}{\varphi(\bar{x}+tw)-\varphi(\bar{x})\over t},$

which is known as Dini upper directional derivative. The lower versions of
these generalized derivatives are

${\rm D}^{-}_{H}\varphi(\bar{x};w)=\liminf_{u\to w\atop t\downarrow
0}{\varphi(\bar{x}+tu)-\varphi(\bar{x})\over t},$

called the Dini-Hadamard lower directional (a.k.a. contingent) derivative of
$\varphi$ at $\bar{x}$, in the direction $w$, and

${\rm D}^{-}_{D}\varphi(\bar{x};w)=\liminf_{t\downarrow
0}{\varphi(\bar{x}+tw)-\varphi(\bar{x})\over t},$

called the Dini lower directional derivative of $\varphi$ at $\bar{x}$, in the
direction $w$.

The set

$\widehat{\partial}\varphi(\bar{x})=\left\\{v\in\mathbb{R}^{n}:\
\liminf_{x\to\bar{x}}{\varphi(x)-\varphi(\bar{x})-\langle
v,x-\bar{x}\rangle\over\|x-\bar{x}\|}\geq 0\right\\}$

is called (Fréchet) regular subdifferential of $\varphi$ at $\bar{x}$.
Whenever $\varphi$ is l.s.c. around $\bar{x}$, it admits the following
representation in terms of Dini-Hadamard lower directional generalized
derivative

(2.4) $\widehat{\partial}\varphi(\bar{x})=\\{v\in\mathbb{R}^{n}:\ \langle
v,w\rangle\leq{\rm D}^{-}_{H}\varphi(\bar{x};w),\quad\forall
w\in\mathbb{R}^{n}\\}.$

Whenever $\varphi$ is Fréchet differentiable at $\bar{x}$, one has
$\widehat{\partial}^{+}\varphi(\bar{x})=\widehat{\partial}\varphi(\bar{x})=\\{\nabla\varphi(\bar{x})\\}$,
where $\nabla\varphi(\bar{x})$ denotes the gradient of $\varphi$ at $\bar{x}$.

Comprehensive discussions from various viewpoints as well as detailed material
about these generalized derivatives can be found in many textbooks devoted to
nonsmooth analysis, among which [7, Chapter I.1], [15, Chapter 1], [18,
Chapter 2], [19, Chapter 8], [20].

###  2.3. Approximation of mappings and bifunctions

A mapping $g:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$ is said to be
$B$-differentiable at $\bar{x}\in\mathbb{R}^{n}$ if there exists a mapping
${\rm
D}_{B}g(\bar{x})\in\mathscr{P}\hskip-2.84544pt\mathscr{H}(\mathbb{R}^{n},\mathbb{R}^{m})$
such that

$\lim_{x\to\bar{x}}{\|g(x)-g(\bar{x})-{\rm
D}_{B}g(\bar{x})(x-\bar{x})\|\over\|x-\bar{x}\|}=0.$

As a consequence of the continuity of ${\rm D}_{B}g(\bar{x})$, it is readily
seen that if $g$ is $B$-differentiable at $\bar{x}$, it is also continuous at
the same point. Notice that, when, in particular, ${\rm
D}_{B}g(\bar{x})\in\mathscr{L}(\mathbb{R}^{n},\mathbb{R}^{m})$, $g$ turns out
to be (Fréchet) differentiable at $\bar{x}$. In such an event, its derivative,
represented by its Jacobian matrix, will be indicated by $\nabla g(\bar{x})$.
Given a nonempty set $K\subseteq\mathbb{R}^{n}$, a bifunction
$f:\mathbb{R}^{n}\times\mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$ is said to
be $B$-differentiable at $\bar{x}\in K$, uniformly on $K$, if there exists a
family $\\{{\rm
D}_{B}f(\bar{x},z)\in\mathscr{P}\hskip-2.84544pt\mathscr{H}(\mathbb{R}^{n},\mathbb{R}^{m}):\
z\in K\\}$ such that for every $\epsilon>0$ $\exists\delta_{\epsilon}>0$ such
that

$\sup_{z\in K}{\|f(x,z)-f(\bar{x},z)-{\rm
D}_{B}f(\bar{x},z)(x-\bar{x})\|\over\|x-\bar{x}\|}<\epsilon,\quad\forall
x\in{\rm B}\left(\bar{x};\delta_{\epsilon}\right).$

It should be clear that the above notion of generalized differentiation for
bifunctions is a kind of partial differentiation, in considering variations of
a mapping with respect to changes of one variable only.

######  Example 2.2.

(i) Separable mappings: let us consider mappings
$f:\mathbb{R}^{n}\times\mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$, which can
be expressed in the form

$f(x,z)=f_{1}(x)+f_{2}(z),$

for proper $f_{1},\,f_{2}:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$.
Whenever $f_{1}$ is $B$-differentiable at $\bar{x}$, with $B$-derivative ${\rm
D}_{B}f_{1}(\bar{x})$, the bifunction $f$ is $B$-differentiable at $\bar{x}$
uniformly on $K$, with $\\{{\rm D}_{B}f(\bar{x},z):\ z\in K\\}=\\{{\rm
D}_{B}f_{1}(\bar{x})\\}$.

(ii) Factorable mappings: whenever a mapping
$f:\mathbb{R}^{n}\times\mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$ can be
factorized as

$f(x,z)=\alpha(z)g(x),$

where $g:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$ is $B$-differentiable at
$\bar{x}$, with $B$-derivative ${\rm D}_{B}g(\bar{x})$, and
$\alpha:\mathbb{R}^{n}\longrightarrow\mathbb{R}$ is bounded on $K$, the
bifunction $f$ is $B$-differentiable at $\bar{x}$ uniformly on $K$, with
$\\{{\rm D}_{B}f(\bar{x},z):\ z\in K\\}=\\{\alpha(z){\rm
D}_{B}g(\bar{x}):z\in\mathbb{R}^{n}\\}$.

(iii) Composition with differentiable mappings: if
$f:\mathbb{R}^{n}\times\mathbb{R}^{n}\longrightarrow\mathbb{R}^{p}$ is
$B$-differentiable at $\bar{x}$ uniformly on $K$ and
$g:\mathbb{R}^{p}\longrightarrow\mathbb{R}^{m}$ is Fréchet differentiable at
each point $f(\bar{x},z)$, with $z\in K$, then their composition $g\circ f$
turns out to be $B$-differentiable at $\bar{x}$ uniformly on $K$, with
$\\{{\rm D}_{B}(g\circ f)(\bar{x},z):\ z\in K\\}=\\{\nabla g(f(\bar{x},z)){\rm
D}_{B}f(\bar{x},z):\ z\in K\\}$.

A stronger notion of uniform $B$-differentiability will be needed for one of
the main results, which is based on strict $B$-differentiability. Given a
nonempty set $K\subseteq\mathbb{R}^{n}$, a bifunction
$f:\mathbb{R}^{n}\times\mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$ is said to
be strictly $B$-differentiable at $\bar{x}\in K$, uniformly on $K$, if there
exists a family $\\{{\rm
D}_{B}f(\bar{x},z)\in\mathscr{P}\hskip-2.84544pt\mathscr{H}(\mathbb{R}^{n},\mathbb{R}^{m}):\
z\in K\\}$ such that for every $\epsilon>0$ $\exists\delta_{\epsilon}>0$ such
that

$\sup_{z\in K}{\|f(x_{1},z)-f(x_{2},z)-{\rm
D}_{B}f(\bar{x},z)(x_{1}-x_{2})\|\over\|x_{1}-x_{2}\|}<\epsilon,\quad\forall
x_{1},\,x_{2}\in{\rm B}\left(\bar{x};\delta_{\epsilon}\right),\ x_{1}\neq
x_{2}.$

###  2.4. Distance from strong vector equilibria

The function $\nu:\mathbb{R}^{n}\longrightarrow[0,+\infty)$, defined by

(2.5) $\nu(x)=\sup_{z\in K}{\rm dist}\left(f(x,z);C\right),$

can be exploited as a natural measure of the distance of a given point
$x\in\mathbb{R}^{n}$ from being a solution to $({\rm VEP})$. Clearly it is
${\mathcal{S}}{\mathcal{E}}=\nu^{-1}(0)\cap K$, while positive values of $\nu$
quantify the violation of the strong equilibrium condition in $({\rm VEP})$.

A local error bound (in terms of $vu$) is said to be valid near
$\bar{x}\in{\mathcal{S}}{\mathcal{E}}$ for problem $({\rm VEP})$ if there
exist positive $\kappa$ and $\delta$ such that

(2.6) ${\rm
dist}\left(x;{\mathcal{S}}{\mathcal{E}}\right)\leq\kappa\nu(x),\quad\forall
x\in{\rm B}\left(\bar{x};\delta\right)\cap K.$

Notice that, whereas for computing ${\rm
dist}\left(x;{\mathcal{S}}{\mathcal{E}}\right)$ one needs to know all the
solutions to $({\rm VEP})$ near $\bar{x}$, the value of $\nu(x)$ can be
computed directly by means of problem data. A study of sufficient conditions
for the error in bound in $(\ref{in:erboSE})$ to hold has been recently
undertaken in [21]. In particular, the following global error bound condition
under an uniform $B$-differentiability assumption on $f$ is known to hold.

######  Proposition 2.3 ([21]).

With reference to a problem $({\rm VEP})$, suppose that:

  * (i)

each function $x\mapsto f(x,z)$ is $C$-u.s.c. on $K$, for every $z\in K$;

  * (ii)

the set-valued mapping $x\leadsto f(x,K)$ takes $C$-bounded values on $K$;

  * (iii)

$K$ is convex;

  * (iv)

$f$ is $B$-differentiable uniformly on $K$ at each point of
$K\backslash{\mathcal{S}}{\mathcal{E}}$;

  * (v)

there exists $\sigma>0$ with the property that for every $x_{0}\in
K\backslash{\mathcal{S}}{\mathcal{E}}$ there is $u_{0}\in{\mathbb{S}}\cap{\rm
cone}\,(K-x_{0})$ such that

${\rm D}_{B}f(x_{0},z)(u_{0})+\sigma{\mathbb{B}}\subseteq C,\quad\forall z\in
K.$

Then, ${\mathcal{S}}{\mathcal{E}}$ is nonempty, closed and the following
estimate holds true

${\rm
dist}\left(x;{\mathcal{S}}{\mathcal{E}}\right)\leq{\nu(x)\over\sigma},\quad\forall
x\in K.$

##  3\. Tangential approximation of ${\mathcal{S}}{\mathcal{E}}$

######  Theorem 3.1 (Inner approximation).

With reference to a problem $({\rm VEP})$, let
$\bar{x}\in{\mathcal{S}}{\mathcal{E}}$. Suppose that:

  * (i)

$f$ is $B$-differentiable at $\bar{x}$, uniformly on $K$, with $\\{{\rm
D}_{B}f(\bar{x},z):\ z\in K\\}$;

  * (ii)

a local error bound such as $(\ref{in:erboSE})$ is valid near $\bar{x}$.

Then, it holds

(3.1) $\bigcap_{z\in K}{\rm D}_{B}f(\bar{x},z)^{-1}(C)\cap{\rm T}_{\rm
r}(\bar{x};K)\subseteq{\rm T}(\bar{x};{\mathcal{S}}{\mathcal{E}}).$

###### Proof.

Let us start with observing that, since it is ${\rm
D}_{B}f(\bar{x},z)\in\mathscr{P}\hskip-2.84544pt\mathscr{H}(\mathbb{R}^{n},\mathbb{R}^{m})$
for every $z\in K$, and $C$ is a cone, each set ${\rm
D}_{B}f(\bar{x},z)^{-1}(C)$ turns out to be a cone containing $\mathbf{0}$, as
well as ${\rm T}_{\rm r}(\bar{x};K)$ does by definition. Thus, if taking
$v=\mathbf{0}\in\bigcap_{z\in K}{\rm D}_{B}f(\bar{x},z)^{-1}(C)\cap{\rm
T}_{\rm r}(\bar{x};K)$, the inclusion $v\in{\rm
T}(\bar{x};{\mathcal{S}}{\mathcal{E}})$ obviously holds as the latter cone is
closed. So, take an arbitrary $v\in\left(\bigcap_{z\in K}{\rm
D}_{B}f(\bar{x},z)^{-1}(C)\cap{\rm T}_{\rm
r}(\bar{x};K)\right)\backslash\\{\mathbf{0}\\}$. Since both the sets in the
inclusion in $(\ref{in:inapproxTangEqui})$ are cones, one can assume without
any loss of generality that $\|v\|=1$. In the light of the characterization
via $(\ref{eq:charTangcone})$, $v$ is proven to belong to ${\rm
T}(\bar{x};{\mathcal{S}}{\mathcal{E}})$ if one shows that

(3.2) $\liminf_{t\downarrow 0}{{\rm
dist}\left(\bar{x}+tv;{\mathcal{S}}{\mathcal{E}}\right)\over t}=0.$

Showing the equality in $(\ref{eq:thesisreform})$ amounts to show that for
every $\tau>0$ and $\epsilon>0$ there exists $t_{0}\in(0,\tau)$ such that

(3.3) ${{\rm dist}\left(\bar{x}+t_{0}v;{\mathcal{S}}{\mathcal{E}}\right)\over
t_{0}}\leq\epsilon.$

So, let us fix ad libitum $\tau$ and $\epsilon$. Hypothesis (ii) ensures the
existence of $\delta,\ \kappa>0$ as in $(\ref{in:erboSE})$. By virtue of
hypothesis (i), corresponding to $\epsilon/\kappa$, there exists
$\delta_{\epsilon}>0$ such that

$f(x,z)\in f(\bar{x},z)+{\rm
D}_{B}f(\bar{x},z)(x-\bar{x})+\kappa^{-1}\epsilon\|x-\bar{x}\|{\mathbb{B}},\quad\forall
x\in{\rm B}\left(\bar{x};\delta_{\epsilon}\right),\ \forall z\in K,$

and hence, in particular,

$f(\bar{x}+tv,z)\in f(\bar{x},z)+t{\rm
D}_{B}f(\bar{x},z)(v)+\kappa^{-1}\epsilon t{\mathbb{B}},\quad\forall
t\in(0,\delta_{\epsilon}),\ \forall z\in K.$

By taking into account that $\bar{x}\in{\mathcal{S}}{\mathcal{E}}$ and
$v\in{\rm D}_{B}f(\bar{x},z)^{-1}(C)$ for every $z\in K$, the above inclusion
implies

$f(\bar{x}+tv,z)\in C+tC+\kappa^{-1}\epsilon t{\mathbb{B}}\subseteq
C+\kappa^{-1}\epsilon t{\mathbb{B}},\quad\forall t\in(0,\delta_{\epsilon}),\
\forall z\in K.$

In terms of the residual function $\nu$ introduced in $(\ref{eq:defnumf})$,
this means

(3.4) $\displaystyle\nu(\bar{x}+tv)=\sup_{z\in K}{\rm
dist}\left(f(\bar{x}+tv,z);C\right)$ $\displaystyle\leq$ $\displaystyle{\rm
exc}(C+\kappa^{-1}\epsilon t{\mathbb{B}};C)={\rm exc}(\kappa^{-1}\epsilon
t{\mathbb{B}};C)$ $\displaystyle\leq$ $\displaystyle\kappa^{-1}\epsilon
t,\quad\forall t\in(0,\delta_{\epsilon}),$

where the second equality holds because $C$ is a convex cone. On the other
hand, according to hypothesis (ii) there exists
$\delta_{0}\in(0,\min\\{\tau,\delta,\delta_{\epsilon}\\})$ such that

(3.5) ${\rm
dist}\left(x;{\mathcal{S}}{\mathcal{E}}\right)\leq\kappa\nu(x),\quad\forall
x\in{\rm B}\left(\bar{x};\delta_{0}\right)\cap K.$

Since it is $v\in{\rm T}_{\rm r}(\bar{x};K)$, for some
$t_{*}\in(0,\delta_{0})$ it happens

$\bar{x}+t_{*}v\in K\cap{\rm B}\left(\bar{x};\delta_{0}\right),$

and therefore, by inequality $(\ref{in:erboEquidelta})$, one obtains

(3.6) $\displaystyle{\rm
dist}\left(\bar{x}+t_{*}v;{\mathcal{S}}{\mathcal{E}}\right)\leq\kappa\nu(\bar{x}+t_{*}v).$

By combining inequalities $(\ref{in:resepst})$ and $(\ref{in:distnuv})$, as it
is $t_{*}<\delta_{0}<\delta_{\epsilon}$, one obtains

${\rm
dist}\left(\bar{x}+t_{*}v;{\mathcal{S}}{\mathcal{E}}\right)\leq\kappa\cdot\kappa^{-1}\epsilon
t_{*}=\epsilon t_{*}.$

The last inequality shows that $(\ref{eq:thesisreform2})$ is true for
$t_{0}=t_{*}\in(0,\tau)$, thereby completing the proof. ∎

The inclusion in $(\ref{in:inapproxTangEqui})$ states that, under proper
assumptions, any solution of the (approximated) problem

(3.7) $\hbox{ find $v\in{\rm T}_{\rm r}(\bar{x};K)$ such that }{\rm
D}_{B}f(\bar{x};z)(v)\in C,\quad\forall z\in K,$

provides a vector, which is tangent to ${\mathcal{S}}{\mathcal{E}}$ at
$\bar{x}$ in the sense of Bouligand. Notice that problem $(\ref{in:HomVEP})$
is almost in the form $({\rm VEP})$ (it would be exactly in the form $({\rm
VEP})$ if ${\rm T}_{\rm r}(\bar{x};K)=K$). Roughly speaking, all of this means
that if the problem data of $({\rm VEP})$ are properly approximated ($K$ by
its radial direction cone, $f$ by its generalized derivatives in the sense of
Bouligand, respectively) near a reference solution $\bar{x}$, then the
solutions of the resulting approximated problem $(\ref{in:HomVEP})$ work as a
first-order approximation of the solution set to the original problem $({\rm
VEP})$. Problem $(\ref{in:HomVEP})$ is typically expected to be easier than
$({\rm VEP})$ by virtue of the structural properties of its data. Basically,
$(\ref{in:HomVEP})$ can be regarded as a cone constrained p.h. vector
inequality system, so its solution set is a cone. Furthermore, if $K$ is
convex and ${\rm
D}_{B}f(\bar{x},z):\mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$ is $C$-concave
for every $z\in K$, the latter meaning that

${\rm D}_{B}f(\bar{x},z)(v_{1})+{\rm
D}_{B}f(\bar{x},z)(v_{2})\leq_{{}_{C}}{\rm
D}_{B}f(\bar{x},z)(v_{1}+v_{2}),\quad\forall v_{1},\,v_{2}\in\mathbb{R}^{n},$

where $\leq_{{}_{C}}$ denotes the partial ordering on $\mathbb{R}^{m}$ induced
in the standard way by the cone $C$, then the solution set to problem
$(\ref{in:HomVEP})$ is a convex cone.

As a further comment to Theorem 3.1, it must be remarked that the inclusion in
$(\ref{in:inapproxTangEqui})$ provides only a one-side approximation of ${\rm
T}(\bar{x};{\mathcal{S}}{\mathcal{E}})$, which may happen to be rather rough.
This fact is illustrated by the next example.

######  Example 3.2 (Inclusion $(\ref{in:inapproxTangEqui})$ may be strict).

Consider the problem $({\rm VEP})$ defined by the following data:
$K=C=\mathbb{R}^{2}_{+}=\\{x=(x_{1},x_{2})\in\mathbb{R}^{2}:\ x_{1}\geq 0,\
x_{2}\geq 0\\}$ and a vector-valued bifunction
$f:\mathbb{R}^{2}\times\mathbb{R}^{2}\longrightarrow\mathbb{R}^{2}$ given by

$f(x_{1},x_{2},z_{1},z_{2})=\left(\begin{array}[]{c}{1\over
2}(-m_{z}^{-}x_{1}+x_{2}+1)^{2}\\\ \\\ {1\over
2}(m_{z}^{+}x_{1}-x_{2}+1)^{2}\end{array}\right),$

where

$m_{z}^{-}=1-{1\over\|z\|^{2}+1}\qquad\hbox{ and }\qquad
m_{z}^{+}=1+{1\over\|z\|^{2}+1},\quad\ z\in\mathbb{R}^{2}.$

Since $f(x,z)\in\mathbb{R}^{2}_{+}$ for every
$(x,z)\in\mathbb{R}^{2}\times\mathbb{R}^{2}$, it is clear that
${\mathcal{S}}{\mathcal{E}}=K=\mathbb{R}^{2}_{+}$. Fix
$\bar{x}=\mathbf{0}\in{\mathcal{S}}{\mathcal{E}}$, so one has

${\rm T}_{\rm r}(\mathbf{0};K)={\rm
T}(\mathbf{0};{\mathcal{S}}{\mathcal{E}})=\mathbb{R}^{2}_{+}.$

In view of the next calculations, it is convenient to observe that

$f(x,z)=(g\circ h)(x,z),$

where the mappings $g:\mathbb{R}^{2}\longrightarrow\mathbb{R}^{2}$ and
$h:\mathbb{R}^{2}\times\mathbb{R}^{2}\longrightarrow\mathbb{R}^{2}$ are given
respectively by

$g(y)=\left(\begin{array}[]{c}y_{1}^{2}/2\\\
y_{2}^{2}/2\end{array}\right)\qquad\hbox{ and }\qquad
h(x,z)=\left(\begin{array}[]{c}-m_{z}^{-}x_{1}+x_{2}+1\\\
m_{z}^{+}x_{1}-x_{2}+1\end{array}\right).$

To check that the bifunction $h$ is $B$-differentiable at $\mathbf{0}$
uniformly on $\mathbb{R}^{2}_{+}$, with

$\left\\{{\rm D}_{B}h(\mathbf{0},z)=\nabla
h(\mathbf{0},z)=\left(\begin{array}[]{rr}-m_{z}^{-}&1\\\
m_{z}^{+}&-1\end{array}\right),\ z\in\mathbb{R}^{2}_{+}\right\\}$

it suffices to observe that

$\displaystyle\|h(x,z)-h(\mathbf{0},z)-{\rm D}_{B}h(\mathbf{0},z)(x)\|$
$\displaystyle=$
$\displaystyle\left\|\left(\begin{array}[]{c}-m_{z}^{-}x_{1}+x_{2}+1\\\
m_{z}^{+}x_{1}-x_{2}+1\end{array}\right)-\left(\begin{array}[]{c}1\\\
1\end{array}\right)-\left(\begin{array}[]{rr}-m_{z}^{-}&1\\\
m_{z}^{+}&-1\end{array}\right)\left(\begin{array}[]{c}x_{1}\\\
x_{2}\end{array}\right)\right\|$ $\displaystyle=$ $\displaystyle
0,\quad\forall z\in\mathbb{R}^{2}_{+}.$

Thus, since $g$ is Fréchet differentiable at each point of $\mathbb{R}^{2}$
and

$\nabla g(y)=\left(\begin{array}[]{rr}y_{1}&0\\\ 0&y_{2}\end{array}\right),$

according to what remarked in Example 2.2(iii), the mapping $f=g\circ h$ turns
out to be $B$-differentiable at $\mathbf{0}$ uniformly on
$\mathbb{R}^{2}_{+}$, with

${\rm D}_{B}f(\mathbf{0},z)=\nabla g(h(\mathbf{0},z))\circ{\rm
D}_{B}h(\mathbf{0},z)=\left(\begin{array}[]{rr}1&0\\\
0&1\end{array}\right)\left(\begin{array}[]{rr}-m_{z}^{-}&1\\\
m_{z}^{+}&-1\end{array}\right)=\left(\begin{array}[]{rr}-m_{z}^{-}&1\\\
m_{z}^{+}&-1\end{array}\right),\ z\in\mathbb{R}^{2}_{+}.$

Notice that a local error bound as in $(\ref{in:erboSE})$ is evidently valid
near $\mathbf{0}$ because it is ${\mathcal{S}}{\mathcal{E}}=K$. Thus, all the
hypotheses of Theorem 3.1 are satisfied.

Now, one readily sees that

${\rm
D}_{B}f(\mathbf{0},z)(v)=\left(\begin{array}[]{c}-m_{z}^{-}v_{1}+v_{2}\\\
m_{z}^{+}v_{1}-v_{2}\end{array}\right)\in\mathbb{R}^{2}_{+}\qquad\hbox{ iff
}\qquad\left\\{\begin{array}[]{c}-m_{z}^{-}v_{1}+v_{2}\geq 0\\\ \\\
m_{z}^{+}v_{1}-v_{2}\geq 0.\end{array}\right.$

This leads to find

${\rm D}_{B}f(\mathbf{0},z)^{-1}(\mathbb{R}^{2}_{+})=\\{v\in\mathbb{R}^{2}:\
m_{z}^{-}v_{1}\leq v_{2}\leq m_{z}^{+}v_{1}\\},\quad\forall
z\in\mathbb{R}^{2}_{+}.$

Since one has

$\lim_{\|z\|\to\infty}m_{z}^{-}=1^{-}=1=1^{+}=\lim_{\|z\|\to\infty}m_{z}^{+},$

it results in

$\bigcap_{z\in\mathbb{R}^{2}_{+}}{\rm
D}_{B}f(\mathbf{0},z)^{-1}(\mathbb{R}^{2}_{+})\cap{\rm T}_{\rm
r}(\mathbf{0};\mathbb{R}^{2}_{+})=\\{v\in\mathbb{R}^{2}_{+}:\
v_{2}=v_{1}\\}\subsetneqq\mathbb{R}^{2}_{+}={\rm
T}(\mathbf{0};{\mathcal{S}}{\mathcal{E}}).$

The above example motivates the interest in outer approximations of
${\mathcal{S}}{\mathcal{E}}$. Below, a result in this direction is presented.

######  Theorem 3.3 (Outer approximation).

With reference to a problem $({\rm VEP})$, let
$\bar{x}\in{\mathcal{S}}{\mathcal{E}}$. Suppose that:

  * (i)

$f$ is strictly $B$-differentiable at $\bar{x}$, uniformly on $K$, with
$\\{{\rm D}_{B}f(\bar{x},z):\ z\in K\\}$;

  * (ii)

the family of mappings $\\{{\rm D}_{B}f(\bar{x},z):\ z\in K\\}$ is
equicontinuous at each point of $\mathbb{R}^{n}$.

Then, it holds

(3.9) ${\rm T}(\bar{x};{\mathcal{S}}{\mathcal{E}})\subseteq\bigcap_{z\in
K}{\rm D}_{B}f(\bar{x},z)^{-1}({\rm T}(f(\bar{x},z);C))\cap{\rm
T}(\bar{x};K).$

###### Proof.

Since it is ${\rm
D}_{B}f(\bar{x},z)\in\mathscr{P}\hskip-2.84544pt\mathscr{H}(\mathbb{R}^{n},\mathbb{R}^{m})$
for every $z\in K$, one has

${\rm D}_{B}f(\bar{x},z)(\mathbf{0})=\mathbf{0}\in{\rm
T}(f(\bar{x},z);C),\quad\forall z\in K.$

Therefore, it clearly holds

$\mathbf{0}\in\bigcap_{z\in K}{\rm D}_{B}f(\bar{x},z)^{-1}({\rm
T}(f(\bar{x},z);C))\cap{\rm T}(\bar{x};K).$

So take an arbitrary $v\in{\rm
T}(\bar{x};{\mathcal{S}}{\mathcal{E}})\backslash\\{\mathbf{0}\\}$. As all the
sets involved in inclusion $(\ref{in:outapproxTangEqui})$ are cones, without
loss of generality it is possible to assume that $\|v\|=1$. According to the
definition of contingent cone, there exist $(v_{n})_{n}$, with
$v_{n}\longrightarrow v$ and $(t_{n})_{n}$, with $t_{n}\downarrow 0$, such
that $\bar{x}+t_{n}v_{n}\in{\mathcal{S}}{\mathcal{E}}\subseteq K$. Notice that
this inclusion in particular implies that $v\in{\rm T}(\bar{x};K)$. What
remains to be shown is that

(3.10) $v\in\bigcap_{z\in K}{\rm D}_{B}f(\bar{x},z)^{-1}({\rm
T}(f(\bar{x},z);C)).$

Fix an arbitrary $\epsilon>0$. By virtue of hypothesis (i), there exists
$\delta_{\epsilon}>0$ such that

$f(x_{1},z)-f(x_{2},z)-{\rm
D}_{B}f(\bar{x},z)(x_{1}-x_{2})\in\epsilon\|x_{1}-x_{2}\|{\mathbb{B}},\quad\forall
z\in K,\ \forall x_{1},\,x_{2}\in{\rm
B}\left(\bar{x};\delta_{\epsilon}\right)$

and hence

(3.11) ${\rm D}_{B}f(\bar{x},z)(x_{1}-x_{2})\in
f(x_{1},z)-f(x_{2},z)+\epsilon\|x_{1}-x_{2}\|{\mathbb{B}},\quad\forall z\in
K,\ \forall x_{1},\,x_{2}\in{\rm B}\left(\bar{x};\delta_{\epsilon}\right).$

Since it is $\bar{x}+t_{n}v_{n}\longrightarrow\bar{x}$ as $n\to\infty$ (as a
converging sequence $(v_{n})_{n}$ must be bounded), for some
$n_{\epsilon}\in\mathbb{N}$ it is true that $\bar{x}+t_{n}v_{n}\in{\rm
B}\left(\bar{x};\delta_{\epsilon}\right)$ for every $n\geq n_{\epsilon}$.
Thus, by taking $x_{1}=\bar{x}+t_{n}v_{n}$ and $x_{2}=\bar{x}$ in
$(\ref{in:fstrBdifx1x2})$, one finds

$t_{n}{\rm D}_{B}f(\bar{x},z)(v_{n})\in
f(\bar{x}+t_{n}v_{n},z)-f(\bar{x},z)+\epsilon
t_{n}\|v_{n}\|{\mathbb{B}},\quad\forall z\in K,\ \forall n\geq n_{\epsilon},$

whence it follows

${\rm D}_{B}f(\bar{x},z)(v_{n})\in{f(\bar{x}+t_{n}v_{n},z)-f(\bar{x},z)\over
t_{n}}+\epsilon\|v_{n}\|{\mathbb{B}},\quad\forall z\in K,\ \forall n\geq
n_{\epsilon}.$

By taking into account that $v_{n}\longrightarrow v$ as $n\to\infty$ and
$\|v\|=1$, one has that $\|v_{n}\|\leq 2$ for all $n\geq n_{\epsilon}$, up to
a proper increase in the value of $n_{\epsilon}$, if needed. Thus, from the
last inclusion one obtains

(3.12) ${\rm
D}_{B}f(\bar{x},z)(v_{n})\in{f(\bar{x}+t_{n}v_{n},z)-f(\bar{x},z)\over
t_{n}}+2\epsilon{\mathbb{B}},\quad\forall z\in K,\ \forall n\geq
n_{\epsilon}.$

By hypothesis (ii) the family $\\{{\rm D}_{B}f(\bar{x},z):\ z\in K\\}$ is
equicontinuous at $v$. This means that there exists $n_{*}\in\mathbb{N}$
(independent of $z$), with $n_{*}\geq n_{\epsilon}$, such that

$\|{\rm D}_{B}f(\bar{x},z)(v_{n})-{\rm
D}_{B}f(\bar{x},z)(v)\|\leq\epsilon,\quad\forall z\in K,\ \forall n\geq
n_{*},$

or, equivalently,

${\rm D}_{B}f(\bar{x},z)(v)\in{\rm
D}_{B}f(\bar{x},z)(v_{n})+\epsilon{\mathbb{B}},\quad\forall z\in K,\ \forall
n\geq n_{*}.$

By recalling $(\ref{in:Bdervnincrrepeps})$, from the last inclusion one gets

${\rm D}_{B}f(\bar{x},z)(v)\in{f(\bar{x}+t_{n}v_{n},z)-f(\bar{x},z)\over
t_{n}}+3\epsilon{\mathbb{B}},\quad\forall z\in K,\ \forall n\geq n_{*}.$

Since it is $\bar{x}+t_{n}v_{n}\in{\mathcal{S}}{\mathcal{E}}$ for every
$n\in\mathbb{N}$, this implies

${\rm D}_{B}f(\bar{x},z)(v)\in{C-f(\bar{x},z)\over
t_{n}}+3\epsilon{\mathbb{B}}\in{\rm
cone}\,(C-f(\bar{x},z))+3\epsilon{\mathbb{B}},\quad\forall z\in K,\ \forall
n\geq n_{*}.$

Since $C$ is convex so ${\rm T}(f(\bar{x},z);C)={\rm cl}\,{\rm
cone}\,(C-f(\bar{x},z)))$, it results in

${\rm D}_{B}f(\bar{x},z)(v)\in{\rm
T}(f(\bar{x},z);C)+3\epsilon{\mathbb{B}},\quad\forall z\in K.$

The arbitrariness of $\epsilon$ and the fact ${\rm T}(f(\bar{x},z);C)$ is
closed allow one to assert that

${\rm D}_{B}f(\bar{x},z)(v)\in{\rm T}(f(\bar{x},z);C),\quad\forall z\in K,$

which proves the validity of $(\ref{in:thesreformoutapprox})$. Thus the proof
is complete. ∎

######  Remark 3.4.

(i) In the case in which ${\rm int}\,C\neq\varnothing$, it is useful to remark
that the formula in $(\ref{in:outapproxTangEqui})$ can be equivalently
rewritten as

${\rm
T}(\bar{x};{\mathcal{S}}{\mathcal{E}})\subseteq\\{\mathbf{0}\\}\cup\left(\bigcap_{z\in
K\cap f^{-1}(\bar{x},\cdot)({\rm bd}\,C)}{\rm D}_{B}f(\bar{x},z)^{-1}({\rm
T}(f(\bar{x},z);C))\cap{\rm T}(\bar{x};K)\right),$

with the convention that an intersection over an empty index set is the empty
set. Indeed, whenever it happens $f(\bar{x},z)\in{\rm int}\,C$, one has ${\rm
T}(f(\bar{x},z);C)=\mathbb{R}^{m}$, with the consequence that ${\rm
D}_{B}f(\bar{x},z)^{-1}({\rm T}(f(\bar{x},z);C))=\mathbb{R}^{n}$.

(ii) It is worth noticing that for all those $z_{0}\in K$ such that
$f(\bar{x},z_{0})=\mathbf{0}$ (if any), the formula in
$(\ref{in:outapproxTangEqui})$ entails

${\rm T}(\bar{x};{\mathcal{S}}{\mathcal{E}})\subseteq{\rm
D}_{B}f(\bar{x},z_{0})^{-1}(C)\cap{\rm T}(\bar{x};K),$

as it is ${\rm T}(f(\bar{x},z_{0});C)={\rm T}(\mathbf{0};C)=C$.

The next example shows that also the outer approximation of ${\rm
T}(\bar{x};{\mathcal{S}}{\mathcal{E}})$ provided by Theorem 3.3 may happen to
be rather rough.

######  Example 3.5 (Inclusion $(\ref{in:outapproxTangEqui})$ may be strict).

Consider the (actually scalar) problem $({\rm VEP})$ defined by the following
data: $K=\mathbb{R}$, $C=[0,+\infty)$,
$f:\mathbb{R}\times\mathbb{R}\longrightarrow\mathbb{R}$ given by

$f(x,z)={x^{2}z\over z^{2}+1}.$

It is clear that ${\mathcal{S}}{\mathcal{E}}=\\{0\\}$. So, fix $\bar{x}=0$. In
order for checking that $f$ is strictly $B$-differentiable at $0$ uniformly on
$\mathbb{R}$, with $\\{{\rm D}_{B}f(0,z)\equiv 0,\ z\in\mathbb{R}\\}$,
according to the definition it suffices to observe that, fixed an arbitrary
$\epsilon>0$, one has

$\displaystyle\sup_{z\in\mathbb{R}}{|f(x_{1},z)-f(x_{2},z)|\over|x_{1}-x_{2}|}$
$\displaystyle=$
$\displaystyle\sup_{z\in\mathbb{R}}{\displaystyle{\left|{x_{1}^{2}z\over
z^{2}+1}-{x_{2}^{2}z\over
z^{2}+1}\right|}\over|x_{1}-x_{2}|}=\sup_{z\in\mathbb{R}}{|z|\over
z^{2}+1}\cdot|x_{1}+x_{2}|\leq|x_{1}|+|x_{2}|$ $\displaystyle\leq$
$\displaystyle\epsilon,\quad\forall x_{1},\,x_{2}\in{\rm
B}\left(0;\epsilon/2\right),\ x_{1}\neq x_{2}.$

As the family $\\{{\rm D}_{B}f(0,z)\equiv 0,\ z\in\mathbb{R}\\}$ is actually
independent of $z\in\mathbb{R}$, also hypothesis (ii) of Theorem 3.3 is
satisfied.

Since $f(0,z)=0$ for every $z\in\mathbb{R}$, so it is ${\rm
T}(f(0,z);[0,+\infty))=[0,+\infty)$, one finds

${\rm D}_{B}f(0,z)^{-1}\left({\rm
T}(f(0,z);[0,+\infty))\right)=\mathbb{R},\quad\forall z\in\mathbb{R}.$

Consequently, in the current case, one obtains

${\rm
T}(0;{\mathcal{S}}{\mathcal{E}})=\\{0\\}\subsetneqq\mathbb{R}\cap\mathbb{R}=\bigcap_{z\in\mathbb{R}}{\rm
D}_{B}f(0,z)^{-1}({\rm T}(f(0,z);[0,+\infty)))\cap{\rm T}(0;\mathbb{R}).$

Relying on both the preceding approximations, the next result singles out a
sufficient condition, upon which one can establish an exact representation of
${\rm T}(\bar{x};{\mathcal{S}}{\mathcal{E}})$.

######  Corollary 3.6.

With reference to a problem $({\rm VEP})$, let
$\bar{x}\in{\mathcal{S}}{\mathcal{E}}$. Suppose that:

  * (i)

$K$ is polyhedral;

  * (ii)

$f(\bar{x},z)=\mathbf{0},\quad\forall z\in K$;

  * (iii)

$f$ is strictly $B$-differentiable at $\bar{x}$, uniformly on $K$, with
$\\{{\rm D}_{B}f(\bar{x},z):\ z\in K\\}$;

  * (iv)

the family of mappings $\\{{\rm D}_{B}f(\bar{x},z):\ z\in K\\}$ is
equicontinuous at each point of $\mathbb{R}^{n}$;

  * (v)

a local error bound such as in $(\ref{in:erboSE})$ is valid near $\bar{x}$.

Then, it holds

${\rm T}(\bar{x};{\mathcal{S}}{\mathcal{E}})=\bigcap_{z\in K}{\rm
D}_{B}f(\bar{x},z)^{-1}(C)\cap{\rm T}(\bar{x};K).$

###### Proof.

The above assumptions enable one to apply both Theorem 3.1 and Theorem 3.3.
From the former one, in the light of Remark 2.1 and hypothesis (i), one
obtains

(3.13) $\bigcap_{z\in K}{\rm D}_{B}f(\bar{x},z)^{-1}(C)\cap{\rm
T}(\bar{x};K)\subseteq{\rm T}(\bar{x};{\mathcal{S}}{\mathcal{E}}).$

From the latter, in the light of hypothesis (ii) and Remark 3.4(ii), one
obtains

(3.14) ${\rm T}(\bar{x};{\mathcal{S}}{\mathcal{E}})\subseteq\bigcap_{z\in
K}{\rm D}_{B}f(\bar{x},z)^{-1}(C)\cap{\rm T}(\bar{x};K).$

By combining inclusions $(\ref{in:innerTang})$ and $(\ref{in:outTang})$ one
gets the equality in the thesis. ∎

##  4\. Applications to constrained optimization

This section deals with first-order optimality conditions for optimization
problems, whose feasible region is formalized as a set of strong vector
equilibria. As such, these problems can be cast in mathematical programming
with equilibrium constraints, a well-recognized topic and active area of
research (see, among others, [11, 12, 14, 17, 22]). Thus, the optimization
problems here considered take the following form

$None$ $\min\vartheta(x)\quad\hbox{ subject to }\quad
x\in{\mathcal{S}}{\mathcal{E}},$

where $\vartheta:\mathbb{R}^{n}\longrightarrow\mathbb{R}$ is the objective
function formalizing the criterion used for comparing variables, while
${\mathcal{S}}{\mathcal{E}}$ is the feasible region of the problem, denoting
as in the previous sections the solution sets to an inner problem $({\rm
VEP})$. Throughout this section $\vartheta$ will be assumed to be continuous
around $\bar{x}$, but possibly nondifferentiable, as well as the bifunction
$f$ defining $({\rm VEP})$.

In constrained nondifferentiable optimization, first-order optimality
conditions are typically obtained by locally approximating the objective
function and the feasible region of a given problem. In this vein, the fact
stated in the next lemma is widely known to hold, which has been used as a
starting point for various, more elaborated, optimality conditions. For a
direct proof see, for instance, [20, Chapter 7.1]. To a deeper view, it can be
restored as a special case of an axiomatic scheme of analysis, which was
developed in [6, 8] (see [6, Theorem 2.1]).

######  Lemma 4.1.

Let $\bar{x}\in{\mathcal{S}}{\mathcal{E}}$ be a local optimal solution to
problem $({\rm MPVEC})$. Then, it holds

(4.1) ${\rm D}^{+}_{D}\vartheta(\bar{x};w)\geq 0,\quad\forall w\in{\rm T}_{\rm
r}(\bar{x};{\mathcal{S}}{\mathcal{E}})$

and

(4.2) ${\rm D}^{+}_{H}\vartheta(\bar{x};w)\geq 0,\quad\forall w\in{\rm
T}(\bar{x};{\mathcal{S}}{\mathcal{E}}).$

######  Remark 4.2.

Since from their very definition one sees that

${\rm D}^{+}_{D}\vartheta(\bar{x};w)\leq{\rm
D}^{+}_{H}\vartheta(\bar{x};w),\quad\forall w\in\mathbb{R}^{n},$

whereas it is ${\rm T}_{\rm
r}(\bar{x};{\mathcal{S}}{\mathcal{E}})\subseteq{\rm
T}(\bar{x};{\mathcal{S}}{\mathcal{E}})$, none of the conditions
$(\ref{in:nocDTr})$ and $(\ref{in:nocDHT})$ can imply in general the other,
unless $\vartheta$ is locally Lipschitz near $\bar{x}$ or it is ${\rm T}_{\rm
r}(\bar{x};{\mathcal{S}}{\mathcal{E}})={\rm
T}(\bar{x};{\mathcal{S}}{\mathcal{E}})$. Thus, the author does not agree with
what asserted in [20, pag. 132]. For the purposes of the present analysis,
only the condition in $(\ref{in:nocDHT})$ will be actually exploited.

######  Theorem 4.3 (Necessary optimality condition).

Let $\bar{x}\in{\mathcal{S}}{\mathcal{E}}$ be a local optimal solution to
problem $({\rm MPVEC})$. Suppose that:

  * (i)

$f$ is $B$-differentiable at $\bar{x}$, uniformly on $K$, with $\\{{\rm
D}_{B}f(\bar{x},z):\ z\in K\\}$;

  * (ii)

a local error bound such as in $(\ref{in:erboSE})$ is valid near $\bar{x}$.

Then, it holds

(4.3) $-\widehat{\partial}^{+}\vartheta(\bar{x})\subseteq{\left(\bigcap_{z\in
K}{\rm D}_{B}f(\bar{x},z)^{-1}(C)\cap{\rm T}_{\rm
r}(\bar{x};K)\right)}^{{}^{\ominus}}.$

###### Proof.

Under the above assumptions, by Theorem 3.1 the inclusion in
$(\ref{in:inapproxTangEqui})$ holds true. Consequently, since
$\bar{x}\in{\mathcal{S}}{\mathcal{E}}$ is a local optimal solution to $({\rm
MPVEC})$, according to condition $(\ref{in:nocDHT})$ it must be

${\rm D}^{+}_{H}\vartheta(\bar{x};w)\geq 0,\quad\forall w\in\bigcap_{z\in
K}{\rm D}_{B}f(\bar{x},z)^{-1}(C)\cap{\rm T}_{\rm r}(\bar{x};K).$

If $\widehat{\partial}^{+}\vartheta(\bar{x})=\varnothing$ the thesis becomes
trivial. Otherwise, by taking into account the representation in
$(\ref{eq:UpsubdDHder})$, which is valid because the function $\vartheta$ is
in particular u.s.c. around $\bar{x}$, for an arbitrary
$v\in\widehat{\partial}^{+}\vartheta(\bar{x})$ one finds

$\langle v,w\rangle\geq 0,\quad\forall w\in\bigcap_{z\in K}{\rm
D}_{B}f(\bar{x},z)^{-1}(C)\cap{\rm T}_{\rm r}(\bar{x};K),$

which amounts to say that

$-v\in{\left(\bigcap_{z\in K}{\rm D}_{B}f(\bar{x},z)^{-1}(C)\cap{\rm T}_{\rm
r}(\bar{x};K)\right)}^{{}^{\ominus}}.$

The arbitrariness of $v\in\widehat{\partial}^{+}\vartheta(\bar{x})$ completes
the proof. ∎

######  Remark 4.4.

To assess the role of the optimality condition formulated in Theorem 4.3,
notice that it does not carry useful information whenever
$\widehat{\partial}\vartheta(\bar{x})=\varnothing$. This happens, for example,
if $\vartheta$ is a convex continuous function, which is nondifferentiable at
$\bar{x}$. Nevertheless, the upper subdifferential is nonempty for large
classes of functions, including the class of semiconcave ones (see [14]). In
all such cases, condition $(\ref{in:NOCMPVEC})$ provides a necessary
optimality condition, which may be more efficient than those expressed in
terms of more traditional lower subdifferentials. This because it requires
that all elements in $-\widehat{\partial}^{+}\vartheta(\bar{x})$ belong to the
set in the right-side of $(\ref{in:NOCMPVEC})$, in contrast to a mere nonempty
intersection requirement, which is typical for the lower subdifferential case.

######  Corollary 4.5.

Under the same assumptions of Theorem 4.3, if the following additional
hypotheses are satisfied:

  * (i)

$K$ is polyhedral;

  * (ii)

${\rm
D}_{B}f(\bar{x},z)\in\mathscr{P}\hskip-2.84544pt\mathscr{H}(\mathbb{R}^{n},\mathbb{R}^{m})$
is $C$-concave for every $z\in K$;

  * (iii)

the qualification condition holds

(4.4) $\bigcap_{z\in K}{\rm D}_{B}f(\bar{x},z)^{-1}(C)\cap{\rm int}\,{\rm
T}(\bar{x};K)\neq\varnothing,$

then the inclusion in $(\ref{in:NOCMPVEC})$ takes the simpler form

$-\widehat{\partial}^{+}\vartheta(\bar{x})\subseteq{\left(\bigcap_{z\in K}{\rm
D}_{B}f(\bar{x},z)^{-1}(C)\right)}^{{}^{\ominus}}+{\rm N}(\bar{x};K).$

###### Proof.

It is well know that if $S_{1}$ and $S_{2}$ are closed convex cones, then
${(S_{1}\cap S_{2})}^{{}^{\ominus}}={\rm
cl}\,({S_{1}}^{{}^{\ominus}}+{S_{2}}^{{}^{\ominus}})$ (see [20, Lemma 2.4.1]).
On the other hand, if $S_{1}-S_{2}=\mathbb{R}^{n}$, then
${S_{1}}^{{}^{\ominus}}+{S_{2}}^{{}^{\ominus}}$ is closed (see [20,
Proposition 2.4.3] If the qualification condition $S_{1}\cap{\rm
int}\,S_{2}\neq\varnothing$ happens to be satisfied, then
$S_{1}-S_{2}=\mathbb{R}^{n}$ (see [20, Lemma 2.4.4]). Thus, since
$\bigcap_{z\in K}{\rm D}_{B}f(\bar{x},z)^{-1}(C)$ and ${\rm T}(\bar{x};K)$ are
closed convex cone, by virtue of $(\ref{in:qcnocMPVEC})$ and the assumption
(i), one obtains

${\left(\bigcap_{z\in K}{\rm D}_{B}f(\bar{x},z)^{-1}(C)\cap{\rm T}_{\rm
r}(\bar{x};K)\right)}^{{}^{\ominus}}={\left(\bigcap_{z\in K}{\rm
D}_{B}f(\bar{x},z)^{-1}(C)\right)}^{{}^{\ominus}}+{{\rm
T}(\bar{x};K)}^{{}^{\ominus}}.$

Then, in order to achieve the inclusion in the thesis it suffices to recall
that ${{\rm T}(\bar{x};K)}^{{}^{\ominus}}={\rm N}(\bar{x};K)$ (see [20, Lemma
11.2.2]). ∎

Now, let us consider sufficient optimality conditions, a topic usually
investigated in a subsequent step of analysis.

The next lemma provides a sufficient optimality condition for $({\rm MPVEC})$
in the case the objective function is locally Lipschitz. For its proof see [7,
Lemma 1.3, Chapter V]. Notice that for the statement of Lemma 4.6, the
hypothesis on the feasible region of the problem to allow a first-order
uniform conical approximation in the sense of Demyanov-Rubinov is not needed
(see [7, Remark 1.6, Chapter V]).

######  Lemma 4.6.

With reference to $({\rm MPVEC})$, suppose that $\vartheta$ is locally
Lipschitz around $\bar{x}\in{\mathcal{S}}{\mathcal{E}}$. If it holds

(4.5) ${\rm D}^{-}_{D}\vartheta(\bar{x};w)>0,\quad\forall w\in{\rm
T}(\bar{x};{\mathcal{S}}{\mathcal{E}})\backslash\\{\mathbf{0}\\},$

then $\bar{x}$ is a strict local solution to $({\rm MPVEC})$.

On the base of the above lemma, one is in a position to establish the next
result.

######  Theorem 4.7 (Sufficient optimality condition).

With reference to $({\rm MPVEC})$, assume that $\vartheta$ is locally
Lipschitz around $\bar{x}\in{\mathcal{S}}{\mathcal{E}}$. Suppose that:

  * (i)

$f$ is strictly $B$-differentiable at $\bar{x}$, uniformly on $K$, with
$\\{{\rm D}_{B}f(\bar{x},z):\ z\in K\\}$;

  * (ii)

the family of mappings $\\{{\rm D}_{B}f(\bar{x},z):\ z\in K\\}$ is
equicontinuous at each point of $\mathbb{R}^{n}$.

If the condition

(4.6) $\mathbf{0}\in\widehat{\partial}\vartheta(\bar{x})+{\rm
int}\,\left[{\left(\bigcap_{z\in K}{\rm D}_{B}f(\bar{x},z)^{-1}({\rm
T}(f(\bar{x},z);C))\cap{\rm T}(\bar{x};K)\right)}^{{}^{\ominus}}\right],$

is satisfied, then $\bar{x}$ is a strict local solution to $({\rm MPVEC})$.

###### Proof.

Observe first that if for a given cone $S\subseteq\mathbb{R}^{n}$ it is
$v\in{\rm int}\,({S}^{{}^{\ominus}})$, then it must be

$\langle v,s\rangle<0,\quad\forall s\in S\backslash\\{\mathbf{0}\\}.$

Indeed, there exists $\delta>0$ such that
$v+\delta{\mathbb{B}}\subseteq{S}^{{}^{\ominus}}$, and therefore it holds

$\langle v+\delta u,s\rangle\leq 0,\quad\forall u\in{\mathbb{B}},\ \forall
s\in S.$

Thus, for any $s\in S\backslash\\{\mathbf{0}\\}$, the last inequality implies

$\sup_{u\in{\mathbb{B}}}\langle v+\delta u,s\rangle=\langle
v,s\rangle+\delta\sup_{u\in{\mathbb{B}}}\langle u,s\rangle=\langle
v,s\rangle+\delta\|s\|\leq 0,$

whence one gets

$\langle v,s\rangle\leq-\delta\|s\|<0.$

Consequently, the condition $(\ref{in:socDlo})$ implies that there exists
$v\in\widehat{\partial}\vartheta(\bar{x})$ such that it is

$\langle v,w\rangle>0,\quad\forall w\in\left[\bigcap_{z\in K}{\rm
D}_{B}f(\bar{x},z)^{-1}({\rm T}(f(\bar{x},z);C))\cap{\rm
T}(\bar{x};K)\right]\backslash\\{\mathbf{0}\\}.$

By recalling the representation of $\widehat{\partial}\vartheta(\bar{x})$ in
$(\ref{eq:FsubdDHder})$, from the last inequality one obtains

${\rm D}^{-}_{D}\vartheta(\bar{x};w)={\rm
D}^{-}_{H}\vartheta(\bar{x};w)>0,\quad\forall w\in\left[\bigcap_{z\in K}{\rm
D}_{B}f(\bar{x},z)^{-1}({\rm T}(f(\bar{x},z);C))\cap{\rm
T}(\bar{x};K)\right]\backslash\\{\mathbf{0}\\}.$

Since under the above assumptions Theorem 3.3 can be applied, then by virtue
of the inclusion in $(\ref{in:outapproxTangEqui})$ one can state that
condition $(\ref{in:socDlolem})$ turns out to be satisfied. Thus, the thesis
of the theorem follows from Lemma 4.6. ∎

######  Remark 4.8.

(i) As it is possible to see by elementary examples (see [15, Chapter 1]),
$\widehat{\partial}\vartheta(\bar{x})$ may happen to be empty even though
$\vartheta$ is locally Lipschitz around $\bar{x}$. In these circumstances, the
condition in $(\ref{in:socDlo})$ can never be satisfied. On the other hand,
whenever the p.h. function ${\rm
D}^{-}_{H}\vartheta(\bar{x};\cdot):\mathbb{R}^{n}\longrightarrow\mathbb{R}$ is
sublinear (and hence continuous), then
$\widehat{\partial}\vartheta(\bar{x})=\partial{\rm
D}^{-}_{H}\vartheta(\bar{x};\cdot)(\mathbf{0})\neq\varnothing$. This happens
e.g. (but not only) when $\vartheta:\mathbb{R}^{n}\longrightarrow\mathbb{R}$
is convex, in which case one has
$\widehat{\partial}\vartheta(\bar{x})=\partial\vartheta(\bar{x})$.

(ii) The local Lipschitz continuity of $\vartheta$ near $\bar{x}$ might lead
to believe that the Clarke subdifferential may come into play in the current
context. Recall that the latter is defined by

${\partial}_{C}\vartheta(\bar{x})=\left\\{v\in\mathbb{R}^{n}:\ \langle
v,w\rangle\leq\limsup_{x\to\bar{x}\atop t\downarrow
0}{\vartheta(x+tw)-\vartheta(x)\over t},\quad\forall
w\in\mathbb{R}^{n}\right\\}.$

Since, if $\vartheta$ is locally Lipschitz around $\bar{x}$, then it is
$\widehat{\partial}\vartheta(\bar{x})\subseteq{\partial}_{C}\vartheta(\bar{x})$
(see, for instance, [15, Chapter 1]), it follows that the condition

(4.7) $\mathbf{0}\in{\partial}_{C}\vartheta(\bar{x})+{\rm
int}\,\left[{\left(\bigcap_{z\in K}{\rm D}_{B}f(\bar{x},z)^{-1}({\rm
T}(f(\bar{x},z);C))\cap{\rm T}(\bar{x};K)\right)}^{{}^{\ominus}}\right]$

does not imply in general the condition in $(\ref{in:socDlo})$.

## References

  * [1] Q. H. Ansari, I.V. Konnov, J.C. Yao, Characterizations of solutions for vector equilibrium problems, J. Optim. Theory Appl. 113 (2002), no. 3, 435–447.
  * [2] Q. H. Ansari, E. Köbis, J.-C. Yao, Vector variational inequalities and vector optimization. Theory and applications. Vector Optimization, Springer, Cham, 2018.
  * [3] Q. H. Ansari, W. Oettli, D. Schläger, A generalization of vectorial equilibria, Math. Methods Oper. Res. 46 (1997), no. 2, 147–152.
  * [4] J.-P. Aubin, H. Frankowska, Set-valued analysis, Birkhäuser Boston, Boston, MA, 2009.
  * [5] M. Bianchi, N. Hadjisavvas, and S. Schaible, Vector equilibrium problems with generalized monotone bifunctions, J. Optim. Theory Appl. 92 (1997), no. 3, 527–542.
  * [6] M. Castellani, M. Pappalardo, First-order cone approximations and necessary optimality conditions, Optimization 35 (1995), no. 2, 113–126.
  * [7] V.F. Demyanov, A.M. Rubinov, Constructive nonsmooth analysis, Peter Lang, Frankfurt am Main, 1995.
  * [8] K.-H. Elster, J. Thierfelder, Abstract cone approximations and generalized differentiability in nonsmooth optimization, Optimization 19 (1988), no. 3, 315–341.
  * [9] X.H. Gong, Strong vector equilibrium problems, J. Global Optim. 36 (2006), no. 3, 339–349.
  * [10] X.H. Gong, K. Kimura, J.-C. Yao, Sensitivity analysis of strong vector equilibrium problems, J. Nonlinear Convex Anal. 9 (2008), no. 1, 83–94.
  * [11] Z.-Q. Luo, J.-S. Pang, D. Ralph, Mathematical programs with equilibrium constraints, Cambridge University Press, Cambridge, 1996.
  * [12] Z.-Q. Luo, J.-S. Pang, D. Ralph, S.-Q. Wu, Exact penalization and stationarity conditions of mathematical programs with equilibrium constraints, Math. Programming 75 (1996), no. 1, Ser. A, 19–76.
  * [13] B.S. Mordukhovich, Variational analysis and generalized differentiation. I. Basic theory, Springer-Verlag, Berlin, 2006.
  * [14] B.S. Mordukhovich, Variational analysis and generalized differentiation. II. Applications, Springer-Verlag, Berlin, 2006.
  * [15] B.S. Mordukhovich, Variational analysis and applications, Springer, Cham, 2018.
  * [16] W. Oettli, A remark on vector-valued equilibria and generalized monotonicity, Acta Math. Vietnam. 22 (1997), no. 1, 213–221.
  * [17] J.V. Outrata, M. Kočvara, J. Zowe, Nonsmooth approach to optimization problems with equilibrium constraints. Theory, applications and numerical results, Nonconvex Optimization and its Applications, 28. Kluwer Academic Publishers, Dordrecht, 1998.
  * [18] J.P. Penot, Calculus without derivatives, Springer, New York, 2013.
  * [19] R.T. Rockafellar and R.J.-B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998.
  * [20] W. Schirotzek, Nonsmooth analysis, Springer, Berlin, 2007.
  * [21] A. Uderzo, Some enhanced existence results for strong vector equlibrium problems, to appear on Pure and Applied Functional Analysis.
  * [22] J.J. Ye, Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints, J. Math. Anal. Appl. 307 (2005), no. 1, 350–369.