assumptionAssumption remarkRemark hypothesisHypothesis claimClaim Optimality
conditions in DC constrained programsR. Correa, M. A. López and P. Pérez-Aros

# Optimality conditions in DC constrained mathematical programming
problems††thanks: Submitted to the editors DATE.The first author was partially
supported by ANID grant: Fondecyt Regular 1190110. The second author has been
partially supported by Grants MTM2014-59179-C2-(1-2)-P from MINECO/MICINN,
Spain, and FEDER , European Union, and by the Australian Research Council,
Project DP180100602. The third author was partially supported by ANID grant:
Fondecyt Regular 1190110 and Fondecyt Regular 1200283.

Rafael Correa Universidad de O’Higgins, Rancagua, Chile and DIM-CMM of
Universidad de Chile, Santiago, Chile<EMAIL_ADDRESS>Marco A. López
Department of Mathematics, University of Alicante, 03071 Alicante, Spain, and
CIAO, Federation University, Australia<EMAIL_ADDRESS>Pedro Pérez-
Aros Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins,
Rancagua, Chile<EMAIL_ADDRESS>
###### Abstract

This paper provides necessary and sufficient optimality conditions for
abstract constrained mathematical programming problems in locally convex
spaces under new qualification conditions. Our approach exploits the
geometrical properties of certain mappings, in particular their structure as
difference of convex functions, and uses techniques of generalized
differentiation (subdifferential and coderivative). It turns out that these
tools can be used fruitfully out of the scope of Asplund spaces. Applications
to infinite, stochastic and semi-definite programming are developed in
separate sections.

###### keywords:

DC functions, DC constrained programming, conic programming, infinite
programming, stochastic programming, semi-definite programming and supremum
function.

90C30, 90C34, 90C26

##  1 Introduction

Mathematical programming has been recognized as one of the fundamental
chapters of applied mathematics since a huge number of problems in
engineering, economics, management science, etc., involve an optimal decision-
making process which gives rise to an optimization model. This fact has
intensely motivated the theoretical foundations of optimization and the study
and development of algorithms.

Among the main issues in mathematical optimization, optimality conditions play
a key role in the theoretical understanding of solutions and their numerical
computation. In the beginning, such conditions were established for linear or
smooth optimization problems. Later on, developments in variational analysis
allowed the researchers to extend this theory to general nonlinear nonsmooth
convex optimization problems defined in infinite-dimensional spaces (see,
e.g., [40]). In the same spirit, generalized differentiation properties became
an intensive field of research with numerous applications to nonsmooth and
nonconvex mathematical programming problems. Nevertheless, in order to provide
such general calculus rules and compute optimal conditions, a certain degree
of smoothness is required, while working either in Banach spaces with the
smooth norm or in Asplund spaces (see, e.g., [26, 27, 35, 28, 29]). In this
paper, we provide necessary and sufficient optimality conditions for a general
class of optimization problems under new qualification conditions which
constitute real alternatives to the well-known Slater constraint
qualification. Our approach is based on the notions of the (regular)
subdifferential and coderivative and we show that these tools work out of the
scope of Asplund spaces, not for the whole family of lower semicontinuous
functions, but for the so-called class of B-DC mappings. This class of
mappings is introduced in Definition 2.3, and constitutes a slight extension
of the concept of DC functions/mappings.

The paper is focused on the study of (necessary and sufficient) optimal
conditions for a constrained programming problem. First, we study the case
where the constraint is of the form, $\Phi(x)\in\mathcal{C}$, where
$\mathcal{C}$ is a nonempty, closed and convex set in a vector space $Y$, and
$\Phi$ is a vector-valued function from the decision space $X$ into $Y$.
Second, we study an abstract conic constraint, that is, the case when
$\mathcal{C}=-\mathcal{K}$, for a nonempty, convex closed cone $\mathcal{K}$.
These abstract representations allow us to cover infinite, stochastic, and
semidefinite programming problems.

With this general aim of establishing necessary and sufficient optimality
conditions, we first introduce an extension of the concept of vector-valued DC
mappings, also called $\delta$-convex mappings, given in [39] (see also [2,
22] for classical notions and further references). Our Definition 2.3
addresses two fundamental aspects in mathematical optimization. First, the
convenience of using functions with extended real values and mappings which
are not defined in the whole space has been widely recognized. This allows us
to handle different classes of constraint systems from an abstract point of
view and, to this purpose, we enlarge the space $Y$ with an extra element
$\infty_{Y}$ (in particular, $\infty_{Y}=+\infty$, whenever $Y=\mathbb{R}$).
Second, we consider specific scalarization sets for the mapping $\Phi$, which
varies along dual directions in respect to the set involved in the constraint;
more specifically, directions in the dual slope of $\mathcal{C}$, or in the
positive polar cone of $\mathcal{K}$, respectively (see definitions below).

The aforementioned notions make it possible for us to exploit the geometrical
properties of mappings (convexity) combined with tools taken from variational
analysis and generalized differentiation (subdifferential and coderivatives).
Using these tools, we obtain necessary and sufficient optimality conditions of
abstract optimization problems defined on general locally convex spaces under
new qualification conditions, which represent alternatives for classical
qualification conditions.

The paper is organized as follows. In Section 2 we introduce the necessary
notations and preliminary results; we give the definition of the set
$\Gamma_{h}(X,Y)$ and introduce the tools of generalized differentiation.
Together, with those notions, we provide the first calculus rules and
machinery needed in the paper, which constitute the working horse in the
formulation of our optimal conditions. In Section 3, we deal with a constraint
of the form $\Phi(x)\in\mathcal{C}$; we transform the problem into an
unconstrained mathematical program, where the objective function is a
difference of two convex functions, and this reformulation yields necessary
and sufficient conditions of global and local optimality. The main result in
this section, concerning global optimality, is Theorem 3.1, and the result for
necessary conditions of local optimality is Theorem 3.9; meanwhile sufficient
conditions are given in Theorem 3.15. Later in Section 4, we confine ourselves
to studying problems with abstract conic constraints given by
$\Phi(x)\in-\mathcal{K}$. In this Section, the cone structure is exploited,
and a set of scalarizations, generating by the positive polar cone
$\mathcal{K}^{+}$ (see Definition 4.1), is used. Appealing to that notion, and
thanks to a suitable reformulation of the problem, we again derived necessary
and sufficient optimality conditions for conic programming problems. In
particular, Theorem 4.4 presents global optimality conditions and Theorems 4.7
and 4.9 are devoted to local optimality. In the final section, we illustrate
our developments by establishing _ad hoc_ optimality conditions for
fundamental problems in applied mathematics such as infinite, stochastic and
semidefinite programming problems.

##  2 Notation and preliminary results

The paper uses the main notations and definitions which are standard in convex
and variational analysis (see, e.g., [26, 27, 35, 28, 29, 40, 3]).

###  2.1 Tools form convex analysis

In this paper $X$ and $Y$ are locally convex (Hausdorff) spaces (lcs, in
brief) with respective topological duals $X^{\ast}$ and $Y^{\ast}$. We denote
by $w(X^{\ast},X)$, $w(Y^{\ast},Y)$ the corresponding weak-topologies on
$X^{\ast}$ and $Y^{\ast}$. We enlarge $Y$ by adding the element $\infty_{Y}$.
The extended real line is denoted $\overline{\mathbb{R}}:=[-\infty,+\infty]$,
and we adopt the convention $+\infty-(+\infty)=+\infty$. We shall represent by
$0_{X^{\ast}}$ the zero vector in $X^{\ast}$ (similar notation for other
spaces $X,\ Y$, etc).

Given a family of sets $\\{A_{i}\\}_{i\in I}$, we denote its union by
brackets:

$\bigcup\left[A_{i}:i\in I\right]:=\bigcup_{i\in I}A_{i}.$

Given a set $A\subset X$ (or $Y$), we denote by $\operatorname*{cl}(A)$,
$\operatorname{int}(A)$, $\operatorname*{co}(A)$, $\operatorname{cone}(A)$ the
_closure_ , the _interior_ , the _convex hull_ and the the _convex cone_
generated by $A$. By $0_{X}$ we represent the zero vector in $X$, similarly
for the spaces $Y,X^{\ast}$ and $Y^{\ast}$. For two sets $A,B\subset X$ and
$\lambda\in\mathbb{R}$ we define the following operations:

$A+B:=\left\\{a+b,\ a\in A\text{ and }b\in B\right\\},\quad\lambda
A:=\\{\lambda a:a\in A\\},$

and

$A\ominus B:=\\{x\in X:x+B\subseteq A\\}.$

In all previous operations we consider the following standard conventions:

$A+\emptyset=\emptyset+A=\emptyset\text{,
}\operatorname*{co}(\emptyset)=\operatorname*{cone}(\emptyset)=\emptyset,\text{
and }\quad 0\emptyset=\\{0_{X}\\}.$

Given a set $T$ we represent the _generalized simplex_ on $T$ by $\Delta(T)$,
which is the set of all the functions $\alpha:T\to[0,1]$ such that
$\alpha_{t}\neq 0$ only for finitely many $t\in T$ and $\sum_{t\in
T}\alpha_{t}=1;$ for $\alpha\in\Delta(T)$ we denote
$\operatorname{supp}\alpha:=\\{t\in T:\alpha_{t}\neq 0\\}$. We also introduce
the symbols $\Delta_{n}:=\Delta(\\{1,2,\ldots,n\\})$,
$\Delta_{n}^{\varepsilon}:=\varepsilon\Delta_{n}.$

If $A$ is convex and $\varepsilon\geq 0$, we define the $\varepsilon$-_normal
set_ to $A$ at $x$ as

(1) $N_{A}^{\varepsilon}(x):=\\{x^{\ast}\in X^{\ast}:\ \langle
x^{\ast},y-x\rangle\leq\varepsilon\text{ for all }y\in A\\},$

if $x\in A,$ and $N_{A}^{\varepsilon}(x)=\emptyset$ if $x\notin A$. If
$\varepsilon=0$, $N_{A}^{0}(x)\equiv N_{A}(x)$ is the so-called _normal cone_
to $A$ at $x.$

If $A\subseteq X$, and $z_{0}\in X$, the _dual slope of_ $A$_with respect to_
$z_{0}$ is given by

$A_{z_{0}}^{\circ}:=\\{x^{\ast}\in X^{\ast}:\ \langle
x^{\ast},y\rangle\leq\langle x^{\ast},z_{0}\rangle+1\text{ for all }y\in
A\\}.$

For $z_{0}=0_{X}$, the above notation coincides with the _classical polar set_
of $A$, simply denoted by $A^{\circ}$. We have
$A_{z_{0}}^{\circ}\equiv(A-z_{0})^{\circ}$. The _positive polar_ _cone_ of $A$
is given by

$A^{+}:=\left\\{x^{\ast}\in X^{\ast}:\langle x^{\ast},x\rangle\geq 0,\text{
for all }x\in A\right\\}.$

Given a mapping $F:X\rightarrow Y\cup\\{{\infty_{Y}\\}}$, the (_effective_)
_domain_ of $F$ is given by $\operatorname*{dom}F:=\\{x\in X:\ F(x)\in Y\\}$.
We say that $F$ is _proper_ if $\operatorname*{dom}F\neq\emptyset$. Given
$y^{\ast}\in Y^{\ast}$, we define $\langle
y^{\ast},F\rangle:X\to\mathbb{R}\cup\\{+\infty\\}$ by

(4) $\displaystyle\langle
y^{\ast},F\rangle(x):=\left\\{\begin{array}[c]{ll}\langle
y^{\ast},F(x)\rangle,&\text{if }x\in\operatorname{\operatorname{dom}}F,\\\
+\infty,&\text{if
}x\notin\operatorname{\operatorname{dom}}F.\end{array}\right.$

We use similar notations for functions
$f:X\rightarrow\mathbb{R}\cup\\{+\infty\\}$. We represent by $\Gamma_{0}(X)$
the set of all functions $h:X\rightarrow\mathbb{R}\cup\\{+\infty\\}$ which are
proper, convex and lower-semicontinuous (lsc, in brief).

The continuity of functions and mappings will be only considered at points of
their domains.

Given the set $A$, the _indicator function_ of $A$ is

$\delta_{A}(x):=\left\\{\begin{array}[c]{cc}0,&\text{if }x\in A,\\\
+\infty,&\text{if }x\notin A.\end{array}\right.$

For $\varepsilon\geq 0$, the _$\varepsilon$ -subdifferential_ (or _approximate
subdifferential_) of a function $f:X\rightarrow\overline{\mathbb{R}}$ at a
point $x\in X$, where $f$ is finite, is the set

$\partial_{\varepsilon}f(x):=\\{x^{\ast}\in X^{\ast}:\ \langle
x^{\ast},y-x\rangle\leq f(y)-f(x)+\varepsilon,\ \forall
y\in\mathbb{R}^{n}\\};$

if $|f(x)|=+\infty$, we set $\partial_{\varepsilon}f(x)=\emptyset$. The
special case $\varepsilon=0$ yields the classical (_Moreau-Rockafellar_)
_convex subdifferential_ , denoted by $\partial f(x)$.

We finish this subsection by recalling the following alternative formulation
of a general constrained optimization problem which uses a maximum function.
Since the proof follows standard argument, we omit its proof.

######  Lemma 2.1.

Given the functions $g,h:X\to\overline{\mathbb{R}}$ and the nonempty set
$C\subseteq X$, let us consider the optimization problem

(5) $\begin{array}[]{cl}&\min g(x)\\\ &\textnormal{s.t. }h(x)\leq 0\\\ &x\in
C.\end{array}$

Assume that the optimal value, $\alpha$, of problem (5) is finite. Then,
$\bar{x}$ is an optimal solution of (5) if and only if $\bar{x}$ is an optimal
solution of the optimization problem

(6) $\begin{array}[]{l}\min\max\\{g(x)-\alpha,h(x)\\}\\\ \textnormal{s.t.
}\hskip 14.22636ptx\in C.\end{array}$

Moreover, the optimal value of problem (6) is zero.

######  Remark 2.2.

The function $H:X\times X\to\overline{\mathbb{R}}$ defined by

$H(x,y):=\max\\{g(x)-g(y),h(x)\\}$

is called standard _improvement function_ (see, e.g.,[4]). Particularly, the
objective function used in problem (6) corresponds to the improvement function
at $y=\bar{x}$.

###  2.2 B-DC functions and basic properties

Next we introduce a new class of DC functions which constitutes the keystone
of this paper. It extends the notion of _DC vector-valued mappings_ introduced
in [16] and is also related to the concept of _delta-convex functions_ in
[39].

######  Definition 2.3.

Let $X$ and $Y$ be lcs spaces and $h\in\Gamma_{0}(X)$.

  1. i)

Consider a nonempty set $B\subseteq Y^{\ast}$. We define the set of all _$B$
-DC mappings with control $h$_, denoted by $\Gamma_{h}(X,Y,B)$, as the set of
all mappings $F:X\to Y\cup\\{\infty_{Y}\\}$ such that
$\operatorname{dom}h\supseteq\operatorname{dom}F$ and

$\langle\lambda^{\ast},F\rangle+h\in\Gamma_{0}(X)\text{ for all
}\lambda^{\ast}\in B.$

In that case we say that $F$ is a _DC mapping with control function $h$
relative to $B$_, or that _$F$ is controlled by $h$ relatively to $B$_.

  2. ii)

We represent by $\Gamma_{h}(X)$ the set of all functions
$f:X\to\mathbb{R}\cup\\{+\infty\\}$ such that
$\operatorname{dom}f\subseteq\operatorname{dom}h$ and $f+h\in\Gamma_{0}(X)$.

######  Remark 2.4.

It is worth mentioning that Definition 2.3 corresponds to a natural extension
of the notion used in [39], with the name delta-convex functions. More
precisely, following [39], for a continuous mapping $F:U\subseteq X\to Y$,
where $X,Y$ are Banach spaces and $U$ is an open convex set, we say:

  1. a)

$F$ is a _weak-DC mapping_ if for every $\lambda^{\ast}\in Y^{\ast}$ there
exists $h_{\lambda^{\ast}}:U\to\mathbb{R}$ convex and continuous such that
$\langle\lambda^{\ast},F\rangle+h_{\lambda^{\ast}}$ is convex and continuous
on $U$.

  2. b)

$F$ is a _DC mapping_ if there exists $h:U\to\mathbb{R}$ convex and continuous
such that $\langle\lambda^{\ast},F\rangle+h$ is convex and continuous for all
$\lambda^{\ast}\in\mathbb{B}_{Y^{\ast}}$, where $\mathbb{B}_{Y^{\ast}}$ is the
closed unit ball in $Y^{\ast}$. In this case it is said that $F$ is a DC
mapping with control function $h$.

Moreover, by [39, Corollary 1.8] both definitions are equivalent when $Y$ is
finite dimensional. In [39], the focus is on analytic properties of vector-
valued mappings defined on convex open sets. Here, according to the tradition
in optimization theory, we deal with mappings which admit _extended values_.
It is important to mention that Definition 2.3 i) reduces to the concept of DC
mapping introduced in [39], when $B$ is the unit ball in the dual space of
$Y$, in the normed space setting. Moreover, if a real-valued function $f$ is
DC mapping with control $h$, then necessarily $-f$ is a DC with control $h$
(see Proposition 2.5 c) below for more details).

The next proposition gathers some elementary properties of the class
$\Gamma_{h}(X,Y,B)$.

######  Proposition 2.5 (Basic properties of B-DC mappings).

  1. a)

Let $F\in\Gamma_{h}(X,Y,B)$, then $\operatorname{dom}F$ is convex.

  2. b)

Let $F_{i}\in\Gamma_{h_{i}}(X,Y,B)$ for $i=1,\ldots p$ with
$\cap_{i=1}^{p}\operatorname{\operatorname{dom}}F_{i}\neq\emptyset$, then
$\sum_{i=1}^{p}F_{i}\in\Gamma_{h}(X,Y,B)$, where $h=\sum_{i=1}^{p}h_{i}$.

  3. c)

Let $F\in\Gamma_{h}(X,Y,B)$ with $\operatorname{dom}F=X$ and suppose that $B$
is symmetric, then $-F\in\Gamma_{h}(X,Y,B)$.

######  Proof 2.6.

_a)_ It follows from the fact that
$\operatorname{dom}F=\operatorname{dom}\left(\langle\lambda^{\ast},F\rangle+h\right)$
for every $\lambda^{\ast}\in B$. _b)_ Let $F:=\sum_{i=1}^{p}F_{i}$ and
$\lambda^{\ast}\in B$. Then, for all $x\in X$ we have that
$\langle\lambda^{\ast},F\rangle+h=\sum_{i=1}^{p}\left(\langle\lambda^{\ast},F_{i}\rangle+h_{i}\right),$
which is a convex proper and lsc function. _c)_ It follows from the fact that
$\langle\lambda^{\ast},-F\rangle+h=\langle-\lambda^{\ast},F\rangle+h$ and
$-\lambda^{\ast}\in B$ due to the symmetry of $B$.

###  2.3 Generalized differentiation and calculus rules

In this subsection, we introduce the necessary notation to distinguish
nonsmooth and nonconvex functions and mappings, and we develop some calculus
rules.

The following notions are based on classical _bornological_ constructions in
Banach spaces (see, e.g., [7, 6, 19, 26] for more details and similar
constructions). Given a locally convex space $X$, we consider $\beta(X)$ as
the family of all bounded sets of $X$ (i.e., those sets such that every
seminorm which generates the topology on $X$ is bounded on them). We simply
write $\beta$, when there is no ambiguity of the space.

######  Definition 2.7.

We say that $g:X\rightarrow\overline{\mathbb{R}}$ is _differentiable at_ $x$,
with $|g(x)|<+\infty$, if there exists $x^{\ast}\in X^{\ast}$ such that

(7) $\lim_{t\rightarrow 0^{+}}\sup_{h\in
S}\left|\frac{g(x+th)-g(x)}{t}-\left\langle
x^{\ast},h\right\rangle\right|=0,\text{ for all }S\in\beta.$

It is not difficult to see that when such $x^{\ast}$ exists, it is unique. In
that case, and following the usual notation, we simply write $\nabla
g(x)=x^{\ast}$. Here it is important to recall that in a general locally
convex space $X$, the differentiability of a function does not imply its
continuity. For instance, the square of the norm in any infinite dimensional
Hilbert space is Fréchet differentiable, but not weak continuous (see, e.g.,
[8]).

######  Definition 2.8.

The _regular (Fréchet) subdifferential_ of
$f:X\rightarrow\overline{\mathbb{R}}$ at $x\in X,$ with $|f(x)|<+\infty$, is
the set, denoted by $\hat{\partial}f(x),$ of all $x^{\ast}$ such that

$\liminf\limits_{t\rightarrow 0^{+}}\sup\limits_{h\in
S}\left(\frac{f(x+th)-f(x)}{t}-\langle x^{\ast},h\rangle\right)\geq 0,\;\text{
for all }S\in\beta.$

For a point $x\in X$, where $|f(x)|=+\infty$ we simply define
$\hat{\partial}f(x)=\emptyset$.

Now, let us formally prove that the regular subdifferential coincides with the
classic convex subdifferential for functions in $\Gamma_{0}(X)$.

######  Lemma 2.9.

Let $f\in\Gamma_{0}(X)$. Then, the regular subdifferential of $f$ coincides
with the classic convex subdifferential of $f$, that is,
$\hat{\partial}f(x)=\partial f(x)$ for every $x\in X$.

######  Proof 2.10.

Since $\partial f(x)\subseteq\hat{\partial}f(x)$ obviously holds, we focus on
the opposite inclusion. Let $x^{\ast}\in\hat{\partial}f(x)$, which implies
that $|f(x)|<+\infty$. Now, consider $y\in X$ and $\varepsilon>0$ arbitrary,
and let $S\in\beta$ such that $h=y-x\in S$. Hence, we have that for small
enough $t\in(0,1)$, the following inequality holds

$\frac{f((1-t)x+ty)-f(x)}{t}-\langle
x^{\ast},y-x\rangle=\frac{f(x+t(y-x))-f(x)}{t}-\langle
x^{\ast},y-x\rangle\geq-\varepsilon,$

so using the convexity of $f$, we yield that $f(y)-f(x)-\langle
x^{\ast},y-x\rangle\geq-\varepsilon,$ then, taking $\varepsilon\to 0$, we have
that $f(y)-f(x)\geq\langle x^{\ast},y-x\rangle$, which from the arbitrariness
of $y\in X$ implies the result.

The following _sum rule_ is applied in the paper, and we provide its proof for
completeness.

######  Lemma 2.11.

Let $x\in X$, and $g:X\to\overline{\mathbb{R}}$ be differentiable at $x$.
Then, for any function $f:X\to\overline{\mathbb{R}}$ we have

(8) $\displaystyle\hat{\partial}(f+g)(x)=\hat{\partial}f(x)+\nabla g(x).$

######  Proof 2.12.

Let us suppose that $x^{\ast}\in\hat{\partial}(f+g)(x)$ and $S\in\beta$.
Hence,

$\displaystyle 0\leq$ $\displaystyle\liminf\limits_{t\rightarrow
0^{+}}\sup\limits_{h\in S}\left(\frac{f(x+th)+g(x+th)-f(x)-g(x)}{t}-\langle
x^{\ast},h\rangle\right)$ $\displaystyle\leq$
$\displaystyle\liminf\limits_{t\rightarrow 0^{+}}\sup\limits_{h\in
S}\left(\frac{f(x+th)-f(x)}{t}-\langle x^{\ast}-\nabla g(x),h\rangle\right)$
$\displaystyle+\lim\limits_{t\rightarrow 0^{+}}\sup\limits_{h\in
S}\left(\frac{g(x+th)-g(x)}{t}-\langle\nabla g(x),h\rangle\right)$
$\displaystyle=$ $\displaystyle\liminf\limits_{t\rightarrow
0^{+}}\sup\limits_{h\in S}\left(\frac{f(x+th)-f(x)}{t}-\langle x^{\ast}-\nabla
g(x),h\rangle\right),$

which shows that $x^{\ast}-\nabla g(x)\in\hat{\partial}f(x)$. To prove the
converse inclusion it is enough to notice that
$\hat{\partial}f(x)=\hat{\partial}(f+g-g)(x)\subseteq\hat{\partial}(f+g)(x)-\nabla
g(x),$ where the final inclusion follows from the previous part, and that ends
the proof.

Next, we employ the regular subdifferential to provide machinery to
differentiate nonsmooth vector-valued mappings.

######  Definition 2.13.

Given a mapping $F:X\rightarrow Y\cup\\{+\infty_{Y}\\}$ and
$x\in\operatorname{\operatorname{dom}}F$ we define the _regular coderivative
of_ $F$ _at_ $x$ by the set-valued map
$\hat{D}^{\ast}F(x):Y^{\ast}\rightrightarrows X^{\ast}$ defined as:

(9) $\hat{D}^{\ast}F(x)(y^{\ast}):=\hat{\partial}\left(\langle
y^{\ast},F\rangle\right)(x),$

where $\langle y^{\ast},F\rangle$ is the function defined in (4).

This operator is positively homogeneous. In particular, definition (9)
coincides with the general construction for the regular coderivative of set-
valued mappings on Banach spaces when $F$ is calm at $x$, that is,

$\left\|F(u)-F(x)\right\|\leq\ell\left\|u-x\right\|,$

for some $\ell>0$ and $u$ close to $x$ (see, e.g., [20, Proposition 1.32]).

The following lemma yields a sum rule for functions in $\Gamma_{h}(X)$.

######  Lemma 2.14.

Let $f_{1},f_{2}\in\Gamma_{h}(X)$ for some function $h\in\Gamma_{0}(X)$.
Suppose that there exists a point in $\operatorname{dom}(f_{2}+h)$ where
$f_{1}+h$ is continuous. Then,

$\displaystyle\hat{\partial}(f_{1}+f_{2})(\bar{x})=\hat{\partial}f_{1}(\bar{x})+\hat{\partial}f_{2}(\bar{x}),$

provided that $h$ is differentiable at $\bar{x}$.

######  Proof 2.15.

Let us compute the subdifferential $\partial(f_{1}+f_{2}+2h)(\bar{x})$. On the
one hand, by Lemma 2.11, we have that
$\partial(f_{1}+f_{2}+2h)(\bar{x})=\hat{\partial}(f_{1}+f_{2})(\bar{x})+2\nabla
h(\bar{x})$. On the other hand, since the convex function $f_{1}+h$ is
continuous at some point in the domain of $f_{2}+h$, we get (see, e.g., [40,
Theorem 2.8.7]) that

$\displaystyle\partial(f_{1}+f_{2}+2h)(\bar{x})$
$\displaystyle=\partial(f_{1}+h)(\bar{x})+\partial(f_{2}+h)(\bar{x})$
$\displaystyle=\hat{\partial}f_{1}(\bar{x})+\nabla
h(\bar{x})+\hat{\partial}f_{2}(\bar{x})+\nabla h(\bar{x}),$

where in the last equality we used Lemma 2.11 again, and that concludes the
proof.

Next, we present some calculus rules for the subdifferential of an extended
real DC function.

######  Proposition 2.16.

_[24, Theorem 1]_ Let $g,h\in\Gamma_{0}(X)$ be such that both are finite at
$x$. Then, for every $\varepsilon\geq 0$

$\partial_{\varepsilon}\left(g-h\right)(x)=\bigcap\limits_{\eta\geq
0}\left(\partial_{\eta+\varepsilon}g(x)\ominus\partial_{\eta}h(x)\right).$

The following result characterizes the $\varepsilon$-subdifferential of the
supremum function of an arbitrary family of functions.

######  Proposition 2.17.

_[31, Proposition 3.1]_ Let $\\{f_{t}:t\in T\\}\subseteq\Gamma_{0}(X)$ and
define $f=\sup_{T}f_{t}$. Then, for every $\varepsilon\geq 0$,

(10)
${\small\partial_{\varepsilon}f(x)=}\bigcap\limits_{\gamma>\varepsilon}\operatorname{cl}^{\ast}\left\\{\sum\limits_{t\in\operatorname{supp}\alpha}\alpha_{t}\partial_{\eta_{t}}f_{t}(x):\begin{array}[]{c}\alpha\in\Delta(T),\;\eta_{t}\geq
0,\;\sum\limits_{t\in\operatorname{supp}\alpha}\alpha_{t}\eta_{t}\in[0,\gamma),\\\
\vspace{-0.2cm}\hfil\\\
\sum\limits_{t\in\operatorname{supp}\alpha}\alpha_{t}f_{t}(x)\geq
f(x)+\sum\limits_{t\in\operatorname{supp}\alpha}\alpha_{t}\eta_{t}-\gamma\end{array}\right\\}{\small,}$

where $\operatorname{cl}^{\ast}$ represents the closure with respect to the
$w^{\ast}$-topology.

The following calculus rules play a key role in our analysis. Given a mapping
$F:X\rightarrow Y\cup\\{\infty_{Y}\\}$, a set $C\subset Y^{\ast}$ and
$\varepsilon\geq 0$ we denote the _$\varepsilon$ -active index set_ at
$x\in\operatorname{\operatorname{dom}}F$ by

$\displaystyle C_{\varepsilon}(x):=\left\\{\lambda^{\ast}\in
C:\sup\limits_{\nu^{\ast}\in
C}\langle\nu^{\ast},F\rangle(x)\leq\langle\lambda^{\ast},F\rangle(x)+\varepsilon\right\\}.$

For $\varepsilon=0$, we simply write $C(x):=C_{0}(x)$.

######  Theorem 2.18.

Let $F:X\to Y\cup\\{\infty_{Y}\\}$ and consider $C$ a convex and compact
subset of $Y^{\ast}$ with respect to the $w^{\ast}$-topology. Let
$g\in\Gamma_{0}(X)$ such that for all $\lambda^{\ast}\in C$ the function
$\langle\lambda^{\ast},F\rangle+g\in\Gamma_{0}(X)$. Then, for every
$\varepsilon\geq 0$ and all $x\in X$, we have

(13) $\displaystyle\partial_{\varepsilon}\left(\sup\limits_{\lambda^{\ast}\in
C}\langle\lambda^{\ast},F\rangle+g\right)(x)=\bigcup\left[\partial_{\eta}\left(\langle\lambda^{\ast},F\rangle+g\right)(x):\begin{array}[]{c}\eta\in[0,\varepsilon]\text{
and }\\\ \lambda^{\ast}\in C_{\varepsilon-\eta}(x)\end{array}\right].$

######  Proof 2.19.

First let us show the inclusion $\supseteq$ in (13). Consider $x^{\ast}$ in
the right-hand side of (13), then there exists $\eta\in[0,\varepsilon]$ and
$\lambda^{\ast}\in C_{\varepsilon-\eta}(x)$ such that
$x^{\ast}\in\partial_{\eta}\left(\langle\lambda^{\ast},F\rangle+g\right)(x)$.
Hence, for all $y\in X$

$\displaystyle\langle x^{\ast},y-x\rangle$
$\displaystyle\leq\langle\lambda^{\ast},F\rangle(y)+g(y)-\langle\lambda^{\ast},F\rangle(x)-g(x)+\eta$
$\displaystyle\leq\sup\limits_{\nu^{\ast}\in
C}\langle\nu^{\ast},F\rangle(y)+g(y)-\sup\limits_{\nu^{\ast}\in
C}\langle\nu^{\ast},F\rangle(x)-g(x)+\varepsilon-\eta+\eta$
$\displaystyle\leq\sup\limits_{\nu^{\ast}\in
C}\langle\nu^{\ast},F\rangle(y)+g(y)-\sup\limits_{\nu^{\ast}\in
C}\langle\nu^{\ast},F\rangle(x)-g(x)+\varepsilon.$

Second, let us consider $T=C$, and the family of functions
$f_{t}=\langle\lambda^{\ast},F\rangle+g$, for $t=\lambda^{\ast}$. Then, by
Proposition 2.17 we have

(17)
$\displaystyle\partial_{\varepsilon}f(x)=\bigcap\limits_{\gamma>\varepsilon}\operatorname{cl}^{\ast}\left(\bigcup\left[\partial_{\eta}\left(\langle\lambda^{\ast},F\rangle+g\right)(x):\begin{array}[]{c}\lambda^{\ast}\in
C,\;\eta\in[0,\gamma),\text{ and }\\\ \vspace{-0.2cm}\hfil\\\
\langle\lambda^{\ast},F\rangle(x)\geq\sup\limits_{\nu^{\ast}\in
C}\langle\nu^{\ast},F\rangle(x)+\eta-\gamma\end{array}\right]\right),$

where we have simplified (10) by convexity of $C$ and linearity of the
application $\lambda^{\ast}\mapsto\langle\lambda^{\ast},F\rangle(w)$ for all
$w\in X$. In fact, for $(\alpha_{t})_{t\in T}\in\Delta(T)$ we have

$\displaystyle\sum\limits_{t\in\operatorname{\textnormal{supp}}\alpha}\alpha_{t}f_{t}(\cdot)$
$\displaystyle=\left(\langle\lambda^{\ast},F\rangle+g\right)(\cdot)$ and
$\displaystyle\sum\limits_{t\in\operatorname{supp}\alpha}\alpha_{t}\partial_{\eta_{t}}f_{t}(x)\subseteq\partial_{\eta}\left(\langle\lambda^{\ast},F\rangle+g\right)(x),$

where $\lambda^{\ast}=\sum_{t\in\operatorname{supp}\alpha}\alpha_{t}t\in C$
and $\eta:=\sum_{t\in T}\alpha_{t}\eta_{t}$. Now, consider
$x^{\ast}\in\partial_{\varepsilon}f(x)$, so that by (17), there exists a net
$\gamma_{\ell}\to\varepsilon$ and $\eta_{\ell}\in[0,\gamma_{\ell})$,
$\lambda^{\ast}_{\ell}\in C$ with

$\langle\lambda^{\ast}_{\ell},F\rangle(x)\geq\sup\limits_{\nu^{\ast}\in
C}\langle\nu^{\ast},F\rangle(x)+\eta_{\ell}-\gamma_{\ell}$

and
${x_{\ell}}^{\ast}\in\partial_{\eta_{\ell}}\left(\langle\lambda_{\ell}^{\ast},F\rangle+g\right)(x)$
such that $x^{\ast}=\lim x_{\ell}^{\ast}$. Hence, by compactness of $C$, we
can assume that $\lambda_{\ell}^{\ast}\to\lambda^{\ast}\in C$, as well as that
$\eta_{\ell}\to\eta\in[0,\varepsilon]$. Furthermore, taking limits
$\langle\lambda^{\ast},F\rangle(x)\geq\sup_{\nu^{\ast}\in
C}\langle\nu^{\ast},F\rangle(x)+\eta-\varepsilon$, so $\lambda^{\ast}\in
C_{\varepsilon-\eta}(x)$. Finally, let us show that
$x^{\ast}\in\partial_{\eta}\left(\langle\lambda^{\ast},F\rangle+g\right)(x)$.
Indeed, for all $y\in X$

$\displaystyle\langle
x^{\ast}_{\ell},y-x\rangle\leq\langle\lambda^{\ast}_{\ell},F(y)\rangle+g(y)-\langle\lambda^{\ast}_{\ell},F(x)\rangle+g(x)+\eta_{\ell},\text{
for all }\ell\in D,$

so taking limits in $\ell$ we conclude that

$\displaystyle\langle
x^{\ast},y-x\rangle\leq\langle\lambda^{\ast},F(y)\rangle+g(y)-\langle\lambda^{\ast},F(x)\rangle+g(x)+\eta,$

from which, and from the arbitrariness of $y\in X$, we conclude that
$x^{\ast}$ belongs to
$\partial_{\eta}\left(\langle\lambda^{\ast},F\rangle+g\right)(x)$.

##  3 DC mathematical programming

This section is devoted to establishing necessary and sufficient conditions
for general DC mathematical programming problems. More precisely, we consider
the following optimization problem

(18) $\min$ | $\varphi(x)$
---|---
s.t. | $\Phi(x)\in\mathcal{C},$

where $\varphi:X\rightarrow\mathbb{R}\cup\\{+\infty\\},$ $\Phi:X\rightarrow
Y\cup\\{\infty_{Y}\\}$ is a vector-valued mapping, and $\mathcal{C}\subseteq
Y$ is a closed convex set. This section has two parts devoted to global and
local optimality conditions.

###  3.1 Global optimality conditions

Let us establish our main result in this section.

######  Theorem 3.1.

Let $\mathcal{C}\subseteq Y$ be a closed and convex set such that there exists
$z_{0}\in\mathcal{C}$ with $\mathcal{C}_{z_{0}}^{\circ}$ is weak∗-compact. Let
$\varphi\in\Gamma_{h}(X)$ and
$\Phi\in\Gamma_{h}(X,Y,\mathcal{C}_{z_{0}}^{\circ})$ for some control function
$h\in\Gamma_{0}(X)$, and suppose that one of the following conditions holds:

  1. a)

the function $x\mapsto\varphi(x)+h(x)$ is continuous at some point of
$\operatorname{dom}\Phi$,

  2. b)

the function
$x\mapsto\sup\\{\langle\lambda^{\ast},\Phi(x)-z_{0}\rangle:\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}\\}+h(x)$
is continuous at some point of $\operatorname{dom}\varphi$.

Then, if $\bar{x}$ is an optimal solution of the optimization problem (18), we
have that

(19)
$\displaystyle\partial_{\eta}h(\bar{x})\subseteq\bigcup\Big{[}\alpha_{1}\partial_{\eta_{1}}(\varphi+h)(\bar{x})+\alpha_{2}\partial_{\eta_{2}}\left(\langle\lambda^{\ast},\Phi\rangle+h\right)(\bar{x})\Big{]},\text{
for all }\eta\geq 0,$

where the union is taken over all $\eta_{1},\eta_{2}\geq 0$,
$(\alpha_{1},\alpha_{2})\in\Delta_{2}$ and
$\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}$ such that

(20)
$\displaystyle\alpha_{1}\eta_{1}+\alpha_{2}(\eta_{2}+1-\langle\lambda^{\ast},\Phi(\bar{x})-z_{0}\rangle)=\eta.$

Conversely, assume that $\bar{x}$ is a feasible point of (18) and that (19)
always holds with $\alpha_{1}>0$, then $\bar{x}$ is a solution of (18)
relative to $\operatorname{dom}\partial h$, that is, $\bar{x}$ is an optimum
of $\min\\{\varphi(x):x\in\Phi^{-1}(\mathcal{C})\cap\operatorname{dom}\partial
h\\}$.

######  Remark 3.2 (before the proof).

According to Banach-Alaouglu-Bourbaki theorem, in order to guarantee that
$\mathcal{C}^{\circ}_{z_{0}}$ is weak∗-compact, it is enough to suppose that
$z_{0}\in\operatorname{int}(\mathcal{C})$. More precisely, [9, Theorem I-14]
establishes that $z_{0}$ belongs to the interior of $\mathcal{C}$ with respect
to the Mackey topology if and only if $\mathcal{C}_{z_{0}}^{\circ}$ is
weak∗-compact. Here, it is important to mention that there are several
relations between the weak∗-compactness of $\mathcal{C}^{\circ}_{z_{0}}$ and
the nonemptiness of the interior of $\mathcal{C}$, with respect to the Mackey
topology (see, e.g. [9, 21] for more details), and they can be connected even
with the classical James’s Theorem (see [32]) and other variational and
geometric properties of functions (see [10, 11, 15]).

######  Proof 3.3.

First let us suppose that $\bar{x}$ is a solution of (18) and let $\alpha$ be
the optimal value of the optimization problem (18). In the first part we prove
two claims.

_Claim 1:_ First we prove that

(21)
$\displaystyle\partial_{\eta}h(\bar{x})\subseteq\partial_{\eta}\psi(\bar{x}),\text{
for all }\eta\geq 0,$

where

(22) $\displaystyle\psi(x)$
$\displaystyle:=\max\left\\{\psi_{1}(x),\psi_{2}(x)\right\\},\;\;$

and

$\displaystyle\psi_{1}(x):=\varphi(x)+h(x)-\alpha,\text{ and
}\psi_{2}(x):=f(x)+h(x),$

with

(23) $\displaystyle
f(x):=\sup\left\\{\langle\lambda^{\ast},\Phi(x)-z_{0}\rangle:\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}\right\\}-1.$

Indeed, first let us notice that, by the bipolar theorem,

$\displaystyle\Phi(x)\in\mathcal{C}$ $\displaystyle\Leftrightarrow$
$\displaystyle\Phi(x)-z_{0}\in\mathcal{C}-z_{0}=(\mathcal{C}-z_{0})^{\circ\circ}=(\mathcal{C}_{z_{0}}^{\circ})^{\circ}$
$\displaystyle\Leftrightarrow$
$\displaystyle\langle\lambda^{\ast},\Phi(x)-z_{0}\rangle\leq 1\text{ for all
}\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}\Leftrightarrow f(x)\leq 0.$

Therefore, by Lemma 2.1, the optimization problem (18) has the same optimal
solutions as the problem

$\displaystyle\min_{x\in
X}\left(\max\\{\varphi(x)+h(x)-\alpha,f(x)+h(x)\\}-h(x)\right).$

Now, by the Fermat rule (i.e.,
$0_{X^{\ast}}\in\partial\left(\psi-h\right)(\bar{x}$)), and using Proposition
2.16, we have that $\bar{x}$ is a solution of (18) if and only if (21) holds.

_Claim 2:_ Next we prove that $\partial_{\eta}\psi(\bar{x})$ is precisely the
right-hand side of (19), and consequently (19) holds.

Let us compute $\partial_{\eta}\psi(\bar{x})$ using the formula for the
$\eta$-subdifferential of the max-function given in [40, Corollary 2.8.15]
(the assumptions are actually satisfied thanks to conditions a) or b)), so we
get

$\displaystyle\partial_{\eta}\psi(\bar{x})=\bigcup\left[\alpha_{1}\partial_{\frac{\varepsilon_{1}}{\alpha_{1}}}\psi_{1}(\bar{x})+\alpha_{2}\partial_{\frac{\varepsilon_{2}}{\alpha_{2}}}\psi_{2}(\bar{x}):\begin{array}[]{c}(\varepsilon_{0},\varepsilon_{1},\varepsilon_{2})\in\Delta_{3}^{\eta},\,(\alpha_{1},\alpha_{2})\in\Delta_{2}\\\
\alpha_{1}\psi_{1}(\bar{x})+\alpha_{2}\psi_{2}(\bar{x})\geq\psi(\bar{x})-\varepsilon_{0}\end{array}\right].$

Now, relabelling $\eta_{1}:={\varepsilon_{1}}/{\alpha_{1}}$,
$\eta_{0}:={\varepsilon_{2}}/{\alpha_{2}}$ for $\alpha_{1}\neq
0\neq\alpha_{2}$ and $\eta_{1}=\eta_{0}=0$ otherwise, and using that
$\psi(\bar{x})=\psi_{1}(\bar{x})=h(\bar{x})$ and
$\psi_{2}(\bar{x})=f(\bar{x})+h(\bar{x})\leq h(\bar{x})$ we get

(26)
$\displaystyle\partial_{\eta}\psi(\bar{x})=\bigcup\left[\alpha_{1}\partial_{\eta_{1}}\psi_{1}(\bar{x})+\alpha_{2}\partial_{\eta_{0}}\psi_{2}(\bar{x}):\begin{array}[]{c}\eta_{1},\eta_{0}\geq
0,\,(\alpha_{1},\alpha_{2})\in\Delta_{2}\\\
\alpha_{1}\eta_{1}+\alpha_{2}(\eta_{0}-f(\bar{x}))\leq\eta\end{array}\right].$

Now, we compute $\partial_{\eta_{0}}\psi_{2}(\bar{x})$ using Theorem 2.18, so

(29)
$\displaystyle\partial_{\eta_{0}}\psi_{2}(\bar{x})=\bigcup\left[\partial_{\eta_{2}}\left(\langle\lambda^{\ast},\Phi\rangle+h\right)(\bar{x}):\begin{array}[]{c}\eta_{2}\in[0,\eta_{0}]\text{
and }\\\
\lambda^{\ast}\in(\mathcal{C}_{z_{0}}^{\circ})_{\eta_{0}-\eta_{2}}(\bar{x})\end{array}\right].$

Hence, combining (26) and (29) we conclude that $\partial_{\eta}\psi(\bar{x})$
is given by the right-hand expression in (19). Finally, by (21) and the last
computation we get that (19) holds by increasing the values of $\eta_{1}$ and
$\eta_{2}$ if it is needed.

Now, to prove the converse, consider $y\in\operatorname{dom}\partial h$ a
feasible point of the optimization problem (18). Then, consider
$x^{\ast}\in\partial h(y)$, we have that

(30) $\displaystyle h(y)-h(\bar{x})+\eta$ $\displaystyle=\langle
x^{\ast},y-\bar{x}\rangle,\text{ and }x^{\ast}\in\partial_{\eta}h(\bar{x}),$

where $\eta:=h^{\ast}(x^{\ast})+h(\bar{x})-\langle x^{\ast},\bar{x}\rangle\geq
0.$ Hence, by (19) we see that there are $\eta_{1},\eta_{2}\geq 0$,
$(\alpha_{1},\alpha_{2})\in\Delta_{2}$ and
$\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}$ satisfying (20) with
$\alpha_{1}\neq 0$, and

$\displaystyle
x^{\ast}\in\alpha_{1}\partial_{\eta_{1}}(\varphi+h)(\bar{x})+\alpha_{2}\partial_{\eta_{2}}\left(\langle\lambda^{\ast},\Phi\rangle+h\right)(\bar{x}).$

Particularly,

$\displaystyle\langle x^{\ast},y-\bar{x}\rangle\leq$
$\displaystyle\alpha_{1}\left(\varphi(y)+h(y)-\varphi(\bar{x})-h(\bar{x})+\eta_{1}\right)$
$\displaystyle+\alpha_{2}\left(\langle\lambda^{\ast},\Phi\rangle(y)+h(y)-\langle\lambda^{\ast},\Phi\rangle(\bar{x})-h(\bar{x})+\eta_{2}\right)$
$\displaystyle\leq$
$\displaystyle\alpha_{1}\left(\varphi(y)-\varphi(\bar{x})\right)+\alpha_{1}\eta_{1}+\alpha_{2}\left(\eta_{2}+1-\langle\lambda^{\ast},\Phi(\bar{x})-z_{0}\rangle\right)+h(y)-h(\bar{x})$
(31) $\displaystyle\leq$
$\displaystyle\alpha_{1}\left(\varphi(y)-\varphi(\bar{x})\right)+h(y)-h(\bar{x})+\eta.$

We conclude, using (30) and (31), that

(32) $\displaystyle 0\leq\alpha_{1}(\varphi(y)-\varphi(\bar{x})),$

which, recalling that $\alpha_{1}>0$, ends the proof.

######  Remark 3.4.

Let us briefly comment on some facts about the last result:

  1. i)

Notice that $\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}$ if and only if
$\lambda^{\ast}\in N^{\eta_{3}}_{\mathcal{C}}(\Phi(\bar{x}))$, where
$\eta_{3}:=1+\langle\lambda^{\ast},z_{0}-\Phi(\bar{x})\rangle\geq 0$. Indeed,
since $\Phi(\bar{x})\in\mathcal{C}$, we have

$\displaystyle\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}$
$\displaystyle\Leftrightarrow\langle\lambda^{\ast},u\rangle\leq\langle\lambda^{\ast},z_{0}\rangle+1,\;\text{
for all }u\in\mathcal{C}$
$\displaystyle\Leftrightarrow\langle\lambda^{\ast},u-\Phi(\bar{x})\rangle\leq\langle\lambda^{\ast},z_{0}-\Phi(\bar{x})\rangle+1\text{
for all }u\in\mathcal{C}$ $\displaystyle\Leftrightarrow\lambda^{\ast}\in
N^{\eta_{3}}_{\mathcal{C}}(\Phi(\bar{x}))\text{ with
}\eta_{3}:=1+\langle\lambda^{\ast},z_{0}-\Phi(\bar{x})\rangle\geq 0.$

Therefore, the existence of multipliers can be equivalently described in terms
of the $\varepsilon$-normal set defined in (1). Nevertheless, we prefer to use
the set $\mathcal{C}_{z_{0}}^{\circ}$ in order to take advantage of the
compactness of this set, which will be exploited later.

  2. ii)

The converse in Theorem 3.1 can be proved assuming that (19) always holds with
multiplier $\alpha_{1}\neq 0$ for all $\eta\in[0,\bar{\eta}]$, where

$\bar{\eta}:=\sup\left\\{h^{\ast}(y^{\ast})+h(\bar{x})-\langle
y^{\ast},\bar{x}\rangle:y^{\ast}\in\partial
h(y),\;y\in\Phi^{-1}(\mathcal{C})\right\\}.$

  3. iii)

It is worth mentioning that, due to the assumption of the existence of a
continuity point of $\varphi+h$, we can prove that $\varphi,h$ are bounded
above on a neighbourhood of a point of their domain. Indeed, let us suppose
that $\varphi+h$ is bounded above on a neighbourhood $U$ of $x_{0}$, that is,
$\varphi(x)+h(x)\leq M$ for all $x\in U$ for some $M\in\mathbb{R}$. Since
$\varphi$ and $h$ are lsc at $x_{0}$, we can assume (shrinking enough the
neighbourhood $U$) that $\inf_{x\in U}\varphi(x)>-m$ and $\inf_{x\in
U}h(x)>-m$, for some constant $m\in\mathbb{R}$, which implies that
$\varphi(x)\leq M+m$ and $h(x)\leq M+m$ for all $x\in U$. Hence, due to the
convexity of the involved functions, we see that $\varphi$ and $h$ are
continuous on $\operatorname{int}\operatorname{dom}\varphi$ and
$\operatorname{int}\operatorname{dom}h$, respectively (see, e.g., [40, Theorem
2.2.9]). Particularly, the latter implies that $\operatorname{dom}\partial
h\supseteq\operatorname{int}\operatorname{dom}h$ is dense on
$\operatorname{dom}\varphi$ (recall
$\operatorname{\operatorname{dom}}\varphi\subset\operatorname{\operatorname{dom}}h$).

The following result shows that the fulfilment of (19) with
$\alpha_{1}\geq\varepsilon_{0}$, for some $\varepsilon_{0}>0$, can be used to
establish that $\bar{x}$ is a solution to problem (18).

######  Theorem 3.5.

In the setting of Theorem 3.1, suppose that $\bar{x}$ is a feasible point of
(18) and that (19) always holds with $\alpha_{1}\geq\varepsilon_{0}$, for some
$\varepsilon_{0}>0$. Then $\bar{x}$ is an optimal solution of (18).

######  Proof 3.6.

The proof follows similar arguments to those given in Theorem 3.1. The only
difference is that, given $y\in\operatorname{dom}h$ and
$\varepsilon^{\prime}\in(0,\varepsilon_{0})$, we take
$x^{\ast}\in\partial_{\varepsilon^{\prime}}h(y)$ (recall that
$\partial_{\varepsilon^{\prime}}h(y)$ is nonempty due to the lsc of the
function $h$). Again we consider $\eta:=h^{\ast}(x^{\ast})+h(\bar{x})-\langle
x^{\ast},\bar{x}\rangle\geq 0$, entailing
$x^{\ast}\in\partial_{\eta}h(\bar{x})$. Therefore, (19) gives rise to the
existence of $\eta_{1},\eta_{2}\geq 0$, $(\alpha_{1},\alpha_{2})\in\Delta_{2}$
and $\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}$ satisfying (20) with
$\alpha_{1}\geq\varepsilon_{0}$, and

$x^{\ast}\in\alpha_{1}\partial_{\eta_{1}}(\varphi+h)(\bar{x})+\alpha_{2}\partial_{\eta_{2}}\left(\langle\lambda^{\ast},\Phi\rangle+h\right)(\bar{x}),$

which itself yields

$\langle
x^{\ast},y-\bar{x}\rangle\leq\alpha_{1}\left(\varphi(y)-\varphi(\bar{x})\right)+h(y)-h(\bar{x})+\eta.$

Then, using that $-h^{\ast}(x^{\ast})+\langle x^{\ast},y\rangle=\langle
x^{\ast},y-\bar{x}\rangle+h(\bar{x})-\eta$, so

$\langle x^{\ast},y\rangle-h^{\ast}(x^{\ast})-h(y)\leq\langle
x^{\ast},y-\bar{x}\rangle+h(\bar{x})-\eta-h(y)\leq\alpha_{1}\left(\varphi(y)-\varphi(\bar{x})\right).$

But now $x^{\ast}\in\partial_{\varepsilon^{\prime}}h(y)$ leads us to to the
inequality $-\varepsilon^{\prime}\leq\alpha_{1}(\varphi(y)-\varphi(\bar{x}))$
instead of (32), hence
$-\frac{\varepsilon^{\prime}}{\varepsilon_{0}}\leq\varphi(y)-\varphi(\bar{x})$
and taking $\varepsilon^{\prime}\rightarrow 0^{+}$, we get the aimed
conclusion.

######  Corollary 3.7.

Let $\mathcal{Q}\subseteq X$ and $\mathcal{C}\subseteq Y$ be closed and convex
sets and suppose that there exists $z_{0}\in\mathcal{C}$ such that
$\mathcal{C}_{z_{0}}^{\circ}$ is weak∗-compact. Let $\varphi\in\Gamma_{h}(X)$
and $\Phi\in\Gamma_{h}(X,Y,\mathcal{C}_{z_{0}}^{\circ})$ for some function
$h\in\Gamma_{0}(X)$. Consider the following optimization problem

(33) $\displaystyle\min\varphi(x)$ $\displaystyle\textnormal{s.t.
}\Phi(x)\in\mathcal{C},\;$ $\displaystyle\hskip 17.07182ptx\in\mathcal{Q}.$

Additionally, assume that there exists a point in $\mathcal{Q}$ such that
$\varphi$, $\Phi$ and $h$ are continuous at this point. Then, if $\bar{x}$ is
an optimal solution of the optimization problem (33) we have that

(34)
$\displaystyle\partial_{\eta}h(\bar{x})\subseteq\bigcup\Big{[}\alpha_{1}\partial_{\eta_{1}}(\varphi+h)(\bar{x})+\alpha_{2}\partial_{\eta_{2}}\left(\langle\lambda^{\ast},\Phi\rangle+h\right)(\bar{x})+N_{\mathcal{Q}}^{\eta_{3}}(\bar{x})\Big{]}\text{
for all }\eta\geq 0,$

where the union is taken over all $\eta_{1},\eta_{2}.\eta_{3}\geq 0$,
$(\alpha_{1},\alpha_{2})\in\Delta_{2}$ and
$\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}$ such that

$\displaystyle\alpha_{1}\eta_{1}+\alpha_{2}(\eta_{2}+1-\langle\lambda^{\ast},\Phi(\bar{x})-z_{0}\rangle)+\eta_{3}=\eta.$

Conversely, assume that $\bar{x}$ is a feasible point of (33) and that (34)
always holds with $\alpha_{1}>0$, then $\bar{x}$ is a solution of (33)
relative to $\operatorname{dom}\partial h$.

######  Proof 3.8.

Let us observe that the optimization problem (33) is equivalent to

(35) $\displaystyle\min\varphi(x)\textnormal{ s.t.
}\Phi_{\mathcal{Q}}(x)\in\mathcal{C},$

where

(38)
$\displaystyle\Phi_{\mathcal{Q}}(x):=\left\\{\begin{array}[]{cc}\Phi(x)&\text{
if }x\in\mathcal{Q}\\\ \infty_{Y}&\text{ if
}x\notin\mathcal{Q}.\end{array}\right.$

Furthermore, it is easy to prove that
$\langle\lambda^{\ast},\Phi_{\mathcal{Q}}\rangle=\langle\lambda^{\ast},\Phi\rangle+\delta_{\mathcal{Q}}$
for every $\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}$, and consequently
$\Phi_{\mathcal{Q}}\in\Gamma_{h}(X,Y,\mathcal{C}_{z_{0}}^{\circ})$. Then we
apply Theorem 3.1 to the optimization problem (35) (notice that $\varphi+h$ is
continuous at some point of
$\operatorname{\operatorname{dom}}\Phi_{\mathcal{Q}}$) and we use the sum rule
for the $\varepsilon$-subdifferential (see, e.g., [40, Theorem 2.8.3]) to
compute the $\varepsilon$-subdifferential of
$\langle\lambda^{\ast},\Phi\rangle+h+\delta_{\mathcal{Q}}$ in terms of the
corresponding subdifferentials of $\langle\lambda^{\ast},\Phi\rangle+h$ and
$\delta_{\mathcal{Q}}$ (here recall that $\langle\lambda^{\ast},\Phi\rangle$
and $h$ are continuous at some point of $\mathcal{Q}$).

###  3.2 Local optimality conditions

In this section we present necessary and sufficient conditions for local
optimality of problem (18). The first result corresponds to a necessary
optimality condition.

######  Theorem 3.9.

In the setting of Theorem 3.1, let $\bar{x}$ be a local optimal solution of
the optimization problem (18) and suppose that $h$ is differentiable at
$\bar{x}$. Then, we have that there are multipliers
$(\alpha_{1},\alpha_{2})\in\Delta_{2}$ and
$\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}$ such that

(39) $\displaystyle
0_{X^{\ast}}\in\alpha_{1}\hat{\partial}\varphi(\bar{x})+\alpha_{2}\hat{D}^{\ast}\Phi(\bar{x})(\lambda^{\ast})\text{
with }\alpha_{2}(1+\langle\lambda^{\ast},z_{0}-\Phi(\bar{x})\rangle)=0.$

In addition, if the following qualification condition holds

(42) $\displaystyle
0_{X^{\ast}}\notin\bigcup\left[\hat{D}^{\ast}\Phi(\bar{x})(\lambda^{\ast}):\begin{array}[]{c}\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}\text{
such that}\\\
\langle\lambda^{\ast},\Phi(\bar{x})\rangle=1+\langle\lambda^{\ast},z_{0}\rangle\end{array}\right],$

then we have

(45) $\displaystyle
0_{X^{\ast}}\in\hat{\partial}\varphi(\bar{x})+\operatorname{cone}\left(\bigcup\left[\hat{D}^{\ast}\Phi(\bar{x})(\lambda^{\ast}):\begin{array}[]{c}\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}\text{
such that}\\\
\langle\lambda^{\ast},\Phi(\bar{x})\rangle=1+\langle\lambda^{\ast},z_{0}\rangle\end{array}\right]\right).$

######  Proof 3.10.

Consider a closed convex neighbourhood $U$ of $\bar{x}$ such that $\bar{x}$ is
a global optimum for the next optimization problem

(46) $\min\varphi(x)\textnormal{ s.t. }\Phi(x)\in\mathcal{C},\;x\in U.$

Following the proof of Theorem 3.1, we can prove that $\bar{x}$ is a solution
to the unconstrained minimization problem

$\displaystyle\min\left(\max\\{\varphi(x)+h(x)-\alpha,f(x)+h(x)\\}+\delta_{U}(x)-h(x)\right),$

where $\alpha$ is the optimal value of (46), and $f$ is defined in (23). Now,
applying the Fermat rule, and using Proposition 2.16 (with
$\varepsilon=\eta=0$), we have that $\bar{x}$ satisfies the following
subdifferential inclusion

$\displaystyle\nabla
h(\bar{x})\in\partial\left(\psi+\delta_{U}\right)(\bar{x}),$

where $\psi$ is the function introduced in (22). Since $U$ is a neighbourhood
of $\bar{x}$, we have that
$\partial\left(\psi+\delta_{U}\right)(\bar{x})=\partial\psi(\bar{x})$.
Moreover, using _Claim 2_ of Theorem 3.1, we conclude the existence
$(\alpha_{1},\alpha_{2})\in\Delta_{2}$ and
$\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}$ with
$\alpha_{2}(1+\langle\lambda^{\ast},z_{0}-\Phi(\bar{x})\rangle)=0$ such that

(47) $\displaystyle\nabla
h(\bar{x})\in\alpha_{1}\partial(\varphi+h)(\bar{x})+\alpha_{2}\partial\left(\langle\lambda^{\ast},\Phi\rangle+h\right)(\bar{x}).$

Moreover, by Lemmas 2.9 and 2.11 (recall that $h$ is differentiable at
$\bar{x}$) we can compute the corresponding subdifferentials as

$\displaystyle\partial(\varphi+h)(\bar{x})$
$\displaystyle=\hat{\partial}(\varphi+h)(\bar{x})=\hat{\partial}\varphi(\bar{x})+\nabla
h(\bar{x}),$
$\displaystyle\partial\left(\langle\lambda^{\ast},\Phi\rangle+h\right)(\bar{x})$
$\displaystyle=\hat{\partial}\left(\langle\lambda^{\ast},\Phi\rangle+h\right)(\bar{x})=\hat{\partial}\langle\lambda^{\ast},\Phi\rangle(\bar{x})+\nabla
h(\bar{x})$ $\displaystyle=\hat{D}^{\ast}\Phi(\bar{x})(\lambda^{\ast})+\nabla
h(\bar{x}).$

Therefore, inclusion (47) reduces to (39). Now, (39) gives us the existence of
$(\alpha_{1},\alpha_{2})\in\Delta_{2}$ and
$\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}$ with
$\alpha_{2}(1+\langle\lambda^{\ast},z_{0}-\Phi(\bar{x})\rangle)=0$ such that
(39) holds. Moreover, by the qualification condition (42), we have that
$\alpha_{1}\neq 0$. Therefore, dividing by $\alpha_{1}$ we get (45).

######  Remark 3.11 (On normality of multiplier $\lambda^{\ast}$).

It is important to emphasize that the conditions
$\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}$ and
$\langle\lambda^{\ast},\Phi(\bar{x})\rangle=1+\langle\lambda^{\ast},z_{0}\rangle$
imply that $\lambda^{\ast}\in N_{\mathcal{C}}(\Phi(\bar{x}))$. Therefore, in
Theorem 3.9 the multiplier $\lambda^{\ast}$ is necessarily a normal vector to
$\mathcal{C}$ at $\Phi(\bar{x})$. Furthermore, the qualification condition
(42) is equivalent to

$\displaystyle
0_{X^{\ast}}\notin\bigcup\left[\hat{D}^{\ast}\Phi(\bar{x})(\lambda^{\ast}):\begin{array}[]{c}\lambda^{\ast}\in
N_{\mathcal{C}}(\Phi(\bar{x}))\text{ such that}\\\
\langle\lambda^{\ast},\Phi(\bar{x})\rangle=1+\langle\lambda^{\ast},z_{0}\rangle\end{array}\right].$

Here, the equality
$\langle\lambda^{\ast},\Phi(\bar{x})\rangle=1+\langle\lambda^{\ast},z_{0}\rangle$
is relevant because, without this condition, $0_{X^{\ast}}$ always belongs to
the set
$\bigcup\left[\hat{D}^{\ast}\Phi(\bar{x})(\lambda^{\ast}):\lambda^{\ast}\in
N_{\mathcal{C}}(\Phi(\bar{x}))\right].$

######  Remark 3.12 (On abstract differentiability).

It is worth mentioning that in Theorem 3.9 above the differentiability and
subdifferentiability can be exchanged for a more general notion using an
abstract concept of subdifferentiability. Indeed, based on the notion of
presubdifferential (see, e.g., [37, 36]), we can adapt the definition there in
the following way: For every $x\in X$ consider a family of functions
$\mathcal{F}_{x}$, which are finite-valued at $x$. Now, consider an operator
${\tilde{\partial}}$ which associates to any lower semicontinuous function
$f:X\to\overline{\mathbb{R}}$ and any $x\in X$, a subset
${\tilde{\partial}}f(x)$ of $X^{\ast}$ with the following properties:

  1. i)

${\tilde{\partial}}f(x)=\emptyset$ for all $x$ where $|f(x)|=+\infty$.

  2. ii)

${\tilde{\partial}}f(x)$ is equal to the convex subdifferential whenever $f$
is proper convex and lower semicontinuous.

  3. iii)

${\tilde{\partial}}\phi(x)$ is single valued for every $x\in X$ and
$\phi\in\mathcal{F}_{x}$. In that case $\phi$ is called
${\tilde{\partial}}$-differentiable at $x$, and we represent by
$\tilde{\nabla}\phi(x)$ the unique point in ${\tilde{\partial}}\phi(x)$.

  4. iv)

For every $x\in\operatorname{dom}f$ and $\phi\in\mathcal{F}_{x}$, we have

${\tilde{\partial}}\left(f+\phi\right)(x)\subseteq{\tilde{\partial}}f(x)+\tilde{\nabla}\phi(x).$

The above notion covers several classes of subdifferentials, for instance:

  1. 1)

Bornological subdifferential (Fréchet, Hadamard, Gateaux, etc) with
$\mathcal{F}_{x}$, the family of differentiable functions at $x$ with respect
to that bornology.

  2. 2)

Viscosity bornological subdifferential with $\mathcal{F}_{x}$, the family of
smooth functions (with respect to that bornology) at $x$ .

  3. 3)

Proximal subdifferential with $\mathcal{F}_{x}$, the family of
$\mathcal{C}^{2}$-functions at $x$.

  4. 4)

Basic subdifferential with $\mathcal{F}_{x}$, the family of
$\mathcal{C}^{1}$-functions at $x$.

Using the above definition, we can define the notation of
${\tilde{\partial}}$-coderivative similar to (9), given by
$\tilde{D}^{\ast}\Phi(x)(y^{\ast}):=\tilde{\partial}\left(\langle
y^{\ast},\Phi\rangle\right)(x).$ Using these tools, it is easy to change the
proof of Theorem 3.9 requiring that the convex function $h$ belongs to
$\mathcal{F}_{\bar{x}}$. In this way, the corresponding inclusion in Theorem
3.9 is given with ${\tilde{\partial}}$ and $\tilde{D}^{\ast}$ replacing the
regular subdifferential and coderivative, respectively.

Therefore, the assumption over the operator ${\tilde{\partial}}$ and the
family $\mathcal{F}_{\bar{x}}$ at the optimal point $\bar{x}$ corresponds to a
trade-off between the differentiability of the data
$h\in\mathcal{F}_{\bar{x}}$ and the robustness of objects ${\tilde{\partial}}$
and $\tilde{D}^{\ast}$, that is, while the notion of differentiability is
weaker, larger is the object for which the optimality condition are presented
(${\tilde{\partial}}$ and $\tilde{D}^{\ast}$). In that case, the reader could
believe that it would be best to directly assume that $h$ satisfies a high
level of smoothness at $\bar{x}$. Nevertheless, for infinite-dimensional
applications, such smoothness does not always hold (see Example 5.4 below).

Similarly to Theorem 3.9, we provide a necessary optimality condition to
problem (33).

######  Corollary 3.13.

In the setting of Corollary 3.7, let $\bar{x}$ be a local optimal solution of
problem (33) and suppose that $h$ is differentiable at $\bar{x}$. Then, there
exist $\eta\geq 0$ and $\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}$ such
that
$\langle\lambda^{\ast},\Phi(\bar{x})\rangle=1+\langle\lambda^{\ast},z_{0}\rangle$
and we have

(48) $\displaystyle
0_{X^{\ast}}\in\hat{\partial}\varphi(\bar{x})+\eta\hat{D}^{\ast}\Phi(\bar{x})(\lambda^{\ast})+N_{\mathcal{Q}}(\bar{x}),$

provided that the following qualification holds

(51) $\displaystyle
0_{X^{\ast}}\notin\bigcup\left[\hat{D}^{\ast}\Phi(\bar{x})(\lambda^{\ast})+N_{\mathcal{Q}}(\bar{x}):\begin{array}[]{c}\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}\text{
such that}\\\
\langle\lambda^{\ast},\Phi(\bar{x})\rangle=1+\langle\lambda^{\ast},z_{0}\rangle\end{array}\right].$

######  Proof 3.14.

Following the proof of Corollary 3.7, we have that the optimization problem
(35) has a local optimal solution at $\bar{x}$. Then, by Theorem 3.9 we get
that

(54) $\displaystyle
0_{X^{\ast}}\in\hat{\partial}\varphi(\bar{x})+\operatorname{cone}\left(\bigcup\left[\hat{D}^{\ast}\Phi_{\mathcal{Q}}(\bar{x})(\lambda^{\ast}):\begin{array}[]{c}\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}\text{
such that}\\\
\langle\lambda^{\ast},\Phi(\bar{x})\rangle=1+\langle\lambda^{\ast},z_{0}\rangle\end{array}\right]\right),$

where $\Phi_{\mathcal{Q}}$ is defined in (38), and provided that, the
following qualification holds:

(57) $\displaystyle
0_{X^{\ast}}\notin\bigcup\left[\hat{D}^{\ast}\Phi_{\mathcal{Q}}(\bar{x})(\lambda^{\ast}):\begin{array}[]{c}\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}\text{
such that}\\\
\langle\lambda^{\ast},\Phi(\bar{x})\rangle=1+\langle\lambda^{\ast},z_{0}\rangle\end{array}\right].$

Now, using Lemma 2.14 we have that (54) and (57) reduce to (48) and (51),
which concludes the proof.

The final result of this section shows that the fulfilment of inclusion (19),
for all small $\eta\geq 0$, is sufficient for a point to be a local optimum of
problem (18).

######  Theorem 3.15.

Let $\bar{x}$ be a feasible point of the optimization problem (18) which
satisfies the subdifferential inclusion (19) for all $\eta$ small enough. In
addition, suppose that there exists $z_{0}\in\mathcal{C}$ such that
$\mathcal{C}_{z_{0}}^{\circ}$ is weak∗-compact, that $h$ is continuous at
$\bar{x}$ and the following qualification condition holds

(60) $\displaystyle\partial
h(\bar{x})\cap\bigcup\left[{\partial}\left(\langle\lambda^{\ast},\Phi\rangle+h\right)(\bar{x}):\begin{array}[]{c}\lambda^{\ast}\in
N_{\mathcal{C}}(\Phi(\bar{x}))\text{ such that}\\\
\langle\lambda^{\ast},\Phi(\bar{x})\rangle=1+\langle\lambda^{\ast},z_{0}\rangle\end{array}\right]=\emptyset.$

Then, $\bar{x}$ is a local solution of (18).

######  Proof 3.16.

First, we claim that $\bar{x}$ satisfies the subdifferential inclusion (19)
with multiplier $\alpha_{1}\neq 0$ for all $\eta\geq 0$ small enough. Indeed,
suppose by contradiction that there are sequences
$\eta_{n},\eta_{n}^{\prime}\to 0^{+}$ and

$x_{n}^{\ast}\in\partial_{\eta_{n}}h(\bar{x})\cap\partial_{\eta^{\prime}_{n}}\left(\langle\lambda_{n}^{\ast},\Phi\rangle+h\right)(\bar{x}),$

where
$\eta^{\prime}_{n}+1-\langle\lambda_{n}^{\ast},\Phi(\bar{x})-z_{0}\rangle\leq\eta_{n}$
and $\lambda_{n}^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}$. Since $h$ is
continuous at $\bar{x}$ we have that $\partial_{\eta_{n}}h(\bar{x})$ is
weak∗-compact (see, e.g., [40, Theorem 2.4.9]). Hence, there exists subnets
(with respect to the weak∗-topology) $x_{n_{\nu}}^{\ast}$ and
$\lambda^{\ast}_{n_{\nu}}$ converging to $x^{\ast}$ and $\lambda^{\ast}$,
respectively. Then, it is easy to see that $x^{\ast}\in\partial
h(\bar{x})\cap\partial\left(\langle\lambda^{\ast},\Phi\rangle+h\right)(\bar{x}),$
which contradicts (60) and proves our claim.

Now, let us denote by $\varepsilon_{0}>0$ a number such that $\bar{x}$
satisfies the subdifferential inclusion (19) with $\alpha_{1}\neq 0$ for all
$\eta\in[0,\varepsilon_{0}]$.

Since, $h$ is locally Lipschitz at $\bar{x}$ there exists a neighbourhood $U$
of $\bar{x}$ such that for all $x,y\in U$ and all $x^{\ast}\in\partial h(x)$

$\displaystyle|\langle x^{\ast},y-x\rangle|+|h(y)-h(x)|\leq\varepsilon_{0}.$

Particularly, for each $y\in U$ (30) holds with $\eta\leq\varepsilon_{0}$.
Now, for $y\in U\cap\Phi^{-1}(\mathcal{C})$, and repeating the arguments given
in the proof of Theorem 3.1, we get that $\varphi(\bar{x})\leq\varphi(y)$,
which ends the proof.

######  Remark 3.17.

Let us notice that when $h$ is differentiable at $\bar{x}$ condition (60)
reduces to

(63) $\displaystyle
0_{X^{\ast}}\notin\bigcup\left[\hat{D}^{\ast}\Phi(\bar{x})(\lambda^{\ast}):\begin{array}[]{c}\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}\text{
such that}\\\
\langle\lambda^{\ast},\Phi(\bar{x})\rangle=1+\langle\lambda^{\ast},z_{0}\rangle\end{array}\right].$

Indeed, if $h$ is differentiable at $\bar{x}$, we can use the sum rule (8) to
get that

${\partial}\left(\langle\lambda^{\ast},\Phi\rangle+h\right)(\bar{x})=\hat{D}^{\ast}\Phi(\bar{x})(\lambda^{\ast})+\nabla
h(\bar{x}).$

Therefore, (60) turns out to be equivalent to (63).

##  4 DC cone-constrained optimization problems

This section addresses to establishing necessary and sufficient conditions for
cone-constrained optimization problems. More precisely, we consider the
following optimization problem

(64) $\displaystyle\min\varphi(x)$ $\displaystyle\textnormal{{s.t.
}}\Phi(x)\in-\mathcal{K},$

where $\varphi:X\rightarrow\mathbb{R}\cup\\{+\infty\\},$ $\Phi:X\rightarrow
Y\cup\\{\infty_{Y}\\}$, and $\mathcal{K}\subset Y$ is a closed convex cone.

The approach in this section is slightly different for the one followed in the
previous section where there was a convex abstract constraint involving a
general closed convex set $\mathcal{C}$. More precisely, we will take
advantage of the particular structure of the cone-constraint
$\Phi(x)\in-\mathcal{K}$ in terms of a more suitable supremum function. In
order to do that we need to introduce the following notion for convex cones.

######  Definition 4.1.

Let $\Theta\subseteq Y^{\ast}$ be a convex closed cone, we say that $\Theta$
is _$w^{\ast}$ -compactly generated_ if there exists a weak${}^{\ast}-$compact
and convex set $\mathcal{B}$ such that

$\Theta=\operatorname{cl}^{\ast}\textnormal{cone}(\mathcal{B}).$

In this case we say that $\Theta$ is _$w^{\ast}$ -compactly generated by_
$\mathcal{B}$.

The next lemma establishes sufficient conditions to ensure that the polar of a
convex cone is $w^{\ast}$-compactly generated.

######  Lemma 4.2.

Let $\mathcal{K}\subseteq Y$ be a convex closed cone. Suppose that one of the
following conditions is satisfied:

  1. _a_)

$Y$ is a normed space.

  2. _b_)

$\mathcal{K}$ has nonempty interior.

Then, $\mathcal{K}^{+}$ is $w^{\ast}$-compactly generated.

######  Proof 4.3.

_a)_ Let us consider the convex compact set
$\mathcal{B}:=\\{x^{\ast}\in\mathcal{K}^{+}:\|x^{\ast}\|\leq 1\\}$. Then,
$\mathcal{K}^{+}$ is weakly∗ compactly generated by $\mathcal{B}$. _b)_
Consider an interior point of $\mathcal{K}$, $y_{0}$, and take a convex
balanced neighbourhood of zero, $V$, such that $y_{0}+V\subseteq\mathcal{K}$.
Therefore $\mathcal{K}^{+}\subseteq\\{x^{\ast}\in Y^{\ast}:\langle
x^{\ast},y_{0}\rangle\geq\sup_{y\in V}\langle x^{\ast},y\rangle\\}$. Then,
consider $\mathcal{B}:=\\{x^{\ast}\in\mathcal{K}^{+}:\sup_{y\in V}\langle
x^{\ast},y\rangle\leq 1\\}$, which is a weak∗-compact (convex) set due to the
Banach-Alaoglu-Bourbaki Theorem. Moreover, for every
$x^{\ast}\in\mathcal{K}^{+}$ we have that $\frac{x^{\ast}}{|\langle
y_{0},x^{\ast}\rangle|+1}\in\mathcal{B}$, so $\mathcal{K}^{+}$ is
$w^{\ast}$-compactly generated by $\mathcal{B}$.

###  4.1 Global optimality conditions

The next theorem gives necessary and sufficient optimality conditions for
problem (64).

######  Theorem 4.4.

Let $\mathcal{K}$ be a closed convex cone such that $\mathcal{K}^{+}$ is
w∗-compactly generated by $\mathcal{B}$, and assume that
$\Phi\in\Gamma_{h}(X,Y,\mathcal{B})$ and $\varphi\in\Gamma_{h}(X)$ for some
function $h\in\Gamma_{0}(X)$. Furthermore, suppose that one of the following
conditions holds:

  1. a)

the function $x\mapsto\varphi(x)+h(x)$ is continuous at some point of
$\operatorname{dom}\Phi$,

  2. b)

the function
$x\mapsto\sup\\{\langle\lambda^{\ast},\Phi(x)\rangle:\lambda^{\ast}\in\mathcal{B}\\}+h(x)$
is continuous at some point of $\operatorname{dom}\varphi$.

Then, if $\bar{x}$ is a minimum of (64), we have that for every $\eta\geq 0$

(65)
$\partial_{\eta}h(\bar{x})\subseteq\bigcup\Big{[}\alpha_{1}\partial_{\eta_{1}}(\varphi+h)(\bar{x})+\alpha_{2}\partial_{\eta_{2}}\left(\langle\lambda^{\ast},\Phi\rangle+h\right)(\bar{x})\Big{]},$

where the union is taken over all $\eta_{1},\eta_{2}\geq 0$,
$(\alpha_{1},\alpha_{2})\in\Delta_{2}$ and $\lambda^{\ast}\in\mathcal{B}$ such
that

$\displaystyle\alpha_{1}\eta_{1}+\alpha_{2}(\eta_{2}-\langle\lambda^{\ast},\Phi(\bar{x})\rangle)=\eta.$

Conversely, assume that $\bar{x}$ is a feasible point of (64) and that (65)
always holds with $\alpha_{1}>0$, then $\bar{x}$ is a solution of (64)
relative to $\operatorname{dom}\partial h$.

######  Proof 4.5.

Suppose that $\bar{x}$ is a minimum of (64). First, let us notice that
$\Phi(x)\in-\mathcal{K}$ if and only if $\sup_{y^{\ast}\in\mathcal{B}}\langle
y^{\ast},\Phi\rangle(x)\leq 0$. Then, by Lemma 2.1 we have that $\bar{x}$ is a
solution of the DC program

$\begin{array}[]{cc}&\min\max\\{\varphi(x)+h(x)-\alpha,\sup\limits_{y^{\ast}\in\mathcal{B}}\langle
y^{\ast},\Phi\rangle(x)+h(x)\\}-h(x).\\\ \end{array}$

where $\alpha$ is the optimal value of (64). Now, using the notation of
Theorem 3.1 we consider

$\psi_{1}(x):=\varphi(x)+h(x)-\alpha,\;\psi_{2}(x):=\sup_{y^{\ast}\in\mathcal{B}}\langle
y^{\ast},\Phi\rangle(x)+h(x)$

and

$\psi(x):=\max\left\\{\psi_{1}(x),\psi_{2}(x)\right\\}.$

Now, mimicking the proof of Theorem 3.1 and taking into account that, instead
of (29), we have

$\displaystyle\partial_{\eta_{0}}\psi_{2}(\bar{x})=\bigcup\left[\partial_{\eta_{2}}\left(\langle\lambda^{\ast},\Phi\rangle+h\right)(\bar{x}):\begin{array}[]{c}\eta_{2}\in[0,\eta_{0}],\lambda^{\ast}\in\mathcal{B}\text{
and }\\\ \vspace{-0.2cm}\hfil\\\ \sup\limits_{y^{\ast}\in\mathcal{B}}\langle
y^{\ast},\Phi\rangle(\bar{x})\leq\langle\lambda^{\ast},\Phi\rangle(\bar{x})+\eta_{0}-\eta_{2}\end{array}\right],$

we have that (65) holds. The converse follows as in the proof of Theorem 3.1,
so we omit the details.

Now, we present a result about optimality conditions of a DC cone-constrained
optimization problem with an extra abstract convex constraint. The proof
follows similar arguments to the proof of Corollary 3.7, but uses Theorem 4.4
instead of Theorem 3.1; so we omit the proof.

######  Corollary 4.6.

Consider the optimization problem

(66) $\displaystyle\min\varphi(x)$ $\displaystyle\textnormal{s.t.
}\Phi(x)\in-\mathcal{K},\;$ $\displaystyle\hskip 17.07182ptx\in\mathcal{Q},$

where $\mathcal{Q}$ is closed and convex, $\mathcal{K}$ is a closed convex
cone such that $\mathcal{K}^{+}$ is weakly∗-compact generated by
$\mathcal{B}$, and $\Phi\in\Gamma_{h}(X,Y,\mathcal{B})$ and
$\varphi\in\Gamma_{h}(X)$ for some function $h\in\Gamma_{0}(X)$. Assume that
there exists a point in $\mathcal{Q}$ such that $\varphi+h$ and $\Phi+h$ are
continuous at this point. Then, if $\bar{x}$ is an optimal solution of problem
(66) we have that

(67)
$\displaystyle\partial_{\eta}h(\bar{x})\subseteq\bigcup\Big{[}\alpha_{1}\partial_{\eta_{1}}(\varphi+h)(\bar{x})+\alpha_{2}\partial_{\eta_{2}}\left(\langle\lambda^{\ast},\Phi\rangle+h\right)(\bar{x})+N_{\mathcal{Q}}^{\eta_{3}}(\bar{x})\Big{]}\text{
for all }\eta\geq 0,$

where the union is taken over all $\eta_{1},\eta_{2},\eta_{3}\geq 0$,
$(\alpha_{1},\alpha_{2})\in\Delta_{2}$ and $\lambda^{\ast}\in\mathcal{B}$ such
that

$\displaystyle\alpha_{1}\eta_{1}+\alpha_{2}(\eta_{2}-\langle\lambda^{\ast},\Phi(\bar{x})\rangle)+\eta_{3}=\eta.$

Conversely, assume that $\bar{x}$ is a feasible point of (66) and that (67)
always holds with $\alpha_{1}>0$, then $\bar{x}$ is a solution of (66)
relative to $\operatorname{dom}\partial h$.

###  4.2 Local optimality conditions

Now, we focus on necessary and sufficient local optimality conditions for DC
cone-constrained optimization problems. The following two results provide
necessary conditions for optimality of problem (64) and for a variant with an
additional abstract convex constraint. The proofs of both results follow
similar arguments to the ones used in Theorem 3.9 and Corollary 3.13,
respectively; accordingly, we omit them.

######  Theorem 4.7.

In the setting of Theorem 4.4, let $\bar{x}$ be a local optimal solution of
problem (64), and suppose that $h$ is differentiable at $\bar{x}$. Then, there
are multipliers $(\alpha_{1},\alpha_{2})\in\Delta_{2}$ and
$\lambda^{\ast}\in\mathcal{B}$ such that

$\displaystyle
0_{X^{\ast}}\in\alpha_{1}\hat{\partial}\varphi(\bar{x})+\alpha_{2}\hat{D}^{\ast}\Phi(\bar{x})(\lambda^{\ast})\text{
and }\alpha_{2}\langle\lambda^{\ast},\Phi(\bar{x})\rangle=0.$

In addition, if the following qualification holds

$\displaystyle
0_{X^{\ast}}\notin\bigcup\left[\hat{D}^{\ast}\Phi(\bar{x})(\lambda^{\ast}):\begin{array}[]{c}\lambda^{\ast}\in\mathcal{B}\text{
such that }\langle\lambda^{\ast},\Phi(\bar{x})\rangle=0\end{array}\right],$

we have

$\displaystyle
0_{X^{\ast}}\in\hat{\partial}\varphi(\bar{x})+\operatorname{cone}\left(\bigcup\left[\hat{D}^{\ast}\Phi(\bar{x})(\lambda^{\ast}):\begin{array}[]{c}\lambda^{\ast}\in\mathcal{B}\text{
such that
}\langle\lambda^{\ast},\Phi(\bar{x})\rangle=0\end{array}\right]\right).$

######  Corollary 4.8.

Under the assumptions of Corollary 4.6, let $\bar{x}$ be a local optimal
solution of problem (66), and suppose that $h$ is differentiable at $\bar{x}$
and $\varphi$ and $\Phi$ are continuous at $\bar{x}$. Then, there exists
$\lambda^{\ast}\in\mathcal{K}^{+}$ such that

$\displaystyle
0_{X^{\ast}}\in\hat{\partial}\varphi(\bar{x})+\hat{D}^{\ast}\Phi(\bar{x})(\lambda^{\ast})+N_{\mathcal{Q}}(\bar{x})\text{
and }\langle\lambda^{\ast},\Phi(\bar{x})\rangle=0,$

provided that the following constraint qualification holds

$\displaystyle
0_{X^{\ast}}\notin\hat{D}^{\ast}\Phi(\bar{x})(\lambda^{\ast})+N_{\mathcal{Q}}(\bar{x}),\text{
for all }\lambda^{\ast}\in\mathcal{B}.$

Similarly to Theorem 3.15, we provide sufficient conditions for local
optimality in terms of (65).

######  Theorem 4.9.

Let $\bar{x}$ be a feasible point problem (64) satisfying the subdifferential
inclusion (65) for all $\eta$ small enough. Additionally, suppose that $h$ is
continuous at $\bar{x}$ and the following qualification holds

(69) $\displaystyle\partial
h(\bar{x})\cap\bigcup\left[{\partial}\left(\langle\lambda^{\ast},\Phi\rangle+h\right)(\bar{x}):\begin{array}[]{c}\lambda^{\ast}\in\mathcal{B}\text{
such that
}\langle\lambda^{\ast},\Phi(\bar{x})\rangle=0\end{array}\right]=\emptyset.$

Then, $\bar{x}$ is a local solution of (18).

######  Remark 4.10.

Notice that when $h$ is differentiable at $\bar{x}$ condition (69) leads us to

$\displaystyle
0_{X^{\ast}}\notin\bigcup\left[\hat{D}^{\ast}\Phi(\bar{x})(\lambda^{\ast}):\begin{array}[]{c}\lambda^{\ast}\in\mathcal{B}\text{
such that }\langle\lambda^{\ast},\Phi(\bar{x})\rangle=0\end{array}\right].$

##  5 Applications to mathematical programs problems

In this section we provide some applications of the theory developed in the
previous sections.

###  5.1 Infinite programming

We consider the optimization problem

(70) $\begin{array}[]{l}\min\varphi(x)\\\ \textnormal{{s.t. }}\phi_{t}(x)\leq
0,\;t\in T,\end{array}$

where $T$ is a locally compact Hausdorff space, the functions
$\phi_{t}:X\to\mathbb{R}$, $t\in T$, are such that, for all $x\in X$, the
function, $t\mapsto\phi_{t}(x)\equiv\phi(t,x)$ is continuous with compact
support, and $\varphi:X\to\overline{\mathbb{R}}$. Problem (70) corresponds to
the class of infinite programming problems (called semi-infinite when $X$ is
finite dimensional); we refer to [23] for more details about the theory.

The space of continuous functions defined on $T$ and with compact support is
denoted by $\mathcal{C}_{c}(T)$. A finite measure
$\mu:{\mathcal{B}}(T)\to[0,+\infty)$, where ${\mathcal{B}}(T)$ is the Borel
$\sigma$-algebra, is called _regular_ if for every $A\in{\mathcal{B}}(T)$

$\displaystyle\mu(A)=\inf\left\\{\mu(V):V\text{ is open and }V\supseteq
A\right\\}=\sup\left\\{\mu(K):K\text{ is compact and }K\subseteq A\right\\}.$

We denote by ${\mathcal{M}}_{+}(T)$ the set of all (finite) regular Borel
measures. Let us recall (e.g., [1, Theorem 14.14]) that the dual of
$\mathcal{C}_{c}(T)$, endowed with the uniform norm, can be identified as the
linear space generated by ${\mathcal{M}}_{+}(T)$.

The following result provides necessary optimality conditions for problem (70)
using a cone representation in the space $\mathcal{C}_{c}(T)$.

######  Theorem 5.1.

Let $X$ be a Banach space and $T$ be a locally compact Hausdorff space.
Suppose that $\varphi,\phi_{t}\in\Gamma_{h}(X)$, $t\in T$, for some function
$h\in\Gamma_{0}(X)$, and assume that the function $x\mapsto\inf_{t\in
T}\phi_{t}(x)$ is locally bounded from below. Let $\bar{x}$ be a local optimum
of problem (70), and assume that $h$ and $\phi_{t}$, $t\in T$, are
differentiable at $\bar{x}$, and that there are $\ell,\varepsilon>0$ such that

(71) $\displaystyle|\phi_{t}(y)-\phi_{t}(\bar{x})|$
$\displaystyle\leq\ell\|y-\bar{x}\|,\quad\text{ for all }(t,y)\in
T\times\mathbb{B}_{\varepsilon}(\bar{x}).$

Then, there exists $\mu\in{\mathcal{M}}_{+}(T)$ such that
$\operatorname{\textnormal{supp}}\mu\subseteq T(\bar{x}):=\\{t\in
T:\phi_{t}(\bar{x})=0\\}$,

$\displaystyle-\int_{T}\nabla\phi(t,\bar{x})\mu(dt)\in\hat{\partial}\varphi(\bar{x})\text{
and }\int_{T}\phi_{t}(\bar{x})\mu(dt)=0,$

provided that

$\displaystyle
0_{X^{\ast}}\notin\bigcup\left\\{\int_{T}\nabla\phi(t,\bar{x})\nu(dt):\begin{array}[]{c}\nu\in{\mathcal{M}}_{+}(T)\text{
such that }\nu(T)=1,\\\ \vspace{-0.3cm}\hfil\\\
\operatorname{\textnormal{supp}}\nu\subseteq T(\bar{x})\text{ and
}\int_{T}\phi_{t}(\bar{x})\nu(dt)=0\end{array}\right\\},$

where the integrals are in the sense of Gelfand (also called
$w^{\ast}$-integrals).

######  Remark 5.2.

(before the proof) It is important to recall that a mapping $x^{\ast}:T\to
X^{\ast}$ is _Gelfand integrable_ if for every $x\in X$, the function
$t\mapsto\langle x^{\ast}(t),x\rangle$ is integrable. In that case, the
integral $\int_{T}x^{\ast}(t)\nu(dt)$ is well-defined as the unique element of
$X^{\ast}$ such that

$\displaystyle X\ni x\mapsto\int_{T}\langle x^{\ast}(t),x\rangle\nu(dt).$

We refer to [17, Chapter II.3, p. 53] for more details.

######  Proof 5.3.

First, let us define $\Phi:X\to\mathcal{C}_{c}(T)$ as the evaluation function
$\Phi(x):T\to\mathbb{R}$ given by $\Phi(x)(t):=\phi_{t}(x)$, which is well-
defined thanks to our assumptions. Given
$\mathcal{K}:=\\{x\in\mathcal{C}_{c}(T):x(t)\geq 0\text{ for all }t\in T\\}$,
by [1, Theorem 14.12] we have $\mathcal{K}^{+}={\mathcal{M}}_{+}(T)$, and it
is easy to see that $\mathcal{K}^{+}$ is $w^{\ast}$-compactly generated by
$\mathcal{B}:=\\{\mu\in{\mathcal{M}}_{+}(T):\mu(T)=1\\}$ (in the sense of
Definition 4.1). Let us first prove two claims.
_Claim 1: The function
$x\mapsto\sup\\{\langle\nu,\Phi(x)\rangle:\nu\in\mathcal{B}\\}+h(x)$ is
continuous, and the mapping $\Phi$ belongs to
$\Gamma_{h}(X,\mathcal{C}_{c}(T),\mathcal{B})$._
To this purpose, fix a measure $\nu\in\mathcal{B}$. By the assumptions, the
function $x\mapsto\phi_{t}(x)+h(x)$ is convex for all $t\in T$, so integration
over $T$ with respect to $\nu$ preserves the convexity on $X$ (see, e.g., [30,
12, 13, 14, 25, 33, 34]); hence, the function $\langle\nu,\Phi\rangle+h$ is
convex. Moreover, consider a sequence $x_{k}\to x$. Since, the function
$x\mapsto\inf_{t\in T}\phi_{t}(x)$ is locally bounded from below and
$h\in\Gamma_{0}(X)$, we can take $\alpha\in\mathbb{R}$ and
$k_{0}\in\mathbb{N}$ such that $\phi_{t}(x_{k})+h(x_{k})\geq\alpha$, for all
$t\in T$ and all $k\geq k_{0}$. Then, Fatou’s lemma and the lower
semicontinuity of the functions $\phi_{t}+h$, $t\in T$, yield

$\displaystyle\langle\nu,\Phi(x)\rangle+h(x)$
$\displaystyle=\int_{T}\left(\phi_{t}(x)+h(x)\right)\nu(dt)\leq\int_{T}\liminf_{k\to\infty}\left(\phi_{t}(x_{k})+h(x_{k})\right)\nu(dt)$
$\displaystyle\leq\liminf_{k\to\infty}\int_{T}\left(\phi_{t}(x_{k})+h(x_{k})\right)\nu(dt)=\liminf_{k\to\infty}\left(\langle\nu,\Phi(x_{k})\rangle+h(x_{k})\right),$

showing the lower semicontinuty of $x\mapsto\langle\nu,\Phi\rangle(x)+h(x)$,
and that, consequently the function $\Phi$ belongs to
$\Gamma_{h}(X,\mathcal{C}_{c}(T),\mathcal{B})$. Finally, since the function
$x\mapsto\sup\\{\langle\nu,\Phi(x)\rangle:\nu\in\mathcal{B}\\}+h(x)$ is convex
lsc and finite valued because $\mathcal{B}$ is $w^{\ast}$-compact, it is also
continuous (recall that $X$ is a Banach space).
_Claim 2: For every $h\in X$ the function
$t\mapsto\langle\nabla\phi_{t}(\bar{x}),h\rangle$ is measurable and, for every
$\nu\in\mathcal{B}$, we have that_

$\hat{D}^{\ast}\Phi(\bar{x})(\nu)\subseteq\left\\{\int_{T}\nabla\phi_{t}(\bar{x})\nu(dt)\right\\}.$

Fix $h\in X$. Since the functions $\phi_{t}$, $t\in T$, are differentiable at
$\bar{x}$, we get that
$\langle\nabla\phi_{t}(\bar{x}),h\rangle=\lim_{k\to\infty}k\left({\phi_{t}(\bar{x}+k^{-1}h)-\phi_{t}(\bar{x})}\right)$.
Particularly, the function $t\mapsto\langle\nabla\phi_{t}(\bar{x}),h\rangle$
is measurable as it is the pointwise limit of a sequence of measurable
functions. Moreover, by (71) we get that
$\langle\nabla\phi_{t}(\bar{x}),h\rangle\leq\ell\|h\|$, for all $t\in T$,
which shows the integrability and, consequently, the Gelfand integral is well-
defined (see, e.g., [1, 17]). Finally, let $x^{\ast}\in
D^{\ast}\Phi(\bar{x})(\nu)$, the definition of regular subdifferential, with
$S=\\{h\\}\in\beta$, implies that

(72) $\displaystyle\langle
x^{\ast},h\rangle\leq\lim\limits_{k\to+\infty}\frac{\langle\nu,\Phi(\bar{x}+k^{-1}h)-\Phi(\bar{x})\rangle}{k^{-1}}=\int_{T}\langle\nabla\phi(t,\bar{x}),h\rangle\nu(dt),\quad\forall
h\in X,$

where in the last equality we use Lebesgue’s dominated convergence theorem,
which can be applied thanks to (71).The proof of this claim ends by
considering $h$ and $-h$ in (72).

Finally, observe that, by [1, Lemma 12.16], any measure
$\nu\in{\mathcal{B}}(T)$ such that $\int_{T}\phi_{t}(\bar{x})\nu(dt)=0$
satisfies that $\operatorname{\textnormal{supp}}\nu\subseteq T(\bar{x})$.
Then, applying Theorem 4.7 we get the desired result.

###  5.2 Stochastic programming

Before introducing our optimization problem in this subsection let us give
some additional notations. In the sequel, $X$ is a separable Banach space, $Y$
a general locally convex space, and $(\Omega,\mathcal{A},\mu)$ a complete
$\sigma$-finite measure space. A set-valued mapping $M:\Omega\rightrightarrows
X$ is said to be _measurable_ if for every open set $U\subseteq X$, we have
$\\{\omega\in\Omega:M(\omega)\cap U\neq\emptyset\\}\in\mathcal{A}$. A function
$\varphi:\Omega\times X\to\mathbb{R}\cup\\{+\infty\\}$ is said to be a _normal
integrand_ provided the set-valued mapping
$\omega\mapsto\operatorname{epi}\varphi_{\omega}:=\\{(x,\alpha)\in
X\times\mathbb{R}:\varphi_{\omega}(x):=\varphi(\omega,x)\leq\alpha\\}$ is
measurable with closed values. In addition, $\varphi$ is a _convex normal
integrand_ if $\varphi_{\omega}$ is convex for all $\omega\in\Omega$.

Given a set-valued mapping $S:\Omega\rightrightarrows X^{\ast}$ we define the
_(Gelfand) integral_ of $S$ by

$\displaystyle\int_{\Omega}S(\omega)\mu(d\omega):=\left\\{\int_{\Omega}x^{\ast}(\omega)\mu(d\omega):\begin{array}[]{c}x^{\ast}\text{
is Gelfand integrable and }\\\ x^{\ast}(\omega)\in S(\omega)\text{ a.e.
}\omega\in\Omega\end{array}\right\\}.$

We refer to [17, 18, 9, 35] for more details about the theory of measurable
multifunctions and integration on Banach spaces.

Given a normal integrand $\varphi:\Omega\times
X\to\mathbb{R}\cup\\{+\infty\\}$ we define the _integral functional_ (also
called _expected functional_) associated to $\varphi$ by
$\mathcal{I}_{\varphi}:X\to\mathbb{R}\cup\\{+\infty,-\infty\\}$ defined as

(73)
$\displaystyle\mathcal{I}_{\varphi}(x):=\int\limits_{\Omega}\varphi_{\omega}(x)\mu(d\omega):=\int\limits_{\Omega}\max\\{\varphi_{\omega}(x),0\\}\mu(d\omega)+\int\limits_{\Omega}\min\\{\varphi_{\omega}(x),0\\}\mu(d\omega),$

with the inf-addition convention $+\infty+(-\infty)=+\infty$.

Finally, a normal integrand $\varphi$ is _integrably differentiable_ at
$\bar{x}$ provided that $\mathcal{I}_{\varphi}$ is differentiable at $\bar{x}$
and the following integral formula holds

(74)
$\displaystyle\nabla\mathcal{I}_{\varphi}(\bar{x})=\int\limits_{\Omega}\nabla\varphi_{\omega}(\bar{x})\mu(d\omega).$

The next example shows that the last notion makes sense for integral mappings
since its smoothness cannot be taken for granted even when all data are
smooth.

######  Example 5.4.

It is important to mention here that the integral functional
$\mathcal{I}_{\varphi}$, for a normal integrand $\varphi$, could fail to be
Fréchet differentiable even when the data functions $\varphi_{\omega}$,
$\omega\in\Omega$, are Fréchet differentiable. Let us consider the measure
space $(\mathbb{N},\mathcal{P}(\mathbb{N}),\mu)$, where the $\sigma$-finite
measure is given by the counting measure $\mu(A):=|A|$, and the Banach space
$X=\ell_{1}$. Next, consider the convex normal integrand function
$\varphi(n,x):=|\langle x,e_{n}\rangle|^{1+\frac{1}{n}}$, where
$\\{e_{1},e_{2},\ldots,e_{n},\ldots\\}$ is the canonical basis of $\ell_{1}$.
It has been shown in [12, Example 2] that $\mathcal{I}_{\varphi}$ is Gateaux
differentiable at any point and the integral formula (74) holds for the
Gateaux derivative. Nevertheless, as it was also proved in that paper, the
function $\mathcal{I}_{\varphi}$ fails to be Fréchet differentiable at zero.

Now, we extend a classical formula for the subdifferential of convex normal
integrand functions to the case of nonconvex normal integrands. This result is
interesting in itself, and for that reason, we present it as an independent
proposition.

######  Proposition 5.5.

Let $\bar{x}\in X$, and $\varphi:\Omega\times X\to\mathbb{R}\cup\\{+\infty\\}$
be a normal integrand. Suppose that
$\varphi_{\omega}\in\Gamma_{h_{\omega}}(X)$ for some convex normal integrand
$h$ such that
$\operatorname{dom}\mathcal{I}_{\varphi}\subseteq\operatorname{dom}\mathcal{I}_{h}$.
Then, $\mathcal{I}_{\varphi}\in\Gamma_{\mathcal{I}_{h}}(X)$ provided that
$\mathcal{I}_{\varphi}$ is proper. In addition, suppose that $h$ is integrably
differentiable at $\bar{x}$ and the functions $\mathcal{I}_{\varphi}$ and
$\varphi_{\omega}$, $\omega\in\Omega$, are continuous at some common point.
Then,

(75)
$\displaystyle\hat{\partial}\mathcal{I}_{\varphi}(\bar{x})=\int_{\Omega}\hat{\partial}\varphi_{\omega}(\bar{x})d\mu+N_{\operatorname{dom}\mathcal{I}_{\varphi}}(\bar{x}).$

######  Proof 5.6.

Let us consider the convex normal integrand $\psi:=\varphi+h$. By our
assumptions
$\operatorname{dom}\mathcal{I}_{\psi}=\operatorname{dom}\mathcal{I}_{\varphi}$
and $\mathcal{I}_{\psi}=\mathcal{I}_{\varphi}+\mathcal{I}_{h}$. Consequently,
$\mathcal{I}_{\psi}$ is proper, entailing that
$\mathcal{I}_{\varphi}\in\Gamma_{\mathcal{I}_{h}}(X)$. Then, by [12, Theorem
2] we have that
$\partial\mathcal{I}_{\psi}(\bar{x})=\int_{\Omega}{\partial}\psi_{\omega}(\bar{x})d\mu+N_{\operatorname{dom}\mathcal{I}_{\psi}}(\bar{x}).$
Now, by Lemmas 2.9 and 2.11, and the integral formula (74) we have that

$\displaystyle\partial\mathcal{I}_{\psi}(\bar{x})$
$\displaystyle=\hat{\partial}\left(\mathcal{I}_{\varphi}+\mathcal{I}_{h}\right)(\bar{x})=\hat{\partial}\mathcal{I}_{\varphi}(\bar{x})+\nabla\mathcal{I}_{h}(\bar{x}),$
$\displaystyle\int_{\Omega}{\partial}\psi_{\omega}(\bar{x})d\mu$
$\displaystyle=\int_{\Omega}\hat{\partial}\varphi_{\omega}(\bar{x})d\mu+\nabla\mathcal{I}_{h}(\bar{x}),$

which implies that (75) holds.

######  Remark 5.7 (On the use of Gelfand integrals).

The above result does not require that $\varphi$ be locally Lipschitzian at
$\bar{x}$ as in other classical results about differentiation of nonconvex
integral functionals (see, e.g., [13, 30] and the references therein).
Consequently, we cannot expect that the formula (75) holds for the _Bochner
integral_ (see, e.g., [1, Definition 11.42]). Indeed, adapting [12, Example
1], let us consider the measure space
$(\mathbb{N},\mathcal{P}(\mathbb{N}),\mu)$, where $\mu(A):=\sum_{j\in
A}2^{-j}$, the Hilbert space $X=\ell_{2}$, and the normal integrand function
$\varphi(n,x):=2^{n}\left(\langle x,e_{n}\rangle\right)^{2}-\|x\|^{2}$, where
$\\{e_{1},e_{2},\ldots,e_{n},\ldots\\}$ is the canonical basis of $\ell_{2}$.
Clearly, the integrand $\varphi$ satisfies all the assumptions of Proposition
5.5 at any point, therefore (75) holds. Nevertheless, the function
$n\mapsto\nabla\phi_{n}(x)=2^{n+1}\left(\langle x,e_{n}\rangle\right)e_{n}-x$
is not always integrable in the Bochner sense because, otherwise, the function
$n\mapsto 2^{n+1}\left(\langle x,e_{n}\rangle\right)e_{n}$ must be integrable
(see, e.g., [1, Theorem 11.44]). Indeed, we always have

$\int_{\mathbb{N}}\|2^{n+1}\left(\langle
x,e_{n}\rangle\right)e_{n}\|d\mu=2\sum_{n\in\mathbb{N}}|x_{n}|.$

Nonetheless, the right-hand side is equal to $+\infty$ for
$x=(\frac{1}{n})_{n\in\mathbb{N}}\in\ell_{2}\backslash\ell_{1}$.

Given the normal integrand $\varphi:\Omega\times
X\to\mathbb{R}\cup\\{+\infty\\}$, the mapping $\Phi:X\to Y$ and the nonempty
convex closed set $\mathcal{C}\subseteq Y$, we consider the problem

(76) $\begin{array}[]{l}\min\mathcal{I}_{\varphi}(x)\\\ \textnormal{s.t.
}\;\Phi(x)\in\mathcal{C}.\end{array}$

######  Theorem 5.8.

Given problem (76), let us assume that there exists $z_{0}\in\mathcal{C}$ such
that $\mathcal{C}_{z_{0}}^{\circ}$ is weak∗-compact, let
$\varphi_{\omega}\in\Gamma_{h_{\omega}}(X)$, $\omega\in\Omega$, for some
convex normal integrand $h$ such that
$\operatorname{dom}{h_{\omega}}=\operatorname{dom}\mathcal{I}_{h}=X$,
$\omega\in\Omega$, and let
$\Phi\in\Gamma_{g}(X,Y,\mathcal{C}_{z_{0}}^{\circ})$ for some function
$g\in\Gamma_{0}(X)$ such that $\operatorname{dom}\Phi=X$. Let
$\bar{x}\in\operatorname{int}\left(\operatorname{dom}\mathcal{I}_{\varphi}\right)$
be a local optimal solution of problem (76) and assume that $h$ is integrably
differentiable at $\bar{x}$ and $g$ is differentiable at $\bar{x}$. Then, we
have

(79) $\displaystyle
0_{X^{\ast}}\in\int_{\Omega}\hat{\partial}\phi_{\omega}(\bar{x})d\mu+\operatorname{cone}\left(\bigcup\left[\hat{D}^{\ast}\Phi(\bar{x})(\lambda^{\ast}):\begin{array}[]{c}\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}\text{
such that}\\\
\langle\lambda^{\ast},\Phi(\bar{x})\rangle=1+\langle\lambda^{\ast},z_{0}\rangle\end{array}\right]\right),$

provided that the following qualification condition holds

$\displaystyle
0_{X^{\ast}}\notin\bigcup\left[\hat{D}^{\ast}\Phi(\bar{x})(\lambda^{\ast}):\begin{array}[]{c}\lambda^{\ast}\in\mathcal{C}_{z_{0}}^{\circ}\text{
such that}\\\
\langle\lambda^{\ast},\Phi(\bar{x})\rangle=1+\langle\lambda^{\ast},z_{0}\rangle\end{array}\right].$

######  Proof 5.9.

Let us define $h_{2}:=\mathcal{I}_{h}+g$, and notice that
$\operatorname{dom}h_{2}=X$ and $\mathcal{I}_{\varphi}\in\Gamma_{h_{2}}(X)$
(see Proposition 5.5), and
$\Phi\in\Gamma_{h_{2}}(X,Y,\mathcal{C}_{z_{0}}^{\circ})$. Moreover, since $X$
is a Banach space and $\operatorname{dom}\Phi=X$, we have that condition b) of
Theorem 3.1 holds. Moreover, $h_{2}$ is continuous
($\operatorname{dom}h_{2}=X$) and differentiable at $\bar{x}$. Then, by
Theorem 3.9 and Propostion 5.5 we have that (79) is satisfied.

###  5.3 Semidefinite Programming

In this subsection, we consider the optimization problem

(80) $\begin{array}[]{l}\min\varphi(x)\\\ \textnormal{s.t. }\Phi(x)\preceq
0,\\\ \hskip 17.07182ptx\in\mathcal{Q}.\end{array}$

where $\varphi:X\to\overline{\mathbb{R}}$,
$\Phi:X\to\mathbb{S}^{p}\cup\\{\infty_{\mathbb{S}^{p}}\\}$, with
$\mathbb{S}^{p}$ being the set of $p\times p$ symmetric (real) matrices, and
$\mathcal{Q}\subseteq X$ a nonempty convex and closed set of $X$. Here,
$A\preceq 0$ ($A\succeq 0$, respectively) means that the matrix $A$ is
negative semidefinite (positive semidefinite, respectively). We recall that
$\mathbb{S}^{p}$ is a Hilbert space with the inner product $\langle
A,B\rangle:=\operatorname{tr}(AB)$, where $\operatorname{tr}$ represents the
trace operator (see, e.g., [5]). We recall that, for any symmetric matrix $A$,
$\operatorname{tr}(A)=\sum_{i=1}^{p}\lambda_{i}(A)$, where
$\lambda_{1}(A)\geq\ldots\geq\lambda_{p}(A)$ are the eigenvalues of $A$ (see,
e.g., [38, Theorem 2.6.6]).

Classical studies of problem (80) suggested imposing some degree of convexity
to the function $\Phi$, more precisely, the so-called _matrix convexity_ (see,
e.g., [5, Section 5.3.2]). This notion is equivalent to assuming that, for
every $v\in\mathbb{R}^{p}$ with $\|v\|=1$, the function $x\mapsto
v^{\top}\Phi(x)v$ is convex, where $v^{\top}$ is the transpose vector of $v$
(see, e.g. [5, Proposition 5.72 ]). In the spirit of DC optimization, a
natural extension of such concept is given by the following notion. We say
that $\Phi:X\to\mathbb{S}^{p}\cup\\{\infty_{\mathbb{S}^{p}}\\}$ is a _DC
matrix-mapping_ , with control function $h\in\Gamma_{0}(X)$, if
$\operatorname{dom}\Phi\subseteq\operatorname{dom}h$ and, for all
$v\in\mathbb{R}^{p}$ with $\sum_{i=1}^{p}v_{i}^{2}=1$, the mapping $x\mapsto
v^{\top}\Phi(x)v+h(x)$ belongs to $\Gamma_{0}(X)$, where

$\displaystyle
v^{\top}\Phi(x)v:=\left\\{\begin{array}[]{cc}v^{\top}\Phi(x)v,&\text{ if
}x\in\operatorname{dom}\Phi,\\\ +\infty,&\text{ if
}x\notin\operatorname{dom}\Phi.\end{array}\right.$

Moreover, it is well-known that problem (80) can be reformulated similarly to
problem (66) involving the cone $\mathbb{S}^{p}_{+}$ of positive semidefinite
matrices. Due to this observation, and the topological structure of
$\mathbb{S}^{p}$, another natural assumption in our framework is the
$\Phi\in\Gamma_{h}(X,\mathbb{S}^{p},\mathcal{B})$, for some convex control
function $h$ and $\mathcal{B}$ being a $w^{\ast}$-compact generator of
$\mathbb{S}^{p}_{+}$ (see Definition 4.1). The following result formally
establishes that both notions, DC matrix-mapping and $\mathcal{B}$-DC
mappings, coincide.

######  Proposition 5.10.

Consider an lcs space $X$,
$\Phi:X\to\mathbb{S}^{p}\cup\\{\infty_{\mathbb{S}^{p}}\\}$, and
$h\in\Gamma_{0}(X)$ with $\operatorname{dom}h\subseteq\operatorname{dom}\Phi$.
Then, the following are equivalent:

  1. a)

$\Phi\in\Gamma_{h}(X,\mathbb{S}^{p},\mathcal{B})$, where
$\mathcal{B}:=\\{A\in\mathbb{S}^{p}_{+}:\operatorname{tr}(A)=1\\}$

  2. b)

$\Phi$ is a DC matrix-mapping with control $h$.

Moreover, in such case, we have that for all $x\in X$, the following equality
holds

(81) $\displaystyle\max\\{\langle
A,\Phi\rangle(x)+h(x):A\in\mathcal{B}\\}=\max\\{v^{\top}\Phi(x)v+h(x):\sum_{i=1}^{p}v_{i}^{2}=1\\}$

######  Proof 5.11.

Let us suppose that $a)$ holds, and fix ${u}\in\mathbb{R}^{p}$ with $\|u\|=1$,
where $\|\cdot\|$ is the Euclidean norm. Then, consider the symmetric matrix
$A=uu^{\top}=(u_{i}u_{j})_{ij}$. We have that $v^{\top}Av=(\langle
u,v\rangle)^{2}$, for all $v\in\mathbb{R}^{p}$, which shows that $A$ is
positive semidefinite and $\operatorname{tr}(A)=\|u\|^{2}=1$; hence,
$A\in\mathcal{B}$. Finally, $\langle A,\Phi\rangle(x)=\langle
A,\Phi(x)\rangle=u^{\top}\Phi(x)u$, for every $x\in\operatorname{dom}\Phi$,
which shows that the function $x\mapsto u^{\top}\Phi(x)u+h(x)$ is convex
proper and lsc because $\Phi\in\Gamma_{h}(X,\mathbb{S}^{p},\mathcal{B})$.

Now, suppose that $b)$ holds, and consider $A\in\mathbb{S}^{p}_{+}$ with
$\operatorname{tr}(A)=1$. Employing its spectral decomposition, we write
$A=PDP^{\top}=\sum_{i=1}^{p}\lambda_{i}(A)v_{i}v_{i}^{\top}$, where $P$ is an
orthogonal matrix whose columns are $v_{i}\in\mathbb{R}^{p}$, $i=1,\ldots,p$,
and $D$ is the diagonal matrix formed by
$\lambda_{1}(A),\ldots,\lambda_{p}(A)$. Then,

$\displaystyle\langle
A,\Phi(x)\rangle=\sum\limits_{i=1}^{p}\lambda_{i}(A)\langle
v_{i}v_{i}^{\top},\Phi(x)\rangle=\sum\limits_{i=1}^{p}\lambda_{i}(A)v_{i}^{\top}\Phi(x)v_{i},\text{
for all }x\in\operatorname{dom}\Phi.$

Next, using the fact that $\operatorname{tr}(A)=1$ and $\lambda_{i}(A)\geq 0$,
we get that

(82) $\langle
A,\Phi\rangle(x)+h(x)=\sum_{i=1}^{p}\lambda_{i}(A)\left(v_{i}^{\top}\Phi(x)v_{i}+h(x)\right),$

which shows the desired convexity as well as the lower semicontinuity of the
function $x\mapsto\langle A,\Phi\rangle(x)+h(x)$.

Finally, (81) remains to be proved. On the one hand, from the fact that
$A=uu^{\top}$ is positive semidefinite, we get that

$v^{\top}\Phi(x)v+h(x)\leq\max\\{\langle
A,\Phi\rangle(x)+h(x):A\in\mathcal{B}\\},$

which proves the inequality $\geq$ in (81). On the other hand, for a given
matrix $A\in\mathcal{B}$, and taking into account (82), we have that

$\displaystyle\langle A,\Phi(x)\rangle+h(x)$
$\displaystyle\leq\sum_{i=1}^{p}\lambda_{i}(A)\max\\{v^{\top}\Phi(x)v+h(x):\sum_{i=1}^{p}v_{i}^{2}=1\\}$
$\displaystyle=\max\\{v^{\top}\Phi(x)v+h(x):\sum_{i=1}^{p}v_{i}^{2}=1\\}.$

and we are done.

The following proposition establishes some sufficient conditions ensuring that
$\Phi$ is a DC matrix-mapping.

######  Proposition 5.12.

Let $\Phi:X\to\mathbb{S}^{p}$ be a mapping with $\Phi(x)=(\phi_{ij}(x))$ for
$\phi_{ij}\in\Gamma_{h_{ij}}(X,\mathbb{R},[-1,1])$ for some control function
$h_{ij}$, $i,j=1,\ldots,p$. Then, $\Phi$ is a DC matrix-mapping with control
$h:=\sum_{ij}h_{ij}$.

######  Proof 5.13.

Let us notice that for every $v\in\mathbb{R}$ with $\|v\|=1$, we have that

$\displaystyle
v^{\top}\Phi(x)v+h(x)=\sum_{i,k=1}^{p}\left(v_{i}v_{k}\phi_{ij}(x)+h_{ij}(x)\right),\text{
for all }x\in X.$

Therefore, the function $x\mapsto v^{\top}\Phi(x)v+h(x)$ is convex and lower
semicontinuous, which yields that $\Phi$ is a DC matrix-mapping with control
$h$.

Although the notion of DC matrix-mapping has a simpler description in terms of
quadratic forms $v^{\top}\Phi(x)v$, the set $\Gamma_{h}(X,Y,\mathcal{B})$ in
Definition 2.3 provides enhanced properties depending on the choice of
scalarizations $\mathcal{B}$ and, consequently, better properties of
operations with the matrix $\Phi$.

Given a mapping $\Phi:X\to\mathbb{S}^{p}\cup\\{\infty_{\mathbb{S}^{p}}\\}$, we
define its _$k$ -th eigenvalue function_ by
$\lambda_{k}^{\Phi}:X\to\overline{\mathbb{R}}$ given by

$\displaystyle\lambda_{k}^{\Phi}(x)$
$\displaystyle:=\left\\{\begin{array}[]{cl}\lambda_{k}(\Phi(x)),&\text{ if
}x\in\operatorname{dom}\Phi,\\\ +\infty,&\text{ if
}x\notin\operatorname{dom}\Phi.\end{array}\right.$

Moreover, we define the sum of the first $k$ eigenvalue functions by
$\Lambda_{k}^{\Phi}:X\to\overline{\mathbb{R}}$ given by
$\Lambda_{k}^{\Phi}(x):=\sum_{j=1}^{k}\lambda_{j}^{\Phi}(x)$.

The following proposition gives sufficient conditions to ensure that the above
functions are DC.

######  Proposition 5.14.

Let $\Phi:X\to\mathbb{S}^{p}\cup\\{\infty_{\mathbb{S}^{p}}\\}$ and
$h\in\Gamma_{0}(X)$.

  1. a)

If $\Phi$ is a DC matrix-mapping with control $h$, then its largest eigenvalue
function, $\lambda_{1}^{\Phi}$, belongs to $\Gamma_{h}(X)$.

  2. b)

If $\Phi\in\Gamma_{h}(X,\mathbb{S}^{p},\mathcal{P}_{k})$, where
$\mathcal{P}_{k}:=\\{P\in\mathbb{S}^{p}:P\text{ has rank }k\text{ and
}P^{2}=P\\}$, then $\Lambda_{k}^{\Phi}\in\Gamma_{h}(X)$.

  3. c)

If $X$ is a Banach space,
$\Phi\in\Gamma_{h}(X,\mathbb{S}^{p},\mathbb{B}_{\mathbb{S}^{p}})$ with
$\operatorname{dom}\Phi=X$, and $\mathbb{B}_{\mathbb{S}^{p}}$ is the unit ball
on $\mathbb{S}^{p}$, then all the eigenvalue functions $\lambda_{k}^{\Phi}$
are a difference of convex functions.

######  Proof 5.15.

Let us recall that $\lambda_{1}^{\Phi}(x)=\sup\\{v^{\top}\Phi(x)v:\|v\|=1\\}$,
which shows statement _a)_. Second, let us notice that

$\Lambda_{k}^{\Phi}(x)+h(x)=\max\\{\langle
P,\Phi(x)\rangle+h(x):P\in\mathcal{P}_{k}\\}$

(see, e.g., [35, Exercise 2.54]), so $\Lambda_{k}^{\Phi}\in\Gamma_{h}(X)$,
which shows _b)_. Finally, to prove _c)_ , we have that
$\lambda_{k}^{\Phi}(x)=\Lambda_{k}^{\Phi}(x)-\Lambda_{k-1}^{\Phi}(x)$ for all
$k\geq 2$, so it is a difference of convex functions.

Finally, let us go back to problem (80).

######  Theorem 5.16.

Let $\varphi\in\Gamma_{h}(X)$ and $\Phi$ be a DC matrix-mapping with control
$h$ such that $x\mapsto\varphi(x)+h(x)$ is continuous at some point of
$\operatorname{dom}\Phi$. Let $\bar{x}$ be a local optimal solution of the
optimization of problem (80) and suppose that $h$ is differentiable at
$\bar{x}$. Then, there exists $A\in\mathbb{S}^{p}_{+}$ with
$\operatorname{tr}(A)=1$ such that

(83) $\displaystyle
0_{X^{\ast}}\in\hat{\partial}\varphi(\bar{x})+\hat{D}^{\ast}\Phi(\bar{x})(A)+N_{\mathcal{Q}}(\bar{x}),\text{
and }v^{\top}\Phi(\bar{x})v=0\text{ for each eigenvector }v\text{ of }A,$

provided that the following qualification holds

(84) $\displaystyle
0_{X^{\ast}}\notin\hat{D}\Phi(\bar{x})(A)+N_{\mathcal{Q}}(\bar{x}),\text{ for
all }A\in\mathbb{S}^{p}_{+}\text{ with }\operatorname{tr}(A)=1.$

######  Proof 5.17.

First, let us notice that by Proposition 5.10 the mapping $\Phi$ belongs to
$\Gamma_{h}(X,\mathbb{S}^{p},\mathcal{B})$, where
$\mathcal{B}:=\\{A\in\mathbb{S}^{p}_{+}:\operatorname{tr}(A)=1\\}$. Then,
applying Corollary 4.8 we get the result.

Consider normed spaces $X,Y$ and recall that a function $F:X\to Y$ is called
$\mathcal{C}^{1,+}$ at $\bar{x}$ if there exists a neighbourhood $U$ of
$\bar{x}$ such that $F$ is Fréchet differentiable on $U$ and its gradient is
Lipschitz continuous on $U$.

######  Corollary 5.18.

Let $\bar{x}$ be a local optimal solution of problem (80). Suppose that $X$ is
a Hilbert space and $\varphi$ and that $\Phi$ are $\mathcal{C}^{1,+}$ at
$\bar{x}$. Then, there exist $v_{i}\in\mathbb{R}^{p}$ with $\|v_{i}\|=1$,
$i=1,\ldots,p$, and $(\lambda_{i})_{i=1}^{p}\in\Delta_{p}$ such that
$v_{i}^{\top}\Phi(\bar{x})v_{i}=0$ and

$\displaystyle
0_{X}\in\hat{\partial}\varphi(\bar{x})+\sum_{i=1}^{p}\lambda_{i}v_{i}^{\top}\nabla\Phi(\bar{x})v_{i}+N_{\mathcal{Q}}(\bar{x}),$

provided that the following qualification holds

(85) $\displaystyle
0_{X}\notin\sum_{i=1}^{p}\lambda_{i}v_{i}^{\top}\nabla\Phi(\bar{x})v_{i}+N_{\mathcal{Q}}(\bar{x}),\text{
for all }(\lambda_{i})_{i=1}^{p}\in\Delta_{p}\text{ and
}v_{i}\in\mathbb{R}^{d}\text{ with }\|v_{i}\|=1,$

where $v_{i}^{\top}\nabla\Phi(\bar{x})v_{i}$ is the gradient of $x\mapsto
v^{\top}_{i}\Phi(x)v_{i}$ at $\bar{x}$.

######  Proof 5.19.

Let us consider a closed and convex neighbourhood $U$ of $\bar{x}$ and
$\rho>0$ such that the functions $\varphi(x)+\rho\|x\|^{2}$ and $\langle
A,\Phi(x)\rangle+\rho\|x\|^{2}$ are convex over $U$ for all
$A\in\mathbb{S}^{p}_{+}$ with $\operatorname{tr}(A)=1$ (see, e.g, [39,
Proposition 1.11]). Hence, for $h(x):=\rho\|x\|^{2}$,
$\varphi_{U}:=\varphi+\delta_{U}\in\Gamma_{h}(X)$. Furthermore, by Proposition
5.10 the mapping

$\displaystyle\Phi_{U}(x):=\left\\{\begin{array}[]{cc}\Phi(x),&\text{ if }x\in
U,\\\ \infty_{\mathbb{S}^{p}},&\text{ if }x\notin U,\end{array}\right.$

is a DC matrix-mapping with control $h$. Then, it is easy to see that
$\bar{x}$ is also a local solution of
$\min\\{\varphi_{U}(x):\Phi_{U}(x)\preceq 0,\;x\in\mathcal{Q}\\}$. Let us
notice that for every matrix $A\in\mathbb{S}^{p}_{+}$, and its spectral
decomposition $A=\sum_{i=1}^{p}\lambda_{i}u_{i}u_{i}^{\top}$, we get

$\displaystyle\hat{D}^{\ast}\Phi(\bar{x})(A)=\sum_{i=1}^{p}\lambda_{i}u_{i}^{\top}\nabla\Phi(\bar{x})u_{i}.$

Hence, condition (85) implies (84). Therefore, Theorem 5.16 implies the
existence of $A\in\mathbb{S}^{p}_{+}$ with $\operatorname{tr}(A)=1$ such that
(83) holds. Now, consider $\lambda_{i}:=\lambda_{i}(A)$, and associated
eigenvalues $v_{i}$, $i=1,\ldots,p$. Then, $(\lambda_{i})\in\Delta_{p}$ and
$v_{i}^{\top}\Phi(x)v_{i}=0$, and that ends the proof.

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