# Regular Subgradients of Marginal Functions with Applications to Calculus and
Bilevel Programming

Le Phuoc Hai Instituto de Alta Investigación (IAI), Universidad de Tarapacá,
Arica, Chile. E-mail<EMAIL_ADDRESS>Felipe Lara Instituto de Alta
Investigación (IAI), Universidad de Tarapacá, Arica, Chile. E-mail:
<EMAIL_ADDRESS>web: felipelara.cl. Research of this author was
partially supported by ANID–Chile under project Fondecyt Regular 1241040.
Boris S. Mordukhovich Department of Mathematics, Wayne State University,
Detroit, Michigan, USA. E-mail<EMAIL_ADDRESS>Research of this author was
partly supported by the US National Science Foundation under grant
DMS-2204519, by the Australian Research Council under Discovery Project
DP190100555, and by the Project 111 of China under grant D21024.

Abstract. The paper addresses the study and applications of a broad class of
extended-real-valued functions, known as optimal value or marginal functions,
which are frequently appeared in variational analysis, parametric
optimization, and a variety of applications. Functions of this type are
intrinsically nonsmooth and require the usage of tools of generalized
differentiation. The main results of this paper provide novel evaluations and
exact calculations of regular/Fréchet subgradients and their singular
counterparts for general classes of marginal functions via their given data.
The obtained results are applied to establishing new calculus rules for such
subgradients and necessary optimality conditions in bilevel programming.
Keywords: Variational analysis, generalized differentiation, marginal
functions, calmness, variational calculus, bilevel programming
Mathematics Subject Classification (2020) 49J52, 49J53, 90C31

##  1 Introduction

Our main interest in this paper is attracted to the class of optimal
value/marginal functions given in the form

$\mu(x):=\inf\big{\\{}\varphi(x,y)\;\big{|}\;y\in G(x)\big{\\}},$ (1.1)

where $\varphi:X\times Y\rightarrow{\mathbb{\bar{R}}}$ is a cost/objective
function defined on the product of normed spaces $X$ and $Y$ and taking values
in the extended real line $\bar{\mathbb{R}}:=(-\infty,\infty]$, and where
$G:X\rightrightarrows Y$ is a constraint set-valued mapping. The
solution/argminimum mapping $M:X\rightrightarrows Y$ associated with (1.1) is
defined by

$M(x):=\big{\\{}y\in G(x)\;\big{|}\;\;\varphi(x,y)=\mu(x)\big{\\}}.$ (1.2)

Over the years, it has been well recognized in variational analysis and
optimization that the class of marginal functions (1.1) plays a highly
important role in many aspects of variational theory and applications; see,
e.g., the books [5, 10, 11, 13, 14, 18, 20, 21] with the discussions and
references therein. It follows from the definition that (1.1) can be viewed as
the optimal value in the parameterized problem of minimizing the cost function
$\varphi(x,y)$ with respect to the decision variable $y$ subject to the
parameter-dependent constraint set $G(x)$. Besides this obvious
interpretation, marginal functions arise in various settings that may not be
related to optimization; see the books mentioned above.

A characteristic feature of (1.1) is the intrinsic nondifferentiability of
this function even in the case of smooth objectives and simple convex
constraint sets. Therefore, sensitivity analysis of marginal functions
requires the usage of suitable machinery of generalized differentiation, which
has been done at some levels for various generalized derivatives under diverse
assumptions; see, e.g., the books [10, 11, 14, 18, 21] and the bibliographies
therein for more details.

In this paper, we focus on the study and applications of regular/Fréchet
subgradients and singular regular subgradients of marginal functions given in
the general form (1.1). These subdifferential constructions for marginal
functions seem to be much less investigated in comparison with the classical
subdifferential of convex analysis, the generalized gradient of Clarke, and
the limiting and singular subdifferentials of Mordukhovich. Some results for
regular subgradients of marginal functions can be found in [6, 16] and are
discussed in what follows. The results of our paper significantly improve the
known ones while providing also completely new conditions for lower and upper
estimates and exact calculation formulas for both regular and singular regular
subdifferentials of marginal functions in general settings of normed spaces.
Moreover, we present applications of the results obtained for marginal
functions to deriving new calculus rules for regular subgradients and their
singular counterparts with further developing the marginal function approach
to optimistic bilevel programming.

The rest of the paper is organized as follows. Section 2 contains some
preliminaries from variational analysis and generalized differentiation
together with definitions of the major constructions and notions that are
largely used to establish our main results in the subsequent sections.

In Section 3, we derive new results on lower and upper estimates for both
regular and singular regular subdifferentials of the general class of marginal
functions (1.1) on normed spaces that are expressed via the given data of
(1.1). To the best of our knowledge, the lower subdifferential estimates for
marginal functions have never been obtained in the literature outside of
equalities. Our results in this direction are established under the
alternative assumptions of either the newly defined notion of nearly-isolated
calmness for constraint mappings, or upper Lipschitzian selections of the
solution mapping (1.2).

Section 4 establishes exact calculation formulas for the regular and singular
regular subdifferentials of marginal functions on arbitrary normed spaces.
Besides the aforementioned assumptions of Section 3, the obtained exact
subdifferential calculations require either the Fréchet differentiability of
the cost function, or certain metric qualification conditions imposed on the
initial data of (1.1). All these assumptions are illustrated by examples. It
has been well realized in variational analysis that developing variational
calculus rules is crucial for applications of any subdifferential
construction. Not much has been done in this direction for regular
subgradients and their singular counterparts.

Section 5 is devoted to applications of the subdifferential formulas for
marginal functions obtained in Section 4 to deriving exact/equality type
chain, sum, product, and quotient rules for regular and singular regular
subgradients of functions on normed spaces. Some of the calculus results of
this type for regular subgradients were obtained in [15] under the local
Lipschitz continuity of the functions involved. Besides establishing novel
calculus rules, we systematically replace the Lipschitz continuity assumptions
in the results of [15] by the corresponding calmness properties, which lead us
to refined calculus.

Section 6 develops the marginal function approach to deriving optimality
conditions in problems of optimistic bilevel programming in Asplund spaces,
which constitutes a broad class of Banach spaces that includes, in particular,
all reflexive ones. Involving subdifferentiation of marginal functions and
calculus rules for the regular and limiting subdifferentials together with the
aforementioned metric qualification conditions, we establish new necessary
optimality conditions for local optimal solutions to optimistic bilevel
programs in generally non-Lipschitzian settings, which significantly improve
those obtained in [12] for problems with fully Lipschitzian data in finite-
dimensional spaces under additional conditions and more restrictive constraint
qualifications.

In Section 7, we present concluding remarks and discuss some directions of our
future research and applications.

##  2 Preliminaries and Basic Definitions

Throughout the paper, we use standard notation of variational analysis and
generalized differentiation; see, e.g., [10]. Given a normed space $X$ endowed
with the norm $\|\cdot\|$, the topologically dual space of $X$ is denoted by
$X^{*}$, where $w^{*}$ stands for the weak∗ topology of $X^{*}$ and
$\langle\cdot,\cdot\rangle$ indicates the canonical pairing between $X$ and
$X^{*}$. The symbols $\mathbb{B}(\bar{x},r)$ and
${\mathbb{\overline{B}}}_{X^{*}}$ signify the open ball centered at $\bar{x}$
with radius $r>0$ and the closed unit ball of $X^{*}$, respectively. We also
use the notations $\mathbb{R}_{+}:=[0,\infty)$, $\mathbb{R}_{-}:=(-\infty,0]$
and $x\xrightarrow{S}\bar{x}$ when $x\to\bar{x}$ with $x\in S$.

For a set-valued mapping $F:X\rightrightarrows X^{*}$, the symbol

$\displaystyle\mathop{\rm
Limsup}\nolimits\limits_{x\rightarrow\bar{x}}\,F(x):=\big{\\{}x^{*}\in
X^{*}\;\big{|}\;\exists\,x_{k}\to\bar{x},\,\,\,x^{*}_{k}\xrightarrow{w^{*}}x^{*}\,\,\,\text{with}\,\,\,x^{*}_{k}\in
F(x_{k}),\;k\in\mathbb{N}\big{\\}}$

stands for the (Painlevé-Kuratowski) sequential outer limit of $F$ as
$x\to\bar{x}$ with respect to the norm topology of $X$ and the weak∗ topology
of the dual space $X^{*}$, where $\mathbb{N}:=\\{1,2,\ldots\\}$ signifies the
collection of natural numbers.

Given a set-valued mapping/multifunction $F:X\rightrightarrows Y$ between
normed spaces, the domain and graph of $F$ are defined by

$\operatorname{dom}F:=\big{\\{}x\in
X\;\big{|}\;F(x)\neq\emptyset\big{\\}},\;\mathop{\rm gph}\nolimits
F:=\big{\\{}(x,y)\in X\times Y\;\big{|}\;x\in\operatorname{dom}F,\;y\in
F(x)\big{\\}},$

respectively. Unless otherwise stated, the norm on the product space $X\times
Y$ is defined by $\|(x,y)\|:=\|x\|+\|y\|$ for any $(x,y)\in X\times Y$.

Given an extended-real-valued function $f:X\to\mathbb{\bar{R}}$ on a normed
space $X$, its domain and epigraph are defined, respectively, by

$\operatorname{dom}f:=\big{\\{}x\in X\;\big{|}\;f(x)\neq\emptyset\\}\;\mbox{
and }\;\mathop{\rm epi}\nolimits f:=\big{\\{}(x,\alpha)\in
X\times\mathbb{R}\;\big{|}\;\alpha\geq f(x)\big{\\}}.$

Following the terminology of [20], the (Fréchet) regular subdifferential
$\hat{\partial}f(\bar{x})$ of $f$ at $\bar{x}\in\operatorname{dom}f$ is the
collection of subgradients $x^{*}\in X^{*}$ such that for every
$\varepsilon>0$ there exists $\gamma>0$ satisfying

$\displaystyle f(x)-f(\bar{x})-\langle
x^{*},x-\bar{x}\rangle\geq-\varepsilon\|x-\bar{x}\|\;\mbox{ whenever
}\;x\in\mathbb{B}(\bar{x},\gamma).$ (2.1)

Along with $\widehat{\partial}f(\bar{x})$, we consider also the upper regular
subdifferential of $f\colon X\to\mathbb{R}\cup\\{-\infty\\}$ at $\bar{x}$ with
$|f(\bar{x})|<\infty$ defined symmetrically as

$\widehat{\partial}^{+}f(\bar{x}):=-\widehat{\partial}(-f)(\bar{x}).$ (2.2)

As well known [10, Proposition 1.87], the sets $\widehat{\partial}f(\bar{x})$
and $\widehat{\partial}^{+}f(\bar{x})$ are nonempty simultaneously if and only
if $f$ is Fréchet differentiable at $\bar{x}$.

The corresponding regular normal cone to a set $S\subset X$ at $\bar{x}\in S$
is given by

$\displaystyle\widehat{N}(\bar{x};S):=\Big{\\{}x^{*}\in
X^{*}\;\Big{|}\;\limsup\limits_{x\xrightarrow{S}\bar{x}}\Big{\langle}x^{*},\dfrac{x-\bar{x}}{\|x-\bar{x}\|}\Big{\rangle}\leq
0\Big{\\}}$ (2.3)

with $\widehat{N}(\bar{x};S):=\emptyset$ if $\bar{x}\notin S$. Note that the
regular subdifferential (2.1) of $f$ at $\bar{x}\in S$ can be geometrically
expressed via the regular normal cone (2.3) by

$\displaystyle\widehat{\partial}f(\bar{x}):=\big{\\{}x^{*}\in
X^{*}\;\big{|}\;(x^{*},-1)\in\widehat{N}\big{(}(\bar{x},f(\bar{x}));\mathop{\rm
epi}\nolimits f\big{)}\big{\\}}.$

Along with the regular subdifferential (2.1), we consider the singular regular
subdifferential of $f$ at $\bar{x}\in\operatorname{dom}f$ defined by

$\displaystyle\widehat{\partial}^{\infty}f(\bar{x}):=\big{\\{}x^{*}\in
X^{*}\;\big{|}\;(x^{*},0)\in\widehat{N}\big{(}(\bar{x},f(\bar{x}));\mathop{\rm
epi}\nolimits f\big{)}\big{\\}},$ (2.4)

which can be equivalently described as follows:
$x^{*}\in\widehat{\partial}^{\infty}f(\bar{x})$ if and only if for any
$\varepsilon>0$ there exists $\gamma>0$ such that

$\displaystyle\langle
x^{*},x-\bar{x}\rangle\leq\varepsilon\big{(}\|x-\bar{x}\|+\lvert\beta-f(x)\rvert\big{)}\;\mbox{
whenever }\;\big{(}x,\beta)\in\mathop{\rm epi}\nolimits
f\cap\mathbb{B}((\bar{x},f(\bar{x})),\gamma\big{)}.$

For a set $S\subset X$ with its indicator function $\delta_{S}\colon
X\to\mathbb{R}$ defined by $\delta_{S}(x):=0$ if $x\in S$ and
$\delta_{S}(x):=\infty$ if $x\notin S$, we have the relationships

$\displaystyle\widehat{N}(\bar{x};S)=\widehat{\partial}\delta_{S}(\bar{x})=\widehat{\partial}^{\infty}\delta_{S}(\bar{x})\;\mbox{
for all }\;\bar{x}\in S.$

Now we define, following the scheme from [9], a counterpart of the regular
constructions (2.1) and (2.3) for set-valued mappings between normed spaces.
The regular coderivative of $F:X\rightrightarrows Y$ at $(\bar{x},\bar{y})\in
X\times Y$ is the multifunction $\widehat{D}^{*}F(\bar{x},\bar{y})\colon
Y^{*}\rightrightarrows X^{*}$ with the values

$\displaystyle\widehat{D}^{*}F(\bar{x},\bar{y})(y^{*}):=\big{\\{}x^{*}\in
X^{*}\;\big{|}\;(x^{*},-y^{*})\in\widehat{N}\big{(}(\bar{x},\bar{y});\mathop{\rm
gph}\nolimits F\big{)}\\}$ (2.5)

if $(\bar{x},\bar{y})\in\mathop{\rm gph}\nolimits F$, and
$\widehat{D}^{*}F(\bar{x},\bar{y})(y^{*}):=\emptyset$ if
$(\bar{x},\bar{y})\notin\mathop{\rm gph}\nolimits F$. We omit $\bar{y}$ in the
coderivative notion if $F(\bar{x})$ is a singleton. Observe that

$\displaystyle\widehat{D}^{*}F(\bar{x})(y^{*})=\big{\\{}\nabla
F(\bar{x})^{*}y^{*}\big{\\}}\;\mbox{ for all }\;y^{*}\in Y^{*}$

if $F$ is Fréchet differentiable at $\bar{x}$ via the adjoint derivative
operator $\nabla F(\bar{x})^{*}$.

Recall that a mapping $h:D\subset X\to Y$ is upper Lipschitzian [19] at
$\bar{x}\in D$ or calm [20] at this point if there exist numbers $r>0$ and
$\ell\geq 0$ such that

$\displaystyle\|h(x)-h(\bar{x})\|\leq\ell\|x-\bar{x}\|\;\mbox{ for all
}\;x\in\mathbb{B}(\bar{x},r)\cap D.$ (2.6)

This notion is definitely weaker than the the Lipschitz continuity of $h$
around $\bar{x}$ as is illustrated by the simple example of
$h:\mathbb{R}\to\mathbb{R}$ given by $h(x):=x^{3/2}\sin(1/x)$ for all
$x\in\mathbb{R}\setminus\\{0\\}$ and $h(0):=0$. This function is calm at
$\bar{x}=0$ while not Lipschitz continuous around this point.

It is not difficult to deduce from the definitions (see, e.g., the proof of
[15, Proposition 3.5]) that the calmness of $h\colon X\to Y$ at $\bar{x}$
yields the regular coderivative scalarization formula

$\widehat{D}^{*}h(\bar{x})(y^{*})=\widehat{\partial}\langle
y^{*},h\rangle(\bar{x})\;\mbox{ for all }\;y^{*}\in Y^{*}$ (2.7)

in any normed spaces $X$ and $Y$.

Given further a set-valued mapping $F:X\rightrightarrows Y$ between normed
spaces, we say as in [5] that $F$ enjoys the isolated calmness property at
$(\bar{x},\bar{y})\in\mathop{\rm gph}\nolimits F$ with modulus $L\geq 0$ if
there exists $\gamma>0$ such that

$F(u)\cap\mathbb{B}_{Y}(\bar{y},\gamma)\subset\\{\bar{y}\\}+L\|u-\bar{x}\|{\mathbb{\overline{B}}}_{Y}\;\mbox{
for all }\;u\in\mathbb{B}_{X}(\bar{x},\gamma).$ (2.8)

Next we formulate and discuss certain metric qualification conditions the
importance of which has been well realized in different aspects of variational
analysis, generalized differential calculus, and optimization theory; see
below.

######  Definition 2.1 (metric qualification conditions).

Let $C,D$ be subsets of a normed space $X$, and let $\bar{x}\in C\cap D$. It
is said that these sets satisfy:

(i)

The metric qualification condition ($Q_{1}$) at $\bar{x}$ if there exist
$a,r>0$ such that

$\displaystyle d({x},C\cap D)\leq ad({x},C)+ad({x},D)\;\mbox{ for all
}\;x\in\mathbb{B}(\bar{x},r),$ ($Q_{1}$)

where $d(x,S)$ stands for the distance from $x$ to the set $S$.

(ii)

The metric qualification condition ($Q_{2}$) at $\bar{x}$ with respect to some
subdifferential $\partial^{\textasteriskcentered}$ if there exists $a>0$ such
that

$\displaystyle\partial^{\textasteriskcentered}d(\bar{x},C\cap D)\subset
a\partial^{\textasteriskcentered}d(\bar{x},C)+a\partial^{\textasteriskcentered}d(\bar{x},D).$
($Q_{2}$)

The terminology of Definition 2.1 is taken from [6] while such conditions are
traced back to [7] in the case of ($Q_{1}$) and to [8] in the case of
($Q_{2}$). Observe that condition ($Q_{2}$), in contrast to ($Q_{1}$), depends
on the choice of the subdifferential $\partial^{\textasteriskcentered}$. In
[6, 7, 8, 17, 18, 21] among others, the reader can find sufficient conditions
for the fulfillment of ($Q_{2}$) with respect to different subdifferential and
their applications. On the other hand, the subgradient-free condition
($Q_{1}$), which is easily to deal with, yields the fulfillment of ($Q_{2}$)
in some important subgradient settings; see, e.g., [6] and the references
therein.

We conclude this section the three lemmas broadly used in what follows. The
first one, taken from [10, Corollary 1.96], provides relationships between
regular normals to sets and regular subgradients of the associated distance
functions.

######  Lemma 2.2 (normals to sets and subgradients of distance functions).

Let $S\subset X$ be a nonempty subset of a normed space. Then for every
$\bar{x}\in S$ we have the regular normal-subgradient relationships

$\displaystyle\widehat{\partial}d(\cdot,S)(\bar{x})=\mathbb{B}_{X^{*}}\cap{\widehat{N}}(\bar{x};S)\;\mbox{
and
}\;{\widehat{N}}(\bar{x},S)=\bigcup_{\lambda>0}\lambda\widehat{\partial}d(\cdot,S)(\bar{x}).$

The next result is a consequence of Lemma 2.2 for the epigraphical set
$S:=\operatorname{epi}f$ with the replacement of $\bar{x}$ by
$(\bar{x},f(\bar{x}))$.

######  Lemma 2.3 (normals and subgradients for epigraphs).

Let $f\colon X\to\bar{\mathbb{R}}$ be defined on a normed space, and let
$\bar{x}\in S$. Then we have

$\displaystyle\widehat{N}\big{(}(\bar{x},f(\bar{x});\mathop{\rm epi}\nolimits
f)\big{)}=\bigcup_{\lambda>0}\lambda\widehat{\partial}d(\cdot,\mathop{\rm
epi}\nolimits f)\big{(}\bar{x},f(\bar{x})\big{)}.$

The last lemma, taken from [10, Proposition 1.88], provides a smooth
variational description of regular subgradients for general functions on
normed spaces.

######  Lemma 2.4 (smooth variational description of regular subgradients).

Let $f\colon X\to\bar{\mathbb{R}}$ be defined on a normed space $X$, and let
$\bar{x}\in\operatorname{dom}f$. Then the inclusion
$x^{*}\in\widehat{\partial}f(\bar{x})$ for regular subgradients is equivalent
to the existence of a neighborhood $U$ of $\bar{x}$ and a function $s\colon
U\to\mathbb{R}$ such that $s(\cdot)$ is Fréchet differentiable at $\bar{x}$,
that $s(\bar{x})=f(\bar{x})$, and that $s(x)\leq f(x)$ for all $x\in U$.

##  3 Estimates for Regular and Singular Regular Subgradients of Marginal
Functions

In this section, we establish lower and upper estimates for regular
subgradients and their singular counterparts of general marginal functions
(1.1). Without further mentioning, all the spaces under consideration are
arbitrarily normed.

The lower estimates for both regular and singular regular subgradients of
marginal functions, new in the literature, are obtained below under the
alternative conditions formulated in the following two definitions.

The first notion, used in [16] to derive equalities for regular (while not
singular regular) subgradients of marginal functions, employs the upper
Lipschitzian/calmness property of single-valued mappings defined in (2.6).

######  Definition 3.1 (upper Lipschitzian selections).

A set-valued mapping $F:D\rightrightarrows Y$ defined on some set $D\subset
X$, admits an upper Lipschitzian selection at $(\bar{x},\bar{y})\in\mathop{\rm
gph}\nolimits F$ if there exists a single-valued mapping $h:D\to Y$, which is
calm at $\bar{x}$ satisfying $h(\bar{x})=\bar{y}$ and $h(x)\in F(x)$ for all
$x\in D$ in a neighborhood of $\bar{x}$.

The next definition introduces two new well-posedness properties of set-valued
mappings that are related to isolated calmness in (2.8).

######  Definition 3.2 (nearly-isolated calmness).

Given a set-valued mapping $F:X\rightrightarrows Y$ with
$(\bar{x},\bar{y})\in\mathop{\rm gph}\nolimits F$, consider the following
properties:

(i)

$F$ is locally nearly-isolated calm at $(\bar{x},\bar{y})$ with modulus $L>0$
if there exists $\gamma>0$ such that

$\displaystyle
F(u)\cap\mathbb{B}_{Y}(\bar{y},\gamma)\subset\\{\bar{y}\\}+L\|u-\bar{x}\|{\mathbb{\overline{B}}}_{Y}\;\mbox{
for all }\;u\in\mathbb{B}_{X}(\bar{x},\gamma)\setminus\\{\bar{x}\\}.$ (3.1)

(ii)

$F$ is semilocally nearly-isolated calm at $(\bar{x},\bar{y})$ with modulus
$L>0$ if there exists $\gamma>0$ such that

$\displaystyle
F(u)\subset\\{\bar{y}\\}+L\|u-\bar{x}\|{\mathbb{\overline{B}}}_{Y}\;\mbox{ for
all }\;u\in\mathbb{B}_{X}(\bar{x},\gamma)\setminus\\{\bar{x}\\}.$

Clearly, the introduced properties are weaker than the isolated calmness (2.8)
even in simple examples as for $F\colon\mathbb{R}\rightrightarrows\mathbb{R}$
defined by $F(x):=\mathbb{R}$ if $x=0$ and $F(x):=\emptyset$. This mapping
satisfies (3.1) but not (2.8) at $(0,0)$.

Our first theorem presents both lower and upper estimates of regular
subgradients (2.1) of marginal functions in the general normed space setting.

######  Theorem 3.3 (estimates for regular subgradients).

Let the marginal function $\mu(\cdot)$ in (1.1) be finite at some
$\bar{x}\in\operatorname{dom}M$ from (1.2), and let $\bar{y}\in M(\bar{x})$.
The following assertions are satisfied:

(i)

If $\widehat{\partial}^{+}\varphi(\bar{x},\bar{y})\neq\emptyset$ for the upper
regular subdifferential (2.2), then

$\displaystyle\widehat{\partial}\mu(\bar{x})\subset\bigcap_{(x^{\ast},y^{\ast})\in\widehat{\partial}^{+}\varphi(\bar{x},\bar{y})}\big{[}x^{\ast}+\widehat{D}^{*}G(\bar{x},\bar{y})(y^{\ast})\big{]}.$
(3.2)

(ii)

We have the lower subgradient estimate

$\displaystyle\widehat{\partial}\mu(\bar{x})\supset\bigcup_{(x^{\ast},y^{\ast})\in\widehat{\partial}\varphi(\bar{x},\bar{y})}\big{[}x^{\ast}+\widehat{D}^{*}G(\bar{x},\bar{y})(y^{\ast})\big{]}$
(3.3)

provided that at least one of the conditions below holds:

(a)

$G$ is semilocally nearly-isolated calm at $(\bar{x},\bar{y})$.

(b)

$G$ is locally nearly-isolated calm at $(\bar{x},\bar{y})$, and for every
$\varepsilon>0$ there exists $\gamma>0$ such that

$\displaystyle\mu(x)=\inf_{y\in G(x)}\varphi(x,y)=\inf_{y\in
G(x)\cap\mathbb{B}(\bar{y},\varepsilon)}\varphi(x,y)\;\mbox{ for
}\;x\in\mathbb{B}(\bar{x},\gamma).$ (3.4)

(c)

The solution mapping $M:\operatorname{dom}G\rightrightarrows Y$ admits an
upper Lipschitzian selection at $(\bar{x},\bar{y})$.

###### Proof.

The upper estimate in (i) is taken [16, Theorem 1] formulated there in the
Banach space setting, while the given proof does not require the completeness
of the normed spaces in question. Let us now verify the lower estimate of
regular subgradients in (ii) in all the three cases therein.

$\bullet$ Case (a), where $G$ is semilocally nearly-isolated calm at
$(\bar{x},\bar{y})$ with modulus $L$. Fix an arbitrary number $\varepsilon>0$
and pick $u^{\ast}$ in the right-hand side of (3.3). Then there exists
${(x^{\ast},y^{\ast})\in\widehat{\partial}\varphi(\bar{x},\bar{y})}$ such that
$u^{\ast}-{x}^{\ast}\in\widehat{D}^{*}G(\bar{x},\bar{y})(y^{\ast})$, and hence
$(u^{\ast}-{x}^{\ast},-y^{\ast})\in\widehat{N}((\bar{x},\bar{y});\mathop{\rm
gph}\nolimits G)$. The inclusion
${(x^{\ast},y^{\ast})\in\widehat{\partial}\varphi(\bar{x},\bar{y})}$ allows us
to choose $\delta_{1}>0$ so that for every
$(u,v)\in\mathbb{B}((\bar{x},\bar{y}),\delta_{1})$ we get

$\langle x^{\ast},u-\bar{x}\rangle+\langle
y^{\ast},v-\bar{y}\rangle\leq\varphi(u,v)-\varphi(\bar{x},\bar{y})+\varepsilon(\|u-\bar{x}\|+\|v-\bar{y}\|).$
(3.5)

By Lemma 2.4, there exists a function $s:X\times Y\rightarrow\mathbb{R}$
Fréchet differentiable at $(\bar{x},\bar{y})$ with $\nabla
s(\bar{x},\bar{y})=(u^{\ast}-x^{\ast},-y^{\ast})$ and

$s(u,v)\leq s(\bar{x},\bar{y})=0\;\mbox{ for all }\;(u,v)\in\mathop{\rm
gph}\nolimits G\;\mbox{ near }\;(\bar{x},\bar{y}).$ (3.6)

Thus there exists $0<\delta_{2}<\delta_{1}$ such that

$\arrowvert
s(u,v)-s(\bar{x},\bar{y})-\langle(u^{\ast}-x^{\ast},-y^{\ast}),(u-\bar{x},v-\bar{y})\rangle\lvert\leq\varepsilon(\|u-\bar{x}\|+\|v-\bar{y}\|)$
(3.7)

for every $(u,v)\in\mathbb{B}((\bar{x},\bar{y}),\delta_{2})$, Combining the
latter with (3.6) yields

$\langle u^{\ast},u-\bar{x}\rangle\leq\langle
x^{\ast},u-\bar{x}\rangle+\langle
y^{\ast},v-\bar{y}\rangle+\varepsilon(\|u-\bar{x}\|+\|v-\bar{y}\|)$ (3.8)

for every $(u,v)\in\mathbb{B}((\bar{x},\bar{y}),\delta_{2})\cap\mathop{\rm
gph}\nolimits G$. The imposed semilocal nearly-isolated calmness of $G$ at
$(\bar{x},\bar{y})$ gives us $\delta_{3}>0$ with
$\max\\{\delta_{3},L\delta_{3}\\}<\delta_{2}$ satisfying

$G(u)\subset\\{\bar{y}\\}+L\|u-\bar{x}\|{\mathbb{\overline{B}}}_{Y}\subset\mathbb{B}(\bar{y},\delta_{2})\;\mbox{
whenever }\;u\in\mathbb{B}(\bar{x},\delta_{3})\setminus\\{\bar{x}\\}.$ (3.9)

Observe that $(u,v)\in\mathbb{B}((\bar{x},\bar{y}),\delta_{2})\cap\mathop{\rm
gph}\nolimits G$ whenever
$u\in\mathbb{B}(\bar{x},\delta_{3})\setminus\\{\bar{x}\\}$ and $v\in G(u)$.
Thus it follows from (3.5), (3.8), and (3.9) that

$\displaystyle\langle
u^{\ast},u-\bar{x}\rangle\leq\varphi(u,v)-\varphi(\bar{x},\bar{y})+2\varepsilon(L+1)\|u-\bar{x}\|\;\mbox{
for all }\;~{}v\in G(u),$

and therefore we arrive at the estimate

$\displaystyle\langle u^{\ast},u-\bar{x}\rangle$
$\displaystyle\leq\mu(u)-\mu(\bar{x})+2\varepsilon(L+1)(\|u-\bar{x}\|)\;\mbox{
when }\;u\in\mathbb{B}(\bar{x},\delta_{3}).$

This tells us that $u^{\ast}\in\widehat{\partial}\mu(\bar{x})$ since
$\varepsilon>0$ was chosen arbitrarily.

$\bullet$ Case (b), where $G$ is locally nearly-isolated calm at
$(\bar{x},\bar{y})$. The proof in this case is similar to (a). Indeed, by
taking $\delta_{1},\delta_{2}$ as in (a), it follows from the local nearly-
isolated calmness of $G$ at $(\bar{x},\bar{y})$ with modulus $L_{0}$ that
there exists $\delta_{3}>0$ with
$\max\\{\delta_{3},\delta_{3}L_{0}\\}<\delta_{2}$ such that for all
$u\in\mathbb{B}(\bar{x},\delta_{3})\setminus\\{\bar{x}\\}$ we have

$\displaystyle
G(u)\cap\mathbb{B}(\bar{y},\delta_{3})\subset\\{\bar{y}\\}+L_{0}\|u-\bar{x}\|{\mathbb{\overline{B}}}_{Y}\subset\mathbb{B}(\bar{y},\delta_{2}).$
(3.10)

Furthermore, the choice of
$u\in\mathbb{B}(\bar{x},\delta_{3})\setminus\\{\bar{x}\\}$ and $v\in
G(u)\cap\mathbb{B}(\bar{y},\delta_{3})$ yields
$(u,v)\in\mathbb{B}((\bar{x},\bar{y}),\delta_{2})\cap\mathop{\rm gph}\nolimits
G$. Combining (3.5), (3.8), and (3.10) ensures that that for every
$u\in\mathbb{B}(\bar{x},\delta_{3})\setminus\\{\bar{x}\\}$ we get the estimate

$\displaystyle\langle
u^{\ast},u-\bar{x}\rangle\leq\varphi(u,v)-\varphi(\bar{x},\bar{y})+2\varepsilon(L_{0}+1)\|u-\bar{x}\|$
(3.11)

whenever $v\in G(u)\cap\mathbb{B}(\bar{y},\delta_{3})$. On the other hand, it
follows from assumption $\eqref{H}$ that there exists
$\bar{\delta}\leq\delta_{3}$ such that

$\mu(u)=\inf_{y\in G(u)}\varphi(u,y)=\inf\limits_{y\in
G(u)\cap\mathbb{B}(\bar{y},\delta_{3})}\varphi(u,y)\;\mbox{ for all
}\;u\in\mathbb{B}(\bar{x},\bar{\delta}).$

Unifying the latter with (3.11) brings us to the implication

$\displaystyle\langle
u^{\ast},u-\bar{x}\rangle\leq\mu(u)-\mu(\bar{x})+2\varepsilon(L_{0}+1)\|u-\bar{x}\|\;\mbox{
if }\;u\in\mathbb{B}(\bar{x},\bar{\delta})\setminus\\{\bar{x}\\}$
$\displaystyle\Longrightarrow\langle
u^{\ast},u-\bar{x}\rangle\leq\mu(u)-\mu(\bar{x})+2\varepsilon(L_{0}+1)\|u-\bar{x}\|\;\mbox{
if }\;u\in\mathbb{B}(\bar{x},\bar{\delta}),$

which means that $u^{*}\in\widehat{\partial}\mu(\bar{x})$ verifying the lower
estimate (3.3) in case (b).

$\bullet$ Case (c), where the solution mapping
$M:\operatorname{dom}G\rightrightarrows Y$ admits a local upper Lipschitzian
selection $h$ at $(\bar{x},\bar{y})$. In this case, we have
$\mu(x)=\varphi(x,h(x))$ for all $x$ from some neighborhood $U$ of $\bar{x}$.
As in case (a), there exist $0<\delta_{3}<\delta_{2}$ such that
$h(\bar{x})=\bar{y}$, $h(x)\in M(x)$, and

$\displaystyle\|h(x)-h(\bar{x})\|<\ell\|x-\bar{x}\|<\delta_{2}\;\mbox{ for all
}\;x\in\mathbb{B}(\bar{x},\delta_{3})\subset U(\bar{x})$ (3.12)

with the calmness modulus $\ell$ from (2.6). Then for every
$u\in\mathbb{B}(\bar{x},\delta_{3})$, we have
$(u,h(u))\in\mathbb{B}((\bar{x},\bar{y}),\delta_{2})\cap\mathop{\rm
gph}\nolimits G$, which implies by (3.5) and (3.8) that

$\displaystyle\langle u^{\ast},u-\bar{x}\rangle$
$\displaystyle\leq\varphi(u,h(u))-\varphi(\bar{x},\bar{y})+2\varepsilon(\|u-\bar{x}\|+\|h(u)-\bar{y}\|)$
$\displaystyle\leq\mu(u)-\mu(\bar{x})+2\varepsilon(\ell+1)\|u-\bar{x}\|\;\mbox{
whenever }\;u\in\mathbb{B}(\bar{x},\delta_{3}).$

Since $\varepsilon>0$ was chosen arbitrarily, we conclude that
$u^{*}\in\widehat{\partial}\mu(\bar{x})$, which verifies (3.3) and completes
the proof the theorem. ∎

The next theorem provides both lower and upper estimates for the singular
regular subgradients (2.4) of marginal functions defined on normed spaces. The
only result on upper estimates of $\widehat{\partial}^{\infty}\mu(\bar{x})$
known to us is given in [6, Theorem 4.1], where a different estimate is
obtained in Asplund spaces via the singular subdifferential and coderivative
by Mordukhovich under the metric qualification conditions (Q1) and (Q2) from
Definition 2.1. We are not familiar with any lower estimate of
$\widehat{\partial}^{\infty}\mu(\bar{x})$ available in the literature.

######  Theorem 3.4 (estimates for singular regular subgradients).

Let $\bar{y}\in M(\bar{x})$ for the solution mapping (1.2) associated with
(1.1) , where $\varphi$ is finite at $\bar{x}$. The following upper and lower
estimates for $\widehat{\partial}^{\infty}\mu(\bar{x})$ are satisfied:

(i)

If $\varphi$ is calm at $(\bar{x}$,$\bar{y}$) as in (2.6), then we have the
upper estimate

$\displaystyle\widehat{\partial}^{\infty}\mu(\bar{x})\subset\widehat{D}^{*}G(\bar{x},\bar{y})(0).$
(3.13)

(ii)

The lower estimate for singular regular subgradients

$\displaystyle\widehat{\partial}^{\infty}\mu(\bar{x})\supset\bigcup_{(x^{*},y^{*})\in\widehat{\partial}^{\infty}\varphi(\bar{x},\bar{y})}x^{*}+\widehat{D}^{*}G(\bar{x},\bar{y})(y^{*})$
(3.14)

holds if at least one of the conditions below is fulfilled:

(a)

$G$ is semilocally nearly-isolated calm at $(\bar{x},\bar{y})$.

(b)

$G$ is locally nearly-isolated calm at $(\bar{x},\bar{y})$, and for every
$\varepsilon>0$ there exists $\gamma>0$ such that (3.4) is satisfied.

(c)

The solution mapping $M:\operatorname{dom}G\rightrightarrows Y$ admits a local
upper Lipschitzian selection at $(\bar{x},\bar{y})$,

###### Proof.

To verify the upper estimate (3.13) in (i), observe that

$\displaystyle\mathop{\rm epi}\nolimits(\varphi+\delta_{\mathop{\rm
gph}\nolimits G})=\mathop{\rm epi}\nolimits\varphi\cap[\mathop{\rm
gph}\nolimits G\times\mathbb{R}].$

Picking $u^{*}\in\widehat{\partial}^{\infty}\mu(\bar{x})$, we get by [6,
Proposition 4.1] that
$(u^{*},0)\in\widehat{\partial}^{\infty}(\varphi+\delta_{\mathop{\rm
gph}\nolimits G})(\bar{x},\bar{y})$, which is equivalent to
$(u^{*},0,0)\in\widehat{N}\left((\bar{x},\bar{y},\varphi(\bar{x},\bar{y}));\mathop{\rm
epi}\nolimits\varphi\cap[\mathop{\rm gph}\nolimits G\times\mathbb{R}]\right)$,
and hence
$(u^{*},0,0)\in\widehat{N}\left((\bar{x},\bar{y},\varphi(\bar{x},\bar{y}));\mathop{\rm
gph}\nolimits\varphi\cap[\mathop{\rm gph}\nolimits G\times\mathbb{R}]\right)$.
Thus for every $\varepsilon>0$ there exists $\delta>0$ such that

$\displaystyle\langle
u^{*},x-\bar{x}\rangle\leq\varepsilon(\|x-\bar{x}\|+\|y-\bar{y}\|+\varphi(x,y)-\varphi(\bar{x},\bar{y})|)$

whenever $(x,y,\varphi(x,y))\in(\mathop{\rm
gph}\nolimits\varphi\cap[\mathop{\rm gph}\nolimits
G\times\mathbb{R}])\cap\mathbb{B}(\bar{x},\bar{y},\varphi(\bar{x},\bar{y}),\delta)$.
Employing the assumed calmness property of $\varphi$ gives us $\kappa\geq 0$
ensuring that

$\displaystyle\langle
u^{*},x-\bar{x}\rangle\leq\varepsilon(\kappa+1)(\|x-\bar{x}\|+\|y-\bar{y}\|)$
(3.15)

for all $(x,y)\in\mathbb{B}((\bar{x},\bar{y}),\delta)$ and $y\in G(x)$.
Remembering that $\varepsilon>0$ was chosen arbitrarily and using (3.15), we
get the relationship

$\displaystyle\limsup\limits_{(x,y)\xrightarrow{\mathop{\rm gph}\nolimits
G}(\bar{x},\bar{y})}\dfrac{\langle u^{*},x-\bar{x}\rangle-\langle
0,y-\bar{y}\rangle}{\|x-\bar{x}\|+\|y-\bar{y}\|}\leq 0$

meaning that $u^{*}\in\widehat{D}^{*}G(\bar{x},\bar{y})(0)$, i.e.,
$\widehat{\partial}^{\infty}\mu(\bar{x})\subset\widehat{D}^{*}G(\bar{x},\bar{y})(0)$.
This justifies the claimed upper subdifferential estimate (3.13).

Now we derive the lower subdifferential estimate in (ii). Fix an arbitrary
number $\varepsilon>0$, pick $u^{\ast}$ from the right-hand side of (3.14),
and
${(x^{\ast},y^{\ast})\in{\widehat{\partial}^{\infty}}\varphi(\bar{x},\bar{y})}$
with $u^{\ast}-{x}^{\ast}\in\widehat{D}^{*}G(\bar{x},\bar{y})(y^{\ast})$,
i.e.,
$(u^{\ast}-{x}^{\ast},-y^{\ast})\in\widehat{N}((\bar{x},\bar{y});\mathop{\rm
gph}\nolimits G)$. It follows from
${(x^{\ast},y^{\ast})\in{\widehat{\partial}^{\infty}}\varphi(\bar{x},\bar{y})}$
that

$(x^{\ast},y^{\ast},0)\in\widehat{N}\big{(}(\bar{x},\bar{y}),\varphi(\bar{x},\bar{y});\mathop{\rm
epi}\nolimits\varphi\big{)},$

which yields the existence of $\delta_{1}>0$ such that for every triple
$(u,v,\alpha)\in\mathbb{B}(((\bar{x},\bar{y}),\varphi(\bar{x},\bar{y})),\delta_{1})\cap\mathop{\rm
epi}\nolimits\varphi$ we have the estimate

$\displaystyle\langle x^{\ast},u-\bar{x}\rangle+\langle
y^{\ast},v-\bar{y}\rangle\leq\varepsilon(\|u-\bar{x}\|+\|v-\bar{y}\|+|\alpha-\varphi(\bar{x},\bar{y})|).$
(3.16)

Lemma 2.4 tells us that there exists a function $s:X\times
Y\rightarrow\mathbb{R}$, which is Fréchet differentiable at
$(\bar{x},\bar{y})$ while satisfying $\nabla
s(\bar{x},\bar{y})=(u^{\ast}-x^{\ast},-y^{\ast})$ and (3.6). This implies in
turn that there exists $0<\delta_{2}<\delta_{1}$ ensuring the fulfillment of
(3.7) and (3.8). Therefore, for every
$(u,v)\in\mathbb{B}((\bar{x},\bar{y}),\delta_{2})\cap\mathop{\rm gph}\nolimits
G$ we have

$\langle u^{\ast},u-\bar{x}\rangle\leq\langle
x^{\ast},u-\bar{x}\rangle+\langle
y^{\ast},v-\bar{y}\rangle+\varepsilon(\|u-\bar{x}\|+\|v-\bar{y}\|).$ (3.17)

Now we are ready to verify (3.14) in all the three cases listed in (ii).

$\bullet$ Case (a), where $G$ is semilocally nearly-isolated calm at
$(\bar{x},\bar{y})$ with modulus $L$. Taking into account the discussions
above and the imposed nearly-isolated calmness condition, we find
$\delta_{3}>0$ with $\max\\{\delta_{3},L\delta_{3}\\}<\delta_{2}$ such that
semilocal calmness condition (3.9) is satisfied. Observing that
$(u,v)\in\mathbb{B}((\bar{x},\bar{y}),\delta_{2})\cap\mathop{\rm gph}\nolimits
G$ for each vectors $u\in\mathbb{B}(\bar{x},\delta_{3})\setminus\\{\bar{x}\\}$
and $v\in G(u)\cap\mathbb{B}(\bar{y},\delta_{3})$ allows us to deduce from
(3.9) (3.16), and (3.17) that

$\displaystyle\langle u^{\ast},u-\bar{x}\rangle\leq
2\varepsilon(\|u-\bar{x}\|+\|v-\bar{y}\|+|\alpha-\varphi(\bar{x},\bar{y})|)$
(3.18)

for every
$(u,v,\alpha)\in[(\mathbb{B}(\bar{x},\delta_{3})\setminus\\{\bar{x}\\}\times\mathbb{B}(\bar{y},\delta_{2}))\cap\mathop{\rm
gph}\nolimits
G\times\mathbb{B}_{\mathbb{R}}(\mu(\bar{x}),\delta_{1})]\cap\mathop{\rm
epi}\nolimits\varphi$. Then we claim the fulfillment of the estimate

$\displaystyle\langle u^{\ast},u-\bar{x}\rangle\leq
2\varepsilon(L+1)(\|u-\bar{x}\|+|\alpha-\mu(\bar{x})|).$ (3.19)

To verify (3.19), take $\alpha>\mu(u)$ and a sequence of $\\{v_{k}\\}$
satisfying

$\varphi(u,v_{k})\xrightarrow{\\{v_{k}\\}\subset
G(u)}\mu(u)\quad\text{as}\quad k\to\infty.$

Then $\alpha\geq\varphi(u,v_{k})$ for $k$ sufficiently large, and thus

$(u,v_{k},\alpha)\in[((\mathbb{B}(\bar{x},\delta_{3})\setminus\\{\bar{x}\\})\times\mathbb{B}(\bar{y},\delta_{2}))\cap\mathop{\rm
gph}\nolimits
G\times\mathbb{B}_{\mathbb{R}}(\mu(\bar{x}),\delta_{1})]\cap\mathop{\rm
epi}\nolimits\varphi.$

If follows from (3.9) and (3.18) that

$\displaystyle\langle u^{\ast},u-\bar{x}\rangle$ $\displaystyle\leq
2\varepsilon(\|u-\bar{x}\|+\|v_{k}-\bar{y}\|+|\alpha-\mu(\bar{x})|)$
$\displaystyle\leq 2\varepsilon(L+1)(\|u-\bar{x}\|+|\alpha-\mu(\bar{x})|)$

for large $k$. If $\alpha=\mu(u)$ for all $\gamma>0$ satisfying
$\alpha+\gamma\in\mathbb{B}_{\mathbb{R}}(\mu(\bar{x}),\delta_{3})$, then
$\alpha+\gamma>\mu(u)$, and the same device as above brings us to

$\displaystyle\langle u^{\ast},u-\bar{x}\rangle\leq
2\varepsilon(L+1)(\|u-\bar{x}\|+|\alpha+\gamma-\mu(\bar{x})|).$

Since $\gamma>0$ was chosen arbitrarily small, this justifies the claimed
estimate (3.19). The arbitrary choice of $\varepsilon>0$ ensures that
$(u^{*},0)\in\widehat{N}((\bar{x},\mu(\bar{x}));\mathop{\rm
epi}\nolimits\mu)$, and thus we get
$u^{\ast}\in\widehat{\partial}^{\infty}\mu(\bar{x})$ justifying (3.14).

$\bullet$ Case (b), where $G$ is locally nearly-isolated calm at
$(\bar{x},\bar{y})$ under the fulfillment of (3.4). The proof of (3.14) in
this case is similar to the above case (a) with taking into account the
arguments in case (b) of Theorem 3.3.

$\bullet$ Case (c), where the solution mapping
$M:\operatorname{dom}G\rightrightarrows Y$ admits an upper Lipschitzian
selection at $(\bar{x},\bar{y})$. Let $h$ be such a selection with modulus
$\ell>0$ in (2.6). We have $\mu(x)=\varphi(x,h(x))$ for all $x\in U$ for a
neighborhood $U$ of $\bar{x}$. As in the proof of case (c) in Theorem 3.3,
find $0<\delta_{3}<\delta_{2}$ such that $h(\bar{x})=\bar{y}$, $h(x)\in M(x)$,
and condition (3.12) is satisfied. Furthermore,
$(u,h(u))\in\mathbb{B}((\bar{x},\bar{y}),\delta_{2})\cap\mathop{\rm
gph}\nolimits G$ for every $u\in\mathbb{B}(\bar{x},\delta_{3})$, and thus it
follows from the estimates in (3.16) and (3.17) that

$\displaystyle\langle u^{\ast},u-\bar{x}\rangle\leq
2\varepsilon(\|u-\bar{x}\|+\|h(u)-\bar{y}\|+|\alpha-\varphi(\bar{x},\bar{y})|)$

for all
$(u,h(u),\alpha)\in[(\mathbb{B}((\bar{x},\delta_{3})\times\mathbb{B}(\bar{y},\delta_{2}))\cap\mathop{\rm
gph}\nolimits
G\times\mathbb{B}_{\mathbb{R}}(\mu(\bar{x}),\delta_{1})]\cap\mathop{\rm
epi}\nolimits\varphi$. By (3.12) and $\mu(u)=\varphi(u,h(u))$ for all
$u\in\mathbb{B}(\bar{x},\delta_{3})$, we get

$\displaystyle\langle u^{\ast},u-\bar{x}\rangle\leq
2\varepsilon(\ell+1)(\|u-\bar{x}\|+|\alpha-\mu(\bar{x})|)$

whenever
$(u,\alpha)\in[\mathbb{B}(\bar{x},\delta_{3})\times(\mathbb{B}_{\mathbb{R}}(\mu(\bar{x}),\delta_{3})]\cap\mathop{\rm
epi}\nolimits\mu$. This tells us, by the arbitrary choice of $\varepsilon>0$,
that $(u^{*},0)\in\widehat{N}((\bar{x},\mu(\bar{x}));\mathop{\rm
epi}\nolimits\mu)$, and hence
$u^{\ast}\in\widehat{\partial}^{\infty}\mu(\bar{x})$, which completes the
proof of theorem. ∎

We conclude this section with an example illustrating the subdifferential
estimates obtained in Theorems 3.3 and 3.4.

######  Example 3.5 (illustrating subdifferential estimates for marginal
functions).

Consider the marginal function $\mu(\cdot)$ in (1.1) with $X=Y=\mathbb{R}$,

$\varphi(x,y):=\lvert x\rvert-y,~{}~{}~{}{\rm
and}~{}~{}~{}G(x)=\left\\{\begin{array}[]{ll}\\{x\\}&{\rm if}~{}x>0,\\\
(-\infty,0]&{\rm if}~{}x=0,\\\ \emptyset,&{\rm otherwise}.\end{array}\right.$

Here $G$ is locally nearly-isolated calm at $(\bar{x},\bar{y}):=(0,0)$ for
every $L>0$,

$\displaystyle\widehat{D}^{*}G(0,0)(y^{*})=\left\\{\begin{array}[]{ll}\emptyset&{\rm
if}~{}y^{*}>0,\\\ (-\infty,y^{*}]&{\rm if}~{}y^{*}\leq 0,\end{array}\right.$

$\widehat{\partial}\varphi(0,0)=[-1,1]\times\\{-1\\}$,
$\widehat{\partial}^{\infty}\varphi(0,0)=\\{(0,0)\\}$, and
$\widehat{\partial}^{+}\varphi(0,0)=\emptyset$. Due to the latter fact, the
upper estimate of $\widehat{\partial}\mu(0)$ in Theorem 3.3(i) cannot be
applied. On the other hand, Theorem 3.3(ii) gives us a lower estimate of
$\widehat{\partial}\mu(0)$ by

$\displaystyle\widehat{\partial}\mu(0)\supset\bigcup_{(x^{\ast},y^{\ast})\in\widehat{\partial}\varphi(0,0)}\big{[}x^{\ast}+\widehat{D}^{*}G(0,0)(y^{\ast})\big{]}=(-\infty,0].$

Furthermore, it follows from inclusions (3.13) and (3.14) of Theorem 3.4 that

$\displaystyle\widehat{\partial}^{\infty}\mu(0)=\widehat{D}^{*}G(0,0)(0)=(-\infty,0].$

##  4 Exact Calculations of Regular and Singular Regular Subdifferentials of
Marginal Functions

The results of this section provide sufficient conditions ensuring exact
formulas (i.e., equalities) for calculating both regular and singular regular
subdifferentials of marginal functions. We present two theorems of this type:
one for marginal functions with Fréchet differential costs, and the other one
for fully nonsmooth data under additional qualification conditions. As in the
previous section, all the spaces in question are arbitrarily normed.

The first theorem concerns the case where the cost function in (1.1) is
Fréchet differentiable. Formula (4.1) for the regular subdifferential can be
found in [16, Theorem 2] under condition (a) formulated in Theorem 3.4. The
other results of the following theorem are, to the best of our knowledge, new.

######  Theorem 4.1 (marginal functions with differentiable costs).

Let $\mu(\cdot)$
from (1.1) be finite at some $\bar{x}\in\operatorname{dom}M$, and let
$\bar{y}\in M(\bar{x})$. Suppose that $\varphi$ is Fréchet differentiable at
$(\bar{x},\bar{y})$. Then we have the calculation formulas

$\displaystyle\widehat{\partial}\mu(\bar{x})=x^{\ast}+\widehat{D}^{*}G(\bar{x},\bar{y})(y^{\ast})\;\mbox{
with }\;(x^{\ast},y^{\ast}):=\nabla\varphi(\bar{x},\bar{y}),$ (4.1)
$\displaystyle\widehat{\partial}^{\infty}\mu(\bar{x})=\widehat{D}^{*}G(\bar{x},\bar{y})(0)$
(4.2)

provided that at least one of the conditions (a)–(c) of Theorem 3.4 is
satisfied.

###### Proof.

Since $\varphi$ is Fréchet differentiable at $(\bar{x},\bar{y})$, we have

$\widehat{\partial}^{+}\varphi(\bar{x},\bar{y})=\widehat{\partial}\varphi(\bar{x},\bar{y})=\nabla\varphi(\bar{x},\bar{y})=\big{\\{}(x^{\ast},y^{\ast})\big{\\}}.$

Therefore, equality (4.1) follows from (3.2) and (3.3) of Theorem 3.3, while
equality (4.2) is a consequence of (3.13) and (3.14) in Theorem 3.4. ∎

The following example shows that Theorem 4.1 can be applied to ensure the
equalities in (4.1) and (4.2) under the near-isolated calmness of $G$, while
the solution mapping $M$ fails to admit an upper Lipschitzian selection,

######  Example 4.2 (equalities under nearly-isolated calmness).

Let $X=Y=\mathbb{R}$ and $(\bar{x},\bar{y})=(0,0)$ in the setting of Theorem
4.1 with

$\varphi(x,y):=(x-y^{2})^{2}~{}~{}~{}{\rm
and}~{}~{}~{}G(x):=\left\\{\begin{array}[]{ll}\mathbb{R}&{\rm if}~{}x=0,\\\
\emptyset&{\rm otherwise}.\end{array}\right.$

Then we can directly calculate that

$\mu(x)=\left\\{\begin{array}[]{ll}0&{\rm if}~{}x=0,\\\ \infty&{\rm
otherwise}\end{array}\right.~{}~{}~{}{\rm
and}~{}~{}~{}M(x)=\left\\{\begin{array}[]{ll}\\{0\\}&{\rm if}~{}x=0,\\\
\emptyset&{\rm otherwise}.\end{array}\right.$

Therefore, the solution mapping $M(\cdot)$ does not admit an upper
Lipschitzian selection at $(0,0)$ telling us that [16, Theorem 2] cannot be
applied. Nevertheless, since $G$ is (semilocally) nearly-isolated calm at
$(0,0)$, we deduce the claimed equalities (4.1) and (4.2) from Theorem 4.1
under the nearly-isolated calmness.

The next example illustrates that neither of the assumptions of Theorem 4.1 is
necessary for the subdifferential equalities therein.

######  Example 4.3 (equalities without nearly-isolated calmness).

Let $X=Y=\mathbb{R}$ and $(\bar{x},\bar{y})=(0,0)$ in the setting of Theorem
4.1 with

$\varphi(x,y):=(x-y^{2})^{2}~{}~{}~{}{\rm and}~{}~{}~{}G(x):=\mathbb{R}.$

This example is taken from [16, Example 2(ii)], where it is shown that the
solution mapping $M$ does not admit an upper Lipschitzian selection, while the
regular subdifferential equality (4.1) holds. In this case, it is easy to see
that the constraint mapping $G$ fails to be nearly-isolated calm at $(0,0)$,
and hence Theorem 4.1 cannot be applied. Nevertheless, we have here that

$\displaystyle\widehat{\partial}^{\infty}\mu(0)=\widehat{\partial}\mu(0)=\\{0\\},~{}~{}\nabla\varphi(0,0)=\\{(0,0)\\},~{}~{}\widehat{D}^{*}G(0,0)(0)=\\{0\\},$

and thus both equalities (4.1) and (4.2) are satisfied.

Now we establish new calculation formulas for both regular and singular
subdifferentials of marginal functions with nondifferentiable costs.
Furnishing this involves a metric qualification condition on the data of
(1.1).

######  Theorem 4.4 (marginal functions with nondifferentiable costs).

Let $\mu(\cdot)$ the marginal function from (1.1) be finite at some
$\bar{x}\in\operatorname{dom}M$, and let $\bar{y}\in M(\bar{x})$. Suppose that
the following conditions are fulfilled:

($\bf\mathcal{A}_{1}$)

The sets ${\rm epi}\,\varphi$ and $\mathop{\rm gph}\nolimits
G\times\mathbb{R}$ satisfy the metric qualification condition ($Q_{2}$) for
$\partial^{\textasteriskcentered}=\widehat{\partial}$ at
$(\bar{x},\bar{y},\varphi(\bar{x},\bar{y}))$.

($\bf\mathcal{A}_{2}$)

At least one of the conditions (a)–(c) of Theorem 3.4 holds.

Then we have the subdifferential calculation formulas

${\widehat{\partial}}\mu(\bar{x})=\bigcup_{(x^{\ast},y^{\ast})\in{\widehat{\partial}}\varphi(\bar{x},\bar{y})}\big{[}x^{\ast}+\widehat{D}^{*}G(\bar{x},\bar{y})(y^{\ast})\big{]},$
(4.3)
${\widehat{\partial}^{\infty}}\mu(\bar{x})=\bigcup_{(x^{\ast},y^{\ast})\in{\widehat{\partial}^{\infty}}\varphi(\bar{x},\bar{y})}\big{[}x^{\ast}+\widehat{D}^{*}G(\bar{x},\bar{y})(y^{\ast})\big{]}.$
(4.4)

###### Proof.

Denoting $C:=\mathop{\rm epi}\nolimits\varphi$ and $D:=\mathop{\rm
gph}\nolimits G\times\mathbb{R}$, we intend to show that

$\displaystyle(x^{*},\beta)\in\widehat{N}\big{(}(\bar{x},\mu(\bar{x});\mathop{\rm
epi}\nolimits\mu\big{)}\Longrightarrow(x^{*},0,\beta)\in\widehat{N}\big{(}(\bar{x},\bar{y},\varphi(\bar{x},\bar{y}));C\cap
D\big{)}.$ (4.5)

Indeed, it follows from Lemma 2.4 that there exists a Fréchet differentiable
function $s:X\times\mathbb{R}\rightarrow\mathbb{R}$ such that $\nabla
s(\bar{x},\mu(\bar{x}))=(x^{\ast},\beta)$ and

$s(x,r)\leq s(\bar{x},\mu(\bar{x}))=0\;\mbox{ whenever
}\;(x,r)\in\text{epi}\,\mu.$

Fix $\varepsilon>0$ and use the Fréchet differentiability of $s(\cdot)$ to
find $\delta_{1}>0$ such that

$\displaystyle|s(x,r)-\langle(x^{\ast},\beta),(x-\bar{x},r-\mu(\bar{x}))\rangle\lvert\leq\varepsilon(\|x-\bar{x}\|+|r-\mu(\bar{x})|)$
$\displaystyle\Longrightarrow\,\langle(x^{\ast},\beta),(x-\bar{x},r-\mu(\bar{x}))\rangle\leq\varepsilon(\|x-\bar{x}\|+|r-\mu(\bar{x})|)$

for every $(x,r)\in\mathbb{B}((\bar{x},\mu(\bar{x})),\delta_{1})$. This yields
the estimate

$\displaystyle\langle x^{\ast},x-\bar{x}\rangle$
$\displaystyle\leq\langle-\beta,r-\mu(\bar{x})\rangle+\varepsilon(\|x-\bar{x}\|+|r-\mu(\bar{x})|)$

whenever $(x,r)\in\mathbb{B}((\bar{x},\mu(\bar{x})),\delta_{1})\cap\mathop{\rm
epi}\nolimits\mu$. Since $\mu(\bar{x})=\varphi(\bar{x},\bar{y})$ and
$\mu(x)\leq\varphi(x,y)\leq r$ for every
$(x,y,r)\in\mathbb{B}((\bar{x},\bar{y},\varphi(\bar{x},\bar{y})),\delta_{1})\cap
C\cap D$, we have

$\displaystyle\langle x^{\ast},x-\bar{x}\rangle\leq$
$\displaystyle\,\langle-\beta,r-\varphi(\bar{x},\bar{y})\rangle+\varepsilon(\|x-\bar{x}\|+\|y-\bar{y}\|+|r-\varphi(\bar{x},\bar{y})|)$
$\displaystyle\mbox{for all
}\;(x,y,r)\in\mathbb{B}((\bar{x},\bar{y},\varphi(\bar{x},\bar{y})),\delta_{1})\cap
C\cap D,$

which therefore justifies the implication in (4.5).

To verify (4.3), pick $p^{*}\in\widehat{\partial}\mu(\bar{x})$ and get
$(p^{*},-1)\in\widehat{N}(\bar{x},\mu(\bar{x});\mathop{\rm epi}\nolimits\mu)$.
By Lemma 2.3, there exist
$(x^{*},\beta)\in\widehat{\partial}d(\bar{x},\mu(\bar{x}),\mathop{\rm
epi}\nolimits\mu)$ and $\lambda>0$ such that $p^{*}=\lambda x^{*}$ and
$-1=\lambda\beta$, where $\lambda\geq 1$ by Lemma 2.2. Since $\bar{y}\in
M(\bar{x})$, we claim that

$\displaystyle(x^{*},\beta)\in\widehat{\partial}d(\bar{x},\mu(\bar{x}),\text{epi}\,\mu)\Longrightarrow(x^{*},0,\beta)\in\widehat{\partial}d((\bar{x},\bar{y},\varphi(\bar{x},\bar{y})),C\cap
D).$

Indeed, it follows from Lemma 2.2 that

$\displaystyle(x^{*},\beta)\in\widehat{\partial}d(\bar{x},\mu(\bar{x}),\mathop{\rm
epi}\nolimits\mu)=\mathbb{B}_{X^{*}\times\mathbb{R}}\cap\widehat{N}((\bar{x},\mu(\bar{x}));\mathop{\rm
epi}\nolimits\mu),$

which implies that $\|(x^{*},\beta)\|\leq 1$ and
$(x^{*},\beta)\in\widehat{N}((\bar{x},\mu(\bar{x}));\text{epi}\,\mu)$. By
using (4.5), we get
$(x^{*},0,\beta)\in\widehat{N}((\bar{x},\bar{y},\varphi(\bar{x},\bar{y}));C\cap
D)$ and $\|(x^{*},0,\beta)\|\leq 1$. Employing again Lemma 2.2 tells us that

$\displaystyle(x^{*},0,\beta)\in\widehat{\partial}d((\bar{x},\bar{y},\varphi(\bar{x},\bar{y})),C\cap
D).$ (4.6)

On the other hand, assumption ($\mathcal{A}_{1}$) ensures the subdifferential
estimate

$\displaystyle\widehat{\partial}d((\bar{x},\bar{y},\varphi(\bar{x},\bar{y})),C\cap
D)$ $\displaystyle\subset
a\big{[}\widehat{\partial}d((\bar{x},\bar{y},\varphi(\bar{x},\bar{y})),C)+\widehat{\partial}d((\bar{x},\bar{y},\varphi(\bar{x},\bar{y})),D)\big{]},$

which yields in turn together with (4.6) the existence of the triples
$(z^{*},y^{*},\gamma)\in\widehat{\partial}d((\bar{x},\bar{y},\varphi(\bar{x},\bar{y})),\mathop{\rm
epi}\nolimits\varphi)$ and
$(u^{*},v^{*})\in\widehat{\partial}d((\bar{x},\bar{y});\mathop{\rm
gph}\nolimits G)$ satisfying

$\displaystyle\hskip
59.75095pt\big{[}x^{*}=a(z^{*}+u^{*}),~{}~{}0=a(y^{*}+v^{*}),~{}~{}\beta=a\gamma\big{]}$
$\displaystyle\Longrightarrow\;\big{[}p^{*}=\lambda x^{*}=\lambda
a(z^{*}+u^{*}),~{}~{}\lambda ay^{*}=-\lambda
av^{*},~{}~{}-1=\lambda\beta=a\lambda\gamma\big{]}.$

Therefore, $(\lambda az^{*},\lambda
ay^{*})\in\widehat{\partial}\varphi(\bar{x},\bar{y})$, $p^{*}\in\lambda
az^{*}+\widehat{D}^{*}G(\bar{x},\bar{y})(\lambda ay^{*})$, and so

$\displaystyle\widehat{\partial}\mu(\bar{x})\subset\bigcup_{(x^{*},y^{*})\in\widehat{\partial}\varphi(\bar{x},\bar{y})}\big{[}x^{*}+\widehat{D}^{*}G(\bar{x},\bar{y})(y^{*})\big{]}.$

Unifying the latter with with the opposite inclusion (3.3) in Theorem 3.3
justifies the claimed equality (4.3) for regular subgradients of marginal
functions.

It remains to verify equality (4.4) for singular regular subgradients. In
fact, the upper singular subdifferential estimate

$\displaystyle\widehat{\partial}^{\infty}\mu(\bar{x})\subset\bigcup_{(x^{*},y^{*})\in\widehat{\partial}^{\infty}\varphi(\bar{x},\bar{y})}x^{*}+\widehat{D}^{*}G(\bar{x},\bar{y})(y^{*})$

can be justified by using the same device as above for
$\widehat{\partial}\mu(\bar{x})$, while the lower estimate for
$\widehat{\partial}^{\infty}\mu(\bar{x})$ is (3.14) obtained in Theorem 3.4. ∎

Finally in this section, we construct an example of a convex marginal function
for which the results of Theorem 4.4 give us the exact calculation formulas
for both regular and singular regular subdifferentials, while known results in
this direction (obtained for the convex subdifferential of marginal functions)
are not applicable. Although regular subgradients and normals for convex
functions and sets reduce to the classical ones of convex analysis, we will
keep the hat-notation for them for notation consistency.

######  Example 4.5 (subgradient calculations for convex marginal functions).

Let $X=Y=\mathbb{R}^{2}$ and
$(\bar{x},\bar{y})=(0_{\mathbb{R}^{2}},0_{\mathbb{R}^{2}})$. Define

$f(y):=\left\\{\begin{array}[]{ll}0&{\rm if}~{}y=0_{\mathbb{R}^{2}},\\\
\infty&{\rm otherwise}\end{array}\right.$

and consider the marginal function $\mu(\cdot)$ in (1.1) with the data

$G(x):=\left\\{\begin{array}[]{ll}\mathbb{R}_{+}\times\\{0\\}&{\rm
if}~{}x=0_{\mathbb{R}^{2}},\\\ \emptyset&{\rm
otherwise}\end{array}\right.~{}~{}~{}{\rm
and}~{}~{}~{}\varphi(x,y):=f(y),\quad(x,y)\in X\times Y.$

Since
$\operatorname{dom}\varphi=\mathbb{R}^{2}\times\\{0_{\mathbb{R}^{2}}\\}$,
$\mathop{\rm gph}\nolimits
G=\\{0_{\mathbb{R}^{2}}\\}\times\mathbb{R}_{+}\times\\{0\\}$ with $\mathop{\rm
int}\nolimits(\mathop{\rm gph}\nolimits G)=\emptyset$, and $\varphi$ is
discontinuous at any point $(x_{0},y_{0})\in\mathop{\rm gph}\nolimits G$, the
result of [2, Theorem 4.2] cannot be applied. We also are not able to use [1,
Theorems 3.3 and 3.4] due to

$\operatorname{dom}\varphi-\mathop{\rm gph}\nolimits
G=\mathbb{R}^{2}\times\mathbb{R}_{-}\times\\{0\\}\neq X\times Y.$

Moreover, observe that [14, Theorem 4.56] is not applicable because $\varphi$
is discontinuous, the set
$\mathbb{R}_{+}(\operatorname{dom}\varphi-\mathop{\rm gph}\nolimits G)$ is not
a closed subspace of $X\times Y$, and we have $\mathop{\rm
ri}\nolimits(\operatorname{dom}\varphi)\cap\mathop{\rm
ri}\nolimits(\mathop{\rm gph}\nolimits G)=\emptyset$.

It is easy to see that the solution mapping (1.2) in this problem is

$\displaystyle M(x)=\left\\{\begin{array}[]{ll}\\{0_{\mathbb{R}^{2}}\\}&{\rm
if}~{}x=0_{\mathbb{R}^{2}},\\\ \emptyset&{\rm otherwise},\end{array}\right.$

which does not admit a local upper Lipschitzian selection at
$(0_{\mathbb{R}^{2}},0_{\mathbb{R}^{2}})$. Thus we are not familiar with any
result in the literature that can be applied to calculate the convex
subdifferential of the marginal function under consideration.

On the other hand, the direct calculations show that

$\displaystyle\text{ $\mathop{\rm
epi}\nolimits\varphi=\mathbb{R}^{2}\times\\{0_{\mathbb{R}^{2}}\\}\times\mathbb{R}_{+}$},$
$\displaystyle\widehat{N}((0_{\mathbb{R}^{2}},0_{\mathbb{R}^{2}},0);\mathop{\rm
gph}\nolimits
G\times\mathbb{R})=\mathbb{R}^{2}\times\mathbb{R}_{-}\times\mathbb{R}\times\\{0\\},$
$\displaystyle\widehat{N}((0_{\mathbb{R}^{2}},0_{\mathbb{R}^{2}},0);\mathop{\rm
epi}\nolimits\varphi)=\\{0_{\mathbb{R}^{2}}\\}\times\mathbb{R}^{2}\times\mathbb{R}_{-},$
$\displaystyle\widehat{N}((0_{\mathbb{R}^{2}},0_{\mathbb{R}^{2}},0);\mathop{\rm
epi}\nolimits\varphi\cap(\mathop{\rm gph}\nolimits
G\times\mathbb{R}))=\widehat{N}((0_{\mathbb{R}^{2}},0_{\mathbb{R}^{2}},0);\\{0_{\mathbb{R}^{2}}\\}\times\\{0_{\mathbb{R}^{2}}\\}\times\mathbb{R}_{+})$
$\displaystyle=\mathbb{R}^{2}\times\mathbb{R}^{2}\times\mathbb{R}_{-}.$

Observe furthermore the representation

$\displaystyle\hskip
42.67912pt\widehat{N}((0_{\mathbb{R}^{2}},0_{\mathbb{R}^{2}},0);\mathop{\rm
epi}\nolimits\varphi\cap(\mathop{\rm gph}\nolimits G\times\mathbb{R}))$
$\displaystyle=\widehat{N}((0_{\mathbb{R}^{2}},0_{\mathbb{R}^{2}},0);\mathop{\rm
epi}\nolimits\varphi)+\widehat{N}((0_{\mathbb{R}^{2}},0_{\mathbb{R}^{2}},0);\mathop{\rm
gph}\nolimits G\times\mathbb{R}),$

which ensures by [6, Theorem 3.3] that assumption $({\cal A}_{1})$ is
satisfied. Since $G$ clearly enjoys the semilocal nearly-isolated calmness
property at $(0_{\mathbb{R}^{2}},0_{\mathbb{R}^{2}})$ for every modulus $L>0$,
assumption ($\mathcal{A}_{2}$) holds too. Hence the equalities in (4.3) and
(4.4) follow from Theorem 4.4. We can also confirm this by the direct
calculations, which show that

$\displaystyle\mu(x)=\inf\big{\\{}f(y)\;\big{|}\;y\in
G(x)\big{\\}}=\left\\{\begin{array}[]{ll}0&{\rm if}~{}x=0_{\mathbb{R}^{2}},\\\
\infty&{\rm otherwise}\end{array}\right.$

and
$\widehat{\partial}^{\infty}\mu(0_{\mathbb{R}^{2}})=\widehat{\partial}\mu(0_{\mathbb{R}^{2}})=\mathbb{R}^{2}$
due to

$\widehat{N}(0_{\mathbb{R}^{2}};\operatorname{epi}\mu)=\widehat{N}(0_{\mathbb{R}^{2}};\\{0_{\mathbb{R}^{2}}\\}\times\mathbb{R}_{+})=\mathbb{R}^{2}\times\mathbb{R}_{-},$
$\widehat{\partial}^{\infty}\varphi(0_{\mathbb{R}^{2}},0_{\mathbb{R}^{2}})=\widehat{\partial}\varphi(0_{\mathbb{R}^{2}},0_{\mathbb{R}^{2}})=\\{0_{\mathbb{R}^{2}}\\}\times\mathbb{R}^{2},\;\mbox{
and}$
$\displaystyle\widehat{D}^{*}G(0_{\mathbb{R}^{2}},0_{\mathbb{R}^{2}})(y^{*}_{1},y^{*}_{2})=\left\\{\begin{array}[]{ll}\mathbb{R}^{2}&{\rm
if}~{}y^{*}_{1}\geq 0,\\\ \emptyset&{\rm otherwise}.\end{array}\right.$

##  5 Applications to Variational Calculus

This section develops new calculus rules as equalities for regular
subgradients and their singular counterparts in general normed spaces.
Needless to saying that such a variational subdifferential calculus plays a
crucial role in applications of variational analysis to structured problems of
optimization and related areas.

The obtained calculus rules are mainly derived from the calculation results
for marginal functions established in Section 4 with the systematical usage of
the calmness and metric qualification conditions. The calculus rules for both
regular and singular regular subgradients obtained below under the metric
qualification conditions seem to be completely new, while some of the results
for regular subgradients involving calmness (2.6) can be found in [15] under
the more restrictive local Lipschitz continuity. Note that the chain rules of
the inclusion type derived in [6] under metric qualification conditions in
Asplund spaces are fully different from ours while providing upper estimated
of regular and singular regular subdifferentials in terms of the limiting ones
by Mordukhovich.

When the constraint mapping $G:=g:X\to Y$ in (1.1) is single-valued, the
marginal function $\mu(\cdot)$ reduces to the (generalized) composition

$\displaystyle\mu(x)=\left(\varphi\circ
g\right)(x):=\varphi\big{(}x,g(x)\big{)},\quad x\in X,$ (5.1)

which is the standard one $\varphi(g(x))$ when $\varphi$ does not depend on
$x$.

First we present chain rules for regular and singular regular subdifferentials
of generalized compositions (5.1).

######  Theorem 5.1 (subdifferentiation of generalized compositions).

Let $\varphi:X\times Y\to\mathbb{\bar{R}}$ be finite at $(\bar{x},\bar{y})$
with $\bar{y}:=g(\bar{x})$, and let $g:X\to Y$ be calm at $\bar{x}$. Assume
that $\varphi$ is Fréchet differentiable at $(\bar{x},\bar{y})$ with
$\nabla\varphi(\bar{x},\bar{y}):=(\nabla_{x}\varphi(\bar{x},\bar{y}),\nabla_{y}\varphi(\bar{x},\bar{y}))$.
Then the following chain rules hold for generalized compositions (5.1):

$\begin{array}[]{ll}\widehat{\partial}\left(\varphi\circ
g\right)(\bar{x})&=\nabla_{x}\varphi(\bar{x},\bar{y})+\widehat{D}^{*}g(\bar{x})\left(\nabla_{y}\varphi(\bar{x},\bar{y})\right)\\\
&=\nabla_{x}\varphi(\bar{x},\bar{y})+\widehat{\partial}\langle\nabla_{y}\varphi(\bar{x},\bar{y}),g\rangle(\bar{x}),\end{array}$
$\widehat{\partial}^{\infty}\left(\varphi\circ g\right)(\bar{x})=0.$

###### Proof.

This follows from Theorem 4.1 specified for (5.1) with observing that the
calmness condition (2.6) imposed on $G=g$ yields the nearly-isolated calmness
in (a) and (b) of Theorem 3.4, while ensuring the regular coderivative
scalarization (2.7) and thus the fulfillment of both claimed formulas by
(4.2). ∎

As consequences of Theorem 5.1, we get the following product and quotient
rules under the calmness assumptions.

######  Corollary 5.2 (products and quotient rules).

Let the functions $\varphi_{1},\varphi_{2}:X\to\mathbb{\bar{R}}$ be calm at
$\bar{x}$. The following assertions hold:

(i)

$\widehat{\partial}\left(\varphi_{1}\cdot\varphi_{2}\right)(\bar{x})=\widehat{\partial}(\varphi_{2}(\bar{x})\varphi_{1}+\varphi_{1}(\bar{x})\varphi_{2})(\bar{x})$
and
$\widehat{\partial}^{\infty}\left(\varphi_{1}\cdot\varphi_{2}\right)(\bar{x})=0$.
If in addition $\varphi_{1}$ is Fréchet differentiable at $\bar{x}$, then

$\displaystyle\widehat{\partial}\left(\varphi_{1}\cdot\varphi_{2}\right)(\bar{x})=\nabla\varphi_{1}(\bar{x})\varphi_{2}(\bar{x})+\widehat{\partial}\left(\varphi_{1}(\bar{x})\varphi_{2}\right)(\bar{x}).$

(ii)

Assume that $\varphi_{2}(\bar{x})\neq 0$. Then

$\displaystyle\widehat{\partial}\left(\varphi_{1}/\varphi_{2}\right)(\bar{x})=\dfrac{\widehat{\partial}\left(\varphi_{2}(\bar{x})\varphi_{1}-\varphi_{1}(\bar{x})\varphi_{2}\right)(\bar{x})}{[\varphi_{2}(\bar{x})]^{2}},\quad\widehat{\partial}^{\infty}\left(\varphi_{1}/\varphi_{2}\right)(\bar{x})=0.$

If in addition $\varphi_{1}$ is Fréchet differentiable at $\bar{x}$, then

$\displaystyle\widehat{\partial}\left(\varphi_{1}/\varphi_{2}\right)(\bar{x})=\dfrac{\nabla\varphi_{1}(\bar{x})\varphi_{2}(\bar{x})+\widehat{\partial}\left(-\varphi_{1}(\bar{x})\varphi_{2}\right)(\bar{x})}{[\varphi_{2}(\bar{x})]^{2}}.$

(iii)

Let $\varphi:X\to\mathbb{\bar{R}}$ be calm at $\bar{x}$ with
$\varphi(\bar{x})\neq 0$. Then

$\displaystyle\widehat{\partial}(1/\varphi)(\bar{x})=-\dfrac{\widehat{\partial}^{+}\varphi(\bar{x})}{[\varphi(\bar{x})]^{2}},\quad\widehat{\partial}^{\infty}\left(1/\varphi_{2}\right)(\bar{x})=0.$

###### Proof.

To verify (i), observe that $\varphi_{1}\cdot\varphi_{2}$ is represented as
composition (5.1) with $\varphi:\mathbb{R}^{2}\to\mathbb{R}$ and
$g:X\to\mathbb{R}^{2}$ defined by $\varphi(y_{1},y_{2}):=y_{1}\cdot y_{2}$ and
$g(x):=(\varphi_{1}(x),\varphi_{2}(x))$. Thus Theorem 5.1 yields the first
equalities in (i), which implies the second one due to the elementary sum rule
from [10, Proposition 1.107(i)].

The proof of (ii) is similar with $\varphi(y_{1},y_{2}):=y_{1}/y_{2}$ and the
same mapping $g$ in (5.1), while (iii) is a special case of (ii) with
$\varphi_{1}:=1$ and $\varphi_{2}:=\varphi$. ∎

The next results provide new sum rules for both regular and singular regular
subdifferentials of nonsmooth functions under the simultaneous fulfillment of
the calmness and metric qualification conditions.

######  Theorem 5.3 (sum rules for nonsmooth functions).

Let $\varphi_{1},\varphi_{2}\colon X\to\mathbb{\bar{R}}$ be extended-real-
valued functions such that $\varphi_{2}$ is calm at $\bar{x}$ and that
$\varphi(x,y):=\varphi_{1}(x)+y$ for all $(x,y)\in X\times\mathbb{R}$ is
finite at $(\bar{x},\varphi_{2}(\bar{x}))$. Suppose in addition that the sets
${\rm epi}\,\varphi$ and $\mathop{\rm
gph}\nolimits\varphi_{2}\times\mathbb{R}$ satisfy the metric qualification
condition ($Q_{2}$) for $\partial^{\textasteriskcentered}=\widehat{\partial}$
at
$(\bar{x},\varphi_{2}{(\bar{x})},\varphi_{1}(\bar{x})+\varphi_{2}(\bar{x}))$.
Then we have

$\displaystyle\widehat{\partial}(\varphi_{1}+\varphi_{2})(\bar{x})=\widehat{\partial}\varphi_{1}(\bar{x})+\widehat{\partial}\varphi_{2}(\bar{x}),$
$\displaystyle\widehat{\partial}^{\infty}(\varphi_{1}+\varphi_{2})(\bar{x})=\widehat{\partial}^{\infty}\varphi_{1}(\bar{x}).$

###### Proof.

It follows from (5.1) that

$\mu(x)=\left(\varphi\circ\varphi_{2}\right)(x)=\varphi(x,\varphi_{2}(x))=\varphi_{1}(x)+\varphi_{2}(x),$

and thus the result follows by applying Theorem 4.4 with $G:=\varphi_{2}$. ∎

######  Theorem 5.4 (chain rules for subdifferentials).

Let $h:Y\to\mathbb{\bar{R}}$ be finite at $\bar{y}$ with $\bar{y}:=g(\bar{x})$
and $g:X\to Y$ be locally calm at $\bar{x}$. Suppose that the sets
$X\times\mathop{\rm epi}\nolimits h$ and $\mathop{\rm gph}\nolimits
g\times\mathbb{R}$ satisfies condition ($Q_{2}$) for $\widehat{\partial}$ at
$(\bar{x},\bar{y},h(\bar{y})$. Then

$\displaystyle{\widehat{\partial}}\left(h\circ
g\right)(\bar{x})=\bigcup_{y^{\ast}\in{\widehat{\partial}}h(\bar{y})}\widehat{\partial}\langle
y^{\ast},g\rangle(\bar{x}),$ (5.2)
$\displaystyle{\widehat{\partial}^{\infty}}\left(h\circ
g\right)(\bar{x})=\bigcup_{y^{\ast}\in{\widehat{\partial}^{\infty}}h(\bar{y})}\widehat{\partial}\langle
y^{\ast},g\rangle(\bar{x}).$ (5.3)

###### Proof.

For the marginal function $\mu(\cdot)$ in (1.1), we consider $\varphi:X\times
Y\to\mathbb{R}\cup\\{\infty\\}$ defined by $\varphi(x,y):=h(y)$ for every
$y\in Y$, and $G:=g$. Then both chain rules (5.2) and (5.3) follow from the
corresponding results of Theorem 4.4. ∎

To get an effective consequence of Theorem 5.4 in the case of real-valued
functions $g$, the next lemma of its own interest is useful. The first
equivalence below can be deduced from [10, Theorem 1.86] while we give
another, variational proof. The second equivalence seems never been mentioned
in the literature.

######  Lemma 5.5 (regular normals to epigraphs and graphs).

Let $f:X\to\mathbb{\bar{R}}$ be finite at $\bar{x}$, let
$(\bar{x},\alpha)\in\mathop{\rm epi}\nolimits f$, and let
$0\neq\lambda\in\mathbb{R}$. Then the following hold:

$(x^{*},-\lambda)\in\widehat{N}\big{(}(\bar{x},\alpha);\mathop{\rm
epi}\nolimits
f\big{)}\Longleftrightarrow\lambda>0,\,\alpha=f(\bar{x}),\,x^{*}\in\widehat{\partial}(\lambda
f)(\bar{x}),$ (5.4)
$(x^{*},-\lambda)\in\widehat{N}\big{(}(\bar{x},f(\bar{x}));\mathop{\rm
gph}\nolimits f)\Longleftrightarrow x^{*}\in\widehat{\partial}\left(\lambda
f\right)(\bar{x}\big{)}.$ (5.5)

###### Proof.

Observe that implication “$\Longleftarrow$” is straightforward in both cases
of (5.4) and (5.5). To verify “$\Longrightarrow$” in (5.4), take
$(x^{*},-\lambda)\in\widehat{N}((\bar{x},\alpha);\mathop{\rm epi}\nolimits f)$
and find by Lemma 2.4 a Fréchet differentiable function
$s:X\times\mathbb{R}\rightarrow\mathbb{R}$ with $\nabla
s(\bar{x},\alpha)=(x^{\ast},-\lambda)$ and $s(x,r)\leq s(\bar{x},\alpha)=0$
for all $(x,r)\in\mathop{\rm epi}\nolimits f$. It follows from the Fréchet
differentiability of $s(\cdot)$ that for every $\varepsilon>0$ there exists
$0<\delta_{1}$ such that

$|s(x,r)-\langle(x^{\ast},-\lambda),(x-\bar{x},r-\alpha)\rangle\lvert\leq\varepsilon(\|x-\bar{x}\|+|r-\alpha|)$

for every $(x,r)\in\mathbb{B}((\bar{x},\alpha),\delta_{1})$. This yields

$\langle(x^{\ast},-\lambda),(x-\bar{x},r-\alpha)\rangle\leq\varepsilon(\|x-\bar{x}\|+|r-\alpha|)$
(5.6)

whenever $(x,r)\in\mathbb{B}((\bar{x},\alpha),\delta_{1})\cap\mathop{\rm
epi}\nolimits f$. Note that $(\bar{x},\beta+\alpha)\in\mathop{\rm
epi}\nolimits f$ for any $\beta\geq 0$, and so for all small $\beta$ we have
$(\bar{x},\beta+\alpha)\in\mathbb{B}((\bar{x},\alpha),\delta_{1})\cap\mathop{\rm
epi}\nolimits f$ and $\langle\beta,-\lambda\rangle\leq\varepsilon\beta$. Thus
$-\lambda\leq\varepsilon$, and hence $\lambda>0$ since $\varepsilon>0$ was
chosen arbitrarily.

Next we show that $\alpha=f(\bar{x})$. Indeed, suppose on the contrary that
$\alpha>f(\bar{x})$. Then for any $\beta>0$ satisfying $\beta<\delta_{1}$ and
$\alpha-\beta>f(\bar{x})$, we have
$(\bar{x},\alpha-\beta)\in\mathbb{B}((\bar{x},\alpha),\delta_{1})\cap\mathop{\rm
epi}\nolimits f$, and thus it follows from (5.6) that
$\lambda\beta\leq\varepsilon\beta$. This brings us to the contradiction
$\lambda\leq 0$ by the arbitrary choice of $\varepsilon>0$.

To verify now implication “$\Longrightarrow$” in (5.5), pick
$(x^{*},-\lambda)\in\widehat{N}((\bar{x},f(\bar{x}));\mathop{\rm gph}\nolimits
f)$ with $\lambda\neq 0$. It follows from the definition that for each
$\varepsilon>0$ there exists $\delta>0$ such that, whenever
$x\in\mathbb{B}(\bar{x},\delta)$, we have

$\displaystyle\langle
x^{*},x-\bar{x}\rangle+\langle-\lambda,f(x)-f(\bar{x})\rangle\leq\varepsilon\left(\|x-\bar{x}\|+\|f(x)-f(\bar{x})\|\right).$
(5.7)

To proceed further, consider the following two cases.

Case 1, where $\lambda>0$. Suppose without loss of generality that
$\varepsilon<\lambda$. For any $x\in\mathbb{B}(\bar{x},\delta)$ with
$(x,\alpha)\in\mathop{\rm epi}\nolimits f$, we deduce from (5.7) that

$\displaystyle\langle
x^{*},x-\bar{x}\rangle-\langle\lambda,\alpha-f(\bar{x})\rangle$
$\displaystyle\leq-\langle\lambda,\alpha-f(x)\rangle+\varepsilon\left(\|x-\bar{x}\|+\|f(x)-f(\bar{x})\|\right)$
$\displaystyle\leq\varepsilon\left(\|x-\bar{x}\|+\|\alpha-f(\bar{x})\|\right).$

This yields $(x^{*},-\lambda)\in\widehat{N}((\bar{x},f(\bar{x}));\mathop{\rm
epi}\nolimits f)$, and thus it follows from (5.4) that
$x^{*}\in\widehat{\partial}\left(\lambda f\right)(\bar{x})$, which justifies
(5.5) in this case.

Case 2, where $\lambda<0$. We may assume that $\varepsilon<-\lambda$. For any
$x\in\mathbb{B}(\bar{x},\delta)$ with $(x,\alpha)\in\mathop{\rm
epi}\nolimits(-f)$, we derive from (5.7) that

$\displaystyle\langle
x^{*},x-\bar{x}\rangle-\langle-\lambda,\alpha-(-f)(\bar{x})\rangle$
$\displaystyle\leq\langle\lambda,\alpha-(-f)(x)\rangle+\varepsilon\left(\|x-\bar{x}\|+\|f(x)-f(\bar{x})\|\right)$
$\displaystyle\leq\varepsilon\left(\|x-\bar{x}\|+\|\alpha-(-f)(\bar{x})\|\right).$

This yields $(x^{*},\lambda)\in\widehat{N}((\bar{x},(-f)(\bar{x}));\mathop{\rm
epi}\nolimits(-f))$, which implies in turn that
$x^{*}\in\widehat{\partial}\left(\lambda f\right)(\bar{x})$ due to (5.4) and
thus completes the proof of the lemma. ∎

Finally, we are ready to derive, as a consequence of Theorem 5.4, the desired
chain rules for both regular and singular regular subdifferentials of
compositions with real-valued inner functions defined on normed spaces.

######  Corollary 5.6 (chain rules with real-valued inner functions).

Let $h:\mathbb{R}\to\mathbb{\bar{R}}$ be finite at $\bar{y}$ with
$\bar{y}:=g(\bar{x})$, and let $g:X\to\mathbb{R}$ be calm at $\bar{x}$.
Suppose that the sets $X\times\mathop{\rm epi}\nolimits h$ and $\mathop{\rm
gph}\nolimits g\times\mathbb{R}$ satisfy condition ($Q_{2}$) for
$\widehat{\partial}$ at $(\bar{x},\bar{y},h(\bar{y}))$. Then

$\displaystyle~{}~{}{\widehat{\partial}}\left(h\circ
g\right)(\bar{x})=\left(\bigcup_{\lambda\in{\widehat{\partial}}h(\bar{y}),\,\lambda\geq
0}\lambda\widehat{\partial}g(\bar{x})\right)\bigcup\left(\bigcup_{\lambda\in{\widehat{\partial}}h(\bar{y}),\,\lambda<0}\lambda\widehat{\partial}^{+}g(\bar{x})\right),$
(5.8) $\displaystyle{\widehat{\partial}^{\infty}}\left(h\circ
g\right)(\bar{x})=\left(\bigcup_{\lambda\in{\widehat{\partial}^{\infty}}h(\bar{y}),\,\lambda\geq
0}\lambda\widehat{\partial}g(\bar{x})\right)\bigcup\left(\bigcup_{\lambda\in{\widehat{\partial}^{\infty}}h(\bar{y}),\,\lambda<0}\lambda\widehat{\partial}^{+}g(\bar{x})\right).$
(5.9)

###### Proof.

Observe that, by the scalarization formula (2.7) under the imposed calmness
assumption, the subgradient set standing on the right-hand sides of (5.2) and
(5.3) reduces to $D^{*}g(\bar{x})(y^{*})$ with $y^{*}:=\lambda\in\mathbb{R}$
due to $Y=\mathbb{R}$ in the setting of the corollary. The coderivative
definition (2.5) tells us that $x^{*}\in\widehat{D}^{*}g(\bar{x})(\lambda)$ if
and only if $(x^{*},-\lambda)\in\widehat{N}((\bar{x},\bar{y});\mathop{\rm
gph}\nolimits g)$. Consider the following two cases.

Case 1, where $\lambda=0$. Since $g$ is calm at $\bar{x}$, we have $x^{*}=0$
by scalarization (2.7). Then the sets on the right-hand sides of (5.8) and
(5.9) are obviously zeros, while the left-hand sides reduce to zeros by (5.2)
and (5.3) in Theorem 5.4.

Case 2, where $\lambda\neq 0$. In this case, it follows from (5.5) in Lemma
5.5 that $x^{*}\in\widehat{\partial}(\lambda g)(\bar{x})$. Using the latter
inclusion and applying Theorem 5.4 with taking into account definition (2.2)
of the upper regular subdifferential, we arrive at both formulas (5.8), (5.9)
and thus complete the proof of the corollary. ∎

##  6 Bilevel Programming in Asplund Spaces

This section is mainly devoted to deriving necessary optimality conditions in
nonsmooth models of optimistic bilevel programming. Developing the marginal
functions approach to bilevel programs and having in hands the results
obtained above, we invoke here the following two major ingredients that are
new in comparison with the previous sections. The first one is dealing with
not general normed spaces while with the class of Asplund spaces, which is
fairly broad (containing, in particular, every reflexive Banach spaces) and
has been well recognized and applied in variational analysis. The second major
ingredient of our developments is involving, along with regular
normal/subdifferential constructions, the limiting normals and subgradients by
Mordukhovich.

Recall that a Banach space $X$ is Asplund if any separable subspace of $X$ has
a separable dual. This class of spaces have been comprehensively investigated
in geometric theory of Banach spaces admitting various characterizations and a
great many applications to different aspects of variational analysis,
optimization, and optimal control; see both volumes of [10] and the references
therein. Unless otherwise stated, all the spaces considered below are assumed
to be Asplund.

Next we recall the limiting generalized differential constructions for general
sets and functions introduced by the third author and largely used below; see,
e.g., the books [10, 18, 20, 21] and their bibliographies.

######  Definition 6.1 (limiting normals and subgradients).

Consider a nonempty set $S\subset X$ and a function
$f:X\rightarrow\bar{\mathbb{R}}$ with ${\rm dom}\,f<\infty$. Then:

  * (i)

The limiting normal cone to $S$ at $\bar{x}\in S$ is defined by

$\displaystyle N(\bar{x};S):=\mathop{\rm
Limsup}\nolimits\limits_{x\rightarrow\bar{x}}\widehat{N}({x};S).$ (6.1)

  * (ii)

The limiting subdifferential of $f$ at $\bar{x}\in{\rm dom}\,f$ is defined by

$\displaystyle\partial f(\bar{x}):=\big{\\{}x^{*}\in
X^{*}\;\big{|}\;(x^{*},-1)\in N\big{(}(\bar{x},f(\bar{x})\big{)};\mathop{\rm
epi}\nolimits f)\big{\\}}.$ (6.2)

  * (iii)

The singular limiting subdifferential of $f$ at
$\bar{x}\in\operatorname{dom}f$ is defined by

$\displaystyle\partial^{\infty}f(\bar{x}):=\big{\\{}x^{*}\in
X^{*}\;\big{|}\;(x^{*},0)\in N\big{(}(\bar{x},f(\bar{x}));\mathop{\rm
epi}\nolimits f\big{)}\big{\\}}.$ (6.3)

Definition 6.1 presents geometric constructions of limiting and singular
limiting subgradients. There are several equivalent representations of the
subdifferentials (6.2) and (6.3) that can be founded in the aforementioned
books but are not employed in what follows. It is important to emphasize that
the sets in (6.1)–(6.3) are often nonconvex while enjoying comprehensive
calculus rules based on variational/extremal principles of variational
analysis.

Now we proceed with formulating the bilevel programming model of our study.
The lower-level problem of parametric optimization is the following:

$\displaystyle\min\varphi(x,y)\quad\text{subject to $y\in G(x)$},$ (6.4)

where the lower-level cost function $\varphi:X\times Y\to\mathbb{\bar{R}}$
depend on the decision variable $y\in Y$ and parameter $x\in X$, and where
$G:X\rightrightarrows Y$ is the parameter-dependent constraint mapping. The
lower-level optimal value function for (6.4) is the marginal function
$\mu(\cdot)$ defined in (1.1), and the corresponding lower-level solution
mapping $M(\cdot)$ is taken from (1.2).

The upper-level problem is formulated as follows:

$\displaystyle\min\psi(x,y)\quad\text{subject to $y\in M(x)$ for each
$x\in\Omega\subset X$},$

where $\psi:X\times Y\to\mathbb{R}$ is the upper-level cost function, and
where the underlying, most involved constraint $y\in M(x)$ is formed by the
solution mapping $M(\cdot)$ associated with the lower-level problem (6.4).

Using the marginal function approach, we finally define the hierarchical
optimization problem known as the optimistic bilevel model (see, e.g., [3, 4,
12, 22] with the discussions and references therein):

$\displaystyle\min\vartheta(x)\quad\text{subject to $x\in\Omega$, where
$\vartheta(x):=\inf\left\\{\psi(x,y)\,\,|\,\,y\in M(x)\right\\}$}.$ (6.5)

For simplicity, we consider below the case of (6.5) with $\Omega=X$ and

$\displaystyle G(x):=\big{\\{}y\in Y\;\big{|}\;g_{i}(x,y)\leq 0\;\mbox{ for
all }\;i=1,\ldots,p\big{\\}}.$ (6.6)

Having this in mind, we can rewrite (6.5) in the following form:

$\displaystyle\min\psi(x,y)\quad\text{subject to $g_{i}(x,y)\leq 0$,
$i=1,\ldots,p$, and $\varphi(x,y)\leq\mu(x)$,}$ (6.7)

where the marginal function $\mu(\cdot)$ is intrinsically nonsmooth and
generally nonconvex even for nice initial data $\varphi$, $\psi$, and $g_{i}$.
It has been realized that standard constraint qualifications fail to fulfill
in optimal solutions of (6.7). In this setting, a new constraint qualification
was introduced in [22] under the name of “partial calmness”. To define it,
consider the perturbed version of (6.7) given by

$\begin{array}[]{ll}\min\psi(x,y)\quad\text{subject to $g_{i}(x,y)\leq 0$,
$i=1,\ldots,p$,}\\\ \text{and $\varphi(x,y)-\mu(x)+\nu=0$ as
$\nu\in\mathbb{R}$.}\end{array}$ (6.8)

######  Definition 6.2 (partial calmness).

Problem (6.7) is partially calm at its local optimal solution
$(\bar{x},\bar{y})$ with modulus $\kappa>0$ if there exists a neighborhood $U$
of $(\bar{x},\bar{y},0)\in X\times Y\times\mathbb{R}$ such that

$\displaystyle\psi(x,y)-\psi(\bar{x},\bar{y})+\kappa|\nu|\geq 0$

for all the triples $(x,y,\nu)\in U$ feasible to (6.8).

We refer the reader to [3, 4, 11, 12, 22] for various explicit conditions
ensuring partial calmness. The essence of partial calmness is the possibility
of exact penalization of the troublesome constraint $\varphi(x)\leq\mu(x)$ in
(6.7), which was observed in [22] in finite-dimensional spaces. The following
statement is taken from [12, Proposition 7.8.12], where the proof holds in
general normed spaces.

######  Lemma 6.3 (exact penalization).

Let (6.7) be partially calm at its local optimal solution $(\bar{x},\bar{y})$
with modulus $\kappa>0$, and let $\psi$ be continuous at $(\bar{x},\bar{y})$.
Then $(\bar{x},\bar{y})$ is a local optimal solution to the penalized problem

$\begin{array}[]{ll}&\min\psi(x,y)+\kappa\big{(}\varphi(x,y)-\mu(x)\big{)}\\\
&{\rm subject\;to}\;g_{i}(x,y)\leq 0,\;i=1,\ldots,p.\end{array}$

The next lemma, valid in normed spaces, is taken from [15, Proposition 4.1].

######  Lemma 6.4 (optimality conditions for difference functions).

Let $\varphi_{1},\varphi_{2}:X\to\mathbb{\bar{R}}$ be functions finite at
$\bar{x}$, and let $\widehat{\partial}\varphi_{2}(\bar{x})\neq\emptyset$. Then
any local minimizer $\bar{x}$ of the difference function
$\varphi_{1}-\varphi_{2}$ satisfies the regular subdifferential inclusion

$\displaystyle\widehat{\partial}\varphi_{1}(\bar{x})\subset\widehat{\partial}\varphi_{2}(\bar{x}).$
(6.9)

The last lemma here, taken from [10, Theorems 3.8 and 3.86], gives us a useful
representation of limiting normals to sets defined by inequality constraints.

######  Lemma 6.5 (limiting normals to inequality constraints).

Let $g_{i}\colon X\times Y$ be locally Lipschitzian around
$({\bar{x}},\bar{y})$, and let $G_{i}:=\\{(x,y)\in X\times
Y\,\,|\,\,g_{i}(x,y)\leq 0\\}$ for all $i=1,\ldots,p$. Denote
$I(\bar{x},\bar{y}):=\\{i\in\\{1,\ldots,p\\}\,\,|\,\,g_{i}(\bar{x},\bar{y})=0\\}$
and impose the constraint qualification condition

$\begin{array}[]{ll}\displaystyle\Big{[}\sum_{i\in
I(\bar{x},\bar{y})}\lambda_{i}x^{*}_{i}=0,\,x_{i}^{*}\in\partial
g_{i}(\bar{x},\bar{y}),\,\lambda_{i}\geq 0,\,i\in I(\bar{x},\bar{y})\Big{]}\\\
\Longrightarrow\big{[}\lambda_{i}=0\;\mbox{ for all }\;i\in
I(\bar{x},\bar{y})\big{]}.\end{array}$ (6.10)

Then for any $x^{*}\in N\left(\bar{x};\bigcap_{i=1}^{p}\Omega_{i}\right)$ we
have the inclusion

$\displaystyle x^{*}\in\sum_{i\in I(\bar{x},\bar{y})}\lambda_{i}\partial
g_{i}(\bar{x},\bar{y})$

with $\lambda_{i}\geq 0$ whenever $i\in I(\bar{x},\bar{y})$.

Now we are ready to derive necessary optimality conditions for optimistic
bilevel programs in arbitrary Asplund spaces.

######  Theorem 6.6 (necessary optimality conditions for bilevel programs).

Let $(\bar{x},\bar{y})$ be a local optimal solution to the bilevel program
(6.7), where $\varphi$ is lower semicontinuous while $g_{i}$ for
$i=1,\ldots,p$ are locally Lipschitzian around $(\bar{x},\bar{y})$, where
$\psi$ is Fréchet differentiable at this point, and where (6.7) is partially
calm at $(\bar{x},\bar{y})$ with modulus $\kappa>0$. Suppose in addition that
the constraint qualification (6.10) holds, that
$\widehat{\partial}\mu(\bar{x})\neq\emptyset$, and that

  * $(\cal{A}$)

the sets ${\rm epi}\,\varphi$ and $\mathop{\rm gph}\nolimits
G\times\mathbb{R}$ for $G$ from (6.6) satisfy the metric qualification
condition ($Q_{2}$) for $\partial^{\textasteriskcentered}=\partial$ at
$(\bar{x},\bar{y},\varphi(\bar{x},\bar{y}))$.

Then for each $u^{*}\in\widehat{\partial}\mu(\bar{x})$, there exist
multipliers $\lambda_{i}$ and $\beta_{i}$ with $i=1,\ldots,p$ such that we
have the following necessary optimality conditions:

$\displaystyle(u^{*},0)\in\partial\varphi(\bar{x},\bar{y})+\sum_{i=1}^{p}\beta_{i}\partial
g_{i}(\bar{x},\bar{y}),$ (6.11)
$\displaystyle(u^{*},0)\in\kappa^{-1}\nabla\psi(\bar{x},\bar{y})+\partial\varphi(\bar{x},\bar{y})+\sum_{i=1}^{p}\lambda_{i}\partial
g_{i}(\bar{x},\bar{y}),$ (6.12) $\displaystyle\lambda_{i},\beta_{i}\geq
0,\quad\lambda_{i}g_{i}(\bar{x},\bar{y})=\beta_{i}g_{i}(\bar{x},\bar{y})=0\;\mbox{
as }\;i=1,\ldots,p.$ (6.13)

###### Proof.

From Lemma 6.3 and the infinite constraint penalization via the indicator
function $\delta(\cdot;\mathop{\rm gph}\nolimits G)$, we get that the pair
$(\bar{x},\bar{y})$ be a local optimal solution to the single-objective
unconstrained problem

$\displaystyle\min\big{[}\psi(x,y)+\kappa\big{(}\varphi(x,y)-\mu(x)\big{)}+\delta(\cdot,\mathop{\rm
gph}\nolimits G)\big{]}.$ (6.14)

Using the necessary optimality condition (6.9) from Lemma 6.4 for the
difference function in (6.14) under $\widehat{\mu}(\bar{x})\neq\emptyset$
together with the elementary sum rule for regular subgradients in [10,
Proposition 1.107(i)], we have

$\begin{array}[]{ll}\big{(}\kappa\widehat{\partial}\mu(\bar{x}),0\big{)}&\subset\nabla\psi(\bar{x},\bar{y})+\widehat{\partial}\big{(}\kappa\varphi(\cdot)+\delta(\cdot,\mathop{\rm
gph}\nolimits G)\big{)}(\bar{x},\bar{y})\\\
&\subset\nabla\psi(\bar{x},\bar{y})+\kappa{\partial}\big{(}\varphi(\cdot)+\delta(\cdot,\mathop{\rm
gph}\nolimits G)\big{)}(\bar{x},\bar{y}).\end{array}$ (6.15)

Moreover, recalling that $\mu(x)=\inf_{y\in
Y}(\varphi(\cdot)+\delta(\cdot,\mathop{\rm gph}\nolimits G))(x,y)$ and then
applying [18, Theorem 4.47] tell us that

$\begin{array}[]{ll}\big{(}\widehat{\partial}\mu(\bar{x}),0\big{)}&\subset\widehat{\partial}\big{(}\varphi(\cdot)+\delta(\cdot,\mathop{\rm
gph}\nolimits G)\big{)}(\bar{x},\bar{y})\\\
&\subset{\partial}\big{(}\varphi(\cdot)+\delta(\cdot,\mathop{\rm gph}\nolimits
G)\big{)}(\bar{x},\bar{y}).\end{array}$ (6.16)

Observing that $\partial\delta(s;S)=N(s;S)$ for any set $S$ with $s\in S$ and
using

$\displaystyle\mathop{\rm epi}\nolimits(\varphi+\delta_{\mathop{\rm
gph}\nolimits G})=\mathop{\rm epi}\nolimits\varphi\cap[\mathop{\rm
gph}\nolimits
G\times\mathbb{R}]\quad\text{and}\quad(\varphi+\delta_{\mathop{\rm
gph}\nolimits G})(\bar{x},\bar{y})=\varphi(\bar{x},\bar{y}),$

we therefore arrive at the equality

$\displaystyle N\big{(}(\bar{x},\bar{y},(\varphi+\delta_{\mathop{\rm
gph}\nolimits G})(\bar{x},\bar{y}));\mathop{\rm
epi}\nolimits\varphi\cap[\mathop{\rm gph}\nolimits G\times\mathbb{R}]\big{)}$
$\displaystyle=N\big{(}(\bar{x},\bar{y},\varphi(\bar{x},\bar{y}));\mathop{\rm
epi}\nolimits\varphi\cap[\mathop{\rm gph}\nolimits G\times\mathbb{R}]\big{)}.$

Since assumption $(\cal A)$ is satisfied, the result of [6, Theorem 3.1]
ensures that

$\displaystyle N\big{(}(\bar{x},\bar{y},\varphi(\bar{x},\bar{y}));\mathop{\rm
epi}\nolimits\varphi\cap[\mathop{\rm gph}\nolimits G\times\mathbb{R}]\big{)}$
$\displaystyle=N\big{(}(\bar{x},\bar{y},\varphi(\bar{x},\bar{y})\big{)};\mathop{\rm
epi}\nolimits\varphi)+N\big{(}(\bar{x},\bar{y},\varphi(\bar{x},\bar{y}));\mathop{\rm
gph}\nolimits G\times\mathbb{R}\big{)}.$

Combining the latter with (6.15) and (6.16) brings us to

$\displaystyle\begin{cases}\big{(}\kappa\widehat{\partial}\mu(\bar{x}),0\big{)}&\subset\nabla\psi(\bar{x},\bar{y})+\kappa{\partial}\varphi(\bar{x},\bar{y})+N\big{(}(\bar{x},\bar{y});\mathop{\rm
gph}\nolimits G\big{)},\\\
\big{(}\widehat{\partial}\mu(\bar{x}),0\big{)}&\subset{\partial}\varphi(\bar{x},\bar{y})+N\big{(}(\bar{x},\bar{y});\mathop{\rm
gph}\nolimits G\big{)}.\end{cases}$

Employing finally Lemma 6.5 under the assumed constraint qualification (6.10),
we readily get the existence of $u^{*}\in\widehat{\partial}\mu(\bar{x})$
together with multipliers $\lambda_{i}$ and $\beta_{i}$ satisfying all the
conditions in (6.11)–(6.13) and thus complete the proof. ∎

Now we present two remarks comparing Theorem 6.6 with previous results in this
direction and discussing the major assumptions imposed.

######  Remark 6.7 (discussions on optimality conditions).

Several types of necessary optimality conditions for optimistic bilevel
programs have been obtained in the literature; see, e.g., [3, 4, 11, 12, 22]
with further references and discussions. The optimality conditions derived
therein are all in finite-dimensional spaces, while Theorem 6.6 is established
in general Asplund space setting. The closest statement to Theorem 6.6 is
given in [12, Theorem 7.9.3], where the same necessary optimality conditions
(6.11)–(6.13) are also obtained while under essentially weaker assumptions
even in finite dimensions. Indeed, we do not require here, in contrast to
[12], that the lower-level cost $\varphi$ is locally Lipschitzian and that the
solution mapping $M$ is inner semicontinuous. Moreover, our assumption (${\cal
A}$) significantly improves the qualification conditions in [12, Theorem
7.9.3]. Observe finally that Theorem 6.6 is free of SNC/PSNC assumptions (see
[10]), which are usually imposed in deriving necessary optimality conditions
for non-Lipschitzian infinite-dimensional problems.

The next remark discusses the two major assumptions of Theorem 6.6.

######  Remark 6.8 (discussing major assumptions).

  * (i)

The metric qualification condition ($Q_{2}$) for limiting subgradients has
been well recognized and largely employed in variational analysis and
optimization; in particular, it significantly improves other qualification
conditions used in subdifferential calculus and optimization theory; see,
e.g., [6, 7, 17, 18, 21] and compare with [10].

  * (ii)

Regarding the condition $\widehat{\partial}\mu(\bar{x})\neq\emptyset$, observe
first that this assumption holds when $\mu$ is convex, but it is way too much
to assume. In particular, it follows from [18, Propositions 4.4 and 4.6] that
if $X$ is finite-dimensional and $\mu(\cdot)$ is tangentially convex as in
[18], then we have $\widehat{\partial}\mu(\bar{x})\neq\emptyset$ if and only
if the function $\mu(\cdot)$ is calm at $\bar{x}$ in the sense of (2.6).
Furthermore, the lower regularity of $\mu(\cdot)$ at $\bar{x}$, in the sense
that $\widehat{\partial}\mu(\bar{x})=\partial\mu(\bar{x})$, ensures the
nonemptiness of $\widehat{\partial}\mu(\bar{x})$ in Asplund spaces under a
certain mild SNC property; see [10, Corollar 2.24]. The latter class of
functions is sufficiently broad including, e.g., maximum, semiconvex,
lower-${\cal C}^{1}$ functions, etc.; see the book [21] with many references
therein.

We conclude this section with the following simplification of Theorem 6.6 in
the case of bilevel programs with differentiable data. Recall that a function
$f\colon Z\to\mathbb{\mathbb{R}}$ defined on a normed space $Z$ is strictly
differentiable at $\bar{z}$ with the strict derivative $\nabla f(\bar{z})$ if

$\lim_{z,u\to\bar{z}}\frac{f(z)-f(u)-\langle\nabla
f(\bar{z}),z-u\rangle}{\|z-u\|}=0.$

It is well known that this notion lies between Fréchet differentiable of $f$
at $\bar{z}$ and continuous differentiability of $f$ around this point. If all
the constraint functions $g_{i},\;i=1,\ldots,p$, in (6.7) are strictly
differentiable at $(\bar{x},\bar{y})$, that the qualification condition (6.10)
reduces to the classical Mangasarian-Fromovitz constraint qualification
(MFCQ).

######  Corollary 6.9 (bilevel programs with differentiable data).

Let $(\bar{x},\bar{y})$ be a local optimal solution to the bilevel program
(6.7) under the fulfillment of $({\cal A})$. Assume that the functions
$g_{i}$, $i=1,\ldots,p$, and $\varphi$ are strictly differentiable at
$(\bar{x},\bar{y})$, that $\psi$ is Fréchet differentiable at
$(\bar{x},\bar{y})$, that (6.7) is partially calm at $(\bar{x},\bar{y})$ with
modulus $\kappa>0$, that MFCQ holds at $(\bar{x},\bar{y})$, and that
$\widehat{\partial}\mu(\bar{x})\neq\emptyset$. Then there exist multipliers
$\lambda_{i}$ and $\beta_{i}$ for $i=1,\ldots,p$ such that

$\begin{array}[]{ll}\displaystyle\nabla_{x}\psi(\bar{x},\bar{y})+\sum_{i=1}^{p}\kappa\left(\lambda_{i}-\beta_{i}\right)\nabla_{x}g_{i}(\bar{x},\bar{y})=0,\\\
\nabla_{y}\psi(\bar{x},\bar{y})+\displaystyle\sum_{i=1}^{p}\kappa\left(\lambda_{i}-\beta_{i}\right)\nabla_{y}g_{i}(\bar{x},\bar{y})=0,\\\
\kappa^{-1}\nabla_{y}\psi(\bar{x},\bar{y})+\nabla_{y}\varphi(\bar{x},\bar{y})+\displaystyle\sum_{i=1}^{p}\lambda_{i}\nabla_{y}g_{i}(\bar{x},\bar{y})=0,\\\
\nabla_{y}\varphi(\bar{x},\bar{y})+\displaystyle\sum_{i=1}^{p}\beta_{i}\nabla_{y}g_{i}(\bar{x},\bar{y})=0,\\\
\lambda_{i}\geq 0,\;\beta_{i}\geq
0,\;\lambda_{i}g_{i}(\bar{x},\bar{y})=\beta_{i}g_{i}(\bar{x},\bar{y})=0\;\mbox{
as }\;i=1,\ldots,p.\end{array}$

###### Proof.

Recall that the limiting subdifferential of a strictly differentiable function
reduce to its strict derivative, and that any such function is locally
Lipschitzian around the point in question. Thus the necessary optimality
conditions formulated in the corollary follow from conditions (6.11)–(6.13) of
Theorem 6.6. ∎

##  7 Conclusions

The paper provides a systematic study of regular subgradients and their
singular counterparts for a general class of marginal functions defined on
normed spaces with applications to variational calculus rules for such
constructions. Further applications are given, together with using the
corresponding limiting normals and subgradients, to deriving necessary
optimality conditions for the optimistic model of bilevel programming in
Asplund spaces via the marginal function approach. Although our developments
are given in infinite-dimensional spaces, the obtained results are new in
finite dimensions. In the future research, we will continue the lines of this
study with applying the newly introduced nearly-isolated calmness properties
to other aspects of variational analysis and optimization and aiming, at
particular, at the pessimistic model of bilevel programming.

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