∎

11institutetext: Duong Thi Kim Huyen 22institutetext: Graduate Training
Center, Institute of Mathematics, Vietnam Academy of Science and Technology,
Hanoi, Vietnam
<EMAIL_ADDRESS>33institutetext: Jen-Chih Yao 44institutetext: Center for
General Education, China Medical University,
Taichung, Taiwan
<EMAIL_ADDRESS>55institutetext: Nguyen Dong Yen, Corresponding author
66institutetext: Institute of Mathematics, Vietnam Academy of Science and
Technology,
Hanoi, Vietnam
<EMAIL_ADDRESS>
# Sensitivity Analysis of a Stationary Point Set Map under Total
Perturbations. Part 1: Lipschitzian Stability

Duong Thi Kim Huyen  Jen-Chih Yao  Nguyen Dong Yen

(Communicated by …

Received: date / Accepted: date)

###### Abstract

By applying some theorems of Levy and Mordukhovich (Math Program 99: 311–327,
2004) and other related results, we estimate the Fréchet coderivative and the
Mordukhovich coderivative of the stationary point set map of a smooth
parametric optimization problem with one smooth functional constraint under
total perturbations. From the obtained formulas we derive necessary and
sufficient conditions for the local Lipschitz-like property of the stationary
point set map. This leads us to new insights into the preceding deep
investigations of Levy and Mordukhovich in the above-cited paper and of Qui (J
Optim Theory Appl 161: 398–429,  2014; J Glob Optim 65: 615–635,  2016).

###### Keywords:

Smooth parametric optimization problem Smooth functional constraint Stationary
point set map Lipschitz-like property Coderivative

###### MSC:

49K4049J5390C3190C20

##  1 Introduction

Appeared at the early stage of optimization theory, smooth programming
problems continue to attract common attention of the optimization community
due to their importance and beauty. Polynomial optimization problems,
including nonconvex quadratic programs, are typical examples of such problems.

The present paper investigates the Lipschitz-like property and the Robinson
stability of the stationary point set map of a smooth parametric optimization
problem with one smooth functional constraint under total perturbations. The
aim is achieved by using some theorems of Levy and Mordukhovich LeMo04  and
other related results from Yen_Yao_NA_2009 , LeeYen11AA , and Qui_JOTA2014 .

Introduced by Aubin (Aubin_1984, , p. 98) under the name pseudo-Lipschitz
property, the local Lipschitz-like property of multifunctions is a fundamental
concept in stability and sensitivity analysis of optimization and equilibrium
problems. It is equivalent to the classical metric regularity of the inverse
map (see Borwein_Zhuang_1988 ; Penot_1989  and B-M_TAMS_1993 ). The local
Lipschitz-like property guarantees the local convergence of some variants of
Newton’s method for generalized equations Célia_Alain_2005 ; Dontchev_1996 ;
DR_2009 . In particular, from (DR_2009, , Theorem 6C.1, p. 328) it follows
that, if a mild approximation condition is satisfied and the solution map
under right-hand-side perturbations is locally Lipschitz-like around a point
in question, then there exists an iterative sequence $Q$-linearly converging
to the solution. Moreover, as shown by Dontchev (Dontchev_1996, , Theorem 1),
the Newton method applied to a generalized equation in a Banach space is
locally convergent uniformly in the canonical parameter if and only if a
certain map is locally Lipschitz-like around the reference point. The author
also proved (see (Dontchev_1996, , Theorem 2)) that the latter property
implies the uniform $Q$-quadratic convergence, provided that the derivative of
the base map is locally Lipschitz.

The Robinson stability of an implicit multifunction, which has been called the
metric regularity in the sense of Robinson by several authors, was introduced
by Robinson Robinson_1976 . This property is a kind of uniform local error
bounds and has numerous applications. Recently, Gfrerer and Mordukhovich
Gfrerer_Mor_SIOPT2016  have given first-order and second-order sufficient
conditions for this stability property of a parametric constraint system and
put it in the relationships with other properties, such as the classical
metric regularity and the local Lipschitz-like property.

The coderivative analysis of composite constraint functions of Levy and
Mordukhovich LeMo04  is based on the rich generalized differentiation calculus
in (Rock_Wets_1998, , Chapter 10). Among other things, it uses the properties
of amenable functions and strongly amenable functions, and the extended chain
rule for subdifferentials (Rock_Wets_1998, , Theorem 10.49). The analysis
allows us to derive sharp upper estimates for the Mordukhovich coderivative of
the stationary point set map, where the limiting second-order subdifferential
is used.

To get lower estimates for the Fréchet and the Mordukhovich coderivatives of
the stationary point set map, we combine the lower estimates of Lee and Yen
LeeYen11AA  with some results of Qui Qui_JOTA2014 ; Qui_JoGO_2016 .

With the above upper and lower coderivative estimates, we can use the
Mordukhovich criterion for the local Lipschitz-like property of locally closed
multifunctions to obtain both necessary and sufficient conditions for this
property. Here, we do not need an additional technical assumption of Qui.
Besides, by invoking a result of Yen_Yao_NA_2009 , we are able to show that
these sufficient conditions also guarantee the Robinson stability of the
stationary point set map.

Our conditions are easy to verify and can be effectively applied to nonconvex
quadratic programming under a possibly nonconvex quadratic constraint. The
results on quadratic programming in this paper extend the preceding ones of
Lee and Yen LeeYen14NA  and Qui and Yen QuiYen_SIOPT2014  to a broader class
of quadratic programs.

Optimization problems under total perturbations have been studied in MNg_2014
; MNg_2016 ; MNR_2015 ; MRS_2013  by different approaches and concepts. But,
our results are very different from those of the cited works.

Solution stability of variational inequalities on fixed or linearly perturbed
polyhedral convex sets, which is closely related to that of optimization
problems under linear constraints, has been investigated intensively; see
Ban_Song_2016 ; DR_1996 ; Henrion_Mordukhovich_Nam_2010 ; Lu_Robinson_2008 ;
Nam_2010 ; Qui_AMV_2011 ; Qui_NA_2011 ; Qui_JMAA_2011 ; Robinson_2007 ;
Trang_2016 ; Yao_Yen_AMV_2009 ; Yao_Yen_PJM_2009 , and the references therein.
The case of nonlinearly perturbed polyhedral convex sets has been considered
in Qui_JOTA_2012 .

It is well known that calmness is weaker than the local Lipschitz-like
property. Calmness of the stationary point set map of a general parametric
optimization problem has been considered, e.g., in (Guo_Lin_Ye_SIOPT2012, ,
Sect. 4).

The paper is divided into two parts. In this part, Sect. 2 recalls some basic
concepts from variational analysis, formulates the problem studied herein, and
presents a series of auxiliary results in a unified form. Sections 3 and 4
present new results on smooth parametric optimization problem with one smooth
functional constraint under total perturbations. Namely, sensitivity analysis
of the stationary point set at the interior points (resp., at the boundary
points) of the progamming variable-parameter domain is given in Sect. 3
(resp., in Sect. 4).

In Part 2, sufficient conditions for the Robinson stability of the stationary
point set map will be established. This allows us to revisit and extend
several stability theorems in indefinite quadratic programming. A comparison
of our results with the ones which can be obtained via another approach will
be also given.

##  2 Preliminaries

The scalar product and the norm in a finite-dimensional Euclidean space are
denoted respectively by $\langle\cdot,\cdot\rangle$ and $\|\cdot\|$. The
symbols $B(x,\rho)$ and $\bar{B}(x,\rho)$ stand for the open (resp., closed)
ball centered at $x\in X$ with radius $\rho>0$. The distance
$\displaystyle\inf_{u\in A}\|x-u\|$ from $x\in X$ to a subset $A\subset X$ is
denoted by $d(x,A)$.

We now recall several basic concepts from variational analysis Rock_Wets_1998
; B-M06  which will be used intensively later on.

The Fréchet normal cone (also called the prenormal cone, or the regular normal
cone) to a set $\Omega\subset I\\!\\!R^{s}$ at $\bar{v}\in\Omega$ is given by

$\displaystyle\widehat{N}_{\Omega}(\bar{v})=\left\\{v^{\prime}\in
I\\!\\!R^{s}\,\mid\,\displaystyle\limsup_{v\xrightarrow{\Omega}\bar{v}}\,\displaystyle\frac{\langle
v^{\prime},v-\bar{v}\rangle}{\|v-\bar{v}\|}\leq 0\right\\},$

where $v\xrightarrow{\Omega}\bar{v}$ means $v\to\bar{v}$ with $v\in\Omega$. By
convention, $\widehat{N}_{\Omega}(\bar{v}):=\emptyset$ when
$\bar{v}\notin\Omega$. Provided that $\Omega$ is locally closed around
$\bar{v}\in\Omega$, one calls

$\displaystyle\begin{array}[]{rl}N_{\Omega}(\bar{v})&=\displaystyle\mathop{{\rm
Lim}\,{\rm sup}}_{v\to\bar{v}}\widehat{N}_{\Omega}(v)\\\
&:=\big{\\{}v^{\prime}\in I\\!\\!R^{s}\,\mid\,\exists\mbox{ sequences
}v_{k}\to\bar{v},\ v_{k}^{\prime}\rightarrow v^{\prime},\\\
&\qquad\qquad\qquad\quad\ \mbox{with
}v_{k}^{\prime}\in\widehat{N}_{\Omega}(v_{k})\,\mbox{ for all
}\,k=1,2,\dots\big{\\}}\end{array}$

the Mordukhovich (or limiting/basic) normal cone to $\Omega$ at $\bar{v}$. If
$\bar{v}\notin\Omega$, then one puts $N_{\Omega}(\bar{v})=\emptyset$.

A multifunction $\Phi:I\\!\\!R^{n}\rightrightarrows I\\!\\!R^{m}$ is said to
be locally closed around a point $\bar{z}=(\bar{x},\bar{y})$ from $\mbox{\rm
gph}\,\Phi:=\\{(x,y)\in I\\!\\!R^{n}\times I\\!\\!R^{m}\,\mid\,y\in\Phi(x)\\}$
if $\mbox{\rm gph}\,\Phi$ is locally closed around $\bar{z}$. Here, the
product space $I\\!\\!R^{n+m}=I\\!\\!R^{n}\times I\\!\\!R^{m}$ is equipped
with the topology generated by the sum norm $\|(x,y)\|=\|x\|+\|y\|$.

For any $\bar{z}=(\bar{x},\bar{y})\in\mbox{gph}\,\Phi$,

$\displaystyle\widehat{D}^{*}\Phi(\bar{z})(y^{\prime}):=\big{\\{}x^{\prime}\in
I\\!\\!R^{n}\,\mid\,(x^{\prime},-y^{\prime})\in\widehat{N}_{\mbox{gph}\,\Phi}(\bar{z})\big{\\}}\quad(y^{\prime}\in
I\\!\\!R^{m})$

are called the Fréchet coderivative values of $\Phi$ at $\bar{z}$. Similarly,
the Mordukhovich coderivative (limiting coderivative) values of $\Phi$ at
$\bar{z}$ are defined by

$\displaystyle D^{*}\Phi(\bar{z})(y^{\prime}):=\big{\\{}x^{\prime}\in
I\\!\\!R^{n}\,\mid\,(x^{\prime},-y^{\prime})\in
N_{\mbox{gph}\,\Phi}(\bar{z})\big{\\}}\quad(y^{\prime}\in I\\!\\!R^{m}).$

Thus, $\widehat{D}^{*}\Phi(\bar{z})$ and $D^{*}\Phi(\bar{z})$ are
multifuntions from $I\\!\\!R^{m}$ to $I\\!\\!R^{n}$. By (B-M06, , Theorem
1.38), if $\Phi$ is strictly Fréchet differentiable at $\bar{x}$, then

$\widehat{D}^{*}\Phi(\bar{x})(y^{\prime})=D^{*}\Phi(\bar{x})(y^{\prime})=\\{\nabla\Phi(\bar{x})^{*}(y^{\prime})\\}$

for any $y^{\prime}\in I\\!\\!R^{m}$.

Suppose that $X$, $Y$, and $Z$ are finite-dimensional Euclidean spaces.
Consider a function $\psi:X\rightarrow\bar{I\\!\\!R}$ with
$|\psi(\bar{x})|<\infty$. The set

$\partial\psi(\bar{x}):=\\{x^{\prime}\in X^{*}\mid(x^{\prime},-1)\in N_{{\rm
epi}\,\psi}(\bar{x},\psi(\bar{x}))\\}$

is the Mordukhovich subdifferential of $\psi$ at $\bar{x}$. We put
$\partial\psi(\bar{x})=\emptyset$ if $|\psi(\bar{x})|=\infty$. The set

$\partial^{\infty}\psi(\bar{x}):=\\{x^{*}\in X^{*}\mid(x^{*},0)\in N_{{\rm
epi}\,\psi}(\bar{x},\psi(\bar{x}))\\}$

is the singular subdifferential of $\psi$ at $\bar{x}$. For a set
$\Omega\subset X$ and a point $\bar{x}\in\Omega$, we have

$N_{\Omega}(\bar{x})=\partial\delta_{\Omega}(\bar{x})=\partial^{\infty}\delta_{\Omega}(\bar{x}),$

where $\delta_{\Omega}(\bar{x})$ is the indicator function of $\Omega$; see
(B-M06, , Proposition 1.79). If $\psi$ depends on two variables $x$ and $y$,
and $|\psi(\bar{x},\bar{y})|<\infty$, then $\partial_{x}\psi(\bar{x},\bar{y})$
denotes the Mordukhovich subdifferential of $\psi(.,\bar{y})$ at $\bar{x}$.
For any $\bar{v}\in\partial\psi(\bar{x})$,

$\partial^{2}\psi(\bar{x}|\bar{v})(u):=D^{*}(\partial\psi)(\bar{x}|\bar{v})(u)\quad(u\in
X^{**}=X)$

is the limiting second-order subdifferential (or the generalized Hessian).

A multifunction $G:Y\rightrightarrows X$ is said to be locally Lipschitz-like
around $(\bar{y},\bar{x})\in{\rm gph}\,G$ if there exists a constant $\ell>0$
and neighborhoods $U$ of $\bar{x}$, $V$ of $\bar{y}$ such that

$G(y^{\prime})\cap U\subset G(y)+\ell\|y^{\prime}-y\|\bar{B}_{X}\quad\forall
y,y^{\prime}\in V,$

where $\bar{B}_{X}$ denotes the closed unit ball in $X$. When $G$ is locally
closed around $(\bar{y},\bar{x})$, the Mordukhovich criterion (see
B-M_TAMS_1993 , (Rock_Wets_1998, , Theorem 9.40), and (B-M06, , Theorem 4.10))
says that $G$ is locally Lipschitz-like around $(\bar{y},\bar{x})$ if and only
if

$D^{*}G(\bar{y},\bar{x})(0)=\\{0\\}.$

For a multifunction $F:X\times Y\rightrightarrows Z$ and a pair
$(\bar{x},\bar{y})\in X\times Y$ satisfying $0\in F(\bar{x},\bar{y})$, we say
that the implicit multifunction $G:Y\rightrightarrows X$ given by
$G(y)=\\{x\in X\mid 0\in F(x,y)\\}$ has the Robinson stability at
$\omega_{0}:=(\bar{x},\bar{y},0)$ if there exist constants $r>0$, $\gamma>0$,
and neighborhoods $U$ of $\bar{x}$, $V$ of $\bar{y}$ such that

$d(x,G(y))\leq rd(0,F(x,y))$

for any $(x,y)\in U\times V$ with $d(0,F(x,y))<\gamma$. Note that the
condition $d(0,F(x,y))<\gamma$ can be omitted if $F$ is inner semicontinuous
at $(\bar{x},\bar{y},0)$; see Huyen_Yen_2016 . Note that, in some cases, the
Robinson stability of $G$ at $(\bar{x},\bar{y},0)$ implies its local
Lipschitz-likeness around $(\bar{y},\bar{x})$; see, e.g., [8]. For the
generalized linear constraint system studied in [11], these properties are
equivalent. In the sequel, we will see that the regularity conditions in use
guarantee for our stationary point set map to have both properties.

Now, let $f_{0}$ and $F$ be twice continuously differentiable real-valued
functions ($C^{2}$-functions for brevity) defined on the product
$I\\!\\!R^{n}\times I\\!\\!R^{d}$ of two Euclidean spaces. For every $w\in
I\\!\\!R^{d}$, we consider the parametric optimization problem

$(P_{w})\quad\ \,{\rm Minimize}\ \,f_{0}(x,w)\ \;{\rm subject\ to}\ x\in
I\\!\\!R^{n}\ \,{\rm and}\ \,F(x,w)\leq 0.$

The constraint set of $(P_{w})$ is $C(w):=\\{x\in I\\!\\!R^{n}\mid F(x,w)\leq
0\\}$. The stationary point set of $(P_{w})$ is defined by

$S(w)=\\{x\in I\\!\\!R^{n}\mid 0\in\nabla_{x}f_{0}(x,w)+N_{C(w)}(x)\\}.$ (2)

When $w$ varies on $I\\!\\!R^{d}$, one has a multifunction
$S:I\\!\\!R^{d}\rightrightarrows I\\!\\!R^{n}$ with $S(w)$ being calculated by
(2). Setting $f(x,w)=g(F(x,w))=(g\circ F)(x,w)$, where
$g(y)=\delta_{I\\!\\!R_{-}}(y)$, i.e., $g(y)=0$ for $y\in(-\infty,0]$ and
$g(y)=+\infty$ for $y>0$, we can rewrite (2) as

$S(w)=\\{x\in I\\!\\!R^{n}\mid
0\in\nabla_{x}f_{0}(x,w)+\partial_{x}f(x,w)\\}.$ (3)

Fix a vector $w=\bar{w}\in I\\!\\!R^{d}$ and suppose that $\bar{x}\in
S(\bar{w})$. Since $(P_{\bar{w}})$ has a single smooth inequality constraint,
the Mangasarian-Fromovitz Constraint Qualification is fulfilled at $\bar{x}\in
C(\bar{w})$ if and only if

$\text{If}\ F(\bar{x},\bar{w})=0,\ \text{then}\
\nabla_{x}F(\bar{x},\bar{w})\neq 0.\quad\quad$ (MFCQ)

In what follows, we assume that (MFCQ) is valid. To study the stability of the
stationary point set map $S$ around the $(\bar{w},\bar{x})$ in ${\rm gph}\,S$,
we compute the Mordukhovich and the Fréchet coderivatives of the partial
subdifferential map $\partial_{x}f:I\\!\\!R^{n}\times
I\\!\\!R^{d}\rightrightarrows I\\!\\!R^{n}$. In general, there is no explicit
formula for the coderivatives of such maps. However, the results of LeMo04
provide us with some tools which allow us to estimate the coderivative value
$D^{*}S(\bar{w}|\bar{x})(x^{\prime})$ for every $x^{\prime}\in I\\!\\!R^{n}$.

The fulfillment of MFCQ at $(\bar{x},\bar{w})$ implies that $g(x,w)=g(F(x,w))$
is a strongly amenable in $x$ at $\bar{x}$ with compatible parameterization in
$w$ at $\bar{w}$. Then, by (Rock_Wets_1998, , Theorem 10.49), for $(x,w)$ near
$(\bar{x},\bar{w})$, we have

$\partial f(x,w)=\nabla F(x,w)^{*}(\partial g(F(x,w)))$ (4)

and

$\partial_{x}f(x,w)=\nabla_{x}F(x,w)^{*}(\partial g(F(x,w)));$ (5)

see (LeMo04, , formulas (14) and (15)).

In order to estimate the limiting second-order subdifferential of $f$, we need
the following result.

###### Lemma 2.1

(see (LeMo04, , Theorem 3.1)) Suppose that $\bar{v}\in\partial
f(\bar{x},\bar{w})$. Then, for any $v^{\prime}\in I\\!\\!R^{n}\times
I\\!\\!R^{d}$,

$\begin{array}[]{rl}&\partial^{2}f((\bar{x},\bar{w})|\bar{v})(v^{\prime})\\\
&\subset\ \displaystyle\bigcup_{\begin{subarray}{c}\bar{y}\in\partial
g(F(\bar{x},\bar{w}))\;\text{with}\\\ \nabla
F(\bar{x},\bar{w})^{*}\bar{y}=\bar{v}\end{subarray}}\bigg{(}\nabla^{2}(\bar{y}\cdot
F)(\bar{x},\bar{w})v^{\prime}+D^{*}(\partial g\circ
F)(\bar{x},\bar{w})|\bar{y})(\nabla
F(\bar{x},\bar{w})v^{\prime})\bigg{)},\end{array}$

where the function $\bar{y}\cdot F:I\\!\\!R^{n+d}\to I\\!\\!R$ is defined by
$(\bar{y}\cdot F)(x,w):=\bar{y}F(x,w)$. If, in addition, at every
$\bar{y}\in\partial g(F(\bar{x},\bar{w}))$ with $\nabla
F(\bar{x},\bar{w})^{*}\bar{y}=\bar{v}$, one has the second-order constraint
qualification

$\partial^{2}g(F(\bar{x},\bar{w})|\bar{y})(0)\cap{\rm ker}\nabla
F(\bar{x},\bar{w})^{*}=\\{0\\},$ (6)

then the estimate above for the second-order subdifferential can be refined by
replacing the coderivative of the multifunction $\partial g\circ F$ via the
inclusion

$D^{*}(\partial g\circ F)((\bar{x},\bar{w})|\bar{y})(\nabla
F(\bar{x},\bar{w})v^{\prime})\subset\nabla
F(\bar{x},\bar{w})^{*}\partial^{2}g(F(\bar{x},\bar{w})|\bar{y})(\nabla
F(\bar{x},\bar{w})v^{\prime}).$

In our problem $(P_{w})$, condition (6) can be omitted. Indeed,
$\bar{y}\in\partial g(F(\bar{x},\bar{w}))$ if and only if $\bar{y}\in
N_{I\\!\\!R_{-}}(F(\bar{x},\bar{w}))$. Hence, $\bar{y}\geq 0$. Clearly,

${\rm gph}\,\partial g=(I\\!\\!R_{-}\times\\{0\\})\cup(\\{0\\}\times
I\\!\\!R_{+}).$

If $F(\bar{x},\bar{w})<0$, then $\bar{y}=0$ and $N_{{\rm gph}\,\partial
g}(F(\bar{x},\bar{w}),\bar{y})=\\{0\\}\times I\\!\\!R.$ It follows that

$\displaystyle\begin{array}[]{rl}\partial^{2}g(F(\bar{x},\bar{w})|\,\bar{y})(0)&=D^{*}(\partial
g(F(\bar{x},\bar{w})|\bar{y}))(0)\\\ &=\\{u^{\prime}\in
I\\!\\!R\mid(u^{\prime},0)\in N_{{\rm gph}\,\partial
g}(F(\bar{x},\bar{w}),\bar{y})\\}=\\{0\\}.\end{array}$

So (6) is satisfied. If $F(\bar{x},\bar{w})=0$, then (MFCQ) implies $\nabla
F(\bar{x},\bar{w})\neq 0$. Hence the linear operator $\nabla
F(\bar{x},\bar{w}):I\\!\\!R^{n}\times I\\!\\!R^{n}\to I\\!\\!R$ is surjective.
Thus ${\rm ker}\,\nabla F(\bar{x},\bar{w})^{*}=\\{0\\}$ by (B-M06, , Lemma
1.18), and we see that (6) is fulfilled. Therefore, applied to $(P_{w})$,
Lemma 2.1 can be reformulated as follows: For any $\bar{v}\in\partial
f(\bar{x},\bar{w})$ and $v^{\prime}\in I\\!\\!R^{n}\times I\\!\\!R^{d}$,

$\partial^{2}f((\bar{x},\bar{w})|\bar{v})(v^{\prime})\subset\displaystyle\bigcup_{\begin{subarray}{c}\bar{y}\in\partial
g(F(\bar{x},\bar{w}))\;\text{with}\\\ \nabla
F(\bar{x},\bar{w})^{*}\bar{y}=\bar{v}\end{subarray}}\Big{(}\nabla^{2}(\bar{y}\cdot
F)(\bar{x},\bar{w})v^{\prime}+\Omega_{1}(\bar{y},v^{\prime})\Big{)},$ (8)

where

$\Omega_{1}(\bar{y},v^{\prime}):=\nabla
F(\bar{x},\bar{w})^{*}\partial^{2}g(F(\bar{x},\bar{w})|\bar{y})(\nabla
F(\bar{x},\bar{w})v^{\prime}).$

###### Remark 2.1

Concerning the paper MR_2012 , observe that the set
$\partial^{2}f((\bar{x},\bar{w})|\bar{v})(v^{\prime})$ in formula (8) is
analogous to the set $\widetilde{\varphi}^{2}_{x}(\bar{x},\bar{w},\bar{y})(u)$
(a value of the extended partial second-order subdifferential) in formula
(3.4) of that work. A careful checking shows that equality (3.4) of MR_2012
implies the upper estimate (8).

In what follows, for any $\bar{v}=(\bar{v}_{x},\bar{v}_{w})\in
I\\!\\!R^{n}\times I\\!\\!R^{d}$, we put ${\rm proj}_{1}\bar{v}=\bar{v}_{x}$.
The upper estimation for the coderivative values of the stationary point set
map $S$ given by Levy and Mordukhovich LeMo04  requires the following
regularity condition: For any $v^{\prime}_{1}\in I\\!\\!R^{n}$,

$0\in\nabla^{2}f_{0}(\bar{x},\bar{w})^{*}(v^{\prime}_{1},0)+\displaystyle\bigcup_{\begin{subarray}{c}\bar{v}\in\partial
f(\bar{x},\bar{w})\;\text{with}\\\ {\rm
proj}_{1}\bar{v}=-\nabla_{x}f_{0}(\bar{x},\bar{w})\end{subarray}}\partial^{2}f((\bar{x},\bar{w})|\bar{v})(v^{\prime}_{1},0)\
\;\Longrightarrow\ \;v^{\prime}_{1}=0$ (9)

(see (LeMo04, , formula (11))). For our problem $(P_{w})$, by the assumption
(MFCQ) and formula (4), we have $\partial f(\bar{x},\bar{w})=\nabla
F(\bar{x},\bar{w})^{*}(\partial g(\bar{x},\bar{w}))$. In addition, it is easy
to show that, for every $\bar{y}\in\partial g(\bar{x},\bar{w})$, ${\rm
proj}_{1}\left(\nabla
F(\bar{x},\bar{w})^{*}\bar{y}\right)=\nabla_{x}F(\bar{x},\bar{w})^{*}\bar{y}$.
Hence

$\begin{array}[]{rl}&\displaystyle\bigcup_{\begin{subarray}{c}\bar{v}\in\partial
f(\bar{x},\bar{w})\;\text{with}\\\ {\rm
proj}_{1}\bar{v}=-\nabla_{x}f_{0}(\bar{x},\bar{w})\end{subarray}}\partial^{2}f((\bar{x},\bar{w})|\bar{v})(v^{\prime}_{1},0)\\\
&=\displaystyle\bigcup_{\begin{subarray}{c}\bar{y}\in\partial
g(F(\bar{x},\bar{w}))\;\text{with}\\\
\nabla_{x}F(\bar{x},\bar{w})^{*}\bar{y}=-\nabla_{x}f_{0}(\bar{x},\bar{w})\end{subarray}}\partial^{2}f((\bar{x},\bar{w})|\nabla
F(\bar{x},\bar{w})^{*}\bar{y})(v^{\prime}_{1},0).\end{array}$

So (9) is equivalent to the following condition:

$0\in\nabla^{2}f_{0}(\bar{x},\bar{w})^{*}(v^{\prime}_{1},0)+\Omega_{2}(v^{\prime}_{1})\
\;\Longrightarrow\ \;v^{\prime}_{1}=0,\quad\quad$ (C0)

where

$\Omega_{2}(v^{\prime}_{1}):=\displaystyle\bigcup_{\begin{subarray}{c}\bar{y}\in\partial
g(F(\bar{x},\bar{w}))\;\text{with}\\\
\nabla_{x}F(\bar{x},\bar{w})^{*}\bar{y}=-\nabla_{x}f_{0}(\bar{x},\bar{w})\end{subarray}}\partial^{2}f((\bar{x},\bar{w})|\nabla
F(\bar{x},\bar{w})^{*}\bar{y})(v^{\prime}_{1},0).$ (10)

The next result from LeMo04  provides us with an upper estimation for the
values of the coderivative map
$D^{*}S(\bar{w}|\bar{x}):I\\!\\!R^{n}\rightrightarrows I\\!\\!R^{d}$.

######  Lemma 2.2 (see (LeMo04, , Corollary 3.1))

If the regularity condition (C0) holds then, for each $x^{\prime}\in
I\\!\\!R^{n}$, the coderivative value $D^{*}S(\bar{w}|\bar{x})(x^{\prime})$ is
contained in the set of $w^{\prime}\in I\\!\\!R^{d}$ for which there exists a
vector $v^{\prime}_{1}\in I\\!\\!R^{n}$ with

$(-x^{\prime},w^{\prime})-\nabla^{2}f_{0}(\bar{x},\bar{w})^{*}(v^{\prime}_{1},0)\in\Omega_{2}(v^{\prime}_{1}).$

Although it is rather difficult to compute the set
$\Omega_{2}(v^{\prime}_{1})$, we can still estimate it by using (8).

Upper estimates for the limiting coderivative values of $S$ can be derived
from a result of Levy and Mordukhovich (LeMo04, , Theorem 2.1). But, a
constraint qualification must be imposed to have these estimates (see
(LeeYen11AA, , p. 1020) for details). Interestingly, due to a result of Lee
and Yen (LeeYen11AA, , Theorem 3.4), sharp lower estimates for the Fréchet
coderivative values of $S$ can be given without any condition. Put
$G(x,w)=\nabla_{x}f_{0}(x,w)$ and $M(x,w)=\partial_{x}f(x,w)$. Then,

$S(w)=\\{x\in I\\!\\!R^{n}\mid 0\in G(x,w)+M(x,w)\\}.$ (11)

Since $\bar{x}\in S(\bar{w})$,
$\bar{\tau}:=(\bar{x},\bar{w},-\nabla_{x}f_{0}(\bar{x},\bar{w}))$ belongs to
${\rm gph}\,M$. Note that ${\rm gph}\,M$ is locally closed around
$\bar{\tau}$. The following result combines the lower estimates with the upper
estimates mentioned above.

###### Lemma 2.3

(see (LeeYen11AA, , Theorem 3.4)) The lower estimates

$\widehat{\Gamma}(x^{\prime})\subset\widehat{D}^{*}S(\bar{w}|\bar{x})(x^{\prime})\subset
D^{*}S(\bar{w}\mid\bar{x})(x^{\prime}),$

where

$\widehat{\Gamma}(x^{\prime}):=\displaystyle\bigcup_{v^{\prime}_{1}\in
I\\!\\!R^{n}}\left\\{w^{\prime}\in
I\\!\\!R^{d}\mid(-x^{\prime},w^{\prime})\in\nabla
G(\bar{x},\bar{w})^{*}v^{\prime}_{1}+\widehat{D}^{*}M(\bar{\tau})(v^{\prime}_{1})\right\\},$
(12)

hold for any $x^{\prime}\in I\\!\\!R^{n}$. If the constraint qualification

$0\in\nabla
G(\bar{x},\bar{w})^{*}v^{\prime}_{1}+D^{*}M(\bar{\tau})(v^{\prime}_{1})\
\;\Longrightarrow\ \;v^{\prime}_{1}=0\quad\quad$ (C1)

is satisfied, then the upper estimate

$D^{*}S(\bar{w}|\bar{x})(x^{\prime})\subset\Gamma(x^{\prime}),$

where

$\Gamma(x^{\prime}):=\displaystyle\bigcup_{v^{\prime}_{1}\in
I\\!\\!R^{n}}\left\\{w^{\prime}\in
I\\!\\!R^{d}\mid(-x^{\prime},w^{\prime})\in\nabla
G(\bar{x},\bar{w})^{*}v^{\prime}_{1}+D^{*}M(\bar{\tau})(v^{\prime}_{1})\right\\},$

is valid for any $x^{\prime}\in I\\!\\!R^{n}$. If, in addition, M is
graphically regular at $\bar{\tau}$, then

$\widehat{\Gamma}(x^{\prime})=\widehat{D}^{*}S(\bar{w}|\bar{x})(x^{\prime})=D^{*}S(\bar{w}|\bar{x})(x^{\prime})=\Gamma(x^{\prime}).$

From Lemma 2.3, for any $x^{\prime}\in I\\!\\!R^{n}$,
$\widehat{\Gamma}(x^{\prime})\subset\widehat{D}^{*}S(\bar{w}|\bar{x})(x^{\prime})$.
This implies that
$\widehat{\Gamma}(0)\subset\widehat{D}^{*}S(\bar{w}|\bar{x})(0)\subset
D^{*}S(\bar{w}|\bar{x})(0)$. If we put $\widetilde{M}(x,w)=G(x,w)+M(x,w)$,
then by the Fréchet coderivative sum rule with equalities (B-M06, , Theorem
1.62),

$\widehat{D}^{*}\widetilde{M}(\omega_{0})(v^{\prime}_{1})=\nabla
G(\bar{x},\bar{w})^{*}v^{\prime}_{1}+\widehat{D}^{*}M(\bar{\tau})(v^{\prime}_{1})$

for any $v^{\prime}_{1}\in I\\!\\!R^{n}$, where
$\omega_{0}:=(\bar{x},\bar{w},0)\in{\rm gph}\,\widetilde{M}$. Therefore, we
can write

$\widehat{\Gamma}(x^{\prime})=\displaystyle\bigcup_{v^{\prime}_{1}\in
I\\!\\!R^{n}}\left\\{w^{\prime}\in
I\\!\\!R^{d}\mid(-x^{\prime},w^{\prime})\in\widehat{D}^{*}\widetilde{M}(\omega_{0})(v^{\prime}_{1})\right\\}.$

Note that $0\in\widehat{\Gamma}(0)$. According to the Mordukhovich criterion,
if $S$ is locally Lipschitz-like around $(\bar{w},\bar{x})$, then
$D^{*}S(\bar{w}|\bar{x})(0)=\\{0\\}$ and $\widehat{\Gamma}(0)=\\{0\\}$ as a
result. In addition, if the constraint qualification (C1) is fulfilled, then
Lemma 2.3 yields
$D^{*}S(\bar{w}|\bar{x})(x^{\prime})\subset\Gamma(x^{\prime})$ for any
$x^{\prime}\in I\\!\\!R^{n}$. In particular,
$D^{*}S(\bar{w}|\bar{x})(0)\subset\Gamma(0)$. Hence, if (C1) is valid and
$\Gamma(0)=\\{0\\}$, then

$D^{*}S(\bar{w}|\bar{x})(0)=\\{0\\}.$

So, due to the Mordukhovich criterion, $S$ is locally Lipschitz-like around
$(\bar{w},\bar{x})$. This idea has been presented in LeeYen11AA  and we will
follow it throughout this paper.

Put $\mathcal{D}=\\{(x,w)\in I\\!\\!R^{n}\times I\\!\\!R^{d}\mid F(x,w)\leq
0\\}$. If $F(\bar{x},\bar{w})<0$, then $(\bar{x},\bar{w})$ is an interior
point of $\mathcal{D}$. If $F(\bar{x},\bar{w})=0$, then $(\bar{x},\bar{w})$ is
a boundary point of $\mathcal{D}$. In the next two sections, we will consider
separately these two possibilities of the reference point $(\bar{x},\bar{w})$.
Remind that $\bar{w}\in I\\!\\!R^{d}$ and $\bar{x}\in S(\bar{w})$ are fixed
and all the notations of this section are kept unchanged.

##  3 Interior points

Suppose that $F(\bar{x},\bar{w})<0$, i.e., $(\bar{x},\bar{w})$ is an interior
point of $\mathcal{D}$. A point $\bar{x}$ belongs to $S(\bar{w})$ if and only
if

$0\in\nabla_{x}f_{0}(\bar{x},\bar{w})+N_{C(\bar{w})}(\bar{x}),$

where $C(\bar{w})=\\{x\in I\\!\\!R^{n}\mid F(x,\bar{w})\leq 0\\}$. Since
$F(\bar{x},\bar{w})<0$, the continuity of $F(.,\bar{w})$ implies that
$\bar{x}\in{\rm int}\,C(\bar{w})$. This yields
$N_{C(\bar{w})}(\bar{x})=\\{0\\}$. Thus, $\bar{x}\in S(\bar{w})$ if and only
if $\nabla_{x}f_{0}(\bar{x},\bar{w})=0$.

The inequality $F(\bar{x},\bar{w})<0$ implies that $\partial
g(F(\bar{x},\bar{w}))=\\{0\\}$. So, $\bar{y}=0$ is the unique element of
$\partial g(F(\bar{x},\bar{w}))$. Since ${\rm gph}\,\partial
g=(I\\!\\!R_{-}\times\\{0\\})\cup(\\{0\\}\times I\\!\\!R_{+})$,

$N_{{\rm gph}\,\partial g}(F(\bar{x},\bar{w}),\bar{y})=\\{0\\}\times
I\\!\\!R.$

Hence, for any $v^{\prime}\in I\\!\\!R^{n}\times I\\!\\!R^{d}$,

$\displaystyle\begin{array}[]{rl}&\partial^{2}g(F(\bar{x},\bar{w})|\,\bar{y})(\nabla
F(\bar{x},\bar{w})v^{\prime})\\\ &=D^{*}(\partial
g(F(\bar{x},\bar{w})|\bar{y}))(\nabla F(\bar{x},\bar{w})v^{\prime})\\\
&=\\{u^{\prime}\in I\\!\\!R\mid(u^{\prime},-\nabla
F(\bar{x},\bar{w})v^{\prime})\in N_{{\rm gph}\,\partial
g}(F(\bar{x},\bar{w}),\bar{y})\\}=\\{0\\}.\end{array}$

Therefore, $\Omega_{1}(\bar{y},v^{\prime})=\\{0\\}$ for any $v^{\prime}\in
I\\!\\!R^{n}\times I\\!\\!R^{d}$. So, from (8) it follows that

$\partial^{2}f((\bar{x},\bar{w})|\bar{v})(v^{\prime})\subset\\{0\\}$

for any $\bar{v}\in\partial f(\bar{x},\bar{w})$ and $v^{\prime}\in
I\\!\\!R^{n}\times I\\!\\!R^{d}$. Since $\nabla_{x}f(\bar{x},\bar{w})=0$,
invoking (4) and the fact that $\partial g(F(\bar{x},\bar{w}))=\\{0\\}$, we
get

$\Omega_{2}(v^{\prime}_{1})=\partial^{2}f((\bar{x},\bar{w})|0)(v^{\prime}_{1},0)\subset\\{0\\}$

for any $v^{\prime}_{1}\in I\\!\\!R^{n}$. Then, condition (C0) is fulfilled if

$0\in\nabla^{2}f_{0}(\bar{x},\bar{w})^{*}(v^{\prime}_{1},0)\
\;\Longrightarrow\ \;v^{\prime}_{1}=0.$

By the symmetry of $\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})$ and the equality
$\nabla^{2}_{xw}f_{0}(\bar{x},\bar{w})^{T}=\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})$,
this is equivalent to

$\begin{cases}\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}=0\\\
\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}=0\end{cases}\
\;\Longrightarrow\ \;v^{\prime}_{1}=0.$

Clearly, the latter means that

${\rm ker}\,\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})\cap{\rm
ker}\,\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})=\\{0\\}.$ (14)

We now suppose that condition (14), which guarantees the validity of (C0), is
satisfied. Then, by Lemma 2.2,

$D^{*}S(\bar{w}|\bar{x})(x^{\prime})\subset\displaystyle\bigcup_{v^{\prime}_{1}\in
I\\!\\!R^{n}}\\{w^{\prime}\in
I\\!\\!R^{d}\mid(-x^{\prime},w^{\prime})-\nabla^{2}f_{0}(\bar{x},\bar{w})^{*}(v^{\prime}_{1},0)\in\Omega_{2}(v^{\prime}_{1})\\},$

for any $x^{\prime}\in I\\!\\!R^{n}$. Since
$\Omega_{2}(v^{\prime}_{1})\subset\\{0\\}$, we have

$\begin{array}[]{rl}&D^{*}S(\bar{w}|\bar{x})(x^{\prime})\\\
&\subset\Gamma_{1}(x^{\prime}):=\displaystyle\bigcup_{v^{\prime}_{1}\in
I\\!\\!R^{n}}\left\\{w^{\prime}\in
I\\!\\!R^{d}\mid\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}=-x^{\prime},\;w^{\prime}=\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}\right\\}.\end{array}$

Note that $0\in D^{*}S(\bar{w}|\bar{x})(0)$. So, if $\Gamma_{1}(0)=\\{0\\}$,
then $D^{*}S(\bar{w}|\bar{x})(0)=\\{0\\}$; as a result, $S$ is locally
Lipschitz-like around $(\bar{w},\bar{x})$ by the Mordukhovich criterion. We
have $\Gamma_{1}(0)=\\{0\\}$ if and only if

$\left\\{w^{\prime}\in I\\!\\!R^{d}\mid\exists v^{\prime}_{1}\in I\\!\\!R^{n}\
{\rm with}\
\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}=0,\;w^{\prime}=\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}\right\\}=\\{0\\}.$

This can be rewritten equivalently as

${\rm ker}\,\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})\subset{\rm
ker}\,\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w}).$ (15)

It is easy to show that (15) and (14) hold simultaneously if and only if

${\rm ker}\,\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})=\\{0\\}.$ (16)

In particular, (16) is a sufficient condition for $S$ being locally Lipschitz-
like around $(\bar{w},\bar{x})$.

###### Example 3.1

Consider the problem $(P_{w})$ with $f_{0}(x,w)=\frac{1}{2}x^{T}Dx+c^{T}x$ and
$F(x,w)=\|x\|^{2}-\rho^{2}$, where $w=(D,c,\rho)$ with $D$ being a $n\times n$
symmetric matrix, $c\in I\\!\\!R^{n}$, and $\rho>0$. Suppose that $\bar{x}\in
S(\bar{w})$ with $\bar{w}:=(\bar{D},\bar{c},\bar{\rho})$ and
$\|\bar{x}\|<\bar{\rho}$. If $\det\bar{D}\neq 0$, then $S$ is locally
Lipschitz-like around $(\bar{w},\bar{x})$ because (16) is satisfied.

Having the sufficient condition (16) for the Lipschitz-likeness of $S$ around
$(\bar{w},\bar{x})$, we want to find a necessary condition for this property.
We know that if $S$ is locally Lipschitz-like around $(\bar{w},\bar{x})$, then
$\widehat{\Gamma}(0)=\\{0\\}$, where the sets have been defined in (12). Since
$F(\bar{x},\bar{w})<0$ and $F$ is continuous, there exit neighborhoods $U$ of
$\bar{x}$ and $W$ of $\bar{w}$ such that $F(x,w)<0$ for any $(x,w)\in U\times
W$. It follows that, for each $w\in W$, the inclusion $x\in{\rm int}\,C(w)$
holds for every $x\in U$. Hence, for each $w\in W$, $N_{C(w)}(x)=\\{0\\}$ for
all $x\in U$. This means that $M(x,w)=\\{0\\}$ for $(x,w)$ in a neighborhood
of $(\bar{x},\bar{w})$. Therefore, from (2),

$S(w)\cap U=\\{x\in I\\!\\!R^{n}\mid\nabla_{x}f_{0}(x,w)=0\\}\cap
U\quad(\forall w\in W).$ (17)

Since $\widehat{D}^{*}M(\bar{\tau})(v^{\prime}_{1})=\\{0\\}$ for any
$v^{\prime}_{1}\in I\\!\\!R^{n}$, one has

$\begin{array}[]{rl}\widehat{\Gamma}(x^{\prime})&=\displaystyle\bigcup_{v^{\prime}_{1}\in
I\\!\\!R^{n}}\left\\{w^{\prime}\in
I\\!\\!R^{d}\mid(-x^{\prime},w^{\prime})\in\nabla
G(\bar{x},\bar{w})^{*}v^{\prime}_{1}\right\\}\\\
&=\displaystyle\bigcup_{v^{\prime}_{1}\in I\\!\\!R^{n}}\left\\{w^{\prime}\in
I\\!\\!R^{d}\mid(-x^{\prime},w^{\prime})\in(\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1},\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1})\right\\}.\end{array}$

It follows that

$\widehat{\Gamma}(0)=\displaystyle\bigcup_{v^{\prime}_{1}\in
I\\!\\!R^{n}}\left\\{w^{\prime}\in
I\\!\\!R^{d}\mid(0,w^{\prime})\in(\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1},\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1})\right\\}.$

Then, $\widehat{\Gamma}(0)=\\{0\\}$ if and only if (15) holds. Thus, condition
(15) is necessary for $S$ being locally Lipschitz-like around
$(\bar{w},\bar{x})$.

Let us consider an illustrative example, where the objective function is
bilinear and the inequality constraint is polynomial.

###### Example 3.2

Consider the problem $(P_{w})$ with
$f_{0}(x,w)=\displaystyle\sum_{i=1}^{n}x_{i}w_{i}$ and

$F(x,w)=\sum_{i=1}^{n}(1-w_{i}^{2})x_{i}^{2}-1$

for all $(x,w)\in I\\!\\!R^{n}\times I\\!\\!R^{n}.$ The stationary point set
of this problem is given by

$S(w)=\\{x\in I\\!\\!R^{n}\mid 0\in\nabla_{x}f_{0}(x,w)+N_{C(w)}(x)\\}=\\{x\in
I\\!\\!R^{n}\mid-w\in N_{C(w)}(x)\\}.$ (18)

In particular, for $\bar{w}=0$ one has $S(\bar{w})=\\{x\in
I\\!\\!R^{n}\mid\|x\|\leq 1\\}$. Let $\bar{x}\in S(\bar{x})$ and
$\|\bar{x}\|<1$. Note that

$\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})=0,\
\,\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})=E,$

where $E$ stands for the unit matrix in $I\\!\\!R^{n\times n}$. Since ${\rm
ker}\,\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})=I\\!\\!R^{n}$, the sufficient
condition (16) fails. However, we can assert that $S$ is not locally
Lipschitz-like around $(\bar{w},\bar{x})$ because the necessary condition (15)
is not satisfied. In fact, we directly prove that $S$ is not locally
Lipschitz-like around $(\bar{w},\bar{x})$. Indeed, from (18) there exist
neighborhoods $W$ of $\bar{w}$ and $U$ of $\bar{x}$ with $S(w)\cap
V=\emptyset$ for all $w\in W\setminus\\{\bar{w}\\}$. This leads us to the
desired result.

###### Remark 3.1

The above arguments show that if $(\bar{x},\bar{w})\in{\rm int}\,D$, then

$D^{*}M(\bar{\tau})(v^{\prime}_{1})=\widehat{D}^{*}M(\bar{\tau})(v^{\prime}_{1})=\\{0\\}$
(19)

for any $v^{\prime}_{1}\in I\\!\\!R^{n}$. This implies that $M$ is graphically
regular at $\bar{\tau}$. According to Lemma 2.3, if the constraint
qualification (C1) is valid, then
$\widehat{\Gamma}(x^{\prime})=D^{*}S(\bar{w}|\bar{x})(x^{\prime})=\Gamma(x^{\prime})$
for any $x^{\prime}\in I\\!\\!R^{n}$. Suppose that (C1) is fulfilled. Then,

$D^{*}S(\bar{w}|\bar{x})(x^{\prime})=\displaystyle\bigcup_{v^{\prime}_{1}\in
I\\!\\!R^{n}}\left\\{w^{\prime}\in
I\\!\\!R^{d}\mid(-x^{\prime},w^{\prime})\in\nabla
G(\bar{x},\bar{w})^{*}v^{\prime}_{1}\right\\}$

for any $x^{\prime}\in I\\!\\!R^{n}$. In particular,

$D^{*}S(\bar{w}|\bar{x})(0)=\left\\{w^{\prime}\in I\\!\\!R^{d}\mid\exists
v^{\prime}_{1}\in I\\!\\!R^{n}\ {\rm with}\ (0,w^{\prime})=\nabla
G(\bar{x},\bar{w})^{*}v^{\prime}_{1}\right\\}.$

According to the Mordukhovich criterion, $S$ is locally Lipschitz-like around
$(\bar{w},\bar{x})$ if and only if $D^{*}S(\bar{w}|\bar{x})(0)=\\{0\\}$. The
latter means that

$\left\\{w^{\prime}\in I\\!\\!R^{d}\mid\exists v^{\prime}_{1}\in I\\!\\!R^{n}\
{\rm with}\
\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}=0,\;w^{\prime}=\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}\right\\}=\\{0\\}.$

Clearly, this is fulfilled if and only if (15) is satisfied. Now let us verify
the constraint qualification (C1). Due to (19), (C1) becomes

$0\in\nabla G(\bar{x},\bar{w})^{*}v^{\prime}_{1}\ \;\Longrightarrow\
\;v^{\prime}_{1}=0,$

or, equivalently,

$0\in\nabla^{2}f_{0}(\bar{x},\bar{w})^{*}(v^{\prime}_{1},0)\
\;\Longrightarrow\ \;v^{\prime}_{1}=0.$

The last condition has been proved to be equivalent to (14). Thus, if (14) is
satisfied, then $S$ is locally Lipschitz-like around $(\bar{w},\bar{x})$ if
and only if (15) holds.

The following theorem summarizes our results for the case of interior points.

###### Theorem 3.1

Suppose that $F(\bar{x},\bar{w})<0$. The following assertions are valid:

(a)

If $S$ is locally Lipschitz-like around $(\bar{w},\bar{x})$, then condition
(15) holds;

(b)

If conditions (14) and (15) are simultaneously fulfilled, then $S$ is locally
Lipschitz-like around $(\bar{w},\bar{x})$;

(c)

If condition (14) is satisfied, then $S$ is locally Lipschitz-like around
$(\bar{w},\bar{x})$ if and only if condition (15) holds.

Thus, if condition (14) fails we can assert nothing about the local Lipschitz-
likeness of $S$ around $(\bar{w},\bar{x})$. The next example shows that $S$
can be locally Lipschitz-like around $(\bar{w},\bar{x})$ when (14) is not
satisfied,

###### Example 3.3

Consider $(P_{w})$ with $f_{0}(x,w)=\frac{1}{3}x^{3}-w^{2}x$ and

$F(x,w)=x^{2}+w^{2}-1$

for all $(x,w)\in I\\!\\!R\times I\\!\\!R$. Put $\bar{w}=0$. The point
$\bar{x}=0$ belongs to $S(\bar{w})$ because $(\bar{x},\bar{w})\in{\rm int}\,D$
and $\nabla_{x}f_{0}(\bar{x},\bar{w})=0$. Since
$\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})=0$ and

$\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})=0,$

condition (14) is invalid. We have known that $S$ is locally defined by

$\begin{array}[]{rl}S(w)&=\\{x\in I\\!\\!R\mid\nabla_{x}f_{0}(x,w)=0\\}.\\\
&=\\{x\in I\\!\\!R\mid|x|=|w|\\}.\end{array}$

Then, for any $x^{\prime}\in I\\!\\!R$,

$\begin{array}[]{rl}D^{*}S(\bar{w}|\bar{x})(x^{\prime})&=\\{w^{\prime}\in
I\\!\\!R\mid(w^{\prime},-x^{\prime})\in N_{{\rm gph}\,S}(0,0)\\}.\\\
&=\\{w^{\prime}\in I\\!\\!R\mid|w^{\prime}|=|x^{\prime}|\\}.\end{array}$

It follows that $D^{*}S(\bar{w}|\bar{x})(0)=\\{0\\}$. Thus, $S$ is locally
Lipschitz-like around $(\bar{w},\bar{x})$.

One referee of the present paper asks: Whether the condition (15) alone can
ensure the local Lipschitz-likeness of $S$ around $(\bar{w},\bar{x})$.
Answering the question, we construct the next example to demonstrate that,
even for polynomial optimization problems, (15) is not sufficient for the
later property to hold.

###### Example 3.4

Consider $(P_{w})$ with $f_{0}(x,w)=\frac{1}{4}x^{4}-w^{2}x$ and

$F(x,w)=x-w-1$

for all $(x,w)\in I\\!\\!R\times I\\!\\!R$. Put $\bar{w}=0$. The point
$\bar{x}:=0$ belongs to $S(\bar{w})$ because $F(\bar{x},\bar{w})<0$ and
$\nabla_{x}f_{0}(\bar{x},\bar{w})=0$. Since
$\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})=\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})=0$,
we see that (14) fails, but (15) is valid. As $S(w)=\\{x\in I\\!\\!R\mid
x^{3}-w^{2}=0\\}=\\{w^{\frac{2}{3}}\\}$, the stationary point set map is not
locally Lipschitz-like around $(\bar{w},\bar{x})=(0,0)$.

###### Remark 3.2

The second assertion of Theorem 3.1 can be obtained by the classical implicit
function theorem. Indeed, if (14) and (15) are simultaneously fulfilled, i.e.,
(16) is satisfied, then by (DR_2009, , Theorem 1B.1) (see also (Rudin_1976, ,
Theorem 9.28)) the implicit multifunction $w\mapsto\\{x\in
I\\!\\!R^{n}\mid\nabla_{x}f_{0}(x,w)=0\\}$ defined by the equation
$\nabla_{x}f_{0}(x,w)=0$ has a single-valued localization (DR_2009, , p. 4)
around $\bar{w}$ for $\bar{x}$ which is continuously differentiable in a
neighborhood of $\bar{w}$. This means that there exist a neighborhood $W$ of
$\bar{w}$ and a neighborhood $U$ of $\bar{x}$ such that for each $w\in W$
there is a unique vector $x=s(w)$ in $U$ satisfying the equation
$\nabla_{x}f_{0}(x,w)=0$ and $s:W\to U$ is continuously differentiable.
Without loss of generality, we can assume that $F(x,w)<0$ for all $(x,w)\in
U\times W$. So, by (17), $S(w)\cap U=\\{s(w)\\}$ for all $w\in W$. Hence, $S$
is locally Lipschitz-like around $(\bar{w},\bar{x})$.

###### Remark 3.3

Consider the extended stationary point set map
$(w,z)\mapsto\widetilde{S}(w,z)$ of $(P_{w})$, which is defined by

$\widetilde{S}(w,z)=\\{x\in I\\!\\!R^{n}\mid
z\in\nabla_{x}f_{0}(x,w)+N_{C(w)}(x)\\}.$

If $F(\bar{x},\bar{w})<0$, then around the point $((\bar{w},0),\bar{x})\in{\rm
gph}\,\widetilde{S}$ one can represent $\bar{S}$ locally as

$\widetilde{S}(w,z)=\\{x\in I\\!\\!R^{n}\mid\widetilde{G}(x,w,z)=0\\},$

where $\widetilde{G}(x,w,z):=\nabla_{x}f_{0}(x,w)-z$. Since
$\nabla_{(w,z)}\widetilde{G}(\bar{x},\bar{w},0)$ has full rank, by (LeMo04, ,
Theorem 2.1) one has

$\begin{array}[]{rcl}D^{*}\widetilde{S}((\bar{w},0),\bar{x})(x^{\prime})=\\{(w^{\prime},z^{\prime})\in
I\\!\\!R^{d}\times
I\\!\\!R^{n}&\mid&w^{\prime}=-\big{(}\nabla_{wx}^{2}f_{0}(\bar{x},\bar{w})\big{)}^{T}z^{\prime},\\\
&&x^{\prime}=\nabla_{xx}^{2}f_{0}(\bar{x},\bar{w})z^{\prime}\\}.\end{array}$

Hence,

$\begin{array}[]{rcl}D^{*}\widetilde{S}((\bar{w},0),\bar{x})(0)=\\{(w^{\prime},z^{\prime})\in
I\\!\\!R^{d}\times
I\\!\\!R^{n}&\mid&\nabla_{xx}^{2}f_{0}(\bar{x},\bar{w})z^{\prime}=0\\\
&&w^{\prime}=-\big{(}\nabla_{wx}^{2}f_{0}(\bar{x},\bar{w})\big{)}^{T}z^{\prime}\\}.\end{array}$

Therefore, $D^{*}\widetilde{S}((\bar{w},0),\bar{x})(0)=\\{0\\}$ if and only if
${\rm ker}\,\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})=\\{0\\}.$ Thanks to
Mordukhovich’s criterion, this shows that condition (16) is necessary and
sufficient for the local Lipschitz-like property of the the extended
stationary point set map $\widetilde{S}$ around $((\bar{w},0),\bar{x})$.

##  4 Boundary points

Suppose that $F(\bar{x},\bar{w})=0$, i.e., $(\bar{x},\bar{w})$ is a boundary
point of $\mathcal{D}$. To obtain a sufficient condition for the Lipschitz-
like property of $S$ around $(\bar{w},\bar{x})$, we will follow the scheme
which has been used in the case of interior points. We first need to find out
a condition which guarantees the fulfillment of (C0). Note that (MFCQ) yields
$\nabla_{x}F(\bar{x},\bar{w})\neq 0$. Since $\partial
g(F(\bar{x},\bar{w}))\subset I\\!\\!R$ and

$\nabla_{x}F(\bar{x},\bar{w})^{*}\gamma=\gamma\nabla_{x}F(\bar{x},\bar{w})$

for any $\gamma\in I\\!\\!R$, there exists only one element
$\lambda\in\partial g(F(\bar{x},\bar{w}))$ satisfying

$\nabla_{x}f_{0}(\bar{x},\bar{w})+\lambda\nabla_{x}F(\bar{x},\bar{w})=0.$ (20)

This element $\lambda$ is the unique Lagrange multiplier for the stationary
point $\bar{x}$ of the minimization problem $\big{(}P_{\bar{w}}\big{)}$. Due
to $\lambda\in\partial g(F(\bar{x},\bar{w}))$, we have $\lambda\geq 0$. Let us
consider two possibilities of the Lagrange multiplier.

4.1 The nondegenerate case

Suppose that $\lambda>0$. Clearly, the equality ${\rm gph}\,\partial
g=(I\\!\\!R_{-}\times\\{0\\})\cup(\\{0\\}\times I\\!\\!R_{+})$ yields

$N_{{\rm gph}\,\partial g}(F(\bar{x},\bar{w}),\lambda)=N_{{\rm gph}\,\partial
g}(0,\lambda)=I\\!\\!R\times\\{0\\}.$

By definition, for any $v^{\prime}\in I\\!\\!R^{n}\times I\\!\\!R^{d}$,

$\displaystyle\begin{array}[]{rl}&\partial^{2}g(F(\bar{x},\bar{w})|\,\lambda)(\nabla
F(\bar{x},\bar{w})v^{\prime})\\\ &=D^{*}(\partial
g)(F(\bar{x},\bar{w})|\lambda)(\nabla F(\bar{x},\bar{w})v^{\prime})\\\
&=\\{u^{\prime}\in I\\!\\!R\mid(u^{\prime},-\nabla
F(\bar{x},\bar{w})v^{\prime})\in N_{{\rm gph}\,\partial
g}(F(\bar{x},\bar{w}),\lambda)\\}.\end{array}$

Hence

$\partial^{2}g(F(\bar{x},\bar{w})|\,\lambda)(\nabla
F(\bar{x},\bar{w})v^{\prime})=\begin{cases}I\\!\\!R&\mbox{ if }\nabla
F(\bar{x},\bar{w})v^{\prime}=0\\\ \ \emptyset&\mbox{ if }\nabla
F(\bar{x},\bar{w})v^{\prime}\neq 0.\end{cases}$

So,

$\Omega_{1}(\lambda,v^{\prime})=\begin{cases}\\{\gamma\nabla
F(\bar{x},\bar{w})\mid\gamma\in I\\!\\!R\\}&\mbox{ if }\nabla
F(\bar{x},\bar{w})v^{\prime}=0\\\ \emptyset&\mbox{ if }\nabla
F(\bar{x},\bar{w})v^{\prime}\neq 0.\end{cases}$

For $\bar{v}:=\nabla F(\bar{x},\bar{w})^{*}\lambda$, the conditions
$\bar{y}\in\partial g(F(\bar{x},\bar{w}))$ and $\nabla
F(\bar{x},\bar{w})^{*}\bar{y}=\bar{v}$ force $\bar{y}=\lambda$. So, by (8) we
get

$\displaystyle\begin{array}[]{rl}\partial^{2}f((\bar{x},\bar{w})|\bar{v})(v^{\prime})&\subset\lambda\nabla^{2}F(\bar{x},\bar{w})v^{\prime}+\Omega_{1}(\lambda,v^{\prime})\\\
&=\lambda\nabla^{2}F(\bar{x},\bar{w})v^{\prime}+\\{\gamma\nabla
F(\bar{x},\bar{w})\mid\gamma\in I\\!\\!R\\}\end{array}$

if $\nabla F(\bar{x},\bar{w})v^{\prime}=0$, and
$\partial^{2}f((\bar{x},\bar{w})|\bar{v})(v^{\prime})=\emptyset$ if $\nabla
F(\bar{x},\bar{w})v^{\prime}\neq 0.$ Since $\bar{y}=\lambda$ is the unique
element satisfying the conditions $\bar{y}\in\partial g(F(\bar{x},\bar{w}))$
and

$\nabla F(\bar{x},\bar{w})^{*}\bar{y}=-\nabla_{x}f_{0}(\bar{x},\bar{w}),$

from (10) it follows that

$\Omega_{2}(v^{\prime}_{1})\subset\left\\{\lambda\nabla^{2}F(\bar{x},\bar{w})(v^{\prime}_{1},0)+\gamma\nabla
F(\bar{x},\bar{w})\mid\gamma\in I\\!\\!R\right\\},$ (23)

for any $v^{\prime}_{1}$ with $\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}=0$.
Besides, if $\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}\neq 0$, then the set
$\Omega_{2}(v^{\prime}_{1})$ is empty. Thus, (C0) is fulfilled if the
following is satisfied: for any $v^{\prime}_{1}\in I\\!\\!R^{n}$, if

$\begin{cases}-\nabla^{2}f_{0}(\bar{x},\bar{w})^{*}(v^{\prime}_{1},0)\in\left\\{\lambda\nabla^{2}F(\bar{x},\bar{w})(v^{\prime}_{1},0)+\gamma\nabla
F(\bar{x},\bar{w})\mid\gamma\in I\\!\\!R\right\\}\\\
\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}=0,\end{cases}$

then $v^{\prime}_{1}=0$. Equivalently, for any $v^{\prime}_{1}\in
I\\!\\!R^{n}$, if

$\begin{cases}0=\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}+\lambda\nabla^{2}_{xx}F(\bar{x},\bar{w})v^{\prime}_{1}+\gamma\nabla_{x}F(\bar{x},\bar{w}),\\\
0=\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}+\lambda\nabla^{2}_{wx}F(\bar{x},\bar{w})v^{\prime}_{1}+\gamma\nabla_{w}F(\bar{x},\bar{w}),\\\
\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}=0,\ \gamma\in I\\!\\!R,\end{cases}$

then $v^{\prime}_{1}=0$. Putting

$A_{1}=\left[\begin{array}[]{ccc}\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})+\lambda\nabla^{2}_{xx}F(\bar{x},\bar{w})&\nabla_{x}F(\bar{x},\bar{w})\end{array}\right]\in
I\\!\\!R^{n\times(n+1)}$

and

$A_{2}=\left[\begin{array}[]{ccc}\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})+\lambda\nabla^{2}_{wx}F(\bar{x},\bar{w})&\nabla_{w}F(\bar{x},\bar{w})\end{array}\right]\in
I\\!\\!R^{d\times(n+1)},$

where $\nabla_{x}F(\bar{x},\bar{w})$ and $\nabla_{w}F(\bar{x},\bar{w})$ are
interpreted as column vectors, we can rewrite the last condition equivalently
as follows:

$\begin{cases}A_{1}\left(\begin{matrix}v^{\prime}_{1}\\\
\gamma\end{matrix}\right)=0,\quad A_{2}\left(\begin{matrix}v^{\prime}_{1}\\\
\gamma\end{matrix}\right)=0\\\
\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}=0,\quad\gamma\in
I\\!\\!R\end{cases}\quad\quad\Longrightarrow\ \;v^{\prime}_{1}=0.$

Since $\nabla_{x}F(\bar{x},\bar{w})\neq 0$, the latter is equivalent to saying
that

${\rm ker}\,A_{1}\cap{\rm ker}\,A_{2}\cap\left({\rm
ker}\,\nabla_{x}F(\bar{x},\bar{w})\times I\\!\\!R\right)=\\{(0,0)\\}.$ (24)

We now suppose that condition (24) is satisfied. Then, by Lemma 2.2, for any
$x^{\prime}\in I\\!\\!R^{n}$,

$D^{*}S(\bar{w}|\bar{x})(x^{\prime})\subset\displaystyle\bigcup_{v^{\prime}_{1}\in
I\\!\\!R^{n}}\left\\{w^{\prime}\in
I\\!\\!R^{d}\mid(-x^{\prime},w^{\prime})-\nabla^{2}f_{0}(\bar{x},\bar{w})^{*}(v^{\prime}_{1},0)\in\Omega_{2}(v^{\prime}_{1})\right\\},$

with $\Omega_{2}(v^{\prime}_{1})$ admitting the upper estimation (23). So, for
any $x^{\prime}\in I\\!\\!R^{n}$,

$D^{*}S(\bar{w}|\bar{x})(x^{\prime})\subset\Gamma_{2}(x^{\prime}),$ (25)

where $\Gamma_{2}(x^{\prime})$ consists of vectors $w^{\prime}\in
I\\!\\!R^{d}$ such that

$\begin{cases}w^{\prime}=\left[\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})+\lambda\nabla^{2}_{wx}F(\bar{x},\bar{w})\right]v^{\prime}_{1}+\gamma\nabla_{w}F(\bar{x},\bar{w}),\\\
\left[\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})+\lambda\nabla^{2}_{xx}F(\bar{x},\bar{w})\right]v^{\prime}_{1}+\gamma\nabla_{x}F(\bar{x},\bar{w})=-x^{\prime},\\\
\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}=0,\ v^{\prime}_{1}\in
I\\!\\!R^{n},\ \gamma\in I\\!\\!R.\end{cases}$

To obtain a lower estimate for the Fréchet coderivative values of $S$, we will
use some results of Qui Qui_JOTA2014 . For any $v^{\prime}_{1}\in
I\\!\\!R^{n}$ satisfying $\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}=0$, the
arguments given in (Qui_JOTA2014, , pp. 410–412) provide us with the inclusion

$\widehat{\Omega}_{M}(v^{\prime}_{1})\subset\widehat{D}^{*}M(\bar{\tau})(v^{\prime}_{1}),$
(26)

where $M$ is the multifunction in (11) and

$\begin{array}[]{rl}\widehat{\Omega}_{M}(v^{\prime}_{1}):=&\\{(x^{\prime},w^{\prime})\in
I\\!\\!R^{n}\times I\\!\\!R^{d}\mid
x^{\prime}=\gamma\nabla_{x}F(\bar{x},\bar{w})+\hat{y}\nabla^{2}_{xx}F(\bar{x},\bar{w})v^{\prime}_{1},\\\
&\qquad\qquad\quad\
\;w^{\prime}=\gamma\nabla_{w}F(\bar{x},\bar{w})+\hat{y}\nabla^{2}_{wx}F(\bar{x},\bar{w})v^{\prime}_{1},\,\gamma\in
I\\!\\!R\\}.\end{array}$

In addition, if $\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}\neq 0$, then by
the reasoning in (Qui_JOTA2014, , pp. 405–406) we have
$\widehat{D}^{*}M(\bar{\tau})(v^{\prime}_{1})=\emptyset$. Note that the
Lagrange multiplier $\lambda$ here coincides with the constant $\mu$ in
(Qui_JOTA2014, , Theorem 3.2) and the above assertions do not require
$\nabla_{w}F(\bar{x},\bar{w})\neq 0$. Then, by (12) and (26),

$\widehat{\Gamma}(x^{\prime})\supset\displaystyle\bigcup_{\begin{subarray}{c}v^{\prime}_{1}\in
I\\!\\!R^{n},\\\
\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}=0\end{subarray}}\left\\{w^{\prime}\in
I\\!\\!R^{d}\mid(-x^{\prime},w^{\prime})\in\nabla
G(\bar{x},\bar{w})^{*}v^{\prime}_{1}+\widehat{\Omega}_{M}(v^{\prime}_{1})\right\\}.$

Since $\nabla
G(\bar{x},\bar{w})^{*}v^{\prime}_{1}=(\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1},\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1})$,
one can easily show that the right-hand-side set equals to
$\Gamma_{2}(x^{\prime})$. Therefore, if (24) is satisfied, then by (25) we
have

$\widehat{\Gamma}(x^{\prime})=\widehat{D}^{*}S(\bar{w}|\bar{x})(x^{\prime})=D^{*}S(\bar{w}|\bar{x})(x^{\prime})=\Gamma_{2}(x^{\prime}).$

Thus, under the assumption (24), $S$ is locally Lipschitz-like around
$(\bar{w},\bar{x})$ if and only if $\Gamma_{2}(0)=\\{0\\}$. Clearly, the set
$\Gamma_{2}(0)$ consists of vectors $w^{\prime}\in I\\!\\!R^{d}$ such that

$\begin{cases}w^{\prime}=\left[\nabla^{2}_{xw}f_{0}(\bar{x},\bar{w})^{T}+\lambda\nabla^{2}_{wx}F(\bar{x},\bar{w})\right]v^{\prime}_{1}+\gamma\nabla_{w}F(\bar{x},\bar{w}),\\\
\left[\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})+\lambda\nabla^{2}_{xx}F(\bar{x},\bar{w})\right]v^{\prime}_{1}+\gamma\nabla_{x}F(\bar{x},\bar{w})=0,\\\
\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}=0,\ v^{\prime}_{1}\in
I\\!\\!R^{n},\ \gamma\in I\\!\\!R.\end{cases}$

Equivalently, $w^{\prime}\in I\\!\\!R^{d}$ belongs to $\Gamma_{2}(0)$ iff

$\begin{cases}w^{\prime}=A_{2}\left(\begin{matrix}v^{\prime}_{1}\\\
\gamma\end{matrix}\right),\\\ A_{1}\left(\begin{matrix}v^{\prime}_{1}\\\
\gamma\end{matrix}\right)=0,\
\,\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}=0,\\\ v^{\prime}_{1}\in
I\\!\\!R^{n},\ \gamma\in I\\!\\!R.\end{cases}$

So, $\Gamma_{2}(0)=\\{0\\}$ if and only if
$A_{2}\left(\begin{matrix}v^{\prime}_{1}\\\ \gamma\end{matrix}\right)=0$ for
any pair $(v^{\prime}_{1},\gamma)\in I\\!\\!R^{n}\times I\\!\\!R$ satisfying

$A_{1}\left(\begin{matrix}v^{\prime}_{1}\\\ \gamma\end{matrix}\right)=0,\
\;\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}=0.$

The latter happens if and only if

${\rm ker}\,A_{1}\cap\left({\rm ker}\,\nabla_{x}F(\bar{x},\bar{w})\times
I\\!\\!R\right)\subset{\rm ker}\,A_{2}.$ (27)

To sum up, we state the following theorem.

###### Theorem 4.1

Suppose that $F(\bar{x},\bar{w})=0$ and the Lagrange multiplier $\lambda$
corresponding to the stationary point $\bar{x}\in S(\bar{w})$ is positive. If
(24) holds, then $S$ is locally Lipschitz-like around $(\bar{w},\bar{x})$ if
and only if (27) is satisfied.

###### Remark 4.1

Combining the conditions (24) and (27), we obtain a sufficient condition for
$S$ being locally Lipschitz-like around $(\bar{w},\bar{x})$, that is

${\rm ker}\,A_{1}\cap\left({\rm ker}\,\nabla_{x}F(\bar{x},\bar{w})\times
I\\!\\!R\right)=\\{0\\}.$ (28)

###### Example 4.1

Consider the problem $(P_{w})$ with $f_{0}(x,w)=-x^{2}+(w-1)x$ and
$F(x,w)=x^{2}+w^{2}-2$ for any $(x,w)\in I\\!\\!R\times I\\!\\!R$. Then, by
(3), the stationary point set map of $(P_{w})$ is defined by

$S(w)=\left\\{x\in I\\!\\!R\mid-2x+w-1+\partial_{x}f(x,w)\right\\},$

with $f(x,w)=(g\circ F)(x,w)$ and $g(y)=\delta_{I\\!\\!R_{-}}(y)$ for any
$y\in I\\!\\!R$. Let $\bar{w}=1$ and $\bar{x}=1$. Since $F(\bar{x},\bar{w})=0$
and $\nabla_{x}F(\bar{x},\bar{w})=2$, condition (MFCQ) is valid. Hence, from
(5) we have

$\begin{array}[]{rl}\partial_{x}f(\bar{x},\bar{w})&=\nabla_{x}F(\bar{x},\bar{w})^{*}N_{\mathbb{R_{-}}}(F(\bar{x},\bar{w}))\\\
&=\nabla_{x}F(\bar{x},\bar{w})^{*}\mathbb{R_{+}}=\mathbb{R_{+}}.\end{array}$

Now, it easy to show that $\bar{x}\in S(\bar{w})$. We have
$\nabla_{x}f_{0}(\bar{x},\bar{w})=-2$ and $\lambda=1$ due to (20). Hence,
$A_{1}=[\begin{matrix}0&2\end{matrix}]$ and ${\rm
ker}\,A_{1}=I\\!\\!R\times\\{0\\}$. Thus, (28) is fulfilled and consequently
$S$ is locally Lipschitz-like around $(\bar{w},\bar{x})$.

###### Remark 4.2

For any stationary point $\bar{x}\in S(\bar{w})$ satisfying (MFCQ), the
corresponding unique multiplier $\lambda$ is defined by the equation (20).
This fact justifies to the following assertion: Given any $w\in I\\!\\!R^{d}$,
if $\nabla_{x}F(x,w)\neq 0$ for all $x$ with $F(x,w)=0$, then one has

$\begin{array}[]{rl}S(w)=\big{\\{}x\in{I\\!\\!R}^{n}\mid&F(x,w)\leq 0,\
\exists\lambda\geq 0\ {\rm s.t.}\ \lambda F(x,w)=0,\\\
&\nabla_{x}f_{0}(x,w)+\lambda\nabla_{x}F(x,w)=0\big{\\}}.\end{array}$

One referee of the present paper asks: Whether the condition (27) alone can
ensure the local Lipschitz-likeness of $S$ around $(\bar{w},\bar{x})$. To
answer the question, let us consider the next example showing that, even for
polynomial optimization problems, (27) is not sufficient for $S$ to be locally
Lipschitz-like around $(\bar{w},\bar{x})$.

###### Example 4.2

Consider $(P_{w})$ with $n=2$, $d=1$,
$f_{0}(x,w)=\frac{1}{4}wx_{1}^{4}-wx_{1}-x_{2}$, and $F(x,w)=wx_{1}+x_{2}-w$
for all $(x,w)=(x_{1},x_{2},w)\in I\\!\\!R^{2}\times I\\!\\!R$. Choose
$\bar{x}=(0,0)$ and $\bar{w}=0$. Using Remark 4.4, one can show that
$S(\bar{w})=I\\!\\!R\times\\{0\\}$ and $S(w)=\\{(0,w)\\}$ for every $w\neq 0$.
Hence the stationary map $S(.)$ is not locally Lipschitz-like around
$(\bar{w},\bar{x})$. Furthermore, since the unique Lagrange multiplier
corresponding to $\bar{x}\in S(\bar{w})$ is $\lambda=1$, one can find that
${\rm ker}\,A_{1}=I\\!\\!R\times I\\!\\!R\times\\{0\\}$, ${\rm
ker}\,A_{2}=I\\!\\!R\times I\\!\\!R\times\\{0\\}$, and ${\rm
ker}\,\nabla_{x}F(\bar{w},\bar{x})=I\\!\\!R\times\\{0\\}$. So (27) is
satisfied, but (24) fails to hold.

4.2 The degenerate case

Consider the second possibility: $\lambda=0$. We have

$N_{{\rm gph}\,\partial g}(F(\bar{x},\bar{w}),\lambda)=N_{{\rm gph}\,\partial
g}(0,0)=(\\{0\\}\times
I\\!\\!R)\cup(I\\!\\!R\times\\{0\\})\cup(I\\!\\!R_{+}\times I\\!\\!R_{-}).$

Hence, for any $v^{\prime}\in I\\!\\!R^{n}\times I\\!\\!R^{d}$,

$\partial^{2}g(F(\bar{x},\bar{w})|\,\lambda)(\nabla
F(\bar{x},\bar{w})v^{\prime})=\begin{cases}I\\!\\!R&\mbox{ if }\nabla
F(\bar{x},\bar{w})v^{\prime}=0,\\\ I\\!\\!R_{+}&\mbox{ if }\nabla
F(\bar{x},\bar{w})v^{\prime}>0,\\\ \\{0\\}&\mbox{ if }\nabla
F(\bar{x},\bar{w})v^{\prime}<0.\end{cases}$

Consequently,

$\Omega_{1}(\lambda,v^{\prime})=\begin{cases}\\{\gamma\nabla
F(\bar{x},\bar{w})\mid\gamma\in I\\!\\!R\\}&\mbox{ if }\nabla
F(\bar{x},\bar{w})v^{\prime}=0,\\\ \\{\gamma\nabla
F(\bar{x},\bar{w})\mid\gamma\in\mathbb{R_{+}}\\}&\mbox{ if }\nabla
F(\bar{x},\bar{w})v^{\prime}>0,\\\ \\{0\\}&\mbox{ if }\nabla
F(\bar{x},\bar{w})v^{\prime}<0.\end{cases}$

In this case, $\bar{y}=\lambda=0$ is the unique element satisfying
$\bar{y}\in\partial g(F(\bar{x},\bar{w}))$ and
$\nabla_{x}F(\bar{x},\bar{w})^{*}\bar{y}=-\nabla_{x}f_{0}(\bar{x},\bar{w})$.
So, by (10) we have

$\Omega_{2}(v^{\prime}_{1})=\partial^{2}f((\bar{x},\bar{w})|0)(v^{\prime}_{1},0).$

Clearly, the conditions $\bar{y}\in\partial g(F(\bar{x},\bar{w}))$ and $\nabla
F(\bar{x},\bar{w})^{*}\bar{y}=0$ imply $\bar{y}=\lambda=0$. So, for
$\bar{v}=0$, from (8) we get

$\partial^{2}f((\bar{x},\bar{w})|0)(v^{\prime})\subset\Omega_{1}(\lambda,v^{\prime})$

for all $v^{\prime}\in I\\!\\!R^{n}\times I\\!\\!R^{d}$. It follows that

$\Omega_{2}(v^{\prime}_{1})\subset\begin{cases}\\{\gamma\nabla
F(\bar{x},\bar{w})\mid\gamma\in I\\!\\!R\\}&\mbox{ if
}\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}=0,\\\ \\{\gamma\nabla
F(\bar{x},\bar{w})\mid\gamma\in\mathbb{R_{+}}\\}&\mbox{ if
}\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}>0,\\\ \\{0\\}&\mbox{ if
}\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}<0.\end{cases}$ (29)

Hence, the condition (C0) is satisfied if the following holds: For any
$v^{\prime}_{1}\in I\\!\\!R^{n}$,

(i)

if $\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}=0$ and

$\left(-\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1},-\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}\right)\in\\{\gamma\nabla
F(\bar{x},\bar{w})\mid\gamma\in I\\!\\!R\\},$

then $v^{\prime}_{1}=0$;

(ii)

if $\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}>0$, then

$\left(-\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1},-\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}\right)\notin\\{\gamma\nabla
F(\bar{x},\bar{w})\mid\gamma\in\mathbb{R_{+}}\\};$

(iii)

if $\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}<0$, then
$\left(\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1},\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}\right)\neq
0$.

It follows that (C0) is fulfilled if the following holds: for any
$v^{\prime}_{1}\in I\\!\\!R^{n}$, if

$\left(-\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1},-\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}\right)\in\\{\gamma\nabla
F(\bar{x},\bar{w})\mid\gamma\in I\\!\\!R\\},$

then $v^{\prime}_{1}=0$. The latter can be written equivalently as

$\begin{cases}A^{\prime}_{1}\left(\begin{matrix}v^{\prime}_{1}\\\
\gamma\end{matrix}\right)=0,\quad
A^{\prime}_{2}\left(\begin{matrix}v^{\prime}_{1}\\\
\gamma\end{matrix}\right)=0\\\ v^{\prime}_{1}\in I\\!\\!R^{n},\quad\gamma\in
I\\!\\!R\end{cases}\quad\quad\Longrightarrow\ \;v^{\prime}_{1}=0,$

where

$A^{\prime}_{1}:=\left[\begin{array}[]{ccc}\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})&\nabla_{x}F(\bar{x},\bar{w})\end{array}\right]\in
I\\!\\!R^{n\times(n+1)},$

and

$A^{\prime}_{2}:=\left[\begin{array}[]{ccc}\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})&\nabla_{w}F(\bar{x},\bar{w})\end{array}\right]\in
I\\!\\!R^{d\times(n+1)}.$

Since $\nabla_{x}F(\bar{x},\bar{w})\neq 0$, this condition is equivalent to

${\rm ker}\,A^{\prime}_{1}\cap{\rm ker}\,A^{\prime}_{2}=\\{0\\}.$ (30)

We now suppose that (30) is valid. Then, by Lemma 2.2 and the upper estimate
(29), for any $x^{\prime}\in I\\!\\!R^{n}$,

$D^{*}S(\bar{w}|\bar{x})(x^{\prime})\subset\Gamma_{3}(x^{\prime}),$ (31)

where $\Gamma_{3}(x^{\prime})$ consists of vectors $w^{\prime}\in
I\\!\\!R^{d}$ for which there exists $v^{\prime}_{1}\in I\\!\\!R^{n}$ with

$\begin{cases}\left(-x^{\prime}-\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1},w^{\prime}-\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}\right)\in\\{\gamma\nabla
F(\bar{x},\bar{w})\mid\gamma\in I\\!\\!R\\},\\\
\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}=0,\end{cases}$

or

$\begin{cases}\left(-x^{\prime}-\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1},w^{\prime}-\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}\right)\in\\{\gamma\nabla
F(\bar{x},\bar{w})\mid\gamma\in I\\!\\!R_{+}\\},\\\
\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}>0,\end{cases}$

or

$\begin{cases}-x^{\prime}-\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}=0,\
\,w^{\prime}-\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}=0,\\\
\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}<0.\end{cases}$

By putting

$\Delta_{1}=\\{(v^{\prime}_{1},\gamma)\in I\\!\\!R^{n}\times
I\\!\\!R\mid\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}>0,\,\gamma\geq 0\\}$

and

$\Delta_{2}=\\{v^{\prime}_{1}\in
I\\!\\!R^{n}\mid\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}<0\\},$

we see that $\Gamma_{3}(x^{\prime})$ consists of vectors $w^{\prime}\in
I\\!\\!R^{d}$ such that

$\begin{cases}w^{\prime}=\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}+\gamma\nabla_{w}F(\bar{x},\bar{w}),\\\
\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}+\gamma\nabla_{x}F(\bar{x},\bar{w})=-x^{\prime},\\\
(v^{\prime}_{1},\gamma)\in{\rm ker}\,\nabla_{x}F(\bar{x},\bar{w})\times
I\\!\\!R,\end{cases}$

or

$\begin{cases}w^{\prime}=\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}+\gamma\nabla_{w}F(\bar{x},\bar{w}),\\\
\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}+\gamma\nabla_{x}F(\bar{x},\bar{w})=-x^{\prime},\\\
(v^{\prime}_{1},\gamma)\in\Delta_{1},\end{cases}$

or

$\begin{cases}w^{\prime}=\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1},\\\
\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}=-x^{\prime},\\\
v^{\prime}_{1}\in\Delta_{2}.\end{cases}$

In particular, the set $\Gamma_{3}(0)$ consists of vectors $w^{\prime}\in
I\\!\\!R^{d}$ such that

$\begin{cases}w^{\prime}=A^{\prime}_{2}\left(\begin{matrix}v^{\prime}_{1}\\\
\gamma\end{matrix}\right),\
\,A^{\prime}_{1}\left(\begin{matrix}v^{\prime}_{1}\\\
\gamma\end{matrix}\right)=0,\\\ (v^{\prime}_{1},\gamma)\in{\rm
ker}\,\nabla_{x}F(\bar{x},\bar{w})\times I\\!\\!R.\end{cases}$

or

$\begin{cases}w^{\prime}=A^{\prime}_{2}\left(\begin{matrix}v^{\prime}_{1}\\\
\gamma\end{matrix}\right),\
\,A^{\prime}_{1}\left(\begin{matrix}v^{\prime}_{1}\\\
\gamma\end{matrix}\right)=0,\\\
(v^{\prime}_{1},\gamma)\in\Delta_{1}.\end{cases}$

or

$\begin{cases}w^{\prime}=\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1},\\\
\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}=0,\,v^{\prime}_{1}\in\Delta_{2}.\end{cases}$

Therefore, $\Gamma_{3}(0)=\\{0\\}$ if and only if the following three
conditions are simultaneously fulfilled

${\rm ker}\,A^{\prime}_{1}\cap\left({\rm
ker}\,\nabla_{x}F(\bar{x},\bar{w})\times I\\!\\!R\right)\subset{\rm
ker}\,A^{\prime}_{2},$ (32) ${\rm
ker}\,A^{\prime}_{1}\cap\Delta_{1}\subset{\rm ker}\,A^{\prime}_{2},$ (33)

and

${\rm ker}\,\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})\cap\Delta_{2}\subset{\rm
ker}\,\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w}).$ (34)

Remember that under condition (30) we have the estimate (31). Therefore, the
fulfillment of (30), (32), (33), and (34) implies
$D^{*}S(\bar{w}|\bar{x})(0)=\\{0\\}$. Applying the Mordukhovich criterion, we
obtain the following sufficient condition for the local Lipschitz-likeness of
$S$ around $(\bar{w},\bar{x})$:

$\begin{cases}{\rm ker}\,A^{\prime}_{1}\cap{\rm
ker}\,A^{\prime}_{2}=\\{0\\},\\\ {\rm ker}\,A^{\prime}_{1}\cap\left({\rm
ker}\,\nabla_{x}F(\bar{x},\bar{w})\times I\\!\\!R\right)\subset{\rm
ker}\,A^{\prime}_{2},\\\ {\rm ker}\,A^{\prime}_{1}\cap\Delta_{1}\subset{\rm
ker}\,A^{\prime}_{2},\\\ {\rm
ker}\,\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})\cap\Delta_{2}\subset{\rm
ker}\,\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w}).\end{cases}$ (35)

Now, by the arguments of Qui (Qui_JOTA2014, , pp. 414–416), the Fréchet
coderivative values of the multifunction $M$ in (11) admit the lower estimate

$\widehat{D}^{*}M(\bar{\tau})(v^{\prime}_{1})\supset\left\\{\gamma\big{(}\nabla_{x}F(\bar{x},\bar{w}),\nabla_{w}F(\bar{x},\bar{w})\big{)}\mid\gamma\geq
0\right\\}$

for any $v^{\prime}_{1}$ with $\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}\geq
0$. If $\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}<0$, then
$\widehat{D}^{*}M(\bar{\tau})(v^{\prime}_{1})=\emptyset$; see (Qui_JOTA2014, ,
p. 412). (It is important to stress that we don’t need the condition
$\nabla_{w}F(\bar{x},\bar{w})\neq 0$ here.) So, by (12) one has
$\widehat{\Gamma}(x^{\prime})\supset\widehat{\Gamma}_{1}(x^{\prime})$ for any
$x^{\prime}\in I\\!\\!R^{n}$, where

$\begin{array}[]{rl}\widehat{\Gamma}_{1}(x^{\prime})&:=\displaystyle\bigcup_{\begin{subarray}{c}v^{\prime}_{1}\in
I\\!\\!R^{n},\\\ \nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}\geq
0\end{subarray}}\Big{\\{}w^{\prime}\in
I\\!\\!R^{d}\mid(-x^{\prime},w^{\prime})\\\ &\qquad\qquad\qquad\in\nabla
G(\bar{x},\bar{w})^{*}v^{\prime}_{1}+\left\\{\gamma\big{(}\nabla_{x}F(\bar{x},\bar{w}),\nabla_{w}F(\bar{x},\bar{w})\big{)}\mid\gamma\geq
0\right\\}\Big{\\}}.\end{array}$

Choosing $v^{\prime}_{1}=0$, we have $0\in\widehat{\Gamma}_{1}(0)$. In
combination with the results in Lemma 2.3, this yields

$\\{0\\}\subset\widehat{\Gamma}_{1}(0)\subset\widehat{\Gamma}(0)\subset\widehat{D}^{*}S(\bar{w}|\bar{x})(0)\subset
D^{*}S(\bar{w}|\bar{x})(0).$

According to the Mordukhovich criterion, if $S$ is locally Lipschitz-like
around $(\bar{w},\bar{x})$, then $D^{*}S(\bar{w}|\bar{x})(0)=\\{0\\}$ and
$\widehat{\Gamma}_{1}(0)=\\{0\\}$ as a result. Since

$\nabla
G(\bar{x},\bar{w})^{*}v^{\prime}_{1}=(\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1},\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}),$

the latter means that

$\begin{cases}w^{\prime}=\nabla^{2}_{wx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}+\gamma\nabla_{w}F(\bar{x},\bar{w})\\\
\nabla^{2}_{xx}f_{0}(\bar{x},\bar{w})v^{\prime}_{1}+\gamma\nabla_{x}F(\bar{x},\bar{w})=0\qquad\quad\
\Longrightarrow\ \,w^{\prime}=0.\\\
\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}\geq 0,\ \,v^{\prime}_{1}\in
I\\!\\!R^{n},\ \,\gamma\in I\\!\\!R_{+}\end{cases}$

For $\Delta_{3}:=\\{(v^{\prime}_{1},\gamma)\in I\\!\\!R^{n}\times
I\\!\\!R\mid\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}\geq 0,\,\gamma\geq
0\\}$, this condition becomes

$\begin{cases}w^{\prime}=A^{\prime}_{2}\left(\begin{matrix}v^{\prime}_{1}\\\
\gamma\end{matrix}\right)\\\
A^{\prime}_{1}\left(\begin{matrix}v^{\prime}_{1}\\\
\gamma\end{matrix}\right)=0\qquad\quad\ \Longrightarrow\ \,w^{\prime}=0.\\\
\left(v^{\prime}_{1},\gamma\right)\in\Delta_{3}\end{cases}$

Clearly, the last property is equivalent to

${\rm ker}\,A^{\prime}_{1}\cap\Delta_{3}\subset{\rm ker}\,A^{\prime}_{2}.$
(36)

Thus, we have shown that (36) is a necessary condition for $S$ being locally
Lipschitz-like around $(\bar{w},\bar{x})$.

Condition (35) implies (36). Indeed, suppose that (35) is fulfilled and
$\left(v^{\prime}_{1},\gamma\right)\in{\rm
ker}\,A^{\prime}_{1}\cap\Delta_{3}$. If
$\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}=0$, then

$\left(v^{\prime}_{1},\gamma\right)\in{\rm ker}\,A^{\prime}_{1}\cap\left({\rm
ker}\,\nabla_{x}F(\bar{x},\bar{w})\times I\\!\\!R\right);$

so $\left(v^{\prime}_{1},\gamma\right)\in{\rm ker}\,A^{\prime}_{2}$ by the
second condition in (35). If $\nabla_{x}F(\bar{x},\bar{w})v^{\prime}_{1}>0$,
then $\left(v^{\prime}_{1},\gamma\right)\in{\rm
ker}\,A^{\prime}_{1}\cap\Delta_{1}$; hence
$\left(v^{\prime}_{1},\gamma\right)\in{\rm ker}\,A^{\prime}_{2}$ by the third
condition in (35). We have thus proved that (35) yields (36).

The above elementary analysis clearly shows how the sufficient condition for
the local Lipschitz-likeness of $S$ around $(\bar{w},\bar{x})$ in the
degenerate case is stronger than the necessary one.

We can summarize our results for the degenerate case as follows.

###### Theorem 4.2

Suppose that $F(\bar{x},\bar{w})=0$ and the Lagrange multiplier $\lambda$
corresponding to the stationary point $\bar{x}\in S(\bar{w})$ is null. The
following assertions are true:

(a)

If $S$ is locally Lipschitz-like around $(\bar{w},\bar{x})$, then condition
(36) holds;

(b)

If condition (35) is fulfilled, then $S$ is locally Lipschitz-like around
$(\bar{w},\bar{x})$.

Let us consider a simple example to see how Theorem 4.2 works for concrete
optimization problems.

###### Example 4.3

Let $f_{0}(x,w)=x^{2}(w-2)$ and $F(x,w)=w(x-1)$ for all $(x,w)\in
I\\!\\!R\times I\\!\\!R$. The stationary point set of $(P_{w})$ is given by

$S(w)=\\{x\in I\\!\\!R\mid 0\in 2x(w-2)+\partial_{x}f(x,w)\\},$

with $f(x,w)=(g\circ F)(x,w)$ and $g(y)=\delta_{I\\!\\!R_{-}}(y)$ for all
$y\in I\\!\\!R$. Let $\bar{w}=1$. Then, the point $\bar{x}=1$ belongs to
$S(\bar{w})$. Indeed, since $F(\bar{x},\bar{w})=0$ and
$\nabla_{x}F(\bar{x},\bar{w})=1$, condition (MFCQ) is valid. Hence, from (5)
we have

$\begin{array}[]{rl
}\partial_{x}f(\bar{x},\bar{w})&=\nabla_{x}F(\bar{x},\bar{w})^{*}N_{\mathbb{R_{-}}}(F(\bar{x},\bar{w}))\\\
&=\nabla_{x}F(\bar{x},\bar{w})^{*}\mathbb{R_{+}}=\mathbb{R_{+}}.\end{array}$

Now it is easy to check that $\bar{x}\in S(\bar{w})$. Here we have
$A^{\prime}_{1}=[\begin{array}[]{ccc}-4&1\end{array}]$,
$A^{\prime}_{2}=[\begin{array}[]{ccc}2&0\end{array}]$, and ${\rm
ker}\,\nabla_{x}F(\bar{x},\bar{w})=\\{0\\}$. Thus, condition (35) is
fulfilled; as a result, $S$ is locally Lipschitz-like around
$(\bar{w},\bar{x})$.

###### Remark 4.3

Looking back to Example 4.3, we see that

$\nabla_{w}F(\bar{x},\bar{w})=\bar{x}-1=0.$

So, the preceding results of Qui (Qui_JoGO_2016, , Theorem 4.2) cannot be
applied for the boundary point $(\bar{x},\bar{w})$ of the set

$\mathcal{D}=\\{(x,w)\mid F(x,w)\leq 0\\}=\\{(x,w)\mid x\leq 1,\,w\geq
0\\}\cup\\{(x,w)\mid x\leq 1,\,w\geq 0\\},$

as it requires that $\nabla_{w}F(\bar{x},\bar{w})\neq 0$.

###### Acknowledgements.

This work was supported by National Foundation for Science $\&$ Technology
Development (Vietnam) and the Grant MOST 105-2115-M-039-002-MY3 (Taiwan). The
authors are grateful to the anonymous referees for their careful readings and
valuable suggestions. Examples 3.4 and 4.2 in this paper present our solutions
to two open questions raised by one of the referees. In addition, Remarks 3.2
and 3.3 are based on some comments of that referee.

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