# Stability for Constrained Minimax Optimization

Yu-Hong Dai111LSEC, ICMSEC, AMSS, Chinese Academy of Sciences, Beijing 100190,
China. Email<EMAIL_ADDRESS>This author was supported by the Natural
Science Foundation of China (Nos. 11991020, 11631013, 11971372 and 11991021)
and the Strategic Priority Research Program of Chinese Academy of Sciences
(No. XDA27000000). 222School of Mathematical Sciences, University of Chinese
Academy of Sciences, Beijing 100049, China. and Liwei Zhang 333Corresponding
author. School of Mathematical Sciences, Dalian University of Technology,
Dalian 116024, China. Email<EMAIL_ADDRESS>This author was supported by
the Natural Science Foundation of China (Nos. 11971089 and 11731013).

###### Abstract

Minimax optimization problems are an important class of optimization problems
arising from both modern machine learning and from traditional research areas.
We focus on the stability of constrained minimax optimization problems based
on the notion of local minimax point by Dai and Zhang (2020). Firstly, we
extend the classical Jacobian uniqueness conditions of nonlinear programming
to the constrained minimax problem and prove that this set of properties is
stable with respect to small ${\cal C}^{2}$ perturbation. Secondly, we provide
a set of conditions, called Property A, which does not require the strict
complementarity condition for the upper level constraints. Finally, we prove
that Property A is a sufficient condition for the strong regularity of the
Kurash-Kuhn-Tucker (KKT) system at the KKT point, and it is also a sufficient
condition for the local Lipschitzian homeomorphism of the Kojima mapping near
the KKT point.

Key words: constrained minimax optimization, Jacobian uniqueness conditions,
strong regularity, strong sufficient optimality condition, Kojima mapping,
local Lipschitzian homeomorphism.
AMS subject classification: 90C30

##  1 Introduction

Let $m$, $n$, $m_{1}$, $m_{2}$, $n_{1}$ and $n_{2}$ be positive integers and
$f:\Re^{n}\times\Re^{m}\rightarrow\Re$,
$h:\Re^{n}\times\Re^{m}\rightarrow\Re^{m_{1}}$,
$g:\Re^{n}\times\Re^{m}\rightarrow\Re^{m_{2}}$,
$H:\Re^{n}\rightarrow\Re^{n_{1}}$ and $G:\Re^{n}\rightarrow\Re^{n_{2}}$ be
given functions. We are interested in the constrained minimax optimization
problem of the form

$\min_{x\in\Phi}\max_{y\in Y(x)}f(x,y),$ (1.1)

where $\Phi\subset\Re^{n}$ is a feasible set of decision variable $x$ defined
by

$\Phi=\\{x\in\Re^{n}:H(x)=0,\,G(x)\leq 0\\}$ (1.2)

and $Y:\Re^{n}\rightrightarrows\Re^{m}$ is a set-valued mapping defined by

$Y(x)=\\{y\in\Re^{m}:h(x,y)=0,\,g(x,y)\leq 0\\}.$ (1.3)

For unconstrained nonconvex-nonconcave minimax optimization, Jin et al. [7]
proposed a proper definition of local minimax point. This definition of local
minimax point is extended in [5] for the constrained minimax optimization
problem (1.1).

###### Definition 1.1

A point $(x^{*},y^{*})\in\Re^{n}\times\Re^{m}$ is said to be a local minimax
point of Problem (1.1) if there exist $\delta_{0}>0$ and a function
$\eta:(0,\delta_{0}]\rightarrow\Re_{+}$ satisfying $\eta(\delta)\rightarrow 0$
as $\delta\rightarrow 0$ such that for any $\delta\in(0,\delta_{0}]$ and any
$(x,y)\in[\textbf{B}_{\delta}(x^{*})\cap\Phi]\times[Y(x^{*})\cap\textbf{B}_{\delta}(y^{*})]$,
we have

$f(x^{*},y)\leq f(x^{*},y^{*})\leq\max_{z}\left\\{f(x,z):z\in
Y(x)\cap\textbf{B}_{\eta(\delta)}(y^{*})\right\\}.$ (1.4)

In [5], we established the first-order optimality, the second-order necessary
and sufficient optimality conditions for Problem (1.1) when the Jacobian
uniqueness conditions are satisfied for the lower level problem and the first-
order necessary optimality conditions when the strong second-order sufficient
optimality condition and the linear independence constraint qualification are
satisfied for the lower level problem.

It is well known that, for nonlinear programming, the Jacobian uniqueness
condition can be used to establish the stability of the ${\cal
C}^{2}$-perturbation (see for instance [6]) and prove that the strong second-
order sufficient optimality condition and the linear independence constraint
qualification are equivalent to the strong regularity of the Kurash-Kuhn-
Tucker (KKT) system (see [11] and [8]). The question naturally arises: What
are the counterparts of these two stability properties for the constrained
minimax optimization problem? The purpose of this paper is to answer this
basic question.

The rest of this paper is organized as follows. In Section 2, we develop a
simplified version for second-order optimality conditions for the constrained
minimax optimization problem, which is suitable for the study of stability
properties. In Section 3, we prove that the proposed Jacobian uniqueness
conditions for Problem (1.1) are kept when a ${\cal C}^{2}$-perturbation of
the original problem occurs. In Section 4, we prove that the proposed Property
A, which does not require the strict complementarity for the upper level
problem, is a sufficient condition for the strong regularity of the KKT system
at the KKT point. Finally, we draw a conclusion in Section 5.

Notation. Scalars and vectors are expressed in lower case letters and matrices
are expressed in upper case letters. For a vector $x$, denote
$\textbf{B}_{\delta}(x)=\\{x^{\prime}:\|x^{\prime}-x\|\leq\delta\\}$. For $a$,
$b\in\Re^{p}$, $a\circ b$ denotes the Hadamard product of $a$ and $b$; namely,
$a\circ b=(a_{1}b_{1},\ldots,a_{p}b_{p})^{T}$. For $a\in\Re^{p}$, $a>0$,
denote $\sqrt{a}={\rm Diag}(\sqrt{a_{1}},\ldots,\sqrt{a_{p}})$. For a convex
set $D\subset\Re^{k}$, we use $\Pi_{D}(w)$ to stand for the projection of $w$
onto $D$. For simplicity, for a function
$F:\Re^{n}\times\Re^{m}\rightarrow\Re$, a mapping
$g:\Re^{n}\times\Re^{m}\rightarrow\Re^{p}$, and a mapping
$y:\Re^{n}\rightarrow\Re^{m}$, we denote

$\begin{array}[]{ll}\nabla_{x}F(x,y(x))=\nabla_{x}F(x,y)|_{y=y(x)},&\quad\nabla_{y}F(x,y(x))=\nabla_{y}F(x,y)|_{y=y(x)},\\\\[4.0pt]
\nabla^{2}_{xx}F(x,y(x))=\nabla^{2}_{xx}F(x,y)|_{y=y(x)},&\quad\nabla^{2}_{xy}F(x,y(x))=\nabla^{2}_{xy}F(x,y)|_{y=y(x)},\\\\[4.0pt]
{\cal J}_{x}g(x,y(x))={\cal J}_{x}g(x,y)|_{y=y(x)},&\quad{\cal
J}_{y}g(x,y(x))={\cal J}_{y}g(x,y)|_{y=y(x)}.\end{array}$

Let $G:\Re^{n}\rightarrow\Re^{m}$ be a locally Lipschitz continuous mapping
over an open set ${\cal O}$. Then $G$ is differentiable almost everywhere in
${\cal O}$. Let ${\cal D}_{G}$ denote the set of differentiable points of $G$
in ${\cal O}$. For a point $x\in{\cal O}$, the B-subdifferential of $G$ at $x$
is defined by

$\partial_{B}G(x)=\left\\{V:\exists x^{k}\in{\cal D}_{G},\,x^{k}\rightarrow
x,\,\displaystyle{\cal J}G(x^{k})\rightarrow V\right\\}$

and the Clarke subdifferential of $G$ at $x$ is defined by

$\partial G(x)={\rm conv}\,\partial_{B}G(x).$

For differential properties of Lipschitz mappings, see the famous book [4].

##  2 Simplified Second-order Optimality Conditions

Consider the case when the Jacobian uniqueness conditions hold at some point
$(x^{*},y^{*},\mu^{*},\lambda^{*})\in\Re^{n}\times\Re^{m}\times\Re^{q}\times\Re^{p}$,
where $(x^{*},y^{*})\in\Re^{n}\times\Re^{m}$ is a point around which $f,h,g$
are twice continuously differentiable. For a point $x\in\Re^{n}$ around
$x^{*}$, we use (${\rm P}_{x}$) to denote the following problem

$\begin{array}[]{cl}\max_{z\in\Re^{m}}&f(x,z)\\\\[4.0pt] {\rm
s.t.}&h(x,z)=0,\\\\[4.0pt] &g(x,z)\leq 0.\end{array}$ (2.1)

The Lagrangian of Problem (${\rm P}_{x}$) is defined by

${\cal L}(x,z,\mu,\lambda)=f(x,z)+\mu^{T}h(x,z)-\lambda^{T}g(x,z).$

###### Definition 2.1

Let $(\mu^{*},\lambda^{*})\in\Re^{m_{1}}\times\Re^{m_{2}}$ be a point. We say
that Jacobian uniqueness conditions of Problem (${\rm P}_{x^{*}}$) are
satisfied at $(y^{*},\mu^{*},\lambda^{*})$ if

  * (a)

The point $(y^{*},\mu^{*},\lambda^{*})$ is a Karush-Kuhn-Tucker point of
Problem (${\rm P}_{x^{*}}$); namely,

$\begin{array}[]{l}\nabla_{y}{\cal
L}(x^{*},y^{*},\mu^{*},\lambda^{*})=0,\\\\[3.0pt] h(x^{*},y^{*})=0,\\\\[3.0pt]
0\leq\lambda^{*}\perp g(x^{*},y^{*})\leq 0.\end{array}$

  * (b)

The linear independence constraint qualification holds at $y^{*}$; namely, the
set of vectors

$\left\\{\nabla_{y}h_{1}(x^{*},y^{*}),\ldots,\nabla_{y}h_{m_{1}}(x^{*},y^{*})\right\\}\cup\left\\{\nabla_{y}g_{i}(x^{*},y^{*}):i\in
I_{x^{*}}(y^{*})\right\\}$

are linearly independent, where
$I_{x^{*}}(y^{*})=\left\\{i:g_{i}(x^{*},y^{*})=0,i=1,\ldots,m_{2}\right\\}$.

  * (c)

The strict complementarity condition holds at $y^{*}$ for $\lambda^{*}$;
namely,

$\lambda^{*}_{i}-g_{i}(x^{*},y^{*})>0,\quad i=1,\ldots,m_{2}.$

  * (d)

The second-order sufficient optimality condition holds at
$(y^{*},\mu^{*},\lambda^{*})$,

$\langle\nabla_{yy}^{2}{\cal
L}(x^{*},y^{*},\mu^{*},\lambda^{*})d_{y},d_{y}\rangle<0\quad\forall
d_{y}\in{\cal C}_{x^{*}}(y^{*}),$

where ${\cal C}_{x^{*}}(y^{*})$ is the critical cone of Problem $({\rm
P}_{x^{*}})$ at $y^{*}$,

${\cal C}_{x^{*}}(y^{*})=\left\\{d_{y}\in\Re^{m}:{\cal
J}_{y}h(x^{*},y^{*})d_{y}=0;\nabla_{y}g_{i}(x^{*},y^{*})d_{y}\leq 0,i\in
I_{x^{*}}(y^{*});\nabla_{y}f(x^{*},y^{*})d_{y}\leq 0\right\\}.$

Let us denote

$\alpha=\Big{\\{}i:g_{i}(x^{*},y^{*})=0,i=1,\ldots,p\Big{\\}},\quad\alpha^{c}=\Big{\\{}i:g_{i}(x^{*},y^{*})<0,i=1,\ldots,p\Big{\\}}.$
(2.2)

###### Lemma 2.1

Let $(x^{*},y^{*})\in\Re^{n}\times\Re^{m}$ be a point around which $f,h,g$ are
twice continuously differentiable. Let
$(\mu^{*},\lambda^{*})\in\Re^{m_{1}}\times\Re^{m_{2}}$ such that Jacobian
uniqueness conditions of Problem $({\rm P}_{x^{*}})$ are satisfied at
$(y^{*},\mu^{*},\lambda^{*})$. Then there exist $\delta_{0}>0$ and
$\varepsilon_{0}>0$, and a twice continuously differentiable mapping
$(y,\mu,\lambda):\textbf{B}_{\delta_{0}}(x^{*})\rightarrow\textbf{B}_{\varepsilon_{0}}(y^{*})\times\textbf{B}_{\varepsilon_{0}}(\mu^{*})\times\textbf{B}_{\varepsilon_{0}}(\lambda^{*})$
such that Jacobian uniqueness conditions of Problem $({\rm P}_{x})$ are
satisfied at $(y(x),\mu(x),\lambda(x))$ when
$x\in\textbf{B}_{\delta_{0}}(x^{*})$. Moreover, for
$x\in\textbf{B}_{\delta_{0}}(x^{*})$,

$\begin{array}[]{l}g_{i}(x,y(x))=0,\ \lambda_{i}(x)>0,\ i\in\alpha,\\\\[4.0pt]
g_{i}(x,y(x))<0,\ \lambda_{i}(x)=0,\ i\in\alpha^{c}.\end{array}$ (2.3)

For $(y(x),\mu(x),\lambda(x))$ given in Lemma 2.1, define the optimal value
function

$\varphi(x)=f(x,y(x)),\quad x\in\textbf{B}_{\delta_{0}}(x^{*})$ (2.4)

and

$K_{\alpha}(x)=\left[\begin{array}[]{ccc}\nabla^{2}_{yy}{\cal
L}(x,y(x),\mu(x),\lambda(x))&{\cal J}_{y}h(x,y(x))^{T}&-{\cal
J}_{y}g_{\alpha}(x,y(x))^{T}\\\\[4.0pt] {\cal J}_{y}h(x,y(x))&0&0\\\\[4.0pt]
-{\cal J}_{y}g_{\alpha}(x,y(x))&0&0\end{array}\right].$ (2.5)

###### Lemma 2.2

Let $(x^{*},y^{*})\in\Re^{n}\times\Re^{m}$ be a point around which $f,h,g$ are
twice continuously differentiable. Let
$(\mu^{*},\lambda^{*})\in\Re^{m_{1}}\times\Re^{m_{2}}$ such that Jacobian
uniqueness conditions of Problem $({\rm P}_{x^{*}})$ are satisfied at
$(x^{*},\mu^{*},\lambda^{*})$. Then $K_{\alpha}(x^{*})$ is nonsingular and
$K_{\alpha}(x)$ is nonsingular when $x\in\textbf{B}_{\delta_{0}}(x^{*})$ for
small $\delta_{0}>0$.

Basing on (2.3), we may simplify the formula in Proposition 2.1 of [5] for the
second-order derivative of $\varphi(x)$.

###### Proposition 2.1

If the assumptions of Lemma 2.1 are satisfied and $\varphi$ is defined by
(2.4), then

$\nabla_{x}\varphi(x)=\nabla_{x}{\cal L}(x,y(x),\mu(x),\lambda(x))$ (2.6)

and

$\nabla^{2}\varphi(x)=\nabla^{2}_{xx}{\cal L}(x,y(x),\mu(x),\lambda(x))\\\
-N_{\alpha}(x)^{T}K_{\alpha}(x)^{-1}N_{\alpha}(x),$ (2.7)

where

$N_{\alpha}(x)=\left[\begin{array}[]{c}\nabla^{2}_{x,y}{\cal
L}(x,y(x)\mu(x),\lambda(x))\\\ {\cal J}_{x}h(x,y(x))\\\ {\cal
J}_{x}g_{\alpha}(x,y(x))\end{array}\right].$ (2.8)

Proof. It is easy to check that $K(x)$ is nonsingular when
$x\in\textbf{B}_{\delta_{0}}(x^{*})$ for small $\delta_{0}>0$. From
Proposition 2.1 of [5], we only need to check

$N(x)^{T}K(x)^{-1}N(x)=N_{\alpha}(x)^{T}K_{\alpha}(x)^{-1}N_{\alpha}(x),$
(2.9)

where $K(x)$ and $N(x)$ are defined in [5] with the following expressions

$K(x)=\left[\begin{array}[]{cccc}\nabla^{2}_{yy}{\cal
L}(x,y(x),\mu(x),\lambda(x))&0&{\cal J}_{y}h(x,y(x))^{T}&{\cal
J}_{y}g(x,y(x))^{T}\\\\[4.0pt] 0&-2{\rm Diag}(\lambda(x))&0&2{\rm
Diag}\left(\sqrt{-g(x,y(x))}\right)\\\\[4.0pt] {\cal
J}_{y}h(x,y(x))&0&0&0\\\\[4.0pt] {\cal J}_{y}g(x,y(x))&2{\rm
Diag}\left(\sqrt{-g(x,y(x))}\right)&0&0\end{array}\right]$

and

$N(x)=\left[\begin{array}[]{c}\nabla^{2}_{x,y}{\cal
L}(x,y(x)\mu(x),\lambda(x))\\\ 0\\\ {\cal J}_{x}h(x,y(x))\\\ {\cal
J}_{x}g(x,y(x))\end{array}\right].$

Define

$P=\left[\begin{array}[]{cccc}I_{n\times n}&0&0&0\\\\[3.0pt] 0&0&I_{p\times
p}&0\\\\[3.0pt] 0&0&0&I_{q\times q}\\\\[3.0pt] 0&I_{p\times p}&0&0\\\\[3.0pt]
\end{array}\right].$

Then $P^{T}P=PP^{T}=I_{n+q+2p}$ and

$K(x)^{-1}=[P^{T}PK(x)P^{T}P]^{-1}=P^{T}[PK(x)P^{T}]^{-1}P.$

Thus

$N(x)^{T}K(x)^{-1}N(x)=N(x)^{T}P^{T}[PK(x)P^{T}]^{-1}PN(x)=[PN(x)]^{T}[PK(x)P^{T}]^{-1}[PN(x)].$

Let $G(x)=\nabla^{2}_{yy}{\cal L}(x,y(x),\mu(x),\lambda(x))$, $D(x)=2{\rm
Diag}\left(\sqrt{-g_{\alpha}(x,y(x))}\right)$ and $E(x)=-2{\rm
Diag}(\lambda_{\alpha}(x))$. We have that

$\begin{array}[]{l}PK(x)P^{T}\\\\[5.0pt]
=\left[\begin{array}[]{cccc}G(x)&{\cal J}_{y}h(x,y(x))^{T}&{\cal
J}_{y}g(x,y(x))^{T}&0\\\\[4.0pt] {\cal J}_{y}h(x,y(x))&0&0&0\\\\[4.0pt] {\cal
J}_{y}g(x,y(x))&0&0&2{\rm Diag}\left(\sqrt{-g(x,y(x))}\right)\\\\[4.0pt]
0&0&2{\rm Diag}\left(\sqrt{-g(x,y(x))}\right)&-2{\rm
Diag}(\lambda(x))\end{array}\right]\\\\[26.0pt]
=\left[\begin{array}[]{cccccc}G(x)&{\cal J}_{y}h(x,y(x))^{T}&{\cal
J}_{y}g(x,y(x))^{T}&0&0&0\\\\[4.0pt] {\cal
J}_{y}h(x,y(x))&0&0&0&0&0\\\\[4.0pt] {\cal
J}_{y}g_{\alpha}(x,y(x))&0&0&0&0&0\\\\[4.0pt] {\cal
J}_{y}g_{\alpha^{c}}(x,y(x))&0&0&0&0&D(x)\\\\[4.0pt] 0&0&0&0&E(x)&0\\\\[4.0pt]
0&0&0&D(x)&0&0\end{array}\right]\end{array}$

Also let

$Q=\left[\begin{array}[]{cccccc}I_{n\times}&0&0&0&0&0\\\\[3.0pt] 0&I_{q\times
q}&0&0&0&0\\\\[3.0pt] 0&0&I_{|\alpha|\times|\alpha|}&0&0&0\\\\[3.0pt]
0&0&0&I_{|\alpha^{c}|\times|\alpha^{c}|}&0&0\\\\[3.0pt]
0&0&0&0&0&I_{|\alpha^{c}|\times|\alpha^{c}|}\\\\[3.0pt]
0&0&0&0&I_{|\alpha|\times|\alpha|}&0\end{array}\right].$

Then $QQ^{T}=Q^{T}Q=I_{n+q+2p}$. Obviously, we have that

$\begin{array}[]{l}QPK(x)P^{T}Q^{T}\\\\[5.0pt]
=\left[\begin{array}[]{cccccc}G(x)&{\cal J}_{y}h(x,y(x))^{T}&{\cal
J}_{y}g(x,y(x))^{T}&0&0&0\\\\[4.0pt] {\cal
J}_{y}h(x,y(x))&0&0&0&0&0\\\\[4.0pt] {\cal
J}_{y}g_{\alpha}(x,y(x))&0&0&0&0&0\\\\[4.0pt] {\cal
J}_{y}g_{\alpha^{c}}(x,y(x))&0&0&0&D(x)&0\\\\[4.0pt] 0&0&0&D(x)&0&0\\\\[4.0pt]
0&0&0&0&0&E(x)\end{array}\right]\end{array}$

and

$QPN(x)=\left[\begin{array}[]{c}\nabla^{2}_{x,y}{\cal
L}(x,y(x)\mu(x),\lambda(x))\\\ {\cal J}_{x}h(x,y(x))\\\ {\cal
J}_{x}g(x,y(x))\\\ 0\end{array}\right].$

Therefore we obtain

$\begin{array}[]{l}N(x)^{T}K(x)^{-1}N(x)=[QPN(x)]^{T}[QPK(x)P^{T}Q^{T}]^{-1}[QPN(x)]\\\\[6.0pt]
=\left[\begin{array}[]{c}N_{\alpha}(x)\\\\[3.0pt] {\cal
J}_{x}g_{\alpha^{c}}(x,y(x))\\\\[3.0pt]
0_{\alpha^{c}}\end{array}\right]^{T}\left[\begin{array}[]{ccc}K_{\alpha}(x)&0&0\\\\[3.0pt]
0&0&D(x)\\\\[3.0pt]
0&D(x)&0\end{array}\right]^{-1}\left[\begin{array}[]{c}N_{\alpha}(x)\\\\[3.0pt]
{\cal J}_{x}g_{\alpha^{c}}(x,y(x))\\\\[3.0pt]
0_{\alpha^{c}}\end{array}\right]\\\\[6.0pt]
=\left[\begin{array}[]{c}N_{\alpha}(x)\\\\[3.0pt] {\cal
J}_{x}g_{\alpha^{c}}(x,y(x))\\\\[3.0pt]
0_{\alpha^{c}}\end{array}\right]^{T}\left[\begin{array}[]{ll}K_{\alpha}(x)^{-1}&0\quad\quad\quad
0\\\\[3.0pt] \begin{array}[]{c}0\\\\[2.0pt]
0\end{array}&\left[\begin{array}[]{cc}0&D(x)\\\\[3.0pt]
D(x)&0\end{array}\right]^{-1}\end{array}\right]\left[\begin{array}[]{c}N_{\alpha}(x)\\\\[3.0pt]
{\cal J}_{x}g_{\alpha^{c}}(x,y(x))\\\\[3.0pt]
0_{\alpha^{c}}\end{array}\right]\\\\[6.0pt]
=N_{\alpha}(x)^{T}K_{\alpha}(x)^{-1}N_{\alpha}(x)+\left[\begin{array}[]{c}{\cal
J}_{x}g_{\alpha^{c}}(x,y(x))\\\\[3.0pt]
0_{\alpha^{c}}\end{array}\right]^{T}\left[\begin{array}[]{cc}0&D(x)\\\\[3.0pt]
D(x)&0\end{array}\right]^{-1}\left[\begin{array}[]{c}{\cal
J}_{x}g_{\alpha^{c}}(x,y(x))\\\\[3.0pt]
0_{\alpha^{c}}\end{array}\right]\\\\[8.0pt]
=N_{\alpha}(x)^{T}K_{\alpha}(x)^{-1}N_{\alpha}(x).\end{array}$

Namely, (2.9) holds. The proof is completed. $\Box$

For $x^{*}\in\Phi$, the Mangasarian-Fromovitz constraint qualification is said
to hold at $x^{*}$ the constraint set $\Phi$ if

  * (a)

The set of vectors $\nabla H_{j}(x^{*}),j=1,\ldots,n_{1}$ are linearly
independent.

  * (b)

There exists a vector $\bar{d}\in\Re^{n}$ such that

$\nabla H_{j}(x^{*})^{T}\bar{d}=0,\,j=1,\ldots,n_{1},\ \nabla
G_{i}(x^{*})^{T}\bar{d}<0,\,i\in I(x^{*}),$

where $I(x^{*})=\\{i:G_{i}(x^{*})=0,i=1,\ldots,n_{2}\\}$.

Define the critical cone at $x^{*}$ by

${\cal C}(x^{*})=\\{d_{x}\in\Re^{n}:{\cal J}H(x^{*})d_{x}=0;\nabla
G_{i}(x^{*})^{T}d_{x}\leq 0,i\in I(x^{*});\varphi^{\prime}(x^{*};d_{x})\leq
0\\}.$ (2.10)

In this case, the critical cone ${\cal C}(x^{*})$ can be expressed as

${\cal C}(x^{*})=\\{d_{x}\in\Re^{n}:{\cal J}H(x^{*})d_{x}=0;\ \nabla
G_{i}(x^{*})^{T}d_{x}\leq 0,i\in I(x^{*});\ \nabla_{x}{\cal
L}(x^{*},y^{*},\mu^{*},\lambda^{*})^{T}d_{x}\leq 0\\}.$ (2.11)

Based on (2.9) we may simplify Theorems 3.1 and 3.2 in [5] as follows.

###### Theorem 2.1

(Necessary Optimality Conditions) Let $(x^{*},y^{*})\in\Re^{n}\times\Re^{m}$
be a point around which $f$, $h$, $g$ are twice continuously differentiable
and $H$, $G$ are twice continuously differentiable around $x^{*}$. Let
$(x^{*},y^{*})$ be a local minimax point of Problem (1.1). Assume that the
linear independence constraint qualification holds at $y^{*}$ for constraint
set $Y(x^{*})$. Then there exists a unique vector
$(\mu^{*},\lambda^{*})\in\Re^{m_{1}}\times\Re^{m_{2}}$ such that

$\begin{array}[]{l}\nabla_{y}{\cal
L}(x^{*},y^{*},\mu^{*},\lambda^{*})=0,\\\\[3.0pt] h(x^{*},y^{*})=0,\\\\[3.0pt]
0\geq\lambda^{*}\perp g(x^{*},y^{*})\leq 0.\end{array}$ (2.12)

For any $d_{y}\in{\cal C}_{x^{*}}(y^{*})$, we have that

$\langle\nabla^{2}_{yy}{\cal
L}(x^{*},y^{*},\mu^{*},\lambda^{*})d_{y},d_{y}\rangle\leq 0.$ (2.13)

Assume further that Problem $({\rm P}_{x^{*}})$ satisfies Jacobian uniqueness
conditions at $(y^{*},\mu^{*},\lambda^{*})$ and the Mangasarian-Fromovitz
constraint qualification holds at $x^{*}$ for the constraint set $\Phi$. Then
there exists $(u^{*},v^{*})\in\Re^{n_{1}}\times\Re^{n_{2}}$ such that

$\begin{array}[]{l}\nabla_{x}{\cal L}(x^{*},y^{*},\mu^{*},\lambda^{*})+{\cal
J}H(x^{*})^{T}u^{*}+{\cal J}G(x^{*})^{T}v^{*}=0,\\\\[3.0pt]
H(x^{*})=0,\\\\[3.0pt] 0\leq v^{*}\perp G(x^{*})\leq 0.\end{array}$ (2.14)

The set of all $(u^{*},v^{*})$ satisfying (2.14), denoted by $\Lambda(x^{*})$,
is nonempty compact convex set. Furthermore, for every $d_{x}\in{\cal
C}(x^{*})$, where ${\cal C}(x^{*})$ is defined by (2.11),

$\begin{array}[]{l}\displaystyle\max_{(u,v)\in\Lambda(x^{*})}\left\\{\left\langle\left[\displaystyle\sum_{j=1}^{n_{1}}u_{i}\nabla^{2}_{xx}H_{j}(x^{*})+\displaystyle\sum_{i=1}^{n_{2}}v_{i}\nabla^{2}_{xx}G_{i}(x^{*})\right]d_{x},d_{x}\right\rangle\right\\}\\\\[16.0pt]
\quad\quad\quad\,+\left\langle\left[\nabla^{2}_{xx}{\cal
L}(x^{*},y^{*},\mu^{*},\lambda^{*})-N_{\alpha}(x^{*})^{T}K_{\alpha}(x^{*})^{-1}N_{\alpha}(x^{*})\right]d_{x},d_{x}\right\rangle\geq
0,\end{array}$ (2.15)

where $K_{\alpha}(x)$ is defined by (2.5) and $N_{\alpha}(x)$ is defined by
(2.8).

We name the first-order necessary optimality conditions (2.14) and (2.12) as
KKT conditions of Problem (1.1) at
$(x^{*},u^{*},v^{*},y^{*},\mu^{*},\lambda^{*})$.

###### Theorem 2.2

(Second-order Sufficient Optimality Conditions) Let
$(x^{*},y^{*})\in\Re^{n}\times\Re^{m}$ be a point around which $f,h,g$ are
twice continuously differentiable and $H$, $G$ are twice continuously
differentiable around $x^{*}$. Assume that $x^{*}\in\Phi$ and $y^{*}\in
Y(x^{*})$. Let $(\mu^{*},\lambda^{*})\in\Re^{m_{1}}\times\Re^{m_{2}}$. Suppose
that Problem $({\rm P}_{x^{*}})$ satisfies Jacobian uniqueness conditions at
$(y^{*},\mu^{*},\lambda^{*})$, $\Lambda(x^{*})\neq\emptyset$, and for every
$d_{x}\in{\cal C}(x^{*})\setminus\emptyset$ (where ${\cal C}(x^{*})$ is
defined by (2.11)),

$\begin{array}[]{l}\displaystyle\sup_{(u,v)\in\Lambda(x^{*})}\left\\{\left\langle\left[\displaystyle\sum_{j=1}^{n_{1}}u_{i}\nabla^{2}_{xx}H_{j}(x^{*})+\displaystyle\sum_{i=1}^{n_{2}}v_{i}\nabla^{2}_{xx}G_{i}(x^{*})\right]d_{x},d_{x}\right\rangle\right\\}\\\\[16.0pt]
\quad\quad\quad\,+\left\langle\left[\nabla^{2}_{xx}{\cal
L}(x^{*},y^{*},\mu^{*},\lambda^{*})-N_{\alpha}(x^{*})^{T}K_{\alpha}(x^{*})^{-1}N_{\alpha}(x^{*})\right]d_{x},d_{x}\right\rangle>0,\end{array}$
(2.16)

where $K_{\alpha}(x)$ is defined by (2.5) and $N_{\alpha}(x)$ is defined by
(2.8). Then there exist $\delta_{1}\in(0,\delta_{0})$,
$\varepsilon_{1}\in(0,\varepsilon_{0})$ (where $\delta_{0}$ and
$\varepsilon_{0}$ are given by Lemma 2.1) and $\gamma_{1}>0$,$\gamma_{2}>0$
such that for $x\in\textbf{B}_{\delta_{1}}(x^{*})\cap\Phi$ and
$y\in\textbf{B}_{\varepsilon_{1}}(y^{*})\cap Y(x^{*})$,

$f(x^{*},y)+\gamma_{1}\|y-y^{*}\|^{2}/2\leq
f(x^{*},y^{*})\leq\displaystyle\sup_{z\in
Y(x)\cap\textbf{B}_{\varepsilon_{0}}(y^{*})}f(x,z)-\gamma_{2}\|x-x^{*}\|^{2}/2,$
(2.17)

which implies that $(x^{*},y^{*})$ is a local minimax point of Problem (1.1).

##  3 Stability under Jacobian Uniqueness Condition

For convenience in stating the stability result about ${\cal C}^{2}$
perturbation of Problem (1.1) when the conditions in Theorem 2.2 are
satisfied, we introduce the following definition.

###### Definition 3.1

If the following conditions are satisfied, we say that Problem (1.1) satisfies
Jacobian uniqueness condition at
$(x^{*},u^{*},v^{*},y^{*},\mu^{*},\lambda^{*})\in\Re^{n}\times\Re^{n_{1}}\times\Re^{n_{2}}\times\Re^{m}\times\Re^{m_{1}}\times\Re^{m_{2}}$.

  * (i)

$x^{*}\in\Phi$ and conditions in (2.14) are satisfied at
$(x^{*},u^{*},v^{*},y^{*},\mu^{*},\lambda^{*})$.

  * (ii)

The set vectors $\big{\\{}\nabla H_{1}(x^{*}),\ldots,\nabla
H_{n_{1}}\big{\\}}\cup\Big{\\{}\nabla G_{i}(x^{*}):i\in I(x^{*})\Big{\\}}$ are
linearly independent, where
$I(x^{*})=\\{i:G_{i}(x^{*})=0,i=1.\ldots,n_{2}\\}$.

  * (iii)

$v^{*}_{i}-G_{i}(x^{*})>0$ for $i\in I(x^{*})$.

  * (iv)

$y^{*}\in Y(x^{*})$ and Problem $({\rm P}_{x^{*}})$ satisfies Jacobian
uniqueness conditions at $(y^{*},\mu^{*},\lambda^{*})$.

  * (v)

For every $d_{x}\in{\cal C}(x^{*})\setminus\\{0\\}$ (where ${\cal C}(x^{*})$
is defined by (2.11)), the second-order sufficient optimality condition (2.16)
is satisfied.

Following Kojima (1980) [9], we define the so-called Kojima mapping for
Problem (1.1),

$F(x,u,w,y,\mu,\xi)=\left[\begin{array}[]{c}\nabla_{x}{\cal
L}(x,y,\mu,\xi^{+})+{\cal J}H(x)^{T}u+{\cal J}G(x)^{T}w^{+}\\\\[3.0pt]
H(x)\\\\[3.0pt] G(x)-w^{-}\\\\[3.0pt] \nabla_{y}{\cal
L}(x,y,\mu,\xi^{+})\\\\[3.0pt] h(x,y)\\\\[3.0pt]
-g(x,y)+\xi^{-}\end{array}\right],$ (3.1)

where $w^{+}_{i}=\max\,\big{\\{}0,w_{i}\big{\\}}$,
$\xi^{-}_{i}=\min\,\big{\\{}0,w_{i}\big{\\}}$, $i=1,\ldots,n_{2}$ for
$w\in\Re^{n_{2}}$ and $\xi^{+}_{i}=\max\,\big{\\{}0,\xi_{i}\big{\\}}$,
$\xi^{-}_{i}=\min\,\big{\\{}0,\xi_{i}\big{\\}}$, $i=1,\ldots,m_{2}$ for
$\xi\in\Re^{m_{2}}$. If $F(x,u,w,y,\mu,\xi)=0$, then, letting $v=w^{+}$ and
$\lambda=\xi^{+}$, $(x,y,u,v,\mu,\lambda)$ satisfies the first-order necessary
optimality conditions of Problem (1.1).

###### Lemma 3.1

Suppose that the Jacobian uniqueness condition of Problem (1.1) in Definition
3.1 is satisfied at
$(x^{*},u^{*},v^{*},y^{*},\mu^{*},\lambda^{*})\in\Re^{n}\times\Re^{n_{1}}\times\Re^{n_{2}}\times\Re^{m}\times\Re^{m_{1}}\times\Re^{m_{2}}$.
Then $F$ is differentiable at $(x^{*},u^{*},w^{*},y^{*},\mu^{*},\xi^{*})$ and
${\cal J}F(x^{*},u^{*},w^{*},y^{*},\mu^{*},\xi^{*})$ is nonsingular for
$w^{*}=v^{*}+G(x^{*})$ and $\xi^{*}=\lambda^{*}+g(x^{*},y^{*})$.

Proof. Since $\lambda^{*}-g(x^{*},y^{*})>0$ and $v^{*}-G(x^{*})>0$, we know
that $\xi^{+}$ and $\xi^{-}$ are differentiable at $\xi^{*}$, and $w^{+}$ and
$w^{-}$ are differentiable at $w^{*}$. Thus $F$ is differentiable at
$(x^{*},y^{*},u^{*},w^{*},\mu^{*},\xi^{*})$. Without loss of generality, we
assume that

$\beta:=I(x^{*})=\\{1,\ldots,r\\},\quad\alpha=\\{1,\ldots,s\\},$

where

$I(x^{*})=\\{i:G_{i}(x^{*})=0,i=1,\ldots,n_{2}\\},\quad\alpha=\\{i:g_{i}(x^{*},y^{*})=0,i=1,\ldots,m_{2}\\}.$

Then, for $\beta^{c}=\\{1,\ldots,n_{2}\\}\setminus\beta$ and
$\alpha^{c}=\\{1,\ldots,m_{2}\\}\setminus\alpha$, we get that

$\beta^{c}=\\{r+1,\ldots,n_{2}\\},\quad\alpha^{c}=\\{s+1,\ldots,m_{2}\\}.$

Thus we obtain

$\begin{array}[]{cclccl}{\cal
J}w^{+}|_{w=w^{*}}&=&\left[\begin{array}[]{cc}I_{r}&0\\\\[3.0pt]
0&0\end{array}\right],&{\cal
J}w^{-}|_{w=w^{*}}&=&\left[\begin{array}[]{cc}0&0\\\\[3.0pt]
0&I_{n_{2}-r}\end{array}\right],\\\\[8.0pt] {\cal
J}\xi^{+}|_{\xi=\xi^{*}}&=&\left[\begin{array}[]{cc}I_{s}&0\\\\[3.0pt]
0&0\end{array}\right],&{\cal
J}\xi^{-}|_{\xi=\xi^{*}}&=&\left[\begin{array}[]{cc}0&0\\\\[3.0pt]
0&I_{m_{2}-s}\end{array}\right].\end{array}$ (3.2)

Denote

$\begin{array}[]{ll}G^{*}_{11}=\nabla^{2}_{xx}{\cal
L}(x^{*},y^{*},\mu^{*},\lambda^{*})+\displaystyle\sum_{j=1}^{n_{1}}u_{i}\nabla^{2}_{xx}H_{j}(x^{*})+\displaystyle\sum_{i=1}^{n_{2}}v_{i}\nabla^{2}_{xx}G_{i}(x^{*}),\\\\[4.0pt]
G^{*}_{12}=\nabla^{2}_{xy}{\cal
L}(x^{*},y^{*},\mu^{*},\lambda^{*}),\\\\[4.0pt]
G^{*}_{22}=\nabla^{2}_{yy}{\cal
L}(x^{*},y^{*},\mu^{*},\lambda^{*}).\end{array}$

For simplicity, we use notations ${\cal J}_{x}h^{*}$ and ${\cal J}_{y}h^{*}$
to represent ${\cal J}_{x}h(x^{*},y^{*})$ and ${\cal J}_{y}h(x^{*},y^{*})$,
respectively. The same notations are also applied to $g_{\alpha}$ and
$g_{\alpha^{c}}$. Then the Jacobian of $F$ at
$(x^{*},u^{*},w^{*},y^{*},\mu^{*},\xi^{*})$ can be expressed as

$\begin{array}[]{l}{\cal
J}F(x^{*},u^{*},w^{*},y^{*},\mu^{*},\xi^{*})\\\\[6.0pt]
=\left[\begin{array}[]{cccccccc}G^{*}_{11}&{\cal J}H(x^{*})^{T}&{\cal
J}G_{\beta}(x^{*})^{T}&0&G^{*}_{12}&{\cal J}_{x}h^{*T}&-{\cal
J}_{x}g_{\alpha}^{*T}&0\\\\[6.0pt] {\cal J}H(x^{*})&0&0&0&0&0&0&0\\\\[6.0pt]
{\cal J}G_{\beta}(x^{*})&0&0&0&0&0&0&0\\\\[6.0pt] {\cal
J}G_{\beta^{c}}(x^{*})&0&0&-I_{n_{2}-r}&0&0&0&0\\\\[6.0pt]
G^{*T}_{12}&0&0&0&G^{*}_{22}&{\cal J}_{y}h^{*T}&-{\cal
J}_{y}g_{\alpha}^{*T}&0\\\\[6.0pt] {\cal J}_{x}h^{*}&0&0&0&{\cal
J}_{y}h^{*}&0&0&0\\\\[6.0pt] -{\cal J}_{x}g_{\alpha}^{*}&0&0&0&-{\cal
J}_{y}g_{\alpha}^{*}&0&0&0\\\\[6.0pt] -{\cal
J}_{x}g_{\alpha^{c}}^{*}&0&0&0&-{\cal
J}_{y}g_{\alpha^{c}}^{*}&0&0&I_{m_{2}-s}\end{array}\right].\end{array}$ (3.3)

The nonsingularity of ${\cal J}F(x^{*},u^{*},w^{*},y^{*},\mu^{*},\xi^{*})$ is
equivalent to the nonsingularity of the following matrix

$\begin{array}[]{l}\left[\begin{array}[]{cccccccc}G^{*}_{11}&{\cal
J}H(x^{*})^{T}&{\cal J}G_{\beta}(x^{*})^{T}&G^{*}_{12}&{\cal
J}_{x}h^{*T}&-{\cal J}_{x}g_{\alpha}^{*T}&0&0\\\\[6.0pt] {\cal
J}H(x^{*})&0&0&0&0&0&0&0\\\\[6.0pt] {\cal
J}G_{\beta}(x^{*})&0&0&0&0&0&0&0\\\\[6.0pt] G^{*T}_{12}&0&0&G^{*}_{22}&{\cal
J}_{y}h^{*T}&-{\cal J}_{y}g_{\alpha}^{*T}&0&0\\\\[6.0pt] {\cal
J}_{x}h^{*}&0&0&{\cal J}_{y}h^{*}&0&0&0&0\\\\[6.0pt] -{\cal
J}_{x}g_{\alpha}^{*}&0&0&-{\cal J}_{y}g_{\alpha}^{*}&0&0&0&0\\\\[6.0pt] {\cal
J}G_{\beta^{c}}(x^{*})&0&0&0&0&0&-I_{n_{2}-r}&0\\\\[6.0pt] -{\cal
J}_{x}g_{\alpha^{c}}^{*}&0&0&-{\cal
J}_{y}g_{\alpha^{c}}^{*}&0&0&0&I_{m_{2}-s}\end{array}\right],\end{array}$

which is equivalent to the nonsingularity of the following matrix

$H=\left[\begin{array}[]{cccccc}G^{*}_{11}&{\cal J}H(x^{*})^{T}&{\cal
J}G_{\beta}(x^{*})^{T}&G^{*}_{12}&{\cal J}_{x}h^{*T}&-{\cal
J}_{x}g_{\alpha}^{*T}\\\\[6.0pt] {\cal J}H(x^{*})&0&0&0&0&0\\\\[6.0pt] {\cal
J}G_{\beta}(x^{*})&0&0&0&0&0\\\\[6.0pt] G^{*T}_{12}&0&0&G^{*}_{22}&{\cal
J}_{y}h^{*T}&-{\cal J}_{y}g_{\alpha}^{*T}\\\\[6.0pt] {\cal
J}_{x}h^{*}&0&0&{\cal J}_{y}h^{*}&0&0\\\\[6.0pt] -{\cal
J}_{x}g_{\alpha}^{*}&0&0&-{\cal J}_{y}g_{\alpha}^{*}&0&0\end{array}\right].$
(3.4)

Therefore, we only need to prove that the matrix $H$ is nonsingular. From
Lemma 2.2, we obtain that $K_{\alpha}(x^{*})$ is nonsingular, where

$K_{\alpha}(x^{*})=\left[\begin{array}[]{ccc}G^{*}_{22}&{\cal
J}_{y}h^{*T}&-{\cal J}_{y}g_{\alpha}^{*T}\\\\[6.0pt] {\cal
J}_{y}h^{*}&0&0\\\\[6.0pt] -{\cal J}_{y}g_{\alpha}^{*}&0&0\end{array}\right].$

So it suffices to prove that $H/K_{\alpha}(x^{*})$ is nonsingular. Noticing
that

$\begin{array}[]{rcl}H/K_{\alpha}(x^{*})&=&\left[\begin{array}[]{ccc}G^{*}_{11}&{\cal
J}H(x^{*})^{T}&{\cal J}G_{\beta}(x^{*})^{T}\\\\[6.0pt] {\cal
J}H(x^{*})&0&0\\\\[6.0pt] {\cal
J}G_{\beta}(x^{*})&0&0\end{array}\right]\\\\[20.0pt]
&&-\left[\begin{array}[]{ccc}G^{*}_{12}&{\cal J}_{x}h^{*T}&-{\cal
J}_{x}g_{\alpha}^{*T}\\\\[6.0pt] 0&0&0\\\\[6.0pt]
0&0&0\end{array}\right]K_{\alpha}(x^{*})^{-1}\left[\begin{array}[]{ccc}G^{*T}_{12}&0&0\\\\[6.0pt]
{\cal J}_{x}h^{*}&0&0\\\\[6.0pt] -{\cal
J}_{x}g_{\alpha}^{*}&0&0\end{array}\right]\\\\[20.0pt]
&=&\left[\begin{array}[]{ccc}G^{*}_{11}-N_{\alpha}(x^{*})^{T}K_{\alpha}(x^{*})N_{\alpha}(x^{*})&{\cal
J}H(x^{*})^{T}&{\cal J}G_{\beta}(x^{*})^{T}\\\\[6.0pt] {\cal
J}H(x^{*})&0&0\\\\[6.0pt] {\cal
J}G_{\beta}(x^{*})&0&0\end{array}\right],\end{array}$ (3.5)

we have from (iii) that ${\cal C}(x^{*})$ is reduced to the following subspace

${\cal C}(x^{*})={\rm Ker}\,\left[\begin{array}[]{l}{\cal
J}H(x^{*})\\\\[6.0pt] {\cal J}G_{\beta}(x^{*})\end{array}\right].$ (3.6)

Now we prove that $H/K_{\alpha}(x^{*})$ is nonsingular via the formula (3.5).
Let $a\in\Re^{n}$, $b\in\Re^{n_{1}}$ and $c\in\Re^{r}$ satisfy

$\left[\begin{array}[]{ccc}G^{*}_{11}-N_{\alpha}(x^{*})^{T}K_{\alpha}(x^{*})N_{\alpha}(x^{*})&{\cal
J}H(x^{*})^{T}&{\cal J}G_{\beta}(x^{*})^{T}\\\\[6.0pt] {\cal
J}H(x^{*})&0&0\\\\[6.0pt] {\cal
J}G_{\beta}(x^{*})&0&0\end{array}\right]\left[\begin{array}[]{c}a\\\\[6.0pt]
b\\\\[6.0pt] c\end{array}\right]=0$

or

$\left\\{\begin{array}[]{l}G^{*}_{11}-N_{\alpha}(x^{*})^{T}K_{\alpha}(x^{*})N_{\alpha}(x^{*})a+{\cal
J}H(x^{*})^{T}b+{\cal J}G_{\beta}(x^{*})^{T}c=0,\\\\[6.0pt] {\cal
J}H(x^{*})a=0,\\\\[6.0pt] {\cal J}G_{\beta}(x^{*})a=0.\end{array}\right.$
(3.7)

It follows from ${\cal J}H(x^{*})a=0$ and ${\cal J}G_{\beta}(x^{*})a=0$ that
$a\in{\cal C}(x^{*})$. Premultiplying $a^{T}$ to the first equation in (3.7),
we obtain

$\begin{array}[]{l}\displaystyle\left\langle\left[\displaystyle\sum_{j=1}^{n_{1}}u^{*}_{i}\nabla^{2}_{xx}H_{j}(x^{*})+\displaystyle\sum_{i=1}^{n_{2}}v^{*}_{i}\nabla^{2}_{xx}G_{i}(x^{*})\right]a,a\right\rangle\\\\[16.0pt]
\quad\quad\quad\,+\left\langle\left[\nabla^{2}_{xx}{\cal
L}(x^{*},y^{*},\mu^{*},\lambda^{*})-N_{\alpha}(x^{*})^{T}K_{\alpha}(x^{*})^{-1}N_{\alpha}(x^{*})\right]a,a\right\rangle=0,\end{array}$

which implies $a=0$ from the condition (2.16). From the first equation in
(3.7) again, we obtain

${\cal J}H(x^{*})^{T}b+{\cal J}G_{\beta}(x^{*})^{T}c=0,$

from which we obtain $b=0$ and $c=0$ from (iii). Therefore
$H/K_{\alpha}(x^{*})$ is nonsingular. The proof is completed. $\Box$

Basing on Lemma 3.1, we may establish the stability on the ${\cal C}^{2}$
perturbation of Problem (1.1) under the Jacobian uniqueness condition at
$(x^{*},u^{*},v^{*},y^{*},\mu^{*},\lambda^{*})$.

Now consider the parameterized constrained minimax optimization problem of the
form

$({\rm
P}_{\vartheta})\quad\quad\quad\min_{x\in\Phi(\vartheta)}\max_{y\in\bar{Y}(x,\vartheta)}\bar{f}(x,y,\vartheta),$
(3.8)

where $\bar{f}:\Re^{n}\times\Re^{m}\times\Re^{l}\rightarrow\Re$,
$\Phi\subset\Re^{n}$ is a feasible set of decision variable $x$ defined by

$\Phi(\vartheta)=\\{x\in\Re^{n}:\bar{H}(x,\vartheta)=0,\,\bar{G}(x,\vartheta)\leq
0\\}$ (3.9)

and $Y:\Re^{n}\times\Re^{l}\rightrightarrows\Re^{m}$ is a set-valued mapping
defined by

$\bar{Y}(x,\vartheta)=\\{y\in\Re^{m}:\bar{h}(x,y,\vartheta)=0,\,\bar{g}(x,y,\vartheta)\leq
0\\}.$ (3.10)

Let $\vartheta_{0}\in\Re^{l}$ be a vector such that

$f(x,y)=\bar{f}(x,y,\vartheta_{0}),\quad
h(x,y)=\bar{h}(x,y,\vartheta_{0}),\quad g(x,y)=\bar{g}(x,y,\vartheta_{0})$

and

$H(x)=\bar{H}(x,\vartheta_{0}),\quad G(x)=\bar{G}(x,\vartheta_{0}).$

###### Definition 3.2

We say Problem (3.8) is a local ${\cal C}^{2}$ perturbation of Problem (1.1)
around $(x^{*},y^{*})$ if there exist open sets ${\cal O}_{1}\subset\Re^{n}$,
${\cal O}_{2}\subset\Re^{m}$ and $\Theta\subset\Re^{l}$ satisfying
$\vartheta_{0}\in\Theta$, $x^{*}\in{\cal O}_{1}$, $y^{*}\in{\cal O}_{2}$ and
$\bar{f}$, $\bar{h}$, $\bar{g}$ are twicely smooth over ${\cal
O}_{1}\times{\cal O}_{2}\times\Theta$, and $\bar{H}$, $\bar{G}$ are twicely
smooth over ${\cal O}_{1}\times\Theta$.

The Kojima mapping for Problem $({\rm{P}}_{\vartheta})$ is the following
function

$\bar{F}(x,u,w,y,\mu,\xi;\vartheta)=\left[\begin{array}[]{c}\nabla_{x}\bar{\cal
L}(x,y,\mu,\xi^{+};\vartheta)+{\cal J}H(x,\vartheta)^{T}u+{\cal
J}G(x,\vartheta)^{T}w^{+}\\\\[3.0pt] H(x,\vartheta)\\\\[3.0pt]
G(x,\vartheta)-w^{-}\\\\[3.0pt] \nabla_{y}\bar{\cal
L}(x,y,\mu,\xi^{+};\vartheta)\\\\[3.0pt] h(x,y,\vartheta)\\\\[3.0pt]
-g(x,y,\vartheta)+\xi^{-}\end{array}\right],$ (3.11)

where

$\bar{\cal
L}(x,y,\mu,\lambda;\vartheta)=\bar{f}(x,y,\vartheta)+\langle\mu,\bar{h}(x,y,\vartheta)\rangle-\langle\lambda,\bar{g}(x,y,\vartheta)\rangle.$

###### Theorem 3.1

Suppose that the Jacobian uniqueness condition of Problem (1.1) is satisfied
at $(x^{*}$, $u^{*}$, $v^{*}$, $y^{*}$, $\mu^{*}$,
$\lambda^{*})\in\Re^{n}\times\Re^{n_{1}}\times\Re^{n_{2}}\times\Re^{m}\times\Re^{m_{1}}\times\Re^{m_{2}}$
and (${\rm P}_{\vartheta}$) is a local ${\cal C}^{2}$ perturbation of Problem
(1.1) around $(x^{*},y^{*})$. Then there exist $\varepsilon>0$ and $\delta>0$
such that
$\textbf{B}(\vartheta_{0},\delta)\subset\Theta$,$\textbf{B}(x^{*},\varepsilon)\subset{\cal
O}_{1}$ and $\textbf{B}(y^{*},\varepsilon)\subset{\cal O}_{2}$, and there is a
mapping
$(x(\cdot),y(\cdot),u(\cdot),v(\cdot),\mu(\cdot),\lambda(\cdot)):\textbf{B}(\vartheta_{0},\delta)\rightarrow\textbf{B}(x^{*},\varepsilon)\times\textbf{B}(y^{*},\varepsilon)\times\textbf{B}(u^{*},\varepsilon)\times\textbf{B}(v^{*},\varepsilon)\times\textbf{B}(\mu^{*},\varepsilon)\times\textbf{B}(\lambda^{*},\varepsilon)$
such that, for $v(\vartheta)=w(\vartheta)^{+}$ and
$\lambda(\vartheta)=\xi(\vartheta)^{+}$,

  * (1)

$(x(\vartheta_{0}),y(\vartheta_{0}),u(\vartheta_{0}),v(\vartheta_{0}),\mu(\vartheta_{0}),\lambda(\vartheta_{0}))=(x^{*},y^{*},u^{*},v^{*},\mu^{*},\lambda^{*})$.

  * (2)

For any $\vartheta\in\textbf{B}(\vartheta_{0},\delta)$,
$(x(\cdot),y(\cdot),u(\cdot),v(\cdot),\mu(\cdot),\lambda(\cdot))$ is
continuously differentiable at $\vartheta$.

  * (3)

For any $\vartheta\in\textbf{B}(\vartheta_{0},\delta)$, Problem (${\rm
P}_{\vartheta}$) satisfies the Jacobian uniqueness condition at
$(x(\vartheta),y(\vartheta),$
$u(\vartheta),v(\vartheta),\mu(\vartheta),\lambda(\vartheta))$.

Proof. Let $w^{*}=v^{*}+G(x^{*})$ and $\xi^{*}=\lambda^{*}+g(x^{*},y^{*})$.
From the definitions of $\bar{F}$ and $F$ in (3.1), we have

$\bar{F}(x,u,w,y,\mu,\xi;\vartheta_{0})=F(x,u,w,y,\mu,\xi).$

Thus we get that

$\bar{F}(x^{*},u^{*},w^{*},y^{*},\mu^{*},\xi^{*};\vartheta_{0})=0,\ {\cal
J}_{(x,u,w,y,\mu,\xi)}\bar{F}(x^{*},u^{*},w^{*},y^{*},\mu^{*},\xi^{*};\vartheta_{0})={\cal
J}F(x^{*},u^{*},w^{*},y^{*},\mu^{*},\xi^{*}),$

and in turn ${\cal
J}_{(x,u,w,y,\mu,\xi)}\bar{F}(x^{*},u^{*},w^{*},y^{*},\mu^{*},\xi^{*};\vartheta_{0})$
is nonsingular from Lemma 3.1. From the classical implicit function theorem,
we get that there exist $\varepsilon>0$ and $\delta>0$ such that
$\textbf{B}(\vartheta_{0},\delta)\subset\Theta$,$\textbf{B}(x^{*},\varepsilon)\subset{\cal
O}_{1}$ and $\textbf{B}(y^{*},\varepsilon)\subset{\cal O}_{2}$, and there is a
mapping
$(x(\cdot),u(\cdot),w(\cdot),y(\cdot),\mu(\cdot),\xi(\cdot)):\textbf{B}(\vartheta_{0},\delta)\rightarrow\textbf{B}(x^{*},\varepsilon)\times\textbf{B}(u^{*},\varepsilon)\times\textbf{B}(w^{*},\varepsilon)\times\textbf{B}(y^{*},\varepsilon)\times\textbf{B}(\mu^{*},\varepsilon)\times\textbf{B}(\xi^{*},\varepsilon)$
such that

$(x(\vartheta_{0}),u(\vartheta_{0}),w(\vartheta_{0}),y(\vartheta_{0}),\mu(\vartheta_{0}),\xi(\vartheta_{0}))=(x^{*},u^{*},w^{*},y^{*},\mu^{*},\xi^{*});$

meanwhile, for any $\vartheta\in\textbf{B}(\vartheta_{0},\delta)$,
$(x(\cdot),u(\cdot),w(\cdot),y(\cdot),\mu(\cdot),\xi(\cdot))$ is continuously
differentiable at $\vartheta$, and

$\bar{F}(x(\vartheta),u(\vartheta),w(\vartheta),y(\vartheta),\mu(\vartheta),\xi(\vartheta);\vartheta)=0,\,\,\forall\vartheta\in\textbf{B}(\vartheta_{0},\delta).$
(3.12)

From the continuity of
$(x(\vartheta),u(\vartheta),w(\vartheta),y(\vartheta),\mu(\vartheta),\xi(\vartheta))$
for $\vartheta\in\textbf{B}(\vartheta_{0},\delta)$, we have from
$G_{i}(x(\vartheta))-w_{i}(\vartheta)^{-}=0$ in (3.12) that

$G_{i}(x(\vartheta))<0,\quad w^{+}(\vartheta)_{i}=0,i\in\beta^{c}$

and

$w^{+}_{i}(\vartheta)>0,\quad G_{i}(x(\vartheta))=0,i\in\beta.$

Therefore we have that

$\beta(\vartheta):=\\{i:G_{i}(x(\vartheta))=0,i=1,\ldots,n_{2}\\}=\beta,\quad\\{i:w^{+}_{i}(x(\vartheta))=0,i=1,\ldots,n_{2}\\}=\beta^{c}$

and that $w^{+}_{i}(\cdot)$ is differentiable at $\vartheta$ for $i\in\beta$
and $w^{+}_{i}(\cdot)\equiv 0$ for $i\in\beta^{c}$. In turn
$v(\vartheta)=w^{+}(\vartheta)$ is differentiable over
$\textbf{B}(\vartheta_{0},\delta)$. Using the same arguments as the above, we
obtain

$\alpha(\vartheta):=\\{i:g_{i}(x(\vartheta),y(\vartheta))=0,i=1,\ldots,m_{2}\\}=\alpha,\quad\\{i:\xi^{+}_{i}(x(\vartheta))=0,i=1,\ldots,m_{2}\\}=\alpha^{c}$

and $\lambda(\vartheta)=\xi(\vartheta)^{+}$ is also differentiable over
$\textbf{B}(\vartheta_{0},\delta)$. Hence the assertions (1) and (2) hold.

Now we prove the assertion (3). From the first three equations in (3.12) and
the definition of $(v(\vartheta),\lambda(\vartheta))$, we obtain

$\begin{array}[]{l}\nabla_{x}{\cal
L}(x(\vartheta),y(\vartheta),\mu(\vartheta),\lambda(\vartheta))+{\cal
J}H(x(\vartheta))^{T}u(\vartheta)+{\cal
J}G(x(\vartheta))^{T}v(\vartheta)=0,\\\\[3.0pt] H(x(\vartheta))=0,\\\\[3.0pt]
G(x(\vartheta))-[G(x(\vartheta)+v(\vartheta)]^{-}=0,\\\\[3.0pt] \end{array}$

which are exactly the conditions in (i) of the Jacobian uniqueness condition
of $({\rm P}_{\vartheta})$ from Definition 3.1. From the continuity of
$(x(\vartheta),v(\vartheta))$ and $\beta(\vartheta)=\beta$ claimed just now,
we have that the set vectors $\big{\\{}\nabla
H_{1}(x(\vartheta)),\ldots,\nabla
H_{n_{1}}(x(\vartheta))\big{\\}}\cup\Big{\\{}\nabla
G_{i}(x(\vartheta)):i\in\beta(\vartheta)\Big{\\}}$ are linearly independent,
where $\beta(\vartheta)=\\{i:G_{i}(x(\vartheta))=0,i=1.\ldots,n_{2}\\}$, which
implies (ii) of the Jacobian uniqueness condition of $({\rm P}_{\vartheta})$.
And we have that $v_{i}(\vartheta)-G_{i}(x(\vartheta))>0$ for
$i\in\beta(\vartheta)$; namely, (iii) of the Jacobian uniqueness condition of
$({\rm P}_{\vartheta})$ holds.

Now we check (iv) of the Jacobian uniqueness condition of $({\rm
P}_{x(\vartheta)})$; namely, (${\rm P}_{x(\vartheta)}$) satisfies the Jacobian
uniqueness condition at $(y(\vartheta),\mu(\vartheta),\lambda(\vartheta))$.
From the last three equations in (3.12) and the definition of
$(v(\vartheta),\lambda(\vartheta))$, we obtain

$\begin{array}[]{l}\nabla_{y}{\cal
L}(x(\vartheta),y(\vartheta),\mu(\vartheta),\lambda(\vartheta))=0,\\\\[3.0pt]
h(x(\vartheta),y(\vartheta))=0,\\\\[3.0pt]
g(x(\vartheta),y(\vartheta))-[g(x(\vartheta),y(\vartheta))+\lambda(\vartheta)]^{-}=0,\\\\[3.0pt]
\end{array}$

which are just KKT conditions of $({\rm P}_{x(\vartheta)})$ at
$(y(\vartheta)$, $\mu(\vartheta)$, $\lambda(\vartheta))$. Since the continuity
of $(x(\vartheta)$, $y(\vartheta)$, $\mu(\vartheta)$, $\lambda(\vartheta))$
and $\alpha(\vartheta)=\alpha$, we have that the set vectors

$\big{\\{}\nabla_{y}h_{1}(x(\vartheta),y(\vartheta)),\ldots,\nabla_{y}h_{n_{1}}(x(\vartheta),y(\vartheta))\big{\\}}\cup\Big{\\{}\nabla_{y}g_{i}(x(\vartheta),y(\vartheta)):i\in\alpha(\vartheta)\Big{\\}}$

are linearly independent when $\delta>0$ is small enough; namely, the linear
independence constraint qualification of $({\rm P}_{x(\vartheta)})$ at
$y(\vartheta)$ is satisfied. We also have
$\lambda_{i}(\vartheta)-g_{i}(x(\theta),y(\vartheta))>0$; namely, the strict
complementarity condition of $({\rm P}_{x(\vartheta)})$ holds at
$(y(\vartheta),\lambda(\vartheta))$. Until now, for the Jacobian uniqueness
condition of $({\rm P}_{x(\vartheta)})$, only the second-order sufficient
optimality condition is left to prove. It can be proved in the same way as
that for (v) of the Jacobian uniqueness condition of $({\rm P}_{\vartheta})$
from Definition 3.1. We omit it here.

Finally, we prove (v) of the Jacobian uniqueness condition of $({\rm
P}_{\vartheta})$ at $(x(\vartheta)$, $y(\vartheta)$, $u(\vartheta)$,
$v(\vartheta)$, $\mu(\vartheta)$, $\lambda(\vartheta))$ from Definition 3.1.
From $\alpha(\vartheta)=\alpha$, we have that

${\cal C}_{x(\vartheta)}(y(\vartheta))={\rm ker}\,{\cal
J}_{y}h(x(\vartheta),y(\vartheta))\cap{\rm ker}\,{\cal
J}_{y}g_{\alpha}(x(\vartheta),y(\vartheta))$ (3.13)

is a subspace of $\Re^{m}$. For proving (v) of the Jacobian uniqueness
condition of $({\rm P}_{\vartheta})$ at $(x(\vartheta),y(\vartheta),$
$u(\vartheta),v(\vartheta),\mu(\vartheta),\lambda(\vartheta))$ from Definition
3.1, we only need to construct a matrix
$Z(\vartheta)\in\Re^{m,m-m_{1}-|\alpha|}$ such that

${\rm Range}\,Z(\vartheta)={\rm ker}\,{\cal
J}_{y}h(x(\vartheta),y(\vartheta))\cap{\rm ker}\,{\cal
J}_{y}g_{\alpha}(x(\vartheta),y(\vartheta))$ (3.14)

and

$Z(\vartheta)^{T}\Psi(\vartheta)Z(\vartheta)\succ 0,$ (3.15)

where

$\begin{array}[]{ll}\Psi(\vartheta)&=\displaystyle\sum_{j=1}^{n_{1}}u_{i}(\vartheta)\nabla^{2}_{xx}H_{j}(x(\vartheta))+\displaystyle\sum_{i=1}^{n_{2}}v_{i}(\vartheta)\nabla^{2}_{xx}G_{i}(x(\vartheta))\\\\[16.0pt]
&\quad\quad\quad\,+\nabla^{2}_{xx}{\cal
L}(x(\vartheta),y(\vartheta),\mu(\vartheta),\lambda(\vartheta))-N_{\alpha}(x(\vartheta))^{T}K_{\alpha}(x(\vartheta))^{-1}N_{\alpha}(x(\vartheta)).\end{array}$

To do this, define $A\in\Re^{m\times m}$ by

$A(\vartheta)=\left[\begin{array}[]{l}{\cal
J}_{y}h(x(\vartheta),y(\vartheta))\\\\[3.0pt] {\cal
J}_{y}g_{\alpha}(x(\vartheta),y(\vartheta))\\\\[3.0pt]
\bar{A}\end{array}\right],$

where $\bar{A}\in\Re^{m-m_{1}-|\alpha|}$ is chosen such that
$A(\vartheta_{0})$ is nonsingular. Then if $\vartheta$ is close to
$\vartheta_{0}$ enough, we have that $A(\vartheta)$ is nonsingular as well. By
applying the standard Gram-Schmidt orthogonalization procedure to the columns
of $A(\vartheta)$, we obtain an orthogonal matrix
$P(\vartheta)=[P_{1}(\vartheta)\,\,P_{2}(\vartheta)]\in\Re^{m\times m}$ with
$P_{1}(\vartheta)\in\Re^{m\times
m_{1}+|\alpha|},P_{2}(\vartheta)\in\Re^{m-m_{1}-|\alpha|}$. Then
$P(\vartheta)$ is a continuous function over
$\textbf{B}(\vartheta_{0},\delta)$. Let $Z(\vartheta)=P_{2}(\vartheta)$
satisfy (3.14) and $Z(\vartheta)$ be continuous over
$\textbf{B}(\vartheta_{0},\delta)$. Then $Z(\vartheta)$ satisfies (3.14) and
(3.15) when $\vartheta\in\textbf{B}(\vartheta_{0},\delta)$ for small
$\delta>0$, which comes from the fact that

$Z(\vartheta_{0})^{T}\Psi(\vartheta_{0})Z(\vartheta_{0})\succ 0,$

from (2.16). The proof is completed. $\Box$

##  4 Strong Regularity without Strict Complementarity

In the Jacobian uniqueness condition of Problem (1.1) by Definition 3.1, a
critical condition is the strict complementarity for the upper level problem.
In this section, we consider the case when this condition does not hold.

Let $(x^{*},u^{*},v^{*},y^{*},\mu^{*},\lambda^{*})$ be a KKT point of Problem
(1.1); namely, it satisfies the following conditions

$\begin{array}[]{l}\nabla_{x}{\cal L}(x,y,\mu,\lambda)+{\cal J}H(x)^{T}u+{\cal
J}G(x)^{T}v=0,\\\\[3.0pt] H(x)=0,\quad 0\leq v\perp G(x)\leq 0,\\\\[3.0pt]
\nabla_{y}{\cal L}(x,y,\mu,\lambda)=0,\\\\[3.0pt] h(x,y)=0,\quad
0\geq\lambda\perp g(x,y)\leq 0.\end{array}$ (4.1)

Let $z:=(x,u,v,y,\mu,\lambda)$ and define

${\cal
K}=\Re^{n}\times\Re^{n_{1}}\times\Re^{n_{2}}_{+}\times\times\Re^{m}\times\Re^{m_{1}}\times\Re^{m_{2}}_{+}$

and

${\cal H}(z)=\left[\begin{array}[]{c}\nabla_{x}{\cal L}(x,y,\mu,\lambda)+{\cal
J}H(x)^{T}u+{\cal J}G(x)^{T}v\\\\[3.0pt] H(x)\\\\[3.0pt] -G(x)\\\\[3.0pt]
\nabla_{y}{\cal L}(x,y,\mu,\lambda)\\\\[3.0pt] h(x,y)\\\\[3.0pt]
-g(x,y)\end{array}\right].$ (4.2)

The KKT conditions above can be expressed as the following generalized
equation

$0\in{\cal H}(z)+N_{\cal K}(z).$ (4.3)

For $\eta=(\eta_{x};\eta_{H};\eta_{G};\eta_{y};\eta_{h};\eta_{g})$, it is easy
to see that the perturbed generalized equation

$\eta\in{\cal H}(z)+N_{\cal K}(z)$ (4.4)

represents the KKT conditions for the following canonical perturbation of
Problem (1.1),

$\min_{x\in\Phi(\eta_{H},\eta_{G})}\max_{y\in
Y(x,\eta_{h},\eta_{g})}f(x,y)-\langle\eta_{x},x\rangle-\langle\eta_{y},y\rangle,$
(4.5)

where $f:\Re^{n}\times\Re^{m}\rightarrow\Re$,
$\Phi:\Re^{n_{1}}\times\Re^{n_{2}}\rightrightarrows\Re^{n}$ is a set-valued
mapping defined by

$\Phi(\eta_{H},\eta_{G})=\Big{\\{}x\in\Re^{n}:H(x)-\eta_{H}=0,G(x)+\eta_{G}\leq
0\Big{\\}}$ (4.6)

and $Y:\Re^{n}\times\Re^{m_{1}}\times\Re^{m_{2}}\rightrightarrows\Re^{m}$ is a
set-valued mapping defined by

$Y(x,\eta_{h},\eta_{g})=\Big{\\{}y\in\Re^{m}:h(x,y)-\eta_{h}=0,g(x,y)+\eta_{g}\leq
0\Big{\\}}.$ (4.7)

Robinson [11] introduced the concept of strong regularity for a solution of
the generalized equation (4.3).

###### Definition 4.1

Let $z^{*}$ be a solution of the generalized equation (4.3). We say that
$z^{*}$ is a strongly regular solution of the generalized equation (4.3) if
there exist positive numbers $\delta$ and $\varepsilon>0$ such that for every
$\eta\in\textbf{B}(0,\delta)$, the following linearized generalized equation

$\eta\in{\cal J}{\cal H}(z^{*})(z-z^{*})+N_{\cal K}(z)$ (4.8)

has a unique solution in $\textbf{B}(z^{*},\varepsilon)$, denoted by
$\widehat{z}(\eta)$, and the mapping
$\widehat{z}:\textbf{B}(0,\delta)\rightarrow\textbf{B}(z^{*},\varepsilon)$ is
Lipschitz continuous.

It follows from [11] or [1] that if $z^{*}$ is a strongly regular solution of
the generalized equation (4.3), then there exist positive numbers $\delta$ and
$\varepsilon>0$ such that for every $\eta\in\textbf{B}(0,\delta)$, the
following generalized equation

$\eta\in{\cal H}(z)+N_{\cal K}(z)$ (4.9)

has a unique solution in $\textbf{B}(z^{*},\varepsilon)$, denoted by
$z(\eta)$, and the mapping
$z:\textbf{B}(0,\delta)\rightarrow\textbf{B}(z^{*},\varepsilon)$ is Lipschitz
continuous over $\textbf{B}(0,\delta)$.

To study the strong regularity of the KKT system at
$(x^{*},u^{*},v^{*},y^{*},\mu^{*},\lambda^{*})$, we introduce the following
definition.

###### Definition 4.2

We say that Problem (1.1) satisfies Property A at
$(x^{*},u^{*},v^{*},y^{*},\mu^{*},\lambda^{*})\in\Re^{n}\times\Re^{n_{1}}\times\Re^{n_{2}}\times\Re^{m}\times\Re^{m_{1}}\times\Re^{m_{2}}$
if

  * (i)

$x^{*}\in\Phi$ and conditions in (2.14) are satisfied at
$(x^{*},u^{*},v^{*},y^{*},\mu^{*},\lambda^{*})$.

  * (ii)

The set vectors $\big{\\{}\nabla H_{1}(x^{*}),\ldots,\nabla
H_{n_{1}}\big{\\}}\cup\Big{\\{}\nabla G_{i}(x^{*}):i\in I(x^{*})\Big{\\}}$ are
linearly independent, where
$I(x^{*})=\\{i:G_{i}(x^{*})=0,i=1.\ldots,n_{2}\\}$.

  * (iv)

$y^{*}\in Y(x^{*})$ and Problem $({\rm P}_{x^{*}})$ satisfies Jacobian
uniqueness conditions at $(y^{*},\mu^{*},\lambda^{*})$.

  * (v)

For every $d_{x}\in{\rm Aff}\,{\cal C}(x^{*})\setminus\\{0\\}$ (where ${\cal
C}(x^{*})$ is defined by (2.11)),

$\begin{array}[]{l}\displaystyle\sup_{(u,v)\in\Lambda(x^{*})}\left\\{\left\langle\left[\displaystyle\sum_{j=1}^{n_{1}}u_{i}\nabla^{2}_{xx}H_{j}(x^{*})+\displaystyle\sum_{i=1}^{n_{2}}v_{i}\nabla^{2}_{xx}G_{i}(x^{*})\right]d_{x},d_{x}\right\rangle\right\\}\\\\[16.0pt]
\quad\quad\quad\,+\left\langle\left[\nabla^{2}_{xx}{\cal
L}(x^{*},y^{*},\mu^{*},\lambda^{*})-N_{\alpha}(x^{*})^{T}K_{\alpha}(x^{*})^{-1}N_{\alpha}(x^{*})\right]d_{x},d_{x}\right\rangle>0,\end{array}$
(4.10)

where $K_{\alpha}(x)$ is defined by (2.5) and $N_{\alpha}(x)$ is defined by
(2.8).

It can be checked that the perturbed Kojima mapping of the form

$F(x,u,w,y,\mu,\xi)=\left[\begin{array}[]{c}\nabla_{x}{\cal
L}(x,y,\mu,\xi^{+})+{\cal J}H(x)^{T}u+{\cal J}G(x)^{T}w^{+}\\\\[3.0pt]
H(x)\\\\[3.0pt] G(x)-w^{-}\\\\[3.0pt] \nabla_{y}{\cal
L}(x,y,\mu,\xi^{+})\\\\[3.0pt] h(x,y)\\\\[3.0pt]
-g(x,y)+\xi^{-}\end{array}\right]-\left[\begin{array}[]{c}\eta_{x}\\\\[3.0pt]
\eta_{H}\\\\[3.0pt] -\eta_{G}\\\\[3.0pt] \eta_{y}\\\\[3.0pt]
\eta_{h}\\\\[3.0pt] \eta_{g}\end{array}\right]$ (4.11)

is the Kojima mapping of the canonical perturbation Problem (4.5), where
$\Phi:\Re^{n_{1}}\times\Re^{n_{2}}\rightrightarrows\Re^{n}$ is defined by
(4.6) and
$Y:\Re^{n}\times\Re^{m_{1}}\times\Re^{m_{2}}\rightrightarrows\Re^{m}$ is
defined by (4.7). Thus we have that $F(x,u,w,y,\mu,\xi)=\eta$ with
$\eta=(\eta_{x};\eta_{H};-\eta_{G};\eta_{y};\eta_{h};\eta_{g})$ corresponds to
KKT conditions for Problem (4.5).

###### Lemma 4.1

Suppose that Property A of Problem (1.1) is satisfied at
$(x^{*},u^{*},v^{*},y^{*},\mu^{*},\lambda^{*})\in\Re^{n}\times\Re^{n_{1}}\times\Re^{n_{2}}\times\Re^{m}\times\Re^{m_{1}}\times\Re^{m_{2}}$.
Then for $w^{*}=v^{*}+G(x^{*})$ and $\xi^{*}=\lambda^{*}+g(x^{*},y^{*})$, any
element of $\partial F(x^{*},u^{*},w^{*},y^{*},\mu^{*},\xi^{*})$ is
nonsingular.

Proof. Since $\lambda^{*}-g(x^{*},y^{*})>0$, we know that $\xi^{+}$ and
$\xi^{-}$ are differentiable at $\xi^{*}$, and $w^{+}$ and $w^{-}$ are
differentiable at $w^{*}$. Since $w^{+}$ and $w^{-}$ are strongly semi-smooth
in the sense of [10], $F$ is strongly semi-smooth at
$(x^{*},y^{*},u^{*},w^{*},\mu^{*},\xi^{*})$. Without loss of generality, we
assume that

$\beta:=I(x^{*})=\\{1,\ldots,r\\},\quad\alpha=\\{1,\ldots,s\\},$

where

$I(x^{*})=\\{i:G_{i}(x^{*})=0,i=1,\ldots,n_{2}\\},\quad\alpha=\\{i:g_{i}(x^{*},y^{*})=0,i=1,\ldots,m_{2}\\}.$

Let

$\beta_{+}=\Big{\\{}i\in\beta:v^{*}_{i}>0\Big{\\}},\quad\beta_{0}=\Big{\\{}i\in\beta:v^{*}_{i}=0\Big{\\}}$

and assume

$\beta_{+}=\Big{\\{}1,\ldots,r_{1}\Big{\\}}.$

Then, for $\beta^{c}=\\{1,\ldots,n_{2}\\}\setminus\beta$ and
$\alpha^{c}=\\{1,\ldots,m_{2}\\}\setminus\alpha$, we get that

$\beta_{+}=\Big{\\{}1,\ldots,r_{1}\Big{\\}},\quad\beta_{0}=\Big{\\{}r_{1}+1,\ldots,r\Big{\\}},\quad\\{\beta^{c}=\\{r+1,\ldots,n_{2}\\},\quad\alpha^{c}=\\{s+1,\ldots,m_{2}\\}.$

Thus we obtain

${\cal J}\xi^{+}|_{\xi=\xi^{*}}=\left[\begin{array}[]{cc}I_{s}&0\\\\[3.0pt]
0&0\end{array}\right],\quad{\cal
J}\xi^{-}|_{\xi=\xi^{*}}=\left[\begin{array}[]{cc}0&0\\\\[3.0pt]
0&I_{m_{2}-s}\end{array}\right]$ (4.12)

and

$\begin{array}[]{l}\partial
w^{+}|_{w=w^{*}}=\left\\{\left[\begin{array}[]{ccc}I_{r_{1}}&0&0\\\\[3.0pt]
0&\omega_{\beta_{0}}&0\\\\[3.0pt]
0&0&0\end{array}\right]:\omega_{\beta_{0}}={\rm
Diag}[\omega_{r_{1}+1},\cdot,\omega_{r}],\
\omega_{i}\in[0,1],\,i\in\beta_{0}\right\\},\\\\[18.0pt] \partial
w^{-}|_{w=w^{*}}=\Big{\\{}I_{n_{2}}-\omega:\omega\in\partial
w^{+}|_{w=w^{*}}\Big{\\}}.\end{array}$ (4.13)

Denote

$\begin{array}[]{ll}G^{*}_{11}=\nabla^{2}_{xx}{\cal
L}(x^{*},y^{*},\mu^{*},\lambda^{*})+\displaystyle\sum_{j=1}^{n_{1}}u_{i}\nabla^{2}_{xx}H_{j}(x^{*})+\displaystyle\sum_{i=1}^{n_{2}}v_{i}\nabla^{2}_{xx}G_{i}(x^{*}),\\\\[4.0pt]
G^{*}_{12}=\nabla^{2}_{xy}{\cal
L}(x^{*},y^{*},\mu^{*},\lambda^{*}),\,\,G^{*}_{22}=\nabla^{2}_{yy}{\cal
L}(x^{*},y^{*},\mu^{*},\lambda^{*}).\end{array}$

For simplicity, we use notations ${\cal J}_{x}h^{*}$ and ${\cal J}_{y}h^{*}$
to represent ${\cal J}_{x}h(x^{*},y^{*})$ and ${\cal J}_{y}h(x^{*},y^{*})$,
respectively. The same notations are also applied to $g_{\alpha}$ and
$g_{\alpha^{c}}$. Let $V$ be an element of $\partial
F(x^{*},u^{*},w^{*},y^{*},\mu^{*},\xi^{*})$. Then there exists an element
$\omega\in\partial w^{+}|_{w=w^{*}}$ such that

$\begin{array}[]{l}V=\left[\begin{array}[]{ccccccccc}G^{*}_{11}&{\cal
J}H(x^{*})^{T}&{\cal J}G_{\beta_{+}}(x^{*})^{T}&\omega_{\beta_{0}}{\cal
J}G_{\beta_{0}}(x^{*})^{T}&0&G^{*}_{12}&{\cal J}_{x}h^{*T}&-{\cal
J}_{x}g_{\alpha}^{*T}&0\\\\[6.0pt] {\cal J}H(x^{*})&0&0&0&0&0&0&0&0\\\\[6.0pt]
{\cal J}G_{\beta_{+}}(x^{*})&0&0&0&0&0&0&0&0\\\\[6.0pt] {\cal
J}G_{\beta_{0}}(x^{*})&0&0&-I_{|\beta_{0}|}+\omega_{\beta_{0}}&0&0&0&0&0\\\\[6.0pt]
{\cal J}G_{\beta^{c}}(x^{*})&0&0&0&-I_{n_{2}-r}&0&0&0&0\\\\[6.0pt]
G^{*T}_{12}&0&0&0&0&G^{*}_{22}&{\cal J}_{y}h^{*T}&-{\cal
J}_{y}g_{\alpha}^{*T}&0\\\\[6.0pt] {\cal J}_{x}h^{*}&0&0&0&0&{\cal
J}_{y}h^{*}&0&0&0\\\\[6.0pt] -{\cal J}_{x}g_{\alpha}^{*}&0&0&0&0&-{\cal
J}_{y}g_{\alpha}^{*}&0&0&0\\\\[6.0pt] -{\cal
J}_{x}g_{\alpha^{c}}^{*}&0&0&0&0&-{\cal
J}_{y}g_{\alpha^{c}}^{*}&0&0&I_{m_{2}-s}\end{array}\right].\end{array}$ (4.14)

The nonsingularity of $V$ is equivalent to the nonsingularity of the following
matrix

$\begin{array}[]{l}\left[\begin{array}[]{ccccccccc}G^{*}_{11}&{\cal
J}H(x^{*})^{T}&{\cal J}G_{\beta_{+}}(x^{*})^{T}&\omega_{\beta_{0}}{\cal
J}G_{\beta_{0}}(x^{*})^{T}&G^{*}_{12}&{\cal J}_{x}h^{*T}&-{\cal
J}_{x}g_{\alpha}^{*T}&0&0\\\\[6.0pt] {\cal
J}H(x^{*})&0&0&0&0&0&0&0&0\\\\[6.0pt] {\cal
J}G_{\beta_{+}}(x^{*})&0&0&0&0&0&0&0&0\\\\[6.0pt] {\cal
J}G_{\beta_{0}}(x^{*})&0&0&-I_{|\beta_{0}|}+\omega_{\beta_{0}}&0&0&0&0&0\\\\[6.0pt]
G^{*T}_{12}&0&0&0&G^{*}_{22}&{\cal J}_{y}h^{*T}&-{\cal
J}_{y}g_{\alpha}^{*T}&0&0\\\\[6.0pt] {\cal J}_{x}h^{*}&0&0&0&{\cal
J}_{y}h^{*}&0&0&0&0\\\\[6.0pt] -{\cal J}_{x}g_{\alpha}^{*}&0&0&0&-{\cal
J}_{y}g_{\alpha}^{*}&0&0&0&0\\\\[6.0pt] {\cal
J}G_{\beta^{c}}(x^{*})&0&0&0&0&0&0&-I_{n_{2}-r}&0\\\\[6.0pt] -{\cal
J}_{x}g_{\alpha^{c}}^{*}&0&0&0&-{\cal
J}_{y}g_{\alpha^{c}}^{*}&0&0&0&I_{m_{2}-s}\end{array}\right],\end{array}$

which is equivalent to the nonsingularity of the following matrix

$H(\omega)=\left[\begin{array}[]{ccccccc}G^{*}_{11}&{\cal J}H(x^{*})^{T}&{\cal
J}G_{\beta_{+}}(x^{*})^{T}&\omega_{\beta_{0}}{\cal
J}G_{\beta_{0}}(x^{*})^{T}&G^{*}_{12}&{\cal J}_{x}h^{*T}&-{\cal
J}_{x}g_{\alpha}^{*T}\\\\[6.0pt] {\cal J}H(x^{*})&0&0&0&0&0&0\\\\[6.0pt] {\cal
J}G_{\beta_{+}}(x^{*})&0&0&0&0&0&0\\\\[6.0pt] {\cal
J}G_{\beta_{0}}(x^{*})&0&0&-I_{|\beta_{0}|}+\omega_{\beta_{0}}&0&0&0\\\\[6.0pt]
G^{*T}_{12}&0&0&0&G^{*}_{22}&{\cal J}_{y}h^{*T}&-{\cal
J}_{y}g_{\alpha}^{*T}\\\\[6.0pt] {\cal J}_{x}h^{*}&0&0&0&{\cal
J}_{y}h^{*}&0&0\\\\[6.0pt] -{\cal J}_{x}g_{\alpha}^{*}&0&0&0&-{\cal
J}_{y}g_{\alpha}^{*}&0&0\end{array}\right].$ (4.15)

Therefore, we only need to prove that the matrix $H(\omega)$ is nonsingular.
From Lemma 2.2, we obtain that $K_{\alpha}(x^{*})$ is nonsingular, where

$K_{\alpha}(x^{*})=\left[\begin{array}[]{ccc}G^{*}_{22}&{\cal
J}_{y}h^{*T}&-{\cal J}_{y}g_{\alpha}^{*T}\\\\[6.0pt] {\cal
J}_{y}h^{*}&0&0\\\\[6.0pt] -{\cal J}_{y}g_{\alpha}^{*}&0&0\end{array}\right].$

So we only need to prove that $H(\omega)/K_{\alpha}(x^{*})$ is nonsingular.
Notice that

$\begin{array}[]{rcl}H(\omega)/K_{\alpha}(x^{*})&=&\left[\begin{array}[]{cccc}G^{*}_{11}&{\cal
J}H(x^{*})^{T}&{\cal J}G_{\beta_{+}}(x^{*})^{T}&\omega_{\beta_{0}}{\cal
J}G_{\beta_{0}}(x^{*})^{T}\\\\[6.0pt] {\cal J}H(x^{*})&0&0&0\\\\[6.0pt] {\cal
J}G_{\beta_{+}}(x^{*})&0&0&0\\\\[6.0pt] {\cal
J}G_{\beta_{0}}(x^{*})&0&0&-I_{|\beta_{0}|}+\omega_{\beta_{0}}\\\\[6.0pt]
\end{array}\right]\\\\[20.0pt] &&-\left[\begin{array}[]{ccc}G^{*}_{12}&{\cal
J}_{x}h^{*T}&-{\cal J}_{x}g_{\alpha}^{*T}\\\\[3.0pt] 0&0&0\\\\[3.0pt]
0&0&0\\\\[3.0pt]
0&0&0\end{array}\right]K_{\alpha}(x^{*})^{-1}\left[\begin{array}[]{cccc}G^{*T}_{12}&0&0&0\\\\[6.0pt]
{\cal J}_{x}h^{*}&0&0&0\\\\[6.0pt] -{\cal
J}_{x}g_{\alpha}^{*}&0&0&0\end{array}\right]\\\\[20.0pt]
&=&\left[\begin{array}[]{cccc}G^{*}_{11}-N_{\alpha}(x^{*})^{T}K_{\alpha}(x^{*})N_{\alpha}(x^{*})&{\cal
J}H(x^{*})^{T}&{\cal J}G_{\beta_{+}}(x^{*})^{T}&\omega_{\beta_{0}}{\cal
J}G_{\beta_{0}}(x^{*})^{T}\\\\[6.0pt] {\cal J}H(x^{*})&0&0&0\\\\[6.0pt] {\cal
J}G_{\beta_{+}}(x^{*})&0&0&0\\\\[6.0pt] {\cal
J}G_{\beta_{0}}(x^{*})&0&0&-I_{|\beta_{0}|}+\omega_{\beta_{0}}\\\\[6.0pt]
\end{array}\right].\end{array}$ (4.16)

It is easy to check that ${\rm Aff}\,{\cal C}(x^{*})$ is of the following
subspace

${\rm Aff}\,{\cal C}(x^{*})={\rm Ker}\,\left[\begin{array}[]{l}{\cal
J}H(x^{*})\\\\[6.0pt] {\cal J}G_{\beta_{+}}(x^{*})\end{array}\right].$ (4.17)

Now we prove that $H(\omega)/K_{\alpha}(x^{*})$ is nonsingular via the formula
(4.16). Let $a_{1}\in\Re^{n}$, $a_{2}\in\Re^{n_{1}}$, $a_{3}\in\Re^{r_{1}}$
and $a_{4}\in\Re^{r-r_{1}}$ satisfy

$\left[\begin{array}[]{cccc}G^{*}_{11}-N_{\alpha}(x^{*})^{T}K_{\alpha}(x^{*})N_{\alpha}(x^{*})&{\cal
J}H(x^{*})^{T}&{\cal J}G_{\beta_{+}}(x^{*})^{T}&\omega_{\beta_{0}}{\cal
J}G_{\beta_{0}}(x^{*})^{T}\\\\[6.0pt] {\cal J}H(x^{*})&0&0&0\\\\[6.0pt] {\cal
J}G_{\beta_{+}}(x^{*})&0&0&0\\\\[6.0pt] {\cal
J}G_{\beta_{0}}(x^{*})&0&0&-I_{|\beta_{0}|}+\omega_{\beta_{0}}\\\\[6.0pt]
\end{array}\right]\left[\begin{array}[]{c}a_{1}\\\\[6.0pt] a_{2}\\\\[6.0pt]
a_{3}\\\\[6.0pt] a_{4}\end{array}\right]=0_{n+n_{1}+r}$

or

$\displaystyle[G^{*}_{11}-N_{\alpha}(x^{*})^{T}K_{\alpha}(x^{*})N_{\alpha}(x^{*})]a_{1}+{\cal
J}H(x^{*})^{T}a_{2}+{\cal
J}G_{\beta_{+}}(x^{*})^{T}a_{3}+\omega_{\beta_{0}}{\cal
J}G_{\beta_{0}}(x^{*})^{T}a_{4}=0,$ (4.18) $\displaystyle{\cal
J}H(x^{*})a_{1}=0,$ (4.19) $\displaystyle{\cal J}G_{\beta_{+}}(x^{*})a_{1}=0,$
(4.20) $\displaystyle{\cal
J}G_{\beta_{0}}(x^{*})a_{1}+[-I_{|\beta_{0}|}+\omega_{\beta_{0}}]a_{4}=0.$
(4.21)

It follows from (4.19) and (4.20) that $a_{1}\in{\rm Aff}\,{\cal C}(x^{*})$.
Premultiplying $a^{T}_{1}$ to (4.18), we obtain

$a_{1}^{T}[G^{*}_{11}-N_{\alpha}(x^{*})^{T}K_{\alpha}(x^{*})N_{\alpha}(x^{*})]a_{1}+a_{1}^{T}\omega_{\beta_{0}}{\cal
J}G_{\beta_{0}}(x^{*})^{T}a_{4}=0.$ (4.22)

From the relation (4.21), we get that

$a_{1}^{T}\omega_{\beta_{0}}{\cal
J}G_{\beta_{0}}(x^{*})^{T}a_{4}=\displaystyle\sum_{i\in\beta_{0}:0<w_{i}<1}\displaystyle\frac{\omega_{i}}{1-\omega_{i}}[\nabla
G_{i}(x^{*})^{T}a_{1}]^{2}\geq 0.$ (4.23)

If follows from (4.18) and (4.23) that

$\begin{array}[]{l}\displaystyle\left\langle\left[\displaystyle\sum_{j=1}^{n_{1}}u^{*}_{i}\nabla^{2}_{xx}H_{j}(x^{*})+\displaystyle\sum_{i=1}^{n_{2}}v^{*}_{i}\nabla^{2}_{xx}G_{i}(x^{*})\right]a_{1},a_{1}\right\rangle\\\\[16.0pt]
\quad\quad\quad\,+\left\langle\left[\nabla^{2}_{xx}{\cal
L}(x^{*},y^{*},\mu^{*},\lambda^{*})-N_{\alpha}(x^{*})^{T}K_{\alpha}(x^{*})^{-1}N_{\alpha}(x^{*})\right]a_{1},a_{1}\right\rangle\leq
0,\end{array}$

which implies $a_{1}=0$ from the condition (2.16). Let
$\bar{\beta}_{0}=\\{i\in\beta_{0}:\omega_{i}=1\\}$ and
$\bar{\beta}_{0}^{c}=\beta_{0}\setminus\bar{\beta}_{0}$. Then from (4.21),
$a_{1}=0$ and (4.18), we can get that

$[a_{4}]_{\bar{\beta}_{0}^{c}}=0$ (4.24)

and

${\cal J}H(x^{*})^{T}a_{2}+{\cal J}G_{\beta_{+}}(x^{*})^{T}a_{3}+{\cal
J}G_{\bar{\beta}_{0}}(x^{*})^{T}[a_{4}]_{\bar{\beta}_{0}}=0.$ (4.25)

From (iii), we obtain $a_{2}=0$,$a_{3}=0$ and $[a_{4}]_{\bar{\beta}_{0}}=0$.
Combining with (4.24), we have that $a_{2}=0$, $a_{3}=0$ and $a_{4}=0$.
Therefore $H(\omega)/K_{\alpha}(x^{*})$ is nonsingular. The proof is
completed. $\Box$

Now we are in a position to establish the main result about the strong
regularity of the KKT system for Problem (1.1).

###### Theorem 4.1

Let $(x^{*},y^{*})\in\Re^{n}\times\Re^{m}$ be a point around which $f$, $h$
and $g$ are twice differentiable, and $H$ and $G$ are twice continuously
differentiable. Assume that there exists
$(u^{*},v^{*},\mu^{*},\lambda^{*})\in\Re^{n_{1}}\times\Re^{n_{2}}\times\Re^{m_{1}}\times\Re^{m_{2}}$
such that $(x^{*},u^{*},v^{*},y^{*},\mu^{*},\lambda^{*})$ satisfies Karush-
Kuhn-Tucker conditions for Problem (1.1). Consider the following four
statements

  * (a)

Property A holds at $(x^{*},u^{*},v^{*},y^{*},\mu^{*},\lambda^{*})$.

  * (b)

For $w^{*}=v^{*}+G(x^{*})$ and $\xi^{*}=\lambda^{*}+g(x^{*},y^{*})$, any
element of $\partial F(x^{*},u^{*},w^{*},y^{*},\mu^{*},\xi^{*})$ is
nonsingular.

  * (c)

$F$ is a locally Lipschitz homeomorphism near
$(x^{*},u^{*},w^{*},y^{*},\mu^{*},\xi^{*})$.

  * (d)

The point $(x^{*},u^{*},v^{*},y^{*},\mu^{*},\lambda^{*})$ is a strongly
regular solution of the generalized equation (4.3).

Then it holds that (a) $\Longrightarrow$ (b) $\Longrightarrow$ (c)
$\Longleftrightarrow$ (d).

Proof. From Lemma 4.1, we obtain (a)$\Longrightarrow$ (b). By Clarke’s inverse
function theorem (Clarke [3, 4]), $F$ is a locally Lipschitz homeomorphism
near $(x^{*},u^{*},w^{*},y^{*},\mu^{*},\xi^{*})$ and hence we get
(b)$\Longrightarrow$ (c). Noting that the generalized equation

$\eta\in{\cal H}(z)+N_{\cal K}(z)$

represents the KKT conditions for Problem (4.5) and $F(x,u,w,y,\mu,\xi)=\eta$
with $\eta=(\eta_{x}$; $\eta_{H}$; $-\eta_{G}$; $\eta_{y}$; $\eta_{h}$;
$\eta_{g})$ corresponds to KKT conditions for Problem (4.5), we obtain the
equivalence between (c) and (d). The proof is completed. $\Box$

##  5 Some Concluding Remarks

In this paper, we have analyzed the stability properties of the Kurash-Kuhn-
Tucker (KKT) system for Problem (1.1). Firstly, we proposed the definition of
Jacobian uniqueness condition of Problem (1.1) and proved that this property
is stable with respect to a small ${\cal C}^{2}$-perturbation. Secondly,
comparing with the Jacobian uniqueness condition, we proposed Property A by
eliminating the strict complementarity condition for the outer level
constraints and adopting the strong second-order sufficiency optimality
condition. We proved that the strong regularity of the KKT system at the KKT
point is equivalent to the local Lipschitz homeomorphism of the Kojima mapping
near the KKT point. Finally, we proved that Property A is a sufficient
condition for the strong regularity of the KKT system at the KKT point.

There are many problems about the stability of constrained minimax
optimization left to us. For instance, in our analysis, even Property A
requires the Jacobian uniqueness condition for the inner level problem. Is it
possible to weaken this condition? A closely related problem is how to obtain
the second-order optimality conditions for the constrained minimax problem
when the Jacobian uniqueness condition for the inner level problem fails. This
might be a difficult problem. Another question is, when the constraints are
linear and the objective function is even convex-concave, can we have a sharp
theoretical result like the result for linear semidefinte programming in [2]?

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