††thanks: The first author was supported by the National Natural Science
Foundation of China (grant 11971422) and CAS “Light of West China" Program,
and by Jointed Key Project of Yunnan Provincial Science and Technology
Department and Yunnan University [No. 2018FY001014] and Program for Innovative
Research Team (in Science and Technology) in Universities of Yunnan Province
[No. C176240111009].††thanks: Research of the second author benefited from the
support of the FMJH Program PGMO and from the support of EDF

# Characterizations of Stability of Error Bounds for Convex Inequality
Constraint Systems

Zhou Wei  Department of Mathematics, Hebei University, Baoding 071002,
People’s Republic of China<EMAIL_ADDRESS>Department of Mathematics,
Yunnan University, Kunming 650091, People’s Republic of China
<EMAIL_ADDRESS>Michel Théra  XLIM UMR-CNRS 7252, Université de Limoges
Limoges, France
Federation University Australia, Ballarat<EMAIL_ADDRESS>Jen-Chih Yao
Research Center for Interneural Computing, China Medical University Hospital
China Medical University, Taichung 40402, Taiwan<EMAIL_ADDRESS>
###### Abstract

In this paper, we mainly study error bounds for a single convex inequality and
semi-infinite convex constraint systems, and give characterizations of
stability of error bounds via directional derivatives. For a single convex
inequality, it is proved that the stability of local error bounds under small
perturbations is essentially equivalent to the non-zero minimun of the
directional derivative at a reference point over the sphere, and the stability
of global error bounds is proved to be equivalent to the strictly positive
infimum of the directional derivatives, at all points in the boundary of the
solution set, over the sphere as well as some mild constraint qualification.
When these results are applied to semi-infinite convex constraint systems,
characterizations of stability of local and global error bounds under small
perturbations are also provided. In particular such stability of error bounds
is proved to only require that all component functions in semi-infinite convex
constraint systems have the same linear perturbation. Our work demonstrates
that verifying the stability of error bounds for convex inequality constraint
systems is, to some degree, equivalent to solving the convex
optimization/minimization problems (defined by directional derivatives) over
the sphere.

###### keywords:

Local and global error bounds, Stability, Convex inequality, Semi-infinite
convex constraint systems, Directional derivative

##  1 Introduction

Our main goal in this paper is to study error bounds of a single convex
inequality and semi-infinite convex constraint systems and to provide
characterizations of stability of local and global error bounds under
perturbations. Theory of error bounds can be traced back to the pioneering
work by Hoffman [22] for systems of affine functions in which it has been
proved that for a given matrix $A$ and a vector $b$, the distance from $x$ to
the polyhedral set $\\{u:Au\leq b\\}$ is bounded above by some scalar constant
(depending on $A$ only) times the norm of the residual error $\|(Ax-b)_{+}\|$,
where for any vector $z$, $(z)_{+}$ denotes the positive part of $z$.

Hoffman’s result was extensively and intensively studied by Robinson [48],
Mangasarian [40], Auslender and Crouzeix [2], Pang [44], Lewis and Pang [36],
Klatte and Li [30], Jourani [29], and there have been important developments
of various aspects of error bounds for convex and nonconvex functions in
recent years. We refer the readers to bibliographies [3, 4, 6, 9, 14, 15, 17,
18, 19, 23, 26, 34, 37, 41, 43, 45, 52] and references therein for the summary
of the theory of error bounds and their various applications for more details.

Error bounds have been applied to the sensitivity analysis of linear programs
(cf. [47, 49]) and to the convergence analysis of descent methods for linearly
constrained minimization (cf. [20, 21, 28, 38, 51]). In addition, it is proved
that error bounds play an important role in the feasibility problem of finding
a point in the intersection of a finite collection of closed convex sets (cf.
[7, 8, 9]) and have an application in the domain of image reconstruction (cf.
[13]). Also, error bounds are extensively discussed in connection with weak
sharp minima of functions and metric regularity/subregularity as well as Aubin
property/calmness of set-valued mappings (cf. [1, 6, 10, 11, 12, 19, 24, 25,
31, 32, 48, 55, 56] and references therein).

Since real-world problems typically have inaccurate data, it is of practical
and theoretical interest to know the behavior of error bounds under data
perturbations. For systems of linear inequalities, this question has been
studied by Luo and Tseng [39] and Azé and Corvellec [5]. Subsequently Deng
[16] studied systems of a finite number of convex inequalities. In 2005, Zheng
and Ng [54] considered the stability of error bounds for systems of conic
linear inequalities in a general Banach space. In 2010, Ngai, Kruger and Théra
[42] studied the stability of error bounds for semi-infinite convex constraint
systems in a Euclidean space and established subdifferential characterizations
of the stability under small perturbations. The infinite dimensional
extensions were considered by Kruger, Ngai and Théra in [33]. In 2012, by
relaxing the convexity assumption, Zheng and Wei [57] discussed the stability
of error bounds for quasi-subsmooth (not necessarily convex) inequalities in a
general Banach space and provided Clarke subdifferential characterizations of
the stability of error bounds. In 2018, Kruger, López and Théra [35] extended
the development in [33, 42] and characterized the stability of error bounds
for convex inequalities in the Banach space setting. From the viewpoint of
infinite dimensional Banach spaces, results on the stability of error bounds
in [33, 35, 42, 57] are dual conditions, and it is a pretty natural idea to
study this issue not involving the dual space since information on the dual
space may be missing. Inspired by this observation, we study characterizations
of stability of local and global error bounds of a single convex inequality
and semi-infinite convex constraint systems via directional derivatives. For a
single convex inequality, we prove that the stability of local error bounds
under small perturbations holds if and only if the minimum of the directional
derivative at a reference point over the sphere is non-zero, and the stability
of global error bounds is proved to be equivalent to the strictly positive
infimum of the directional derivatives, at all points in the boundary of the
solution set, over the sphere as well as some mild constraint qualification.
When these results are applied to semi-infinite convex constraint systems,
characterizations of the stability of local and global error bounds under
small perturbations are also provided. Particularly such stability of error
bounds is proved to only require that all component functions in semi-infinite
convex constraint systems have the same linear perturbation. Our work
demonstrates that verifying the stability of error bounds for convex
inequality constraint systems is, to some degree, equivalent to solving
optimization/minimization problems (defined by directional derivatives) over
the sphere.

The paper is organized as follows. In Section 2, we give some definitions and
preliminary results. Section 3 is devoted to the study on stability of error
bounds for a single convex inequality. In terms of directional derivatives, we
provide characterizations of local and global error bounds for a single convex
inequality under small perturbations(see Section 3.1 and Section 3.2). When
these results are applied to the semi-infinite convex constraint systems in
Section 4, the stability of local and global error bounds can be obtained (see
Section 4 and Section 4). Conclusions of this paper are given in Section 5.

##  2 Preliminaries

In what follows we consider the Euclidean space $\mathbb{R}^{m}$ equipped with
the norm $\|\cdot\|:=\sqrt{\langle\cdot,\cdot\rangle}$. We denote by
$\mathbf{B}^{m}$ the closed unit ball of $\mathbb{R}^{m}$ and following the
standard notation by $\Gamma_{0}(\mathbb{R}^{m})$ the set of extended-real-
valued lower semicontinuous convex functions
$f:\mathbb{R}^{m}\rightarrow\mathbb{R}\cup\\{+\infty\\}$ which are supposed to
be proper, that is such that ${\rm
dom}(f):=\\{x\in\mathbb{R}^{m}:f(x)<+\infty\\}$ is nonempty.

For a subset $D$ of $\mathbb{R}^{m}$, we denote by $d(x,D)$ the distance from
$x$ to $D$ which is defined by

$d(x,D):=\inf\\{\|x-y\|:y\in D\\},$

where we use the convention $\inf\emptyset=+\infty$. We denote by ${\rm
bdry}(D)$ and ${\rm int}(D)$ the boundary and the interior of $D$,
respectively.

Let $f\in\Gamma_{0}(\mathbb{R}^{m})$ and $\bar{x}\in{\rm dom}(f)$. For any
$h\in\mathbb{R}^{m}$, we recall that the directional derivative
$f^{\prime}(\bar{x},h)$ of $f$ at $\bar{x}$ along the direction $h$ is defined
as

$f^{\prime}(\bar{x},h):=\lim\limits_{t\rightarrow
0^{+}}\frac{f(\bar{x}+th)-f(\bar{x})}{t}.$ (1)

It is known from [46, 50] that the function

$t\mapsto\frac{f(\bar{x}+th)-f(\bar{x})}{t}$

is nonincreasing as $t\rightarrow 0^{+}$ and thus

$f^{\prime}(\bar{x},h)=\inf_{t>0}\frac{f(\bar{x}+th)-f(\bar{x})}{t}.$ (2)

We denote by $\partial f(\bar{x})$ the subdifferential of $f$ at $\bar{x}$
which is defined by

$\partial f(\bar{x}):=\\{x^{*}\in\mathbb{R}^{m}:\langle
x^{*},x-\bar{x}\rangle\leq f(x)-f(\bar{x})\ \ {\rm for\ all}\
x\in\mathbb{R}^{m}\\}.$

It is known from [46, 50] that if $\partial f(\bar{x})$ is nonempty, one has

$\partial f(\bar{x})=\\{x^{*}\in\mathbb{R}^{m}:\langle x^{*},h\rangle\leq
f^{\prime}(\bar{x},h)\ \ {\rm for\ all}\ h\in\mathbb{R}^{m}\\}$ (3)

and

$f^{\prime}(\bar{x},h)=\max\\{\langle x^{*},h\rangle:x^{*}\in\partial
f(\bar{x})\\}.$ (4)

We conclude this section with the following lemma which is used in our
analysis. Let $f\in\Gamma_{0}(\mathbb{R}^{m})$ and $\bar{x}\in{\rm dom}(f)$ be
such that $-\infty<\min_{\|h\|=1}f^{\prime}(\bar{x},h)<0$. Then

$-\min_{\|h\|=1}f^{\prime}(\bar{x},h)=d(0,\partial f(\bar{x})).$ (5)

######  Proof 2.1.

Note that $\min_{\|h\|=1}f^{\prime}(\bar{x},h)<0$ and thus $0\not\in\partial
f(\bar{x})$. Let $r:=d(0,\partial f(\bar{x}))$. By the separation theorem,
there exists $h_{0}\in\mathbb{R}^{m}$ with $\|h_{0}\|=1$ such that

$\sup_{x^{*}\in\partial f(\bar{x})}\langle x^{*},h_{0}\rangle+r=0$

and thus by (4), one has

$\min_{\|h\|=1}f^{\prime}(\bar{x},h)\leq f^{\prime}(\bar{x},h_{0})=-r.$ (6)

For any $\varepsilon>0$, we can select
$u^{*}_{\varepsilon}\in(r+\varepsilon)\mathbf{B}^{m}\cap\partial f(\bar{x})$.
Then for any $h\in\mathbb{R}^{m}$ with $\|h\|=1$, one has

$f^{\prime}(\bar{x},h)\geq\langle
u^{*}_{\varepsilon},h\rangle\geq-(r+\varepsilon)$

and consequently

$\min_{\|h\|=1}f^{\prime}(\bar{x},h)\geq-(r+\varepsilon).$

By letting $\varepsilon\rightarrow 0^{+}$, it follows from (6) that (5) holds.
The proof is complete.

##  3 Stability of Error Bounds for A Single Convex Inequality

In this section, we mainly study local and global error bounds for a single
convex inequality, and provide characterizations of stability (in terms of
directional derivatives) of error bounds. We first recall the definition of
error bounds for a single convex inequality.

For a given $f\in\Gamma_{0}(\mathbb{R}^{m})$, we consider the set of solutions
of a single convex inequality:

$S_{f}:=\\{x\in\mathbb{R}^{m}:f(x)\leq 0\\}.$ (7)

Recall that convex inequality (7) is said to have a global error bound if
there exists a constant $\tau\in(0,+\infty)$ such that

$d(x,S_{f})\leq\tau[f(x)]_{+}\ \ \forall x\in\mathbb{R}^{m},$ (8)

where $[f(x)]_{+}:=\max\\{f(x),0\\}$. We denote by
$\tau_{\min}(f):=\inf\\{\tau>0:\eqref{3.2}\ {\rm holds}\\}$ the global error
bound modulus of $S_{f}$.

For $\bar{x}\in{\rm bdry}(S_{f})$, convex inequality (7) is said to have a
local error bound at $\bar{x}$ if there exist constants
$\tau,\delta\in(0,+\infty)$ such that

$d(x,S_{f})\leq\tau[f(x)]_{+}\ \ \forall x\in B(\bar{x},\delta).$ (9)

We denote by $\tau_{\min}(f,\bar{x}):=\inf\\{\tau>0:\ {\rm there\ exists}\
\delta>0\ {\rm such\ that}\ \eqref{3.3}\ {\rm holds}\\}$ the local error bound
modulus of $S_{f}$ at $\bar{x}$.

The following theorem gives characterizations of global and local error
bounds. We refer the readers to [5] for more details. This result is needed in
the sequel.

Let $f\in\Gamma_{0}(\mathbb{R}^{m})$. Then

  * (i)

$S_{f}$ has a global error bound if and only if

$\eta(f):=\inf\\{d(0,\partial f(x)):x\in\mathbb{R}^{m},f(x)>0\\}>0.$

More precisely, $\tau_{\min}(f)=[\eta(f)]^{-1}$.

  * (ii)

$S_{f}$ has a local error bound at $\bar{x}\in{\rm bdry}(S_{f})$ if and only
if

$\eta(f,\bar{x}):=\liminf_{x\rightarrow\bar{x},f(x)>0}d(0,\partial f(x))>0.$

More precisely, $\tau_{\min}(f,\bar{x})=[\eta(f,\bar{x})]^{-1}$.

  * (iii)

The following equality holds:

$\tau_{\min}(f)=\sup_{\bar{x}\in{\rm bdry}S_{f}}\tau_{\min}(f,\bar{x})$

For a mapping $\phi:X\rightarrow Y$ between Banach spaces $X,Y$, we denote by
${\rm Lip}(\phi)$ the Lipschitz constant which is defined by

${\rm Lip}(\phi):=\sup_{u,v\in X,u\not=v}\frac{\|\phi(u)-\phi(v)\|}{\|u-v\|}.$

###  3.1 Stability of Local Error bounds

In this subsection, we mainly study local error bounds for a single convex
inequality and aim to provide equivalent criterion for the stability of local
error bounds for convex inequality (7). We first give a sufficient condition
for the local error bound of convex inequality (7).

######  Proposition 1.

Let $f\in\Gamma_{0}(\mathbb{R}^{m})$ and $\bar{x}\in S_{f}$ such that
$\min_{\|h\|=1}f^{\prime}(\bar{x},h)\not=0$. Then convex inequality (7) has a
local error bound at $\bar{x}$ and moreover

$\tau_{\min}(f,\bar{x})\leq\frac{1}{\Big{|}\min\limits_{\|h\|=1}f^{\prime}(\bar{x},h)\Big{|}}.$
(10)

######  Proof 3.1.

Let $\beta(f,\bar{x}):=\min_{\|h\|=1}f^{\prime}(\bar{x},h)$. Suppose that
$\beta(f,\bar{x})>0$. Then for any $x\not=\bar{x}$, by (2), one can verify
that

$\displaystyle f(x)-f(\bar{x})$ $\displaystyle=$ $\displaystyle
f\Big{(}\bar{x}+\|x-\bar{x}\|\frac{x-\bar{x}}{\|x-\bar{x}\|}\Big{)}-f(\bar{x})$
$\displaystyle\geq$ $\displaystyle
f^{\prime}\Big{(}\bar{x},\frac{x-\bar{x}}{\|x-\bar{x}\|}\Big{)}\|x-\bar{x}\|$
$\displaystyle\geq$
$\displaystyle\beta(f,\bar{x})\|x-\bar{x}\|\geq\beta(f,\bar{x})d(x,S_{f}).$

This means that $\tau_{\min}(f,\bar{x})\leq[\beta(f,\bar{x})]^{-1}$.

Suppose that $\beta(f,\bar{x})<0$. Then Lemma 1 implies that $d(0,\partial
f(\bar{x}))=-\beta(f,\bar{x})$ and by virtue of Section 3, one has

$\tau(f,\bar{x})\leq\frac{1}{-\beta(f,\bar{x})}.$

Hence (10) holds. The proof is complete.

Proposition 3 shows that the solution set $S_{f}=\\{\bar{x}\\}$ if
$\min_{\|h\|=1}f^{\prime}(\bar{x},h)>0$ and $f(\bar{x})=0$ since for any
$x\not=\bar{x}$, one has

$f(x)-f(\bar{x})\geq
f^{\prime}\Big{(}\bar{x},\frac{x-\bar{x}}{\|x-\bar{x}\|}\Big{)}\|x-\bar{x}\|\geq\min_{\|h\|=1}f^{\prime}(\bar{x},h)\|x-\bar{x}\|>0.$

Note that the condition $\min_{\|h\|=1}f^{\prime}(\bar{x},h)\not=0$ is only a
sufficient condition for the existence of a local error bound of (7). Indeed,
let $f(x)\equiv 0$ for all $x\in\mathbb{R}$. Then $S_{f}=\mathbb{R}$ has a
global error bound, while $\min_{\|h\|=1}f^{\prime}(\bar{x},h)=0$ for all
$\bar{x}\in\mathbb{R}$. $\Box$

The following theorem shows that the condition
$\min_{\|h\|=1}f^{\prime}(\bar{x},h)\not=0$ can be used to give
characterizations of stability of the local error bound for the convex
inequality (7). For the sake of completeness, we provide a self-contained
proof of this theorem. Let $f\in\Gamma_{0}(\mathbb{R}^{m})$ and
$\bar{x}\in\mathbb{R}^{m}$ be such that $f(\bar{x})=0$. Then the following
statements are equivalent:

  * (i)

$\min_{\|h\|=1}f^{\prime}(\bar{x},h)\not=0$;

  * (ii)

There exist constants $c,\varepsilon>0$ such that for all
$g\in\Gamma_{0}(\mathbb{R}^{m})$ satisfying $\bar{x}\in S_{g}$ and

$\limsup_{x\rightarrow\bar{x}}\frac{|(f(x)-g(x))-(f(\bar{x})-g(\bar{x}))|}{\|x-\bar{x}\|}\leq\varepsilon,$
(11)

one has $\tau_{\min}(g,\bar{x})\leq c$;

  * (iii)

There exist constants $c,\varepsilon>0$ such that for all
$u^{*}\in\mathbb{R}^{m}$ with $\|u^{*}\|\leq 1$, one has
$\tau_{\min}(g_{u^{*},\varepsilon},\bar{x})\leq c$, where
$g_{u^{*},\varepsilon}(x):=f(x)+\varepsilon\langle u^{*},x-\bar{x}\rangle$ for
all $x\in\mathbb{R}^{m}$.

######  Proof 3.2.

Let $\beta(f,\bar{x}):=\min_{\|h\|=1}f^{\prime}(\bar{x},h)$.

(i) $\Rightarrow$ (ii): Take any $\varepsilon>0$ such that
$\varepsilon<|\beta(f,\bar{x})|$ and let
$c:=(|\beta(f,\bar{x})|-\varepsilon)^{-1}$. For any
$g\in\Gamma_{0}(\mathbb{R}^{m})$ such that $\bar{x}\in S_{g}$ and (11) holds.
If $\beta(f,\bar{x})>0$, then for any $h\in\mathbb{R}^{m}$, one has

$g^{\prime}(\bar{x},h)\geq f^{\prime}(\bar{x},h)-\varepsilon,$

and thus

$\min_{\|h\|=1}g^{\prime}(\bar{x},h)\geq\min_{\|h\|=1}f^{\prime}(\bar{x},h)-\varepsilon\geq\beta(f,\bar{x})-\varepsilon.$

This and Proposition 3 imply that
$\tau_{\min}(g,\bar{x})\leq[\beta(f,\bar{x})-\varepsilon]^{-1}=c$.

If $\beta(f,\bar{x})<0$, then for any $h\in\mathbb{R}^{m}$, one has

$g^{\prime}(\bar{x},h)\leq f^{\prime}(\bar{x},h)+\varepsilon,$

and thus

$\min_{\|h\|=1}g^{\prime}(\bar{x},h)\leq\min_{\|h\|=1}f^{\prime}(\bar{x},h)+\varepsilon\leq\beta(f,\bar{x})+\varepsilon.$

By using Proposition 3 again, one yields that
$\tau_{\min}(g,\bar{x})\leq[-\beta(f,\bar{x})-\varepsilon]^{-1}=c$. Hence (ii)
holds.

Note that the implication (ii) $\Rightarrow$ (iii) is clear and it remains to
prove (iii) $\Rightarrow$ (i).

Suppose on the contrary that there exists $h_{0}\in\mathbb{R}^{m}$ with
$\|h_{0}\|=1$ such that $f^{\prime}(\bar{x},h_{0})=0$. Let $\varepsilon>0$ and
consider the function $g_{\varepsilon}(x):=f(x)+\varepsilon\langle
h_{0},x-\bar{x}\rangle$ for all $x\in\mathbb{R}^{m}$. From
$\beta(f,\bar{x})=0$, one can verify that $f(x)\geq f(\bar{x})$ for any
$x\not=\bar{x}$. Note that $f^{\prime}(\bar{x},h_{0})=0$ and it follows that
there exists a sequence $\\{\delta_{k}\\}$ decreasing to $0$ such that

$f(\bar{x}+\delta_{k}h_{0})<f(\bar{x})+\varepsilon\delta_{k}=\inf_{x\in\mathbb{R}^{m}}f(x)+\varepsilon\delta_{k}.$
(12)

By virtue of the Ekeland variational principle, we can select
$z_{k}\in\mathbb{R}^{m}$ such that
$\|z_{k}-(\bar{x}+\delta_{k}h_{0})\|<\frac{\delta_{k}}{2},f(z_{k})\leq
f(\bar{x}+\delta_{k}h_{0})$ and

$f(x)+2\varepsilon\|x-z_{k}\|>f(z_{k}),\ \forall x\not=z_{k}.$ (13)

This implies that $z_{k}\rightarrow\bar{x}$,
$g_{\varepsilon}(\bar{x})=f(\bar{x})=0$ and

$\displaystyle g_{\varepsilon}(z_{k})$ $\displaystyle=$ $\displaystyle
f(z_{k})+\varepsilon\langle h_{0},z_{k}-\bar{x}\rangle$ $\displaystyle\geq$
$\displaystyle f(\bar{x})+\varepsilon\langle h_{0},z_{k}-\bar{x}\rangle$
$\displaystyle=$ $\displaystyle\varepsilon\langle
h_{0},z_{k}-\bar{x}+\delta_{k}h_{0}\rangle+\varepsilon\delta_{k}$
$\displaystyle>$
$\displaystyle\varepsilon\delta_{k}-\frac{1}{2}\varepsilon\delta_{k}=\frac{1}{2}\varepsilon\delta_{k}>0.$

We claim that

$\min_{\|h\|=1}g_{\varepsilon}^{\prime}(z_{k},h)<0.$ (14)

(Otherwise, $\min_{\|h\|=1}g_{\varepsilon}^{\prime}(z_{k},h)\geq 0$ and then
one has $g_{\varepsilon}(z_{k})=\inf_{x\in\mathbb{R}^{m}}g_{\varepsilon}(x)$,
which contradicts $g_{\varepsilon}(\bar{x})=0$).

For any $h\in\mathbb{R}^{m}$ with $\|h\|=1$ and any $t>0$, by (13), one has

$\displaystyle\frac{g_{\varepsilon}(z_{k}+th)-g_{\varepsilon}(z_{k})}{t}=\frac{f(z_{k}+th)-f(z_{k})}{t}+\varepsilon\langle
h_{0},h\rangle\geq-2\varepsilon\|h\|-\varepsilon=-3\varepsilon$

and consequently

$0\geq\min_{\|h\|=1}g_{\varepsilon}^{\prime}(z_{k},h)\geq-3\varepsilon.$

Thanks to Lemma 1 and Proposition 3, one can obtain that
$\tau_{\min}(g_{\varepsilon},\bar{x})\geq\frac{1}{3\varepsilon}$, which
contradicts (iii) as $\varepsilon$ is arbitrary. The proof is complete.

(a) From [35, 42], the condition (11) means that $g$ is an
$\varepsilon$-perturbation of $f$ near $\bar{x}$, and the condition
$\min_{\|h\|=1}f^{\prime}(\bar{x},h)\not=0$ is proved to be equivalent to the
stability of this $\varepsilon$-perturbation of local error bounds. Further,
it is showed in Section 3.1 that the stability of such
$\varepsilon$-perturbation is essentially equivalent to $\varepsilon$-linear
perturbation.

(b) Section 3.1 can be regarded as the equivalent version of [42, Theorem 2]
since one can prove that

$\min_{\|h\|=1}f^{\prime}(\bar{x},h)\not=0\Longleftrightarrow 0\not\in{\rm
bdry}(\partial f(\bar{x})).$

  * Indeed, suppose that $0\not\in{\rm bdry}(\partial f(\bar{x}))$. For the case that $0\in{\rm int}(\partial f(\bar{x}))$, there is $r>0$ such that $r\mathbf{B}^{m}\subseteq\partial f(\bar{x})$. This and (4) imply that

$\min_{\|h\|=1}f^{\prime}(\bar{x},h)\geq r>0.$

For the case that $0\not\in\partial f(\bar{x})$, by the separation theorem,
there exists $h_{0}\in\mathbb{R}^{m}$ with $\|h_{0}\|=1$ such that

$0>\sup\\{\langle x^{*},h_{0}\rangle:x^{*}\in\partial
f(\bar{x})\\}=f^{\prime}(\bar{x},h_{0})$

and consequently

$\min_{\|h\|=1}f^{\prime}(\bar{x},h)\leq f^{\prime}(\bar{x},h_{0})<0.$

On the other hand, if $0\in{\rm bdry}(\partial f(\bar{x}))$, then
$\min_{\|h\|=1}f^{\prime}(\bar{x},h)\geq 0$ and for any $\varepsilon>0$, we
can select $u^{*}_{\varepsilon}\in\varepsilon\mathbf{B}^{m}\backslash\partial
f(\bar{x})$ and $x_{\varepsilon}\not=\bar{x}$ such that

$\langle
u^{*}_{\varepsilon},x_{\varepsilon}-\bar{x}\rangle>f(x_{\varepsilon})-f(\bar{x}).$

By (2), for any $t\in(0,1)$, one has

$\frac{f(\bar{x}+t(x_{\varepsilon}-\bar{x}))-f(\bar{x})}{t}\leq
f(x_{\varepsilon})-f(\bar{x})<\langle
u^{*}_{\varepsilon},x_{\varepsilon}-\bar{x}\rangle$

and thus

$f^{\prime}\Big{(}\bar{x},\frac{x_{\varepsilon}-\bar{x}}{\|x_{\varepsilon}-\bar{x}\|}\Big{)}\leq\Big{\langle}u^{*}_{\varepsilon},\frac{x_{\varepsilon}-\bar{x}}{\|x_{\varepsilon}-\bar{x}\|}\Big{\rangle}\leq\varepsilon.$

This means that $\min_{\|h\|=1}f^{\prime}(\bar{x},h)\leq\varepsilon\rightarrow
0^{+}$ and so $\min_{\|h\|=1}f^{\prime}(\bar{x},h)=0$. $\Box$

###  3.2 Stability of Global Error Bounds

This subsection is devoted to the study on stability of global error bounds
for a single convex inequality, and the aim is to give sufficient and/or
necessary conditions for the stability via directional derivatives. The
following theorem gives criterion for the stability of global error bounds.

Let $f\in\Gamma_{0}(\mathbb{R}^{m})$ be such that ${\rm bdry}(S_{f})\subseteq
f^{-1}(0)$. Consider the following statements:

  * (i)

There exists $\tau\in(0,+\infty)$ such that

$\inf\left\\{\Big{|}\min_{\|h\|=1}f^{\prime}(\bar{x},h)\Big{|}:\bar{x}\in{\rm
bdry}(S_{f})\right\\}>\tau.$ (15)

  * (ii)

There exist constants $c,\varepsilon\in(0,+\infty)$ such that for all
$g\in\Gamma_{0}(\mathbb{R}^{m})$ satisfying

$S_{f}\subseteq S_{g}\ \ {\it and}\ \ {\rm Lip}(f-g)<\varepsilon,$ (16)

one has $\tau_{\min}(g)\leq c$.

  * (iii)

There exist constants $c,\varepsilon\in(0,+\infty)$ such that for all
$g\in\Gamma_{0}(\mathbb{R}^{m})$ satisfying

${\rm bdry}(S_{f})\cap g^{-1}(0)\not=\emptyset\ \ {\it and}\ \ {\rm
Lip}(f-g)<\varepsilon,$ (17)

one has $\tau_{\min}(g)\leq c$.

Then (i) $\Rightarrow$ (ii) and (iii) $\Rightarrow$ (i).

######  Proof 3.3.

(i) $\Rightarrow$ (ii): If there is some $\bar{x}\in{\rm bdry}(S_{f})$ such
that $\min_{\|h\|=1}f^{\prime}(\bar{x},h)>0$, then one can verify that
$S_{f}=\\{\bar{x}\\}$ and the implication follows as in the proof of Section
3.1. We next consider the case $\min_{\|h\|=1}f^{\prime}(\bar{x},h)\leq 0$ for
all $\bar{x}\in{\rm bdry}(S_{f})$. By virtue of (15) and Section 3, one can
verify that $S_{f}$ has a global error bound with the constant
$\frac{1}{\tau}$; that is,

$d(x,S_{f})\leq\frac{1}{\tau}[f(x)]_{+},\ \ \forall x\in\mathbb{R}^{m}.$ (18)

Take any $\varepsilon\in(0,\tau)$. Suppose that
$g\in\Gamma_{0}(\mathbb{R}^{m})$ satisfies (16). Let $x\in\mathbb{R}^{m}$ be
such that $g(x)>0$. Then $f(x)>0$ as $S_{f}\subseteq S_{g}$. We claim that

$\min_{\|h\|=1}f^{\prime}(x,h)\leq-\tau.$ (19)

Granting this, by ${\rm Lip}(f-g)<\varepsilon$ in (16), one can prove that

$\min_{\|h\|=1}g^{\prime}(x,h)\leq\min_{\|h\|=1}f^{\prime}(x,h)+\varepsilon\leq-(\tau-\varepsilon).$

This and Section 3 imply that $\tau_{\min}(g)\leq(\tau-\varepsilon)^{-1}$.

We next prove the claim (19). Take $z\in{\rm bdry}(S_{f})$ such that
$\|x-z\|=d(x,S_{f})$ and (18) implies that

$f(x)\geq\tau d(x,S_{f})=\tau\|x-z\|.$

Then for any $t\in(0,1)$, one has

$f(x+t(z-x))\leq tf(z)+(1-t)f(x)$

and thus

$\frac{f(x+t(z-x))-f(x)}{t}\leq-f(x)\leq-\tau\|x-z\|.$

This means that

$\min_{\|h\|=1}f^{\prime}(\bar{x},h)\leq
f^{\prime}\Big{(}x,\frac{z-x}{\|x-z\|}\Big{)}\leq-\tau$

Hence (19) holds.

(iii) $\Rightarrow$ (i): Suppose that there exists a sequence
$\\{x_{k}\\}\subseteq{\rm bdry}(S_{f})$ such that

$\alpha_{k}:=\min_{\|h\|=1}f^{\prime}(x_{k},h)\rightarrow 0^{-}\ ({\rm as}\
k\rightarrow\infty).$

Let $\varepsilon>0$ be arbitrary and $k$ be sufficiently large such that

$\frac{3}{2}\alpha_{k}+\frac{\varepsilon}{2}>0.$ (20)

Note that for any $x\not=x_{k}$, one has

$\frac{f(x)-f(x_{k})}{\|x-x_{k}\|}=\frac{f\big{(}x_{k}+\|x-x_{k}\|\cdot\frac{x-x_{k}}{\|x-x_{k}\|}\big{)}-f(x_{k})}{\|x-x_{k}\|}\geq
f^{\prime}\Big{(}x_{k},\frac{x-x_{k}}{\|x-x_{k}\|}\Big{)}\geq\alpha_{k}$

and thus

$f(x)-\alpha_{k}\|x-x_{k}\|\geq f(x_{k}),\ \ \forall x\in\mathbb{R}^{m}.$ (21)

Choose $h_{k}\in\mathbb{R}^{m}$ with $\|h_{k}\|=1$ such that

$f^{\prime}(x_{k},h_{k})=\alpha_{k}.$ (22)

Then we can take $r_{k}\rightarrow 0^{+}$ (as $k\rightarrow\infty$) such that

$f(x_{k}+r_{k}h_{k})<f(x_{k})+(\alpha_{k}+\varepsilon)r_{k}.$ (23)

This and (21) imply that

$f(x_{k}+r_{k}h_{k})-\alpha_{k}\|x_{k}+r_{k}h_{k}-x_{k}\|<\inf_{x\in\mathbb{R}^{m}}(f(x)-\alpha_{k}\|x-x_{k}\|)+\varepsilon
r_{k}.$

Applying Ekeland variational principle, we can select $y_{k}\in\mathbb{R}^{m}$
such that

$\|y_{k}-(x_{k}+r_{k}h_{k})\|<\frac{r_{k}}{2},f(y_{k})-\alpha_{k}\|y_{k}-x_{k}\|\leq
f(x_{k}+r_{k}h_{k})-\alpha_{k}r_{k},$ (24)

and

$f(x)-\alpha_{k}\|x-x_{k}\|+2\varepsilon\|x-y_{k}\|>f(y_{k})-\alpha_{k}\|y_{k}-x_{k}\|,\
\ \forall x\not=y_{k}.$ (25)

This implies that

$\|y_{k}-x_{k}\|>r_{k}-\frac{r_{k}}{2}=\frac{r_{k}}{2}\ {\rm and}\
\|y_{k}-x_{k}\|<r_{k}+\frac{r_{k}}{2}=\frac{3}{2}r_{k},$

and thus $y_{k}\not=x_{k}$. Let us consider a function
$g_{\varepsilon}\in\Gamma_{0}(\mathbb{R}^{m})$ defined by

$g_{\varepsilon}(x):=f(x)+\varepsilon\langle h_{k},x-x_{k}\rangle\ \ {\rm for\
all}\ x\in\mathbb{R}^{m}.$

By virtue of (20), (21), (22) and (25), one has

$\displaystyle g_{\varepsilon}(y_{k})=f(y_{k})+\varepsilon\langle
h_{k},y_{k}-x_{k}\rangle$ $\displaystyle=$ $\displaystyle
f(y_{k})+\varepsilon\langle h_{k},y_{k}-(x_{k}+r_{k}h_{k})\rangle+\varepsilon
r_{k}$ $\displaystyle\geq$
$\displaystyle\alpha_{k}\|y_{k}-x_{k}\|-\varepsilon\|y_{k}-(x_{k}+r_{k}h_{k})\|+\varepsilon
r_{k}$ $\displaystyle\geq$
$\displaystyle\alpha_{k}\cdot\frac{3}{2}r_{k}+\frac{\varepsilon}{2}r_{k}>0.$

If $\min_{\|h\|=1}g_{\varepsilon}^{\prime}(y_{k},h)\geq 0$, then for any
$x\not=y_{k}$, one has

$g_{\varepsilon}(x)-g_{\varepsilon}(y_{k})\geq
g_{\varepsilon}^{\prime}\big{(}y_{k},\frac{x-y_{k}}{\|x-y_{k}\|}\big{)}\|x-y_{k}\|\geq\min_{\|h\|=1}g_{\varepsilon}^{\prime}(y_{k},h)\|x-y_{k}\|\geq
0.$

This and $g_{\varepsilon}(y_{k})>0$ imply that
$S_{g_{\varepsilon}}=\emptyset$, and thus
$\tau_{\min}(g_{\varepsilon})=+\infty$, which contradicts (iii).

Next, we consider the case of
$\min_{\|h\|=1}g_{\varepsilon}^{\prime}(y_{k},h)<0$. For any
$h\in\mathbb{R}^{m}$ with $\|h\|=1$ and $t>0$, by (25), one has

$\displaystyle\frac{g_{\varepsilon}(y_{k}+th)-g_{\varepsilon}(y_{k})}{t}$
$\displaystyle=$
$\displaystyle\frac{f(y_{k}+th)-f(y_{k})}{t}+\varepsilon\langle
h_{k},h\rangle$ $\displaystyle\geq$
$\displaystyle\frac{1}{t}\big{(}\alpha_{k}\|y_{k}+th-
x_{k}\|-\alpha_{k}\|y_{k}-x_{k}\|-2\varepsilon\|y_{k}+th-
y_{k}\|\big{)}+\varepsilon\langle h_{k},h\rangle$ $\displaystyle\geq$
$\displaystyle\alpha_{k}-2\varepsilon-\varepsilon$

and consequently

$0>\min_{\|h\|=1}g_{\varepsilon}^{\prime}(y_{k},h)\geq\alpha_{k}-2\varepsilon-\varepsilon\geq-4\varepsilon.$

Thanks to Lemma 1 and Section 3, we obtain
$\tau_{\min}(g_{\varepsilon})\geq\frac{1}{4\varepsilon}$, which contradicts
(iii) as $\varepsilon$ is arbitrary. The proof is complete.

$(a)$ Compared with [42, Theorem 7] in which a subdifferential
characterization of stability of global error bounds was established with the
aid of the so-called asymptotic qualification condition, Section 3.2 studies
the stability of global error bounds via directional derivatives without
additional hypothesis. It is known from Section 3.2 that the condition (15) is
sufficient for the stability of global error bounds as said in (ii) of Section
3.2, and is necessary for the stability as in (iii) of Section 3.2.

$(b)$ It should be noted that the condition (15) is not sufficient for the
stability of global error bounds as in (iii) of Section 3.2, and the
assumption $S_{f}\subseteq S_{g}$ for the stability as said in (ii) of Section
3.2 is crucial. To see this, let us consider the following example:

Let $f(x):=e^{x}-1$ for all $x\in\mathbb{R}$. Then $S_{f}=(-\infty,0]$, ${\rm
bdry}(S_{f})=\\{0\\}$ and $|\min_{|h|=1}f^{\prime}(0,h)|=1>0$. However, for
any $\varepsilon\in(0,+\infty)$, let us consider the function
$g_{\varepsilon}(x):=f(x)-\varepsilon x$ for all $x\in\mathbb{R}$. Then one
can verify that $g_{\varepsilon}$ has two different zero points which are
denoted by $x_{1}:=\bar{x}<0$ and $x_{2}:=0$ and
$S_{g_{\varepsilon}}=[\bar{x},0]$. Thus $S_{f}\not\subseteq
S_{g_{\varepsilon}}$ and for any $x<\bar{x}$, one has

$\frac{d(x,S_{g_{\varepsilon}})}{g_{\varepsilon}(x)}=\frac{\bar{x}-x}{e^{x}-1-\varepsilon
x}\rightarrow\frac{1}{\varepsilon}\ {\rm as}\ x\rightarrow-\infty.$

This implies that

$\tau_{\min}(g_{\varepsilon})\geq\frac{1}{2\varepsilon},$

and consequently the global stability (for $f$) as said in (iii) of Section
3.2 does not hold as $\varepsilon>0$ is arbitrary. Further, a natural question
arises from the above example:

Does there exist some type of stability of global error bounds that can be
characterized by condition (15)?

We do not have an answer to this question. However, if the answer is
affirmative, we conjecture that such global stability should be strictly
stronger than that of (ii) and weaker than that of (iii) in Section 3.2.

The following theorem gives characterizations of the stability of global error
bounds for a convex inequality as said in (iii) of Section 3.2.

Let $f\in\Gamma_{0}(\mathbb{R}^{m})$ be such that ${\rm bdry}(S_{f})\subseteq
f^{-1}(0)$. Then the following statements are equivalent:

  * (i)

There exists $\tau\in(0,+\infty)$ such that (15) holds and the following
qualification condition is satisfied:
$\mathbf{(QC)}$ For any sequence $\\{z_{k}\\}\subseteq S_{f}\backslash{\rm
bdry}(S_{f})$, one has

$\liminf_{k\rightarrow\infty}\Big{|}\min_{\|h\|=1}f^{\prime}(z_{k},h)\Big{|}>\tau$
(26)

if there is a sequence $\\{x_{k}\\}\subseteq{\rm bdry}(S_{f})$ satisfying
$\lim_{k\rightarrow\infty}\frac{f(z_{k})-f(x_{k})}{\|z_{k}-x_{k}\|}=0$.

  * (ii)

There exist constants $c,\varepsilon\in(0,+\infty)$ such that for all
$g\in\Gamma_{0}(\mathbb{R}^{m})$ satisfying (17), one has $\tau_{\min}(g)\leq
c$;

  * (iii)

There exist constants $c,\varepsilon>0$ such that for any $\bar{x}\in{\rm
bdry}(S_{f})$ and $u\in\mathbb{R}^{m}$ with $\|u\|\leq 1$, one has
$\tau_{\min}(g_{u,\varepsilon})\leq c$, where
$g_{u,\varepsilon}(x):=f(x)+\varepsilon\langle u,x-\bar{x}\rangle$ for all
$x\in\mathbb{R}^{m}$.

######  Proof 3.4.

(i) $\Rightarrow$ (ii): If there is some $\bar{x}\in{\rm bdry}(S_{f})$ such
that $\min_{\|h\|=1}f^{\prime}(\bar{x},h)>0$, then one can verify that
$S_{f}=\\{\bar{x}\\}$ and the implication follows as in the proof of Section
3.1. We next consider the case $\min_{\|h\|=1}f^{\prime}(\bar{x},h)\leq 0$ for
all $\bar{x}\in{\rm bdry}(S_{f})$. We first prove the following claim:

Claim: There exists $\varepsilon_{0}>0$ such that for all $x_{0}\in{\rm
bdry}(S_{f})$, one has

$\inf\left\\{\Big{|}\min_{\|h\|=1}f^{\prime}(z_{0},h)\Big{|}:z_{0}\in\mathbb{R}^{m},f(z_{0})\geq-\varepsilon_{0}\|z_{0}-x_{0}\|\right\\}\geq\tau.$
(27)

Suppose on the contrary that there exist $\varepsilon_{k}\rightarrow 0^{+}$,
$x_{k}\in{\rm bdry}(S_{f})$ and $z_{k}\in\mathbb{R}^{m}$ such that

$f(z_{k})\geq-\varepsilon_{k}\|z_{k}-x_{k}\|\ \ {\rm and}\ \
\Big{|}\min_{\|h\|=1}f^{\prime}(z_{k},h)\Big{|}<\tau\ \ {\rm for\ all}\ k.$
(28)

Then $f(z_{k})\leq 0$ for all $k$ (otherwise, similar to the proof of (19),
one can prove that
$\big{|}\min_{\|h\|=1}f^{\prime}(z_{k},h)\big{|}>\tau$, a contradiction). By
(15), one has $z_{k}\in S_{f}\backslash{\rm bdry}(S_{f})$ and it follows from
(28) that

$0\geq\frac{f(z_{k})-f(x_{k})}{\|z_{k}-x_{k}\|}=\frac{f(z_{k})}{\|z_{k}-x_{k}\|}\geq-\varepsilon_{k}.$

This and the qualification condition in (i) imply that

$\liminf_{k\rightarrow\infty}\Big{|}\min_{\|h\|=1}f^{\prime}(z_{k},h)\Big{|}>\tau,$

which contradicts (28). Hence the claim is proved.∎

Let $\varepsilon>0$ be such that $\varepsilon<\min\\{\varepsilon_{0},\tau\\}$.
Suppose that $g\in\Gamma_{0}(\mathbb{R}^{m})$ satisfies (17). Take any
$\bar{x}\in{\rm bdry}(S_{f})\cap g^{-1}(0)$. Then for any $x\in\mathbb{R}^{m}$
with $g(x)>0$, one has

$f(x)\geq
g(x)+(f(\bar{x})-g(\bar{x}))-\varepsilon\|x-\bar{x}\|>-\varepsilon\|x-\bar{x}\|.$

Using (27), one obtains

$\min_{\|h\|=1}f^{\prime}(x,h)<-\tau$

and thus

$\min_{\|h\|=1}g^{\prime}(x,h)<\min_{\|h\|=1}f^{\prime}(x,h)+\varepsilon<-(\tau-\varepsilon).$

By virtue of Lemma 1 and Section 3 we derive the inequality
$\tau_{\min}(g)\leq\frac{1}{\tau-\varepsilon}$. Note that (ii) $\Rightarrow$
(iii) follows immediately and it remains to prove (iii) $\Rightarrow$ (i).
Suppose on the contrary that (i) does not hold. Based on (iii) $\Rightarrow$
(i) in Section 3.2, we only consider the case that there exist $z_{k}\in
S_{f}\backslash{\rm bdry}(S_{f})$ and $x_{k}\in{\rm bdry}(S_{f})$ such that

$\lim_{k\rightarrow\infty}\frac{f(z_{k})-f(x_{k})}{\|z_{k}-x_{k}\|}=0\ \ {\rm
and}\ \ \alpha_{k}:=\min_{\|h\|=1}f^{\prime}(z_{k},h)\rightarrow 0^{-}.$ (29)

Let $\varepsilon>0$ be arbitrary. Without loss of generality, we can assume
that $\frac{z_{k}-x_{k}}{\|z_{k}-x_{k}\|}\rightarrow h_{0}$ (considering
subsequence if necessary). Then $\|h_{0}\|=1$. Suppose that $k$ is
sufficiently large such that

$\alpha_{k}+\varepsilon>0\ \ {\rm and}\ \
\frac{f(z_{k})-f(x_{k})}{\|z_{k}-x_{k}\|}+\varepsilon\Big{\langle}h_{0},\frac{z_{k}-x_{k}}{\|z_{k}-x_{k}\|}\Big{\rangle}>0.$
(30)

Let us consider a function
$g_{h_{0},\varepsilon}\in\Gamma_{0}(\mathbb{R}^{m})$ defined by

$g_{h_{0},\varepsilon}(x):=f(x)+\varepsilon\langle h_{0},x-x_{k}\rangle\ \
{\rm for\ all}\ x\in\mathbb{R}^{m}.$

Then $g_{h_{0},\varepsilon}(z_{k})=f(z_{k})+\varepsilon\langle
h_{0},z_{k}-x_{k}\rangle>0$ by (30) and thus

$0>\min_{\|h\|=1}g_{h_{0},\varepsilon}^{\prime}(z_{k},h)\geq\min_{\|h\|=1}f^{\prime}(z_{k},h)-\varepsilon=\alpha_{k}-\varepsilon>-2\varepsilon.$

This together with Lemma 1 and Section 3 implies that
$\tau_{\min}(g_{h_{0},\varepsilon})\geq\frac{1}{2\varepsilon}$, which
contradicts (iii) as $\varepsilon$ is arbitrary. The proof is complete.

Note that condition $\mathbf{(QC)}$ in (26) is necessary for the stability of
global error bounds. Consider the example given in Remark 8 again. Let
$f(x):=e^{x}-1$ for all $x\in\mathbb{R}$. Then the stability of global error
bounds for $f$ as said in (iii) of Section 3.2 does not hold. Further, for any
$z_{k}\rightarrow-\infty$, one can verify that

$\Big{|}\min_{|h|=1}f^{\prime}(z_{k},h)\Big{|}=e^{z_{k}}\rightarrow 0\ {\rm
as}\ k\rightarrow\infty,$

which means that $\mathbf{(QC)}$ (for $f$) in (26) fails.

##  4 Stability of error bounds for Semi-infinite Convex Constraint Systems

In this section, we study local and global error bounds for semi-infinite
convex constraint systems, and mainly provide characterizations of stability
of error bounds by directional derivatives. We first recall the definition of
error bounds for semi-infinite convex constraint systems.

For semi-infinite convex constraint systems in $\mathbb{R}^{m}$, we mean the
problem of finding $x\in\mathbb{R}^{m}$ satisfying:

$f_{i}(x)\leq 0\ \ {\rm for\ all}\ i\in I,$ (31)

where $I$ is a compact, possibly infinite, Hausdorff space,
$f_{i}:\mathbb{R}^{m}\rightarrow\mathbb{R},i\in I$, are given convex functions
such that $i\mapsto f_{i}(x)$ is continuous on $I$ for each
$x\in\mathbb{R}^{m}$. It is known from [50, Theorem 7.10] that in this case,
$(i,x)\mapsto f_{i}(x)$ is continuous on $I\times\mathbb{R}^{m}$.

Let $F\in C(I\times\mathbb{R}^{m},\mathbb{R})$ be defined by
$F(i,x):=f_{i}(x)$ for all $(i,x)\in I\times\mathbb{R}^{m}$. We denote the
solution set of system (31) by

$S_{F}:=\\{x\in\mathbb{R}^{m}:f_{i}(x)\leq 0\ {\rm for\ all}\ i\in I\\}.$ (32)

For any $\ x\in\mathbb{R}^{m}$, we set

$f(x):=\max\\{f_{i}(x):i\in I\\}\ \ {\rm and}\ \ I_{f}(x):=\\{i\in
I:f_{i}(x)=f(x)\\}.$ (33)

Recall that system (31) is said to have a global error bound if there exists a
constant $\tau\in(0,+\infty)$ such that

$d(x,S_{F})\leq\tau[f(x)]_{+}\ \ \forall x\in\mathbb{R}^{m}.$ (34)

We denote by $\tau_{\min}(F):=\inf\\{\tau>0:\eqref{4.4}\ {\rm holds}\\}$ the
global error bound modulus of $S_{F}$.

For $\bar{x}\in{\rm bdry}(S_{F})$, system (31) is said to have a local error
bound at $\bar{x}$ if there exist constants $\tau,\delta\in(0,+\infty)$ such
that

$d(x,S_{F})\leq\tau[f(x)]_{+}\ \ \forall x\in B(\bar{x},\delta).$ (35)

We denote by $\tau_{\min}(F,\bar{x}):=\inf\\{\tau>0:\ {\rm there\ exists}\
\delta>0\ {\rm such\ that}\ \eqref{4.5}\ {\rm holds}\\}$ the local error bound
modulus of $S_{F}$ at $\bar{x}$.

We first study stability of local error bounds for semi-infinite convex
constraint system (31) and aim to provide characterizations of the stability
of local error bounds for system (31). To this aim, we need the following
proposition which is of independent interest.

######  Proposition 2.

Let $x\in\mathbb{R}^{m}$. Then for any $h\in\mathbb{R}^{m}$, one has

$f^{\prime}(x,h)=\max_{i\in I_{f}(x)}f_{i}^{\prime}(x,h).$ (36)

######  Proof 4.1.

Let $h\in\mathbb{R}^{m}$. Take any $i\in I_{f}(x)$. Then for any $t>0$, one
has

$\frac{f_{i}(x+th)-f_{i}(x)}{t}\leq\frac{f(x+th)-f(x)}{t}$

and thus $f_{i}^{\prime}(x,h)\leq f^{\prime}(x,h)$. This implies that

$f^{\prime}(x,h)\geq\max_{i\in I_{f}(x)}f_{i}^{\prime}(x,h).$ (37)

By virtue of (4), one has

$f^{\prime}(x,h)=\max_{x\in\partial f(x)}\langle x^{*},h\rangle,$

and thus there is $z^{*}\in\partial f(x)$ such that

$f^{\prime}(x,h)=\langle z^{*},h\rangle.$ (38)

Note that the subdifferential of the function $f$ at a point
$x\in\mathbb{R}^{m}$ is given by (see Ioffe & Tikhomirov [27])

$\partial f(x)={\rm co}\Big{(}\bigcup_{i\in I_{f}(x)}\partial f_{i}(x)\Big{)}$

where “${\rm co}$" denotes the convex hull of a set. Then by (38), there exist
$\lambda_{1},\cdots,\lambda_{N}\geq 0$, $i_{1},\cdots,i_{N}\in I_{f}(x)$ and
$z_{k}^{*}\in\partial f_{i_{k}}(x),k=1,\cdots,N$ such that

$\sum_{k=1}^{N}\lambda_{k}=1\ \ {\rm and}\ \
z^{*}=\sum_{k=1}^{N}\lambda_{k}z_{k}^{*}.$

This and (38) imply that

$f^{\prime}(x,h)=\langle z^{*},h\rangle=\sum_{k=1}^{N}\lambda_{k}\langle
z_{k}^{*},h\rangle\leq\sum_{k=1}^{N}\lambda_{k}f^{\prime}_{i_{k}}(x,h)\leq\max_{i\in
I_{f}(x)}f^{\prime}_{i}(x,h).$

Hence (36) follows from (37) and the above inequality. The proof is complete.

The following theorem gives characterizations (by directional derivatives) of
stability of local error bounds for system (31).

Let $\bar{x}\in\mathbb{R}^{m}$ be such that $f(\bar{x})=0$. Then the following
statements are equivalent:

  * (i)

$\min_{\|h\|=1}f^{\prime}(\bar{x},h)\not=0$.

  * (ii)

There exist constants $c,\varepsilon>0$ such that if

$\displaystyle G\in
C(I\times\mathbb{R}^{m},\mathbb{R}),g_{i}(x):=G(i,x),g_{i}\ {\it is\ convex};$
$\displaystyle g(x):=\max_{i\in I}g_{i}(x),I_{g}(x):=\\{i\in
I:g_{i}(x)=g(x)\\};$ $\displaystyle g(\bar{x})=0;$ $\displaystyle
I_{g}(\bar{x})\subseteq I_{f}(\bar{x})\ {\it whenever}\
\min_{\|h\|=1}f^{\prime}(\bar{x},h)<0;$ $\displaystyle I_{f}(\bar{x})\subseteq
I_{g}(\bar{x})\ {\it whenever}\ \min_{\|h\|=1}f^{\prime}(\bar{x},h)>0;$
$\displaystyle\limsup\limits_{x\rightarrow\bar{x}}\frac{|f_{i}(x)-g_{i}(x)-(f_{i}(\bar{x})-g_{i}(\bar{x}))|}{\|x-\bar{x}\|}\leq\varepsilon,\
\forall i\in I_{f}(\bar{x})\cap I_{g}(\bar{x}),$

then one has $\tau_{\min}(G,\bar{x})\leq c$.

  * (iii)

There exist constants $c,\varepsilon>0$ such that for all
$u^{*}\in\mathbb{R}^{m}$ with $\|u^{*}\|\leq 1$, one has
$\tau_{\min}(G,\bar{x})\leq c$, where $G\in
C(I\times\mathbb{R}^{m},\mathbb{R})$ is defined by

$G(i,x):=f_{i}(x)+\varepsilon\langle u^{*},x-\bar{x}\rangle\ \ {\it for\ all}\
(i,x)\in I\times\mathbb{R}^{m}.$ (39)

######  Proof 4.2.

We set

$\beta(f,\bar{x}):=\min_{\|h\|=1}f^{\prime}(\bar{x},h).$

(i) $\Rightarrow$ (ii): Suppose that $\beta(f,\bar{x})>0$. Then one can verify
that $S_{F}=\\{\bar{x}\\}$. Choose any $\varepsilon\in(0,\beta(f,\bar{x}))$.
Suppose that $G,g_{i}$ and $g$ satisfy all conditions said in (ii). Then for
any $i\in I_{f}(\bar{x})\subseteq I_{g}(\bar{x})$, one has

$g_{i}^{\prime}(\bar{x},h)\geq f_{i}(\bar{x},h)-\varepsilon$

and it follows from Proposition 2 that

$\displaystyle\min_{\|h\|=1}g^{\prime}(\bar{x},h)=\min_{\|h\|=1}\max_{i\in
I_{g}(\bar{x})}g_{i}^{\prime}(\bar{x},h)$ $\displaystyle\geq$
$\displaystyle\min_{\|h\|=1}\max_{i\in
I_{f}(\bar{x})}g_{i}^{\prime}(\bar{x},h)$ $\displaystyle\geq$
$\displaystyle\Big{(}\min_{\|h\|=1}\max_{i\in
I_{f}(\bar{x})}f_{i}^{\prime}(\bar{x},h)\Big{)}-\varepsilon$ $\displaystyle=$
$\displaystyle\min_{\|h\|=1}f^{\prime}(\bar{x},h)-\varepsilon$
$\displaystyle=$ $\displaystyle\beta(f,\bar{x})-\varepsilon>0$

(thanks to $I_{f}(\bar{x})\subseteq I_{g}(\bar{x})$). Applying Proposition 3,
we derive the inequality

$\tau_{\min}(G,\bar{x})=\tau_{\min}(g,\bar{x})\leq\frac{1}{\beta(f,\bar{x})-\varepsilon}.$

Suppose that $\beta(f,\bar{x})<0$. Choose any $\varepsilon>0$ such that
$\beta(f,\bar{x})+\varepsilon<0$. Then for any $i\in I_{g}(\bar{x})\subseteq
I_{f}(\bar{x})$, by (4.14), one has

$g_{i}^{\prime}(\bar{x},h)\leq f_{i}(\bar{x},h)+\varepsilon$

and it follows from Proposition 2 that

$\displaystyle\min_{\|h\|=1}g^{\prime}(\bar{x},h)=\min_{\|h\|=1}\max_{i\in
I_{g}(\bar{x})}g_{i}^{\prime}(\bar{x},h)$ $\displaystyle\leq$
$\displaystyle\min_{\|h\|=1}\max_{i\in
I_{g}(\bar{x})}(f_{i}^{\prime}(\bar{x},h)+\varepsilon)$ $\displaystyle\leq$
$\displaystyle\Big{(}\min_{\|h\|=1}\max_{i\in
I_{f}(\bar{x})}f_{i}^{\prime}(\bar{x},h)\Big{)}+\varepsilon$ $\displaystyle=$
$\displaystyle\min_{\|h\|=1}f^{\prime}(\bar{x},h)+\varepsilon$
$\displaystyle=$ $\displaystyle\beta(f,\bar{x})+\varepsilon<0$

(thanks to $I_{g}(\bar{x})\subseteq I_{f}(\bar{x})$). Applying Proposition 3
again, we obtain the inequality

$\tau_{\min}(G,\bar{x})=\tau_{\min}(g,\bar{x})\leq\frac{1}{-\beta(f,\bar{x})-\varepsilon}.$

(ii) $\Rightarrow$ (iii): The implication follows immediately as
$I_{f}(\bar{x})=I_{g}(\bar{x})$.

(iii) $\Rightarrow$ (i): Let $u^{*}\in\mathbb{R}^{m}$ with $\|u^{*}\|\leq 1$
and $G\in C(I\times\mathbb{R}^{m},\mathbb{R})$ be defined as (39). Note that

$g_{i}(x)=G(i,x)=f_{i}(x)+\varepsilon\langle u^{*},x-\bar{x}\rangle$

and thus

$g(x)=\max_{i\in I}g_{i}(x)=\max_{i\in I}(f_{i}(x)+\varepsilon\langle
u^{*},x-\bar{x}\rangle)=f(x)+\varepsilon\langle u^{*},x-\bar{x}\rangle.$

This means that the implication follows from (iii) $\Rightarrow$ (i) as in
Section 3.1. The proof is complete.

(a) Section 4, given in terms of the directional derivatives, can be regarded
as an equivalent version and a supplement of [42, Theorem 4] in which a
subdifferential characterization of stability of local error bounds for system
(31) was established. Further, in contrast with [42, Theorem 4], the stability
of local error bounds for system (31) only requires that all component
functions in system (31) have the same $\varepsilon$-linear perturbation.

(b) It should be observed that the condition $I_{f}(\bar{x})\subseteq
I_{g}(\bar{x})$ or $I_{g}(\bar{x})\subseteq I_{f}(\bar{x})$ in Section 4 is
crucial. To see this, we consider the following two examples:

  * $\triangle$

Let $f_{i}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be defined by
$f_{i}(x):=|x_{i}|,i=1,2$ for all $x=(x_{1},x_{2})\in\mathbb{R}^{2}$,
$\bar{x}=(0,0)$, $F:=(f_{1},f_{2})$ and $f:=\max\\{f_{1},f_{2}\\}$. Then

$I_{f}(\bar{x})=\\{1,2\\}\ \ {\it and}\ \
\min_{\|h\|=1}f^{\prime}(\bar{x},h)=\frac{\sqrt{2}}{2}>0.$

However, for each $\varepsilon>0$, we define functions $g_{1,\varepsilon}$ and
$g_{2,\varepsilon}$ by

$g_{1,\varepsilon}(x):=|x_{1}|+\varepsilon|x_{2}|,g_{2,\varepsilon}(x):=|x_{2}|-\varepsilon,\
\ {\it for\ all}\ x=(x_{1},x_{2})\in\mathbb{R}^{2}.$

We set $G_{\varepsilon}:=(g_{1,\varepsilon},g_{2,\varepsilon})$ and
$g_{\varepsilon}:=\max\\{g_{1,\varepsilon},g_{2,\varepsilon}\\}$. Then one can
verify that $I_{g}(\bar{x})=\\{1\\}$ and thus $I_{f}(\bar{x})\not\subseteq
I_{g}(\bar{x})$. Note that

$S_{G_{\varepsilon}}=\\{\bar{x}\\}\ \ {\it and}\ \ {\rm
Lip}(f_{1}-g_{1,\varepsilon})\leq\varepsilon.$

For any $\delta\in(0,\varepsilon^{-1})$, we set
$z_{\delta}:=(0,\delta)\in\mathbb{R}^{2}$. Then
$d(z_{\delta},S_{G_{\varepsilon}})=\delta$ and
$g_{\varepsilon}(z_{\delta})=\varepsilon\delta$, which implies that
$\tau_{\min}(G_{\varepsilon},\bar{x})\geq\frac{1}{\varepsilon}$.

  * $\triangle\triangle$

Let $f_{1},f_{2}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be defined by
$f_{1}(x):=x_{1}$ and $f_{1}(x):=-x_{1}+|x_{2}|-1$ for all
$x=(x_{1},x_{2})\in\mathbb{R}^{2}$, $\bar{x}=(0,0)$, $F:=(f_{1},f_{2})$ and
$f:=\max\\{f_{1},f_{2}\\}$. Then

$I_{f}(\bar{x})=\\{1\\}\ \ {\it and}\ \
\min_{\|h\|=1}f^{\prime}(\bar{x},h)=\min_{\|h\|=1}f_{1}^{\prime}(\bar{x},h)=-1<0.$

However, for each $\varepsilon>0$, we define functions $g_{1,\varepsilon}$ and
$g_{2,\varepsilon}$ as

$g_{1,\varepsilon}(x):=x_{1}+\varepsilon|x_{2}|,g_{2,\varepsilon}(x):=-x_{1}+\varepsilon|x_{2}|,\
\ {\rm for\ all}\ x=(x_{1},x_{2})\in\mathbb{R}^{2}.$

We set $G_{\varepsilon}:=(g_{1,\varepsilon},g_{2,\varepsilon})$ and
$g_{\varepsilon}:=\max\\{g_{1,\varepsilon},g_{2,\varepsilon}\\}$. Then one can
verify that $I_{g}(\bar{x})=\\{1,2\\}$ and thus $I_{g}(\bar{x})\not\subseteq
I_{f}(\bar{x})$. Note that

$S_{G_{\varepsilon}}=\\{\bar{x}\\}\ \ {\it and}\ \ {\rm
Lip}(f_{1}-g_{1,\varepsilon})\leq\varepsilon.$

For any $\delta\in(0,\varepsilon^{-1})$, set
$z_{\delta}:=(0,\delta)\in\mathbb{R}^{2}$. Then
$d(z_{\delta},S_{G_{\varepsilon}})=\delta$ and
$g_{\varepsilon}(z_{\delta})=\varepsilon\delta$. This means that
$\tau_{\min}(G_{\varepsilon},\bar{x})\geq\varepsilon^{-1}$.

We now turn our attention to the stability of global error bounds for semi-
infinite constraint system (31) and mainly give equivalent criterion for such
stability. Based on Section 3.2, the following theorem establishes equivalent
conditions for the stability of global error bounds for the system (31). The
following statements are equivalent:

  * (i)

There exists $\tau\in(0,+\infty)$ such that

$\inf\left\\{\Big{|}\min_{\|h\|=1}f^{\prime}(\bar{x},h)\Big{|}:\bar{x}\in{\rm
bdry}(S_{f})\right\\}>\tau,$ (40)

and $\mathbf{(QC)}$ as in Section 3.2 is satisfied.

  * (ii)

There exist constants $c,\varepsilon>0$ such that if

$\displaystyle G\in
C(I\times\mathbb{R}^{m},\mathbb{R}),g_{i}(x):=G(i,x),g_{i}\ {\it is\ convex};$
$\displaystyle g(x):=\max_{i\in I}g_{i}(x),I_{g}(x):=\\{i\in
I:g_{i}(x)=g(x)\\};$ $\displaystyle\big{\\{}z\in{\rm
bdry}(S_{f}):f_{i}(z)=g_{i}(z)\ {\it for\ all}\ i\in
I\big{\\}}\not=\emptyset;$ $\displaystyle\sup_{i\in I}{\rm
Lip}(f_{i}-g_{i})<\varepsilon;$ $\displaystyle I_{g}(x)\subseteq I_{f}(x)\
{\it whenever}\ \min_{\|h\|=1}f^{\prime}(x,h)<0;$ $\displaystyle
I_{f}(x)\subseteq I_{g}(x)\ {\it whenever}\ \min_{\|h\|=1}f^{\prime}(x,h)>0,$

then one has $\tau_{\min}(G)\leq c$.

  * (iii)

There exist constants $c,\varepsilon>0$ such that for all $\bar{x}\in{\rm
bdry}(S_{f})$ and $u^{*}\in\mathbb{R}^{m}$ with $\|u^{*}\|\leq 1$, one has
$\tau_{\min}(G)\leq c$, where $G\in C(I\times\mathbb{R}^{m},\mathbb{R})$ is
defined by

$G(i,x):=f_{i}(x)+\varepsilon\langle u^{*},x-\bar{x}\rangle\ \ {\it for\ all}\
(i,x)\in I\times\mathbb{R}^{m}.$ (41)

######  Proof 4.3.

(i) $\Rightarrow$ (ii): If there is some $\bar{x}\in{\rm bdry}(S_{f})$ such
that $\min_{\|h\|=1}f^{\prime}(\bar{x},h)>0$, then one can verify that
$S_{f}=\\{\bar{x}\\}$ and the implication follows as in the proof of Section
4. We next consider the case $\min_{\|h\|=1}f^{\prime}(\bar{x},h)\leq 0$ for
all $\bar{x}\in{\rm bdry}(S_{f})$. Thanks to the claim given in the proof of
Section 3.2, there exists $\varepsilon_{0}>0$ such that for all $x_{0}\in{\rm
bdry}(S_{f})$, one has

$\inf\left\\{\Big{|}\min_{\|h\|=1}f^{\prime}(x,h)\Big{|}:x\in\mathbb{R}^{m},f(x)\geq-\varepsilon_{0}\|x-x_{0}\|\right\\}\geq\tau.$
(42)

Take any $\varepsilon>0$ such that
$\varepsilon<\min\\{\varepsilon_{0},\tau\\}$. Suppose that $G,g_{i}$ and $g$
satisfy all conditions said in (ii). Let $x\in\mathbb{R}^{m}$ be such that
$g(x)>0$. We claim that

$\min_{\|h\|=1}f^{\prime}(x,h)\leq-\tau.$ (43)

Granting this, one has

$\displaystyle\min_{\|h\|=1}g^{\prime}(x,h)=\min_{\|h\|=1}\Big{(}\max_{i\in
I_{g}(x)}g_{i}^{\prime}(x,h)\Big{)}$ $\displaystyle\leq$
$\displaystyle\min_{\|h\|=1}\Big{(}\max_{i\in
I_{g}(x)}f_{i}^{\prime}(x,h)+\varepsilon\Big{)}$ $\displaystyle\leq$
$\displaystyle\min_{\|h\|=1}\Big{(}\max_{i\in
I_{f}(x)}f_{i}^{\prime}(x,h)\Big{)}+\varepsilon$ $\displaystyle=$
$\displaystyle\min_{\|h\|=1}f^{\prime}(x,h)+\varepsilon$ $\displaystyle\leq$
$\displaystyle-(\tau-\varepsilon)$

(thanks to $\sup_{i\in I}{\rm Lip}(f_{i}-g_{i})<\varepsilon$ and
$I_{g}(x)\subseteq I_{f}(x)$). Applying Lemma 1 and Section 3, we derive the
inequality $\tau_{\min}(G)\leq\frac{1}{\tau-\varepsilon}$. It remains to prove
relation (43). For the case that $f(x)>0$, similar to the proof of (19), one
can verify that (43) holds. Thus we only need to consider the case that
$f(x)\leq 0$.

From the condition in (ii), there is $z_{0}\in{\rm bdry}(S_{f})$ such that
$f_{i}(z_{0})=g_{i}(z_{0})$ for all $i\in I$. Then for any $i\in
I_{g}(x)\subseteq I_{f}(x)$, one has

$\displaystyle f_{i}(x)\geq
g_{i}(x)-(f_{i}(z_{0})-g_{i}(z_{0}))-\varepsilon\|x-z_{0}\|=g(x)-\varepsilon\|x-z_{0}\|>-\varepsilon\|x-z_{0}\|$

and thus $f(x)>-\varepsilon_{0}\|x-z_{0}\|$. This and (42) imply that (43)
holds.

(ii) $\Rightarrow$ (iii): The implication follows immediately since
$I_{f}(x)=I_{g}(x)$ for all $x\in\mathbb{R}^{m}$.

(iii) $\Rightarrow$ (i): Let $\bar{x}\in{\rm bdry}(S_{f})$,
$u^{*}\in\mathbb{R}^{m}$ with $\|u^{*}\|\leq 1$ and $G\in
C(I\times\mathbb{R}^{m},\mathbb{R})$ be defined as (41). Note that

$g_{i}(x)=G(i,x)=f_{i}(x)+\varepsilon\langle u^{*},x-\bar{x}\rangle$

and consequently

$g(x)=\max_{i\in I}g_{i}(x)=\max_{i\in I}(f_{i}(x)+\varepsilon\langle
u^{*},x-\bar{x}\rangle)=f(x)+\varepsilon\langle u^{*},x-\bar{x}\rangle.$

Thus, the implication follows from (iii) $\Rightarrow$ (i) as in Section 3.2.
The proof is complete

##  5 Conclusions

This paper is devoted to the study of stability of local and global error
bounds for convex inequality constraint systems including a single convex
inequality and semi-infinite convex constraint systems. The main results
provide characterizations (in terms of directional derivatives) of stability
of local and global error bounds for a single convex inequality (see Section
3.1 and Section 3.2). When these results are applied to error bounds for semi-
infinite convex constraint systems, characterizations of the stability of
local and global error bounds are also established in terms of directional
derivatives (see Section 4 and Section 4). These results show that the
stability of error bounds for convex inequality constraint systems can be
equivalent to solving the convex optimization/minimization problems defined by
directional derivatives over the sphere.

##  6 Acknowledgments

The authors would like to thank the two anonymous referees whose meticulous
reading helped us to improve the presentation.

## References

  * [1] M. Abbasi and M. Théra. Strongly regular points of mappings. Fixed Point Theory Algorithms Sci. Eng., Paper No. 14, 2021.
  * [2] A. Auslender and J. P.Crouzeix. Global regularity theorems. Math. Oper. Res., 13(2): 243-253, 1988.
  * [3] D. Azé. A survey on error bounds for lower semicontinuous functions. In Proceedings of 2003 MODE-SMAI Conference, volume 13 of ESAIM Proc., pages 1¨C17. EDP Sci., Les Ulis, 2003.
  * [4] D. Azé. A unified theory for metric regularity of multifunctions. J. Convex Anal., 13(2): 225-252, 2006.
  * [5] D. Azé and J.-N. Corvellec. On the sensitivity analysis of Hoffman constants for systems of linear inequalities, SIAM J. Optim., 12(4): 913-927, 2002.
  * [6] D. Azé and J. -N. Corvellec. Characterizations of error bounds for lower semicontinuous functions on metric spaces. ESAIM Control Optim. Calc. Var., 10: 409-425, 2004.
  * [7] H. H. Bauschke and J. M. Borwein. On projection algorithms for solving convex feasibility problems. SIAM Rev., 38(3): 367-426, 1996.
  * [8] A. Beck and M. Teboulle. Convergence rate analysis and error bounds for projection algorithms in convex feasibility problems. Optim. Methods Softw., 18(4): 377-394, 2003.
  * [9] E. M. Bednarczuk and A. Y. Kruger,. Error bounds for vector-valued functions: necessary and sufficient conditions. Nonlinear Anal., 75(3): 1124-1140, 2012.
  * [10] J. V. Burke and S. Deng. Weak sharp minima revisited. I. Basic theory. Control Cybernet., 31(3): 439-469, 2002. Well-Posedness in Optimization and Related Topics (Warsaw, 2001).
  * [11] J. V. Burke and S. Deng. Weak sharp minima revisited. II. Application to linear regularity and error bounds. Math. Program., 104, (2-3 Ser. B): 235-261, 2005.
  * [12] M. J. Cánovas, A. Y. Kruger, M. A. López,, J.Parra, M. A. Théra. Calmness modulus of linear semi-infinite programs. SIAM J. Optim. 24(1): 29-48, 2014.
  * [13] P. L. Combettes. Hilbertian convex feasibility problem: convergence of projection methods. Appl. Math. Optim., 35(3): 311-330, 1997.
  * [14] J.-N. Corvellec and V. V. Motreanu. Nonlinear error bounds for lower semicontinuous functions on metric spaces. Math. Program., 114(2 Ser. A): 291-319, 2008.
  * [15] N. D. Cuong and A. Y. Kruger. Error bounds revisited. arXiv e-prints, arXiv 2012.03941 (2020)
  * [16] S. Deng. Perturbation analysis of a condition number for convex inequality systems and global error bounds for analytic systems. Math. Programming, 83(2 Ser. A ): 263-276, 1998.
  * [17] A. L. Dontchev, A.S. Lewis and R.T. Rockafellar. The radius of metric regularity. Trans. Amer. Math. Soc., 355(2): 493-517, 2003.
  * [18] M.J. Fabian, R. Henrion, A.Y. Kruger and J. V. Outrata. Error bounds: necessary and sufficient conditions. Set-Valued Var. Anal., 18(2):121-149, 2010.
  * [19] H. Gfrerer. First order and second order characterizations of metric subregularity and calmness of constraint set mappings. SIAM J. Optim., 21(4): 1439-1474, 2011.
  * [20] O. Güler, Augmented Lagrangian algorithms for linear programming, J. Optim. Theory Appl., 75(3): 445-478, 1992.
  * [21] R. Hesse and D. R. Luke. Nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problems. SIAM J. Optim., 23(4): 2397-2419, 2013.
  * [22] A. J. Hoffman. On approximate solutions of systems of linear inequalities. J. Res. Nat. Bur. Standards, 49: 263-265, 1952.
  * [23] L. R. Huang and K.F. Ng. On first- and second-order conditions for error bounds. SIAM J. Optim., 14(4): 1057-1073, 2004.
  * [24] A. D. Ioffe. Metric regularity-a survey. Part I. Theory. J. Aust. Math. Soc., 101(2): 188-243, 2016.
  * [25] A. D. Ioffe. Metric regularity-a survey. Part II. Applications. J. Aust. Math. Soc., 101(3): 376-417, 2016.
  * [26] A. D. Ioffe. Variational analysis of regular mappings. Springer Monographs in Mathematics. Springer, Cham, 2017. Theory and applications.
  * [27] A. D. Ioffe and V. M. Tikhomirov. Theory of Extremal Problems, vol. 6 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam, 1979. Translated from the Russian by Karol Makowski.
  * [28] A. N. Iusem and A. R. De Pierro, On the convergence properties of Hildreth’s quadratic programming algorithm, Math. Program., 47: 37-51, 1990.
  * [29] A. Jourani. Hoffman¡¯s error bound, local controllability, and sensitivity analysis. SIAM J. Control Optim., 38(3): 947-970, 2000.
  * [30] D. Klatte and W. Li. Asymptotic constraint qualifications and global error bounds for convex inequalities. Math. Program., 84(1 Ser. A ): 137-160, 1999.
  * [31] A. Y. Kruger. Error bounds and Hölder metric subregularity. Set-Valued Var. Anal., 23(4): 705-736, 2015.
  * [32] A. Y. Kruger. Error bounds and metric subregularity. Optimization, 64(1):49¨C79, 2015.
  * [33] A. Y. Kruger, H. V.Ngai and M. Théra. Stability of error bounds for convex constraint systems in Banach spaces. SIAM J. Optim., 20(6): 3280-3296, 2010.
  * [34] A. Y. Kruger, M. A. López, X. Yang and J. Zhu. Hölder error bounds and Hölder calmness with applications to convex semi-infinite optimization. Set-Valued Var. Anal., 27(4): 995-1023, 2019.
  * [35] A. Y. Kruger, M. A. López and M. A. Théra. Perturbation of error bounds, Math. Program. 168(1-2 Ser. B ): 533-554, 2018.
  * [36] A. S. Lewis and J. S. Pang, Error bounds for convex inequality systems, In Generalized Convexity, Generalized Monotonicity: Recent Results (Luming, 1996), volume 27 of Nonconvex Optim. Appl., pages 75-100. Kluwer Acad. Publ., Dordrecht, 1998.
  * [37] D. R. Luke, Nguyen H. Thao and M.K. Tam. Implicit error bounds for Picard iterations on Hilbert spaces. Vietnam J. Math., 46(2): 243-258, 2018.
  * [38] Z.-Q. Luo and P. Tseng. On a global error bound for a class of monotone affine variational inequality problems. Oper. Res. Lett., 11(3): 159-165, 1992.
  * [39] Z.-Q. Luo and P. Tseng. Perturbation analysis of a condition number for linear systems. SIAM J. Matrix Anal. Appl., 15(2), 636-660, 1994.
  * [40] O. L. Mangasarian. A condition number for differentiable convex inequalities. Math. Oper. Res., 10(2): 175-179, 1985.
  * [41] K. F. Ng and X. Y. Zheng. Error bounds for lower semicontinuous functions in normed spaces. SIAM J. Optim., 12(1): 1-17, 2001.
  * [42] H. V. Ngai, A. Y. Kruger and M. Théra. Stability of error bounds for semi-infinite convex constraint systems. SIAM J. Optim, 20: 2080-2096, 2010.
  * [43] H. V. Ngai, M. Théra, Error bounds for systems of lower semicontinuous functions in Asplund spaces. Math. Program., Ser. B, 116(1-2): 397-427, 2009.
  * [44] J. S. Pang. Error bounds in mathematical programming. Math. Program., Ser. B, 79(1-3): 299-332, 1997.
  * [45] J.-P. Penot. Calculus without derivatives, volume 266 of Graduate Texts in Mathematics. Springer, New York, 2013.
  * [46] R. R. Phelps. Convex functions, Monotone Operators and Differentiability, volume 1364 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, second edition, 1993.
  * [47] S. M. Robinson. Bounds for error in the solution set of a perturbed linear program. Linear Algebra Appl., 6: 69-81, 1973.
  * [48] S. M. Robinson. An application of error bound for convex programming in a linear space SIAM. Control. Optim. 13: 271-273, 1975.
  * [49] S. M. Robinson. A characterization of stability in linear programming, Oper. Res., 25: 435-447, 1977.
  * [50] R. T. Rockafellar. Convex Analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, N.J., 1970.
  * [51] P. Tseng and D. P. Bertsekas. On the convergence of the exponential multiplier method for convex programming. Math. Program., 60(1 Ser. A): 1-19, 1993.
  * [52] Z. Wu and J. J. Ye. On error bounds for lower semicontinuous functions. Math. Program., 92(2, Ser. A): 301-314, 2002.
  * [53] J.-C. Yao and X. Y. Zheng. Error bound and well-posedness with respect to an admissible function. Appl. Anal., 95(5): 1070-1087, 2016.
  * [54] X. Y. Zheng and K. F. Ng. Perturbation analysis of error bounds for systems of conic linear inequalities in Banach spaces. SIAM J. Optim. 15(4): 1026-1041, 2005.
  * [55] X. Y. Zheng and K. F. Ng. Metric subregularity and calmness for nonconvex generalized equations in Banach spaces. SIAM J. Optim., 20(5): 2119-2136, 2010.
  * [56] X. Y. Zheng and K. F. Ng. Metric subregularity for proximal generalized equations in Hilbert spaces. Nonlinear Anal. 75(3): 1686-1699, 2012.
  * [57] X. Y. Zheng and Z. Wei. Perturbation analysis of error bounds for quasi-subsmooth inequalities and semi-infinite constraint systems. SIAM J. Optim., 22(1): 41-65, 2012.