∎

11institutetext: Zhou Wei22institutetext: Hebei Key Laboratory of Machine
Learning and Computational Intelligence & College of Mathematics and
Information Science, Hebei University, Baoding, 071002, China
22email<EMAIL_ADDRESS>
Michel Théra 33institutetext: ORCID 0000-0001-9022-6406 44institutetext: XLIM
UMR-CNRS 7252, Université de Limoges, Limoges, France
and 55institutetext: Federation University Australia, Ballarat
55email<EMAIL_ADDRESS>
Jen-Chih Yao 66institutetext: Research Center for Interneural Computing, China
Medical University Hospital, China Medical University, Taichung, Taiwan
66email<EMAIL_ADDRESS>
# Primal Characterizations of Stability of Error Bounds for Semi-infinite
Convex Constraint Systems in Banach Spaces ††thanks: Research of the first
author was supported by the National Natural Science Foundations of China
(Grant Nos. 11971422 and 12171419), and funded by Science and Technology
Project of Hebei Education Department (No. ZD2022037) and the Natural Science
Foundation of Hebei Province (A2022201002). Research of the second author
benefited from the support of the FMJH Program PGMO and from the support of
EDF.

Zhou Wei  Michel Théra  Jen-Chih Yao

(Received: date / Accepted: date)

###### Abstract

This article is devoted to the stability of error bounds (local and global)
for semi-infinite convex constraint systems in Banach spaces. We provide
primal characterizations of the stability of local and global error bounds
when systems are subject to small perturbations. These characterizations are
given in terms of the directional derivatives of the functions that enter into
the definition of these systems. It is proved that the stability of error
bounds is essentially equivalent to verifying that the optimal values of
several minimax problems, defined in terms of the directional derivatives of
the functions defining these systems, are outside of some neighborhood of
zero. Moreover, such stability only requires that all component functions
entering the system have the same linear perturbation. When these stability
results are applied to the sensitivity analysis of Hoffman’s constants for
semi-infinite linear systems, primal criteria for Hoffman’s constants to be
uniformly bounded under perturbations of the problem data are obtained.

###### Keywords:

Stability Error bound Semi-infinite convex constraint Directional derivative
Hoffman’s constant.

###### MSC:

90C3190C2549J5246B20

##  1 Introduction

This paper is devoted to the study of the stability of error bounds for semi-
infinite convex systems under data perturbations in a Banach space
$\mathbb{X}$. By a semi-infinite convex system, we mean the problem of finding
$x\in\mathbb{X}$ satisfying

$f_{t}(x)\leq 0\ \ {\rm for\ all}\ \ t\in T,$ (1.1)

where $T$ is a compact, possibly infinite Hausdorff space,
$f_{t}:\mathbb{X}\rightarrow\mathbb{R},t\in T$ are convex functions such that
$(t,x)\mapsto f_{t}(x)$ is continuous on $T\times\mathbb{X}$. The aim of this
work is to investigate the behavior of error bounds of system (1.1) under
perturbations of its data.

Research on error bounds can be traced back to Hoffman’s pioneering work in 28
. More precisely, given a $m\times n$ matrix $A$ and a $m$-vector $b$, in
order to estimate the Euclidean distance from a point $x$ to its nearest point
in the polyhedral set $\\{u:Au\leq b\\}$, Hoffman 28  showed that this
distance is bounded from above by some constant (only dependent on $A$) times
the Euclidean norm of the residual error $\|(Ax-b)_{+}\|$, where $(Ax-b)_{+}$
denotes the positive part of $Ax-b$. Hoffman’s work has been well-recognized
and extensively studied and extended by many authors, including Robinson 54 ,
Mangasarian 42 , Auslender & Crouzeix 1 , Pang 52 , Lewis & Pang 41 , Klatte &
Li 34 , Jourani jourani  , Abassi & Théra Abassi-Thera1 ; Abassi-Thera , Ioffe
ioffe-book  and many others. The abundant literature on error bounds shows
that this theory has important applications in the sensitivity analysis of
linear/integer programs (cf. Rob73 ; Rob77 ), the convergence analysis of
descent methods for linearly constrained minimization (cf. Gul92 ; HLu ; IuD90
; TsL92 ; TsB93 ), the feasibility problem of finding a point in the
intersection of finitely many closed convex sets (cf. 5 ; 6 ; 7 ; BurDeng02 )
and the domain of image reconstruction (cf. 16 ). The reader is referred to
bibliographies Aze03 ; Aze06 ; 3 ; 7 ; CKLT ; 18 ; DL ; 22 ; ioffe-JAMS-1 ;
ioffe-JAMS-2 ; Kru15 ; Luke ; 44 ; 48 ; penot-book ; 55 ; ZTY-JCA ; YZ  and
references therein for more details on theoretical research and practical
applications.

Due to inaccurate data originated from real-world problems, it is necessary to
study the behavior of error bounds when perturbing the data of the problem,
and from viewpoints of practical and theoretical interest, the stability of
error bounds under perturbations of the data is expected. In 1994, Luo and
Tseng LT  studied the perturbation analysis of a condition number for linear
system. In 1998, Deng D  further discussed the perturbation analysis of a
condition number for convex inequality systems. In 2002, Azé and Corvellec 2
considered the sensitivity analysis of Hoffman’s constants to semi-infinite
linear constraint systems. In 2010, Ngai, Kruger and Théra 45  studied the
stability of error bounds for semi-infinite convex constraint systems in a
Euclidean space and established subdifferential characterizations for such
stability. Further, Kruger, Ngai & Théra 38  considered the stability of error
bounds for semi-infinite convex constraint systems in a Banach space and gave
its dual characterizations with respect to perturbations of functions entering
in the system. In 2012, by relaxing the convexity assumption, Zheng and Wei ZW
discussed the stability of error bounds for a semi-infinite constraint system
defined by quasi-subsmooth (not necessarily convex) inequalities and provided
Clarke subdifferential characterizations of the stability. In 2018, Kruger,
Lopéz and Théra MP2018  further developed the subdifferential
characterizations of the stability of error bounds given in 38 ; 45 .
Recently, in terms of directional derivatives of convex functions, the authors
WTY  proved characterizations of the stability of error bounds for convex
inequalities constraint systems in a Euclidean space. In this paper, we
continue the study given in WTY  on the stability of error bounds for semi-
infinite convex constraint systems in a Banach space and with an emphasis to
primal characterizations of this stability when perturbing the data of the
problem under consideration. It is proved that to verify the stability of
error bounds for such systems, to some degree, can be equivalent to solving a
class of minimax optimization problems determined by directional derivatives
of the component functions entering in the system. When these stability
results are applied to the sensitivity analysis of Hoffman’s constants for
semi-infinite linear systems, as a result we obtain primal criteria for the
Hoffman’s constants to be uniformly bounded under perturbations on the problem
data.

The paper is organized as follows. In Section 2, we give some definitions and
preliminary results used in the paper. Section 3 is devoted to main results on
stability of local and global error bounds for semi-infinite convex constraint
systems in a Banach space. In terms of directional derivatives of component
functions, we provide primal characterizations of the stability of error
bounds under small perturbations (see Theorems 3.1 and 3.2). In Section 4, we
apply these stability results to the sensitivity analysis of the Hoffman’s
constant for semi-infinite linear systems, and provide primal criteria for
Hoffman’s constants to be uniformly bounded under perturbations (see Theorem
4.1). Concluding remarks and perspectives of this paper are given in Section
5.

##  2 Preliminaries

Let $\mathbb{X}$ be a Banach space with topological dual $\mathbb{X}^{*}$. We
denote by $\mathbb{B}_{\mathbb{X}}$ the closed unit ball of $\mathbb{X}$ and
for any $x\in\mathbb{X}$ and $\delta>0$, by $\mathbf{B}(x,\delta)$ the open
ball with center $x$ and radius $\delta$.

Let $D$ be a subset of $\mathbb{X}$. We denote by ${\rm int}(D)$ and ${\rm
bdry}(D)$ the interior and the boundary of $D$, respectively. The distance
from a point $x\in\mathbb{X}$ to the set $D$ is given by

$\mathbf{d}(x,D):=\inf_{u\in D}\|x-u\|,$

and we use the convention $\mathbf{d}(x,D)=+\infty$ whenever $D=\emptyset$.

Let $\varphi:\mathbb{X}\rightarrow\mathbb{R}\cup\\{+\infty\\}$ be a proper
lower semicontinuous and convex function and $\bar{x}\in{\rm
dom}(\varphi):=\\{x\in\mathbb{X}:\varphi(x)<+\infty\\}$. For any
$h\in\mathbb{X}$, we denote by $d^{+}\varphi(\bar{x},h)$ the directional
derivative of $\varphi$ at $\bar{x}$ along the direction $h$, which is defined
by

$d^{+}\varphi(\bar{x},h):=\lim\limits_{t\rightarrow
0^{+}}\frac{\varphi(\bar{x}+th)-\varphi(\bar{x})}{t}.$ (2.1)

According for example to P  we know that

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

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

$d^{+}\varphi(\bar{x},h)=\inf_{t>0}\frac{\varphi(\bar{x}+th)-\varphi(\bar{x})}{t}.$
(2.2)

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

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

It is known from (P, , Proposition 2.24) that if $\varphi$ is continuous at
$\bar{x}$ then $\partial\varphi(\bar{x})\not=\emptyset$,

$\partial\varphi(\bar{x})=\\{x^{*}\in\mathbb{X}^{*}:\langle x^{*},h\rangle\leq
d^{+}\varphi(\bar{x},h)\ \ {\rm for\ all}\ h\in\mathbb{X}\\},$ (2.3)

and

$d^{+}\varphi(\bar{x},h)=\sup\\{\langle
x^{*},h\rangle:x^{*}\in\partial\varphi(\bar{x})\\}.$ (2.4)

Given a mapping $\phi:\mathbb{X}\rightarrow\mathbb{Y}$ from $\mathbb{X}$ to a
normed linear space  $\mathbb{Y}$, we denote by

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

the Lipschitz constant of $\phi$.

##  3 Stability of Error Bounds for Semi-infinite Convex Constraint Systems

This section is devoted to the stability of local and global error bounds for
semi-infinite convex constraint systems and aims to establish primal
characterizations in terms of directional derivatives. We first recall some
notations and definitions.

Let $T$ be a compact, possibly infinite Hausdorff space, and consider the
space $\mathcal{C}(T\times\mathbb{X},\mathbb{R})$ of continuous functions on
$T\times\mathbb{X}$. Let $F\in\mathcal{C}(T\times\mathbb{X},\mathbb{R})$ be
such that $F(t,\cdot)$ is convex for each $t\in T$. By a semi-infinite convex
constraint system in $\mathbb{X}$, we mean the problem of finding
$x\in\mathbb{X}$ satisfying:

$f_{t}(x)\leq 0\ \ {\rm for\ all}\ t\in T,$ (3.1)

where $f_{t}:=F(t,\cdot)$ for all $t\in T$. Let $S_{F}$ denote the set of
solutions to (3.1); that is,

$S_{F}:=\\{x\in X:f_{t}(x)\leq 0\ \ {\rm for\ all}\ t\in T\\}.$ (3.2)

Set

$f(x):=\max\\{f_{t}(x):t\in T\\}\ \ {\rm and}\ \ T_{f}(x):=\\{t\in
T:f_{t}(x)=f(x)\\},\ \ {\rm for\ each}\ x\in\mathbb{X}.$ (3.3)

Recall that system (3.1) is said to have a global error bound, if there exists
a real $c\in(0,+\infty)$ such that

$c\mathbf{d}(x,S_{F})\leq[f(x)]_{+},\ \ \forall x\in\mathbb{X},$ (3.4)

where $[f(x)]_{+}:=\max\\{f(x),0\\}$. We denote by

${\rm Er}F:=\inf_{x\not\in S_{F}}\frac{f(x)}{\mathbf{d}(x,S_{F})}$

the global error bound modulus of (3.1).

Let $\bar{x}\in{\rm bdry}(S_{F})$. System (3.1) is said to have a local error
bound at $\bar{x}$, if there exist $c,\delta\in(0,+\infty)$ such that

$c\mathbf{d}(x,S_{F})\leq[f(x)]_{+},\ \ \forall
x\in\mathbf{B}(\bar{x},\delta).$ (3.5)

We denote by

${\rm Er}F(\bar{x}):=\liminf_{x\rightarrow\bar{x},x\not\in
S_{F}}\frac{f(x)}{\mathbf{d}(x,S_{F})}$

the local error bound modulus of (3.1) at $\bar{x}$.

Now, we are in a position to present main work on stability of local and
global error bounds for the semi-infinite convex constraint system in a Banach
space. The following theorem gives primal characterizations on the stability
of local error bounds for the semi-infinite convex constraint system (3.1).

###### Theorem 3.1

Let $F\in\mathcal{C}(T\times\mathbb{X},\mathbb{R})$ be such that
$f_{t}:=F(t,\cdot)$ is convex for all $t\in T$, $\bar{x}\in{\rm bdry}(S_{F})$
and $f,T_{f}(x)$ be given as in (3.3). Then the following statements are
equivalent:

  * (i)

The optimal value of minimax problem $\inf\limits_{\|h\|=1}\sup\limits_{t\in
T_{f}(\bar{x})}d^{+}f_{t}(\bar{x},h)$ is non-zero.

  * (ii)

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

$\displaystyle
G\in\mathcal{C}(T\times\mathbb{X},\mathbb{R}),g_{t}(x):=G(t,x),g_{t}\ {\it is\
convex};$ $\displaystyle g(x):=\sup\limits_{t\in T}g_{t}(x),T_{g}(x):=\\{t\in
T:g_{t}(x)=g(x)\\};$ $\displaystyle g(\bar{x})=0;$ $\displaystyle
T_{g}(\bar{x})\subseteq T_{f}(\bar{x})\ {\it whenever}\
\inf\limits_{\|h\|=1}\sup\limits_{t\in
T_{f}(\bar{x})}d^{+}f_{t}(\bar{x},h)<0;$ $\displaystyle
T_{f}(\bar{x})\subseteq T_{g}(\bar{x})\ {\it whenever}\
\inf\limits_{\|h\|=1}\sup\limits_{t\in
T_{f}(\bar{x})}d^{+}f_{t}(\bar{x},h)>0;$
$\displaystyle\limsup\limits_{x\rightarrow\bar{x}}\displaystyle\frac{|f_{t}(x)-g_{t}(x)-(f_{t}(\bar{x})-g_{t}(\bar{x}))|}{\|x-\bar{x}\|}\leq\varepsilon,\
\forall t\in T_{f}(\bar{x})\cap T_{g}(\bar{x}),$

one has ${\rm Er}G(\bar{x})\geq c$.

  * (iii)

There exist $c,\varepsilon>0$ such that for all
$u^{*}\in\mathbb{B}_{\mathbb{X}^{*}}$, one has ${\rm Er}G_{u^{*}}(\bar{x})\geq
c$, where $G_{u^{*}}\in\mathcal{C}(T\times\mathbb{X},\mathbb{R})$ is defined
as follows:

$G_{u^{*}}(t,x):=f_{t}(x)+\varepsilon\langle u^{*},x-\bar{x}\rangle,\ \ {\it
for\ all}\ (t,x)\in T\times\mathbb{X}.$ (3.6)

The following theorem provides primal characterizations of the stability of
the global error bounds for semi-infinite convex constraint system (3.1).

###### Theorem 3.2

Let $F\in\mathcal{C}(T\times\mathbb{X},\mathbb{R})$ be such that
$f_{t}:=F(t,\cdot)$ is convex for all $t\in T$ and $f,T_{f}(x)$ be given as in
(3.3). Then the following statements are equivalent:

  * (i)

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

$\inf\left\\{\Big{|}\inf_{\|h\|=1}\sup_{t\in
T_{f}(\bar{x})}d^{+}f_{t}(\bar{x},h)\Big{|}:\bar{x}\in{\rm
bdry}(S_{F})\right\\}\geq\tau,$ (3.7)

and

$\liminf_{k\rightarrow\infty}\Big{|}\inf_{\|h\|=1}\sup_{t\in
T_{f}(z_{k})}d^{+}f_{t}(z_{k},h)\Big{|}\geq\tau$ (3.8)

holds for all $\\{(z_{k},x_{k})\\}\subseteq{\rm int}(S_{F})\times{\rm
bdry}(S_{F})$ with
$\lim\limits_{k\rightarrow\infty}\frac{f(z_{k})-f(x_{k})}{\|z_{k}-x_{k}\|}=0$.

  * (ii)

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

$\displaystyle
G\in\mathcal{C}(T\times\mathbb{X},\mathbb{R}),g_{t}(x):=G(t,x),g_{t}\ {\it is\
convex};$ $\displaystyle g(x):=\sup\limits_{t\in T}g_{t}(x),T_{g}(x):=\\{t\in
T:g_{t}(x)=g(x)\\};$ $\displaystyle\big{\\{}z\in{\rm
bdry}(S_{F}):f_{t}(z)=g_{t}(z)\ \ {\it for\ all}\ t\in
T\big{\\}}\not=\emptyset;$ $\displaystyle\sup\limits_{t\in T}{\rm
Lip}(f_{t}-g_{t})<\varepsilon;$ $\displaystyle T_{g}(x)\subseteq T_{f}(x)\ \
{\it whenever}\ \ \inf\limits_{\|h\|=1}\sup\limits_{t\in
T_{f}(x)}d^{+}f_{t}(x,h)<0;$ $\displaystyle T_{f}(x)\subseteq T_{g}(x)\ \ {\it
whenever}\ \ \inf\limits_{\|h\|=1}\sup\limits_{t\in
T_{f}(x)}d^{+}f_{t}(x,h)>0,$

one has ${\rm Er}G\geq c$.

  * (iii)

There exist $c,\varepsilon>0$ such that for all $\bar{x}\in{\rm bdry}(S_{F})$
and $u^{*}\in\mathbb{B}_{\mathbb{X}^{*}}$, one has ${\rm
Er}G_{u^{*},\bar{x}}\geq c$, where
$G_{u^{*},\bar{x}}\in\mathcal{C}(T\times\mathbb{X},\mathbb{R})$ is defined as
follows:

$G_{u^{*},\bar{x}}(t,x):=f_{t}(x)+\varepsilon\langle u^{*},x-\bar{x}\rangle,\
\ {\it for\ all}\ (t,x)\in T\times\mathbb{X}.$ (3.9)

Next, our main attention will be drawn to the complete proofs of Theorems 3.1
and 3.2. It is easy to verify that error bounds of system (3.1) is equivalent
to error bounds for $f$ given as in (3.3). Thus to prove Theorems 3.1 and 3.2,
it is necessary to study the stability of error bounds for the continuous
function $f$. Recall that the error bound property for $f$ is defined by the
inequality:

$c\mathbf{d}(x,S_{f})\leq[f(x)]_{+}$ (3.10)

where $S_{f}:=\\{x\in\mathbb{X}:f(x)\leq 0\\}$ denotes the lower level set of
$f$. If inequality (3.10) holds for some $c>0$ and all $x\in\mathbb{X}$, then
$f$ is said to have a global error bound. In this case, we denote by

${\rm Er}f:=\inf_{f(x)>0}\frac{f(x)}{\mathbf{d}(x,S_{f})}$ (3.11)

the global error bound modulus of $f$. Given $\bar{x}\in X$ with
$f(\bar{x})=0$, if inequality (3.10) holds for some $c>0$ at all $x$ near to
$\bar{x}$, then $f$ is said to have a local error bound at $\bar{x}$. In this
case, we denote

${\rm
Er}f(\bar{x}):=\liminf_{x\rightarrow\bar{x},f(x)>0}\frac{f(x)}{\mathbf{d}(x,S_{f})}$
(3.12)

the local error bound modulus of $f$ at $\bar{x}$.

###  3.1 Stability of Error Bounds for $f$

We first prove the following proposition on primal characterizations of error
bounds for the continuous convex function $f$ given as in (3.3). This is a key
tool to prove main results in this paper.

###### Proposition 1

Let $f$ be given in (3.3) such that the lower level set $S_{f}$ is nonempty.

  * (i)

$f$ has a global error bound if and only if

$\beta(f):=\inf_{f(x)>0}\left(-\inf_{\|h\|=1}d^{+}f(x,h)\right)>0.$

Moreover, ${\rm Er}f=\beta(f)$.

  * (ii)

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

$\beta(f,\bar{x}):=\liminf_{x\rightarrow\bar{x},f(x)>0}\left(-\inf_{\|h\|=1}d^{+}f(x,h)\right)>0.$

Moreover, ${\rm Er}f(\bar{x})=\beta(f,\bar{x})$.

Proof. We first prove that

$-\inf_{\|h\|=1}d^{+}f(x,h)=\mathbf{d}(0,\partial f(x)),\ \ \forall x:f(x)>0.$
(3.13)

Let $x\in\mathbb{X}$ be such that $f(x)>0$. Then $\partial f(x)\not=\emptyset$
and $0\not\in\partial f(x)$ (thanks to $f(x)>0$ and $S_{f}\not=\emptyset$). We
denote $r:=\mathbf{d}(0,\partial f(x))>0$. Take any $\varepsilon>0$
sufficiently small in order to have
$(r-\varepsilon)\mathbb{B}_{\mathbb{X}^{*}}\cap\partial f(x)=\emptyset$.
Noting that $\mathbb{B}_{\mathbb{X}^{*}}$ and $\partial f(x)$ are
weak∗-compact, it follows from the separation theorem in (Rudin, , Theorem
3.4) that there exist $h_{0}\in\mathbb{X}$ with $\|h_{0}\|=1$ and
$\gamma_{1}\in\mathbb{R}$ such that

$\sup\\{\langle x^{*},h_{0}\rangle:x^{*}\in\partial
f(x)\\}<\gamma_{1}<\inf\\{\langle
x^{*},h_{0}\rangle:x^{*}\in(r-\varepsilon)\mathbb{B}_{\mathbb{X}^{*}})\\}=-(r-\varepsilon).$

This and (2.4) imply that

$\inf_{\|h\|=1}d^{+}f(x,h)\leq d^{+}f(x,h_{0})<-(r-\varepsilon)\rightarrow-r\
\ (\rm as\ \varepsilon\rightarrow 0^{+}).$ (3.14)

On the other hand, let $\varepsilon>0$. We can select
$u^{*}_{\varepsilon}\in(r+\varepsilon)\mathbb{B}_{\mathbb{X}^{*}}\cap\partial
f(x)$. Then for any $h\in\mathbb{X},\|h\|=1$, one has

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

and consequently

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

Letting $\varepsilon\rightarrow 0^{+}$, one has (3.13) holds by (3.14).

By virtue of (38, , Theorem 1 and Theorem 22), the following equalities on
error bound moduli for $f$ hold:

${\rm Er}f=\inf_{f(x)>0}\mathbf{d}(0,\partial f(x))\ \ {\rm and}\ \ {\rm
Er}f(\bar{x})=\liminf_{x\rightarrow\bar{x},f(x)>0}\mathbf{d}(0,\partial
f(x)).$ (3.15)

This and (3.13) imply that the conclusions in (i) and (ii) follow. The proof
is complete.∎

The following proposition is to present primal criteria of the local error
bound for the function $f$.

###### Proposition 2

Let $f$ be given as in (3.3), $\bar{x}\in{\rm bdry}(S_{F})$ and
$\inf_{\|h\|=1}d^{+}f(\bar{x},h)\not=0$.

  * (i)

If $\inf_{\|h\|=1}d^{+}f(\bar{x},h)>0$, then $S_{F}=\\{\bar{x}\\}$ and ${\rm
Er}f(\bar{x})\geq\inf_{\|h\|=1}d^{+}f(\bar{x},h)$.

  * (ii)

If $\inf_{\|h\|=1}d^{+}f(\bar{x},h)<0$, then ${\rm
Er}f(\bar{x})\geq-\inf_{\|h\|=1}d^{+}f(\bar{x},h)$.

Proof. (i) It is easy to verify that $S_{F}=S_{f}$ and $f(\bar{x})=0$. Let
$x\in\mathbb{X}\backslash\\{\bar{x}\\}$. By (2.2), one has

$\displaystyle f(x)-f(\bar{x})$ $\displaystyle=$ $\displaystyle
f\Big{(}\bar{x}+\|x-\bar{x}\|\frac{x-\bar{x}}{\|x-\bar{x}\|}\Big{)}-f(\bar{x})$
$\displaystyle\geq$ $\displaystyle
d^{+}f\Big{(}\bar{x},\frac{x-\bar{x}}{\|x-\bar{x}\|}\Big{)}\|x-\bar{x}\|>0.$

This implies that $S_{f}=\\{\bar{x}\\}$ and ${\rm
Er}f(\bar{x})\geq\inf_{\|h\|=1}d^{+}f(\bar{x},h)$.

(ii) By virtue of (3.13) and (3.15), one has

${\rm Er}f(\bar{x})=\liminf_{x\rightarrow\bar{x},f(x)>0}\mathbf{d}(0,\partial
f(x))\geq\mathbf{d}(0,\partial f(\bar{x}))=-\inf_{\|h\|=1}d^{+}f(\bar{x},h).$

Thus the conclusion holds. The proof is complete. $\Box$

###### Remark 1

The inequalities in Proposition 2 may hold strictly. For example, let
$\mathbb{X}:=\mathbb{R}$ and $F(t,x)\equiv 0$ for all $(t,x)\in
T\times\mathbb{R}$. Then $f$ has the local error bound at any
$\bar{x}\in\mathbb{R}$ and ${\rm Er}f(\bar{x})=+\infty$. However, one can
verify that $\inf_{\|h\|=1}d^{+}f(\bar{x},h)=0$, and thus ${\rm
Er}f(\bar{x})>\inf_{\|h\|=1}d^{+}f(\bar{x},h)$.

The following theorem shows that the condition

$\inf_{\|h\|=1}d^{+}f(\bar{x},h)\not=0$

is sufficient and necessary for the stability of local error bounds for $f$ at
$\bar{x}$.

###### Theorem 3.3

Let $f$ be as (3.3) and $\bar{x}\in{\rm bdry}(S_{F})$. Then the following
statements are equivalent:

  * (i)

$\inf_{\|h\|=1}d^{+}f(\bar{x},h)\not=0$.

  * (ii)

There exist $c,\varepsilon>0$ such that for all
$g\in\mathcal{C}(\mathbb{X},\mathbb{R})$ satisfying $\bar{x}\in S_{g}$ and

$\limsup_{x\rightarrow\bar{x}}\frac{|(f(x)-g(x))-(f(\bar{x})-g(\bar{x}))|}{\|x-\bar{x}\|}\leq\varepsilon,$
(3.16)

one has ${\rm Er}g(\bar{x})\geq c$.

  * (iii)

There exist $c,\varepsilon>0$ such that for all
$u^{*}\in\mathbb{B}_{\mathbb{X}^{*}}$, one has ${\rm
Er}g_{u^{*},\varepsilon}(\bar{x})\geq c$, where
$g_{u^{*},\varepsilon}(\cdot):=f(\cdot)+\varepsilon\langle
u^{*},\cdot-\bar{x}\rangle$.

Proof. We denote

$\gamma:=\inf_{\|h\|=1}d^{+}f(\bar{x},h).$

(i) $\Rightarrow$ (ii): Let $\varepsilon>0$ be such that
$\varepsilon<|\gamma|$ and $c:=|\gamma|-\varepsilon$. Take any
$g\in\mathcal{C}(\mathbb{X},\mathbb{R})$ satisfying $\bar{x}\in S_{g}$ and
(3.16). If $\gamma>0$, then for any $h\in\mathbb{X}$ with $\|h\|=1$, one has

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

and consequently

$\inf_{\|h\|=1}d^{+}g(\bar{x},h)\geq\inf_{\|h\|=1}d^{+}f(\bar{x},h)-\varepsilon=\gamma-\varepsilon.$

Using Proposition 2, one has ${\rm Er}g(\bar{x})\geq\gamma-\varepsilon$.

If $\gamma<0$, then for any $h\in\mathbb{X}$, one has

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

and thus

$\inf_{\|h\|=1}d^{+}g(\bar{x},h)\leq\inf_{\|h\|=1}d^{+}f(\bar{x},h)+\varepsilon=\gamma+\varepsilon.$

Using Proposition 2 again, one can get ${\rm
Er}g(\bar{x})\geq-\gamma-\varepsilon=c$. Hence (ii) holds.

Note that (ii) $\Rightarrow$ (iii) follows immediately and we next prove (iii)
$\Rightarrow$ (i).

Let $\varepsilon>0$. Suppose on the contrary that $\gamma=0$. Then there is a
sequence $\\{h_{k}\\}\subseteq\mathbb{X}$ such that $\|h_{k}\|=1$ and

$\alpha_{k}:=d^{+}f(\bar{x},h_{k})\rightarrow\gamma=0\ \ {\rm as}\
k\rightarrow\infty.$

Without loss of generality, we can assume that $|\alpha_{k}|<\varepsilon$
holds for all $k$ (considering sufficiently large $k$ if necessary). Then for
any $k$, by the Hahn-Banach theorem, there is $h^{*}_{k}\in\mathbb{X}^{*}$
such that

$\|h^{*}_{k}\|=1\ \ {\rm and}\ \ \langle h_{k}^{*},h_{k}\rangle=\|h_{k}\|.$

We now consider the following function:

$g_{\varepsilon}(x):=f(x)+\varepsilon\langle
h_{k}^{*},x-\bar{x}\rangle,\forall x\in\mathbb{X}.$

Note that $\gamma=0$ and thus for any $x\not=\bar{x}$, one has $f(x)\geq
f(\bar{x})$. By the definition of a directional derivative, there exists
$\delta_{k}\rightarrow 0^{+}$ such that

$f(\bar{x}+\delta_{k}h_{k})<f(\bar{x})+(\varepsilon+\alpha_{k})\delta_{k}=\inf_{x\in\mathbb{X}}f(x)+(\varepsilon+\alpha_{k})\delta_{k}.$
(3.17)

By virtue of Ekeland’s variational principle(cf. E1974 ), there is
$z_{k}\in\mathbb{X}$ such that

$\|z_{k}-(\bar{x}+\delta_{k}h_{k})\|<\frac{\delta_{k}}{2},f(z_{k})\leq
f(\bar{x}+\delta_{k}h_{k})$

and

$f(x)+2(\varepsilon+\alpha_{k})\|x-z_{k}\|>f(z_{k}),\ \forall x:x\not=z_{k}.$
(3.18)

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

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

We claim that

$\inf_{\|h\|=1}d^{+}g_{\varepsilon}(z_{k},h)<0,$ (3.19)

since otherwise, $\inf_{\|h\|=1}d^{+}g_{\varepsilon}(z_{k},h)\geq 0$, and thus
$g_{\varepsilon}(z_{k})=\inf_{x\in\mathbb{X}}g_{\varepsilon}(x)$, which
contradicts $g_{\varepsilon}(\bar{x})=0$.

For any $h\in\mathbb{X}$, $\|h\|=1$ and any $t>0$, by (3.18), one can get

$\displaystyle\frac{g_{\varepsilon}(z_{k}+th)-g_{\varepsilon}(z_{k})}{t}=\frac{f(z_{k}+th)-f(z_{k})}{t}+\varepsilon\langle
h_{k}^{*},h\rangle\geq-2(\varepsilon+\alpha_{k})\|h\|-\varepsilon=-5\varepsilon.$

Thus,

$\inf_{\|h\|=1}d^{+}g_{\varepsilon}(z_{k},h)\geq-5\varepsilon.$

Combining (3.19) with Proposition 1 yields ${\rm
Er}g_{\varepsilon}(\bar{x})\leq 5\varepsilon\rightarrow 0^{+}$, which
contradicts (iii), as $\varepsilon$ is arbitrary. The proof is complete.
$\Box$

###### Remark 2

By (38, , Definition 5), condition (3.16) is called as an
$\varepsilon$-perturbation of $f$ near $\bar{x}$. It is shown in Theorem 3.3
that condition $\inf_{\|h\|=1}d^{+}f(\bar{x},h)\not=0$ is sufficient and
necessary for a local error bound moduli of all $\varepsilon$-perturbations of
$f$ near $\bar{x}$ to be uniformly bounded. Further, such stability under
perturbations on $f$ only requires all the linear $\varepsilon$-perturbations
of $f$ near $\bar{x}$.

The following corollaries are immediate from Theorem 3.3.

###### Corollary 1

Let $f$ be given as in (3.3) and $\bar{x}\in{\rm bdry}(S_{F})$. Then
$\inf\limits_{\|h\|=1}d^{+}f(\bar{x},h)\not=0$ holds if and only if there is
an $\varepsilon>0$ such that

$\inf\big{\\{}{\rm Er}g(\bar{x}):g\in\mathcal{C}(\mathbb{X},\mathbb{R})\ {\it
satisfying}\ \bar{x}\in S_{g}\ {\it and}\ \eqref{3.9}\big{\\}}>0.$

###### Corollary 2

Let $f$ be given as in (3.3), $\bar{x}\in{\rm bdry}(S_{F})$ and
$\varepsilon>0$.

  * (i)

The following inequality holds:

$\inf\big{\\{}{\rm Er}g(\bar{x}):g\in\mathcal{C}(\mathbb{X},\mathbb{R})\ {\it
satisfying}\ \bar{x}\in S_{g}\ {\it and}\ \eqref{3.9}\ \big{\\}}\
\geq\left|\inf_{\|h\|=1}d^{+}f(\bar{x},h)\right|-\varepsilon.$

  * (ii)

If $\inf_{\|h\|=1}d^{+}f(\bar{x},h)=0$, then

$\inf\big{\\{}{\rm Er}g(\bar{x}):g\in\mathcal{C}(\mathbb{X},\mathbb{R})\ {\it
satisfying}\ \bar{x}\in S_{g}\ {\it and}\ \eqref{3.9}\ \big{\\}}=0.$

The following proposition is to give primal criteria for the stability of the
global error bound for the continuous convex function $f$.

###### Theorem 3.4

Let $f$ be given as in (3.3).

  * (i)

If there is $\tau\in(0,+\infty)$ such that

$\inf\left\\{\Big{|}\inf_{\|h\|=1}d^{+}f(\bar{x},h)\Big{|}:\bar{x}\in{\rm
bdry}(S_{f})\right\\}\geq\tau,$ (3.20)

then there exist $c,\varepsilon\in(0,+\infty)$ such that for all
$g\in\mathcal{C}(T\times\mathbb{X},\mathbb{R})$ satisfying

$S_{f}\subseteq S_{g}\ \ {\it and}\ \ {\rm Lip}(f-g)<\varepsilon,$ (3.21)

one has ${\rm Er}g\geq c$.

  * (ii)

If there are $c,\varepsilon\in(0,+\infty)$ such that ${\rm Er}g\geq c$ holds
for all $g\in\mathcal{C}(\mathbb{X},\mathbb{R})$ satisfying

${\rm bdry}(S_{f})\cap g^{-1}(0)\not=\emptyset\ \ {\it and}\ \ {\rm
Lip}(f-g)<\varepsilon,$ (3.22)

then there exists $\tau\in(0,+\infty)$ such that (3.20) holds

Proof. Note that $f$ is continuous and thus ${\rm bdry}(S_{f})\subseteq
f^{-1}(0)$.

(i) Suppose that there is some $\bar{x}\in{\rm bdry}(S_{f})$ such that
$\inf_{\|h\|=1}d^{+}f(\bar{x},h)>0$. By Proposition 2, the lower level set
$S_{f}$ is a singleton and thus $S_{f}=\\{\bar{x}\\}$. This means that the
conclusion in (i) follows by using Theorem 3.3 as well as its proof.

Next, we assume that $\inf_{\|h\|=1}d^{+}f(\bar{x},h)\leq 0$ for all
$\bar{x}\in{\rm bdry}(S_{f})$. We claim that

$\tau\mathbf{d}(x,S_{f})\leq[f(x)]_{+},\ \ \forall x\in\mathbb{X}.$ (3.23)

Let $x\in\mathbb{X}\backslash S_{f}$ and $\nu>0$ sufficiently small. Take
$x_{\nu}\in{\rm bdry}(S_{f})$ such that

$\|x-x_{\nu}\|<(1+\nu)\mathbf{d}(x,S_{f}).$ (3.24)

By virtue of Proposition 1 and (3.13), one has

${\rm Er}f(x_{\nu})=\liminf_{u\rightarrow x_{\nu},u\not\in
S_{f}}\Big{(}-\inf_{\|h\|=1}d^{+}f(u,h)\Big{)}\geq-\inf_{\|h\|=1}d^{+}f(x_{\nu},h)\geq\tau.$

Then there exists $\lambda\in(0,1)$ such that

$\displaystyle\tau\leq\frac{f(\lambda
x_{\nu}+(1-\lambda)x)}{\mathbf{d}(\lambda
x_{\nu}+(1-\lambda)x,S_{f})}+\nu\leq\frac{\lambda
f(x_{\nu})+(1-\lambda)f(x)}{\mathbf{d}(x,S_{f})-\|\lambda
x_{\nu}+(1-\lambda)x-x\|}+\nu,$

and it follows from (3.24) that

$\displaystyle\tau\leq\frac{(1-\lambda)f(x)}{(1-\lambda(1+\nu))\mathbf{d}(x,S_{f})}+\nu=\frac{f(x)}{\mathbf{d}(x,S_{f})}+\frac{\nu\lambda
f(x)}{\mathbf{d}(x,S_{f})}+\nu\rightarrow\frac{f(x)}{\mathbf{d}(x,S_{f})}\
{\rm as\ }\nu\rightarrow 0^{+}.$

This means that (3.23) holds.

Let $\varepsilon\in(0,\tau)$ and $g\in\mathcal{C}(\mathbb{X},\mathbb{R})$
satisfying (3.21). Take any $x\in\mathbb{X}$ such that $g(x)>0$. Then $f(x)>0$
(thanks to $S_{f}\subseteq S_{g}$).

We claim that

$\inf_{\|h\|=1}d^{+}f(x,h)\leq-\tau.$ (3.25)

Granting this, it follows from ${\rm Lip}(f-g)<\varepsilon$ in (3.21) that

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

By Proposition 2, one has that ${\rm Er}g\geq\tau-\varepsilon$.

It remains to prove (3.25). For any $n\in\mathbb{N}$, we can take
$z_{n}\in{\rm bdry}(S_{f})$ such that

$\|x-z_{n}\|<(1+\frac{1}{n})\mathbf{d}(x,S_{f}).$

Then (3.23) implies that

$f(x)\geq\tau\mathbf{d}(x,S_{f})>\frac{n}{n+1}\tau\|x-z_{n}\|.$

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

$f(x+t(z_{n}-x))\leq tf(z_{n})+(1-t)f(x),$

and thus

$\frac{f(x+t(z_{n}-x))-f(x)}{t}\leq-f(x)\leq-\frac{n}{n+1}\tau\|x-z_{n}\|.$

This implies that

$\inf_{\|h\|=1}d^{+}f(x,h)\leq
d^{+}f\Big{(}x,\frac{z_{n}-x}{\|x-z_{n}\|}\Big{)}\leq-\frac{n}{n+1}\tau.$

Hence

$\inf_{\|h\|=1}d^{+}f(x,h)\leq-\frac{n}{n+1}\tau.$

Letting $n\rightarrow\infty$, one has that (3.25) holds.

(ii) Suppose on the contrary that there is a sequence
$\\{x_{k}\\}\subseteq{\rm bdry}(S_{f})$ such that

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

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

$\frac{3}{2}\alpha_{k}+\frac{\varepsilon}{2}>0.$ (3.26)

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

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

and consequently

$f(x)-\alpha_{k}\|x-x_{k}\|\geq f(x_{k}),\ \ \forall x\in\mathbb{X}.$ (3.27)

By the definition of $\alpha_{k}$, there exists $h_{k}\in\mathbb{X}$ such that

$\|h_{k}\|=1\ \ {\rm and}\ \
d^{+}f(x_{k},h_{k})<\alpha_{k}+\frac{\varepsilon}{2}.$ (3.28)

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

$f(x_{k}+r_{k}h_{k})<f(x_{k})+(\alpha_{k}+\varepsilon)r_{k}.$ (3.29)

This and (3.27) imply that

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

Applying Ekeland’s variational principle (cf E1974 ), there exists
$y_{k}\in\mathbb{X}$ such that

$\|y_{k}-(x_{k}+r_{k}h_{k})\|<\frac{r_{k}}{2},f(y_{k})-\alpha_{k}\|y_{k}-x_{k}\|\leq
f(x_{k}+r_{k}h_{k})-\alpha_{k}r_{k},$ (3.30)

and

$f(x)-\alpha_{k}\|x-x_{k}\|+2\varepsilon\|x-y_{k}\|>f(y_{k})-\alpha_{k}\|y_{k}-x_{k}\|,\
\ \forall x:x\not=y_{k}.$ (3.31)

Then

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

and thus $y_{k}\not=x_{k}$.

For any $h_{k}$, by the Hahn-Banach theorem, there exists
$h^{*}_{k}\in\mathbb{X}^{*}$ such that

$\|h^{*}_{k}\|=1\ \ {\rm and}\ \ \langle h_{k}^{*},h_{k}\rangle=\|h_{k}\|.$

We consider the following function $g_{\varepsilon}$:

$g_{\varepsilon}(x):=f(x)+\varepsilon\langle h_{k}^{*},x-x_{k}\rangle\ \
\forall x\in\mathbb{X}.$

Then $g_{\varepsilon}\in\mathcal{C}(\mathbb{X},\mathbb{R})$. By virtue of
(3.26), (3.27), (3.28) and (3.31), 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 $\inf_{\|h\|=1}d^{+}g_{\varepsilon}(y_{k},h)\geq 0$, then for any
$x\not=y_{k}$, one has

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

This together with $g_{\varepsilon}(y_{k})>0$ implies that
$S_{g_{\varepsilon}}=\emptyset$ and thus ${\rm Er}g_{\varepsilon}=0$, which
contradicts (ii).

Next, we consider $\inf_{\|h\|=1}d^{+}g_{\varepsilon}(y_{k},h)<0$. For any
$h\in\mathbb{X}$ with $\|h\|=1$ and any $t>0$, by (3.31), 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

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

Applying Proposition 1, one derives that ${\rm Er}(g_{\varepsilon})\leq
4\varepsilon$, which contradicts (ii), as $\varepsilon$ is arbitrary. The
proof is complete. $\Box$

The following corollary follows from the proof of Theorem 3.4.

###### Corollary 3

Let $f$ be given as in (3.3). Then

${\rm
Er}f\geq\inf\left\\{\Big{|}\inf_{\|h\|=1}d^{+}f(\bar{x},h)\Big{|}:\bar{x}\in{\rm
bdry}(S_{f})\right\\}.$ (3.32)

###### Remark 3

Theorem 3.4 shows that (3.20) is a sufficient condition to ensure the
stability of the global error bound for $f$ as said in (i) and meanwhile it is
a necessary condition for the stability of the global error bound for $f$ as
said in (ii). However, condition (3.20) may not be sufficient for the
stability of the global error bound for $f$ as in (ii). For example, let
$\mathbb{X}:=\mathbb{R}$ and $F(t,x):=e^{x}-1$ for all $(t,x)\in
T\times\mathbb{R}$. Then $f(x)=e^{x}-1$, $S_{f}=(-\infty,0]$, ${\rm
bdry}(S_{f})=\\{0\\}$ and $|\inf_{|h|=1}d^{+}f(0,h)|=1>0$. Let
$\varepsilon\in(0,+\infty)$, and consider
$g_{\varepsilon}(x):=f(x)-\varepsilon x,\forall x\in\mathbb{R}$. Then one can
verify that $g_{\varepsilon}$ has two different zero points $0$ and
$\bar{x}<0$. Further, one can verify that $S_{g_{\varepsilon}}=[\bar{x},0]$
and thus $S_{f}\not\subseteq S_{g_{\varepsilon}}$. However, for any
$x<\bar{x}$, one has

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

This implies that

${\rm Er}g_{\varepsilon}\leq 2\varepsilon,$

and consequently the stability of global error bound for $f$ as in (ii) does
not hold (as $\varepsilon>0$ is arbitrary).

The following theorem shows that (3.20) plus a mild qualification can
characterize the stability of the global error bound for $f$ as said in (ii)
of Theorem 3.4.

###### Theorem 3.5

Let $f$ be as (3.3). Then the following statements are equivalent:

  * (i)

There exists $\tau\in(0,+\infty)$ such that (3.20) holds and

$\liminf_{k\rightarrow\infty}\Big{|}\inf_{\|h\|=1}d^{+}f(z_{k},h)\Big{|}\geq\tau,\
\forall\\{(z_{k},x_{k})\\}\subseteq{\rm int}(S_{f})\times{\rm bdry}(S_{f})\
{\it with}\
\lim\limits_{k\rightarrow\infty}\frac{f(z_{k})-f(x_{k})}{\|z_{k}-x_{k}\|}=0.$
(3.33)

  * (ii)

There exist $c,\varepsilon\in(0,+\infty)$ such that for all
$g\in\mathcal{C}(T\times\mathbb{X},\mathbb{R})$ satisfying (3.22), one has
${\rm Er}g\geq c$;

  * (iii)

There exist $c,\varepsilon>0$ such that for any $\bar{x}\in{\rm bdry}(S_{f})$
and any $u^{*}\in\mathbb{B}_{\mathbb{X}^{*}}$, one has ${\rm
Er}g_{u^{*},\varepsilon}\geq c$, where
$g_{u^{*},\varepsilon}(\cdot):=f(\cdot)+\varepsilon\langle
u^{*},\cdot-\bar{x}\rangle$.

Proof. (i)$\Rightarrow$(ii): If there is $\bar{x}\in{\rm bdry}(S_{f})$ such
that $\inf_{\|h\|=1}d^{+}f(\bar{x},h)>0$, the conclusion follows by
Proposition 2 as well as the proof of (i)$\Rightarrow$ (ii) in Theorem 3.3.

Next, we consider $\inf_{\|h\|=1}d^{+}f(\bar{x},h)\leq 0$ for all
$\bar{x}\in{\rm bdry}(S_{f})$. We first prove the following claim:

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

$\inf\left\\{\Big{|}\inf_{\|h\|=1}d^{+}f(z_{0},h)\Big{|}:z_{0}\in\mathbb{X},f(z_{0})\geq-\varepsilon_{0}\|z_{0}-x_{0}\|\right\\}\geq\tau.$
(3.34)

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

$f(z_{k})\geq-\varepsilon_{k}\|z_{k}-x_{k}\|\ \ {\rm and}\ \
\Big{|}\inf_{\|h\|=1}d^{+}f(z_{k},h)\Big{|}<\tau,\ \ \forall k.$ (3.35)

Then $f(z_{k})\leq 0$ for all $k$, since otherwise, using the proof of (3.25),
one get

$\big{|}\inf_{\|h\|=1}d^{+}f(z_{k},h)\big{|}>\tau,$

which contradicts (3.35).

By virtue of (3.20), one has $z_{k}\in S_{f}\backslash{\rm bdry}(S_{f})$ and
then (3.35) implies

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

By (3.33), one gets

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

which contradicts (3.35). Hence the claim holds.

Let $\varepsilon>0$ be such that $\varepsilon<\min\\{\varepsilon_{0},\tau\\}$
and $g\in\mathcal{C}(\mathbb{X},\mathbb{R})$ be such that (3.22) holds. Take
any $\bar{x}\in{\rm bdry}(S_{f})\cap g^{-1}(0)$. Then for any $x\in\mathbb{X}$
with $g(x)>0$, one has

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

By (3.34), one has

$\inf_{\|h\|=1}d^{+}f(x,h)<-\tau,$

and consequently

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

Then Proposition 2 implies that ${\rm Er}g\geq\tau-\varepsilon$.

Note (ii)$\Rightarrow$(iii) follows immediately and it remains to prove
(iii)$\Rightarrow$(i).

Suppose on the contrary that the conclusions in (i) does not hold. Based on
Theorem 3.4 and its proof, it suffices to assume that (3.33) does not hold for
any $\tau>0$. Then, there exists $(z_{k},x_{k})\in{\rm int}(S_{f})\times{\rm
bdry}(S_{f})$ such that

$\lim_{k\rightarrow\infty}\frac{f(z_{k})-f(x_{k})}{\|z_{k}-x_{k}\|}=0\ \ {\rm
and}\ \ \alpha_{k}:=\inf_{\|h\|=1}d^{+}f(z_{k},h)\rightarrow 0^{-}\ \ {\rm
as}\ k\rightarrow\infty.$ (3.36)

Let $\varepsilon>0$. For any $k$, by the Hahn-Banach theorem, there is
$h_{k}^{*}\in\mathbb{X}^{*}$ such that

$\|h_{k}^{*}\|=1\ \ {\rm and}\ \ \langle
h_{k}^{*},z_{k}-x_{k}\rangle=\|z_{k}-x_{k}\|.$

Then when $k$ is sufficiently large, one has

$\alpha_{k}+\varepsilon>0\ \ {\rm and}\ \
\frac{f(z_{k})-f(x_{k})}{\|z_{k}-x_{k}\|}+\varepsilon\Big{\langle}h_{k}^{*},\frac{z_{k}-x_{k}}{\|z_{k}-x_{k}\|}\Big{\rangle}>0.$
(3.37)

Consider the function $g_{k,\varepsilon}$ defined as follows:

$g_{k,\varepsilon}(x):=f(x)+\varepsilon\langle h_{k}^{*},x-x_{k}\rangle\ \
\forall x\in\mathbb{X}.$

Then $g_{k,\varepsilon}\in\mathcal{C}(\mathbb{X},\mathbb{R})$ and it follows
from (3.37) that

$g_{k,\varepsilon}(z_{k})=f(z_{k})+\varepsilon\langle
h_{k}^{*},z_{k}-x_{k}\rangle>0.$

Thus

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

Applying Proposition 1 yields ${\rm Er}g_{k,\varepsilon}\leq 2\varepsilon$,
which contradicts (iii), as $\varepsilon>0$ is arbitrary. The proof is
complete.∎

###### Remark 4

It should be noted that the stability of the global error bound for $f$ in
Theorem 3.5 may not hold if condition (3.33) is violated. Consider again the
example given in Remark 3. Then $f(x)=e^{x}-1$, $S_{f}=(-\infty,0]$, ${\rm
bdry}(S_{f})=\\{0\\}$ and the stability of the global error bound in Theorem
3.5 is not satisfied. Further, for any $z_{k}\rightarrow-\infty$, one can
verify that

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

which implies that (3.33) does not hold. ∎

###  3.2 Proofs of Theorems 3.1 and 3.2

This subsection is devoted to the proofs of Theorems 3.1 and 3.2. We first
prove Theorem 3.1.

Proof of Theorem 3.1 Applying (SIAM13, , Theorem 4.2.3)(or (S2007, ,
Proposition 4.5.2)), for any $x\in\mathbb{X}$, one has

$d^{+}f(x,h)=\sup_{t\in T_{f}(x)}d^{+}f_{t}(x,h),\ \ \forall h\in\mathbb{X}.$
(3.38)

Denote

$\gamma:=\inf\limits_{\|h\|=1}\sup\limits_{t\in
T_{f}(\bar{x})}d^{+}f_{t}(\bar{x},h).$

(i)$\Rightarrow$(ii): We first consider the case of $\gamma>0$. By virtue of
Proposition 2, one has $S_{F}=\\{\bar{x}\\}$. Let $\varepsilon>0$ be such that
$\varepsilon<\gamma$ and take $c:=\gamma-\varepsilon>0$. Suppose that
$G,g_{t}$ and $g$ satisfies all conditions of (ii). Then for any $t\in
T_{f}(\bar{x})\subseteq T_{g}(\bar{x})$, one has

$d^{+}g_{t}(\bar{x},h)\geq d^{+}f_{t}(\bar{x},h)-\varepsilon.$

By using (3.38) and $T_{f}(\bar{x})\subseteq T_{g}(\bar{x})$, one has

$\displaystyle\inf_{\|h\|=1}d^{+}g(\bar{x},h)=\inf_{\|h\|=1}\sup_{t\in
T_{g}(\bar{x})}d^{+}g_{t}(\bar{x},h)$ $\displaystyle\geq$
$\displaystyle\inf_{\|h\|=1}\sup_{i\in T_{f}(\bar{x})}d^{+}g_{t}(\bar{x},h)$
$\displaystyle\geq$ $\displaystyle\Big{(}\inf_{\|h\|=1}\sup_{i\in
T_{f}(\bar{x})}d^{+}f_{t}(\bar{x},h)\Big{)}-\varepsilon$ $\displaystyle=$
$\displaystyle\inf_{\|h\|=1}d^{+}f(\bar{x},h)-\varepsilon=c>0.$

Using Proposition 2, one gets

${\rm Er}G(\bar{x})={\rm Er}g(\bar{x})\geq c.$

We next consider the case of $\gamma<0$. Let $\varepsilon>0$ be such that
$c:=-\gamma-\varepsilon>0$. Then for any $t\in T_{g}(\bar{x})\subseteq
T_{f}(\bar{x})$, one has

$d^{+}g_{t}(\bar{x},h)\leq d^{+}f_{t}(\bar{x},h)+\varepsilon.$

This and $T_{g}(\bar{x})\subseteq T_{f}(\bar{x})$ imply

$\displaystyle\inf_{\|h\|=1}d^{+}g(\bar{x},h)=\inf_{\|h\|=1}\sup_{t\in
T_{g}(\bar{x})}d^{+}g_{t}(\bar{x},h)$ $\displaystyle\leq$
$\displaystyle\inf_{\|h\|=1}\sup_{t\in
T_{g}(\bar{x})}(d^{+}f_{t}(\bar{x},h)+\varepsilon)$ $\displaystyle\leq$
$\displaystyle\Big{(}\inf_{\|h\|=1}\sup_{i\in
T_{f}(\bar{x})}d^{+}f_{t}(\bar{x},h)\Big{)}+\varepsilon$ $\displaystyle=$
$\displaystyle\inf_{\|h\|=1}d^{+}f(\bar{x},h)+\varepsilon=-c<0.$

Using Proposition 2, yields

${\rm Er}G(\bar{x})={\rm Er}g(\bar{x})\geq c.$

Note (ii)$\Rightarrow$(iii) follows immediately as
$T_{f}(\bar{x})=T_{g}(\bar{x})$. Thus, it remains to prove
(iii)$\Rightarrow$(i).

Let $u^{*}\in\mathbb{B}_{\mathbb{X}^{*}}$ and
$G_{u^{*}}\in\mathcal{C}(T\times\mathbb{X},\mathbb{R})$ be given as in (3.6).
Note that

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

and thus

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

Using the proof in (iii)$\Rightarrow$(i) of Theorem 3.3, one derives the
conclusion. The proof is complete.∎

Proof of Theorem 3.2 (i)$\Rightarrow$(ii): Based on Proposition 2 and the
proof of Theorem 3.1, we only need to consider the case
$\inf_{\|h\|=1}\sup_{t\in T_{f}(\bar{x})}d^{+}f_{t}(\bar{x},h)\leq 0$ for all
$\bar{x}\in{\rm bdry}(S_{F})$.

Using the proof of the claim in (3.34), there exists $\varepsilon_{0}>0$ such
that for any $x_{0}\in{\rm bdry}(S_{F})$, one has

$\inf\left\\{\Big{|}\inf_{\|h\|=1}d^{+}f(x,h)\Big{|}:x\in\mathbb{X},f(x)\geq-\varepsilon_{0}\|x-x_{0}\|\right\\}\geq\tau.$
(3.39)

Let $\varepsilon>0$ be such that $\varepsilon<\min\\{\varepsilon_{0},\tau\\}$.
Suppose that $G,g_{t}$ and $g$ satisfy all conditions as in (ii). Let
$x\in\mathbb{X}$ be such that $g(x)>0$.

We claim that

$\inf_{\|h\|=1}d^{+}f(x,h)\leq-\tau.$ (3.40)

Granting this, by virtue of $\sup_{t\in T}{\rm Lip}(f_{t}-g_{t})<\varepsilon$
and $T_{g}(x)\subseteq T_{f}(x)$, one obtains

$\displaystyle\inf_{\|h\|=1}d^{+}g(x,h)=\inf_{\|h\|=1}\Big{(}\sup_{t\in
T_{g}(x)}d^{+}g_{t}(x,h)\Big{)}$ $\displaystyle\leq$
$\displaystyle\inf_{\|h\|=1}\Big{(}\sup_{t\in
T_{g}(x)}d^{+}f_{t}(x,h)+\varepsilon\Big{)}$ $\displaystyle\leq$
$\displaystyle\inf_{\|h\|=1}\Big{(}\sup_{t\in
T_{f}(x)}d^{+}f_{t}(x,h)\Big{)}+\varepsilon$ $\displaystyle=$
$\displaystyle\inf_{\|h\|=1}d^{+}f(x,h)+\varepsilon$ $\displaystyle\leq$
$\displaystyle-(\tau-\varepsilon).$

Applying Proposition 2, yields ${\rm Er}G={\rm Er}g=\geq\tau-\varepsilon$.

It remains to prove (3.40). If $f(x)>0$, then one has (3.40) by using the
proof of (3.25). We next consider the case of $f(x)\leq 0$.

By the conditions in (ii), there is $z_{0}\in{\rm bdry}(S_{F})$ such that

$f_{t}(z_{0})=g_{t}(z_{0}),\ \ \forall t\in T.$

Then for any $t\in T_{g}(x)\subseteq T_{f}(x)$, one has

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

This implies that $f(x)>-\varepsilon_{0}\|x-z_{0}\|$ and thus it follows from
(3.39) that (3.40) holds.

Note that (ii)$\Rightarrow$(iii) follows immediately since $T_{f}(x)=T_{g}(x)$
for all $x\in\mathbb{X}$. It remains to show that (iii) $\Rightarrow$ (i).

Take any $\bar{x}\in{\rm bdry}(S_{F})$ and any
$u^{*}\in\mathbb{B}_{\mathbb{X}^{*}}$, and let
$G\in\mathcal{C}(T\times\mathbb{X},\mathbb{R})$ be defined as in (3.9). Note
that

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

and consequently

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

Thus the conclusion follows from (iii)$\Rightarrow$(i) in the proof of Theorem
3.5. The proof is complete. ∎

##  4 Sensitivity Analysis of Hoffman’s Constants for Semi-infinite Linear
Systems

In this section, by using results from the preceding section, we study
sensitivity analysis of Hoffman’s constants for semi-infinite linear systems
in a Banach space. The aim is to give primal criteria for the Hoffman
constants to be uniformly bounded under perturbations on the problem data.

Let $T$ be a compact, possibly infinite metric space (with metric $\rho$) and
$a^{*}:T\rightarrow\mathbb{X}^{*},b:T\rightarrow\mathbb{R}$ be continuous
functions on $T$. We consider now semi-infinite linear systems in $\mathbb{X}$
defined by

$\langle a^{*}(t),x\rangle\leq b(t),\ \ {\rm for\ all}\ t\in T.$ (4.1)

We denote by $\mathcal{S}_{a^{*},b}$ the set of solutions to system (4.1). We
use the following notations:

$\displaystyle f_{a^{*},b}(x):=\max_{t\in T}(\langle a^{*}(t),x\rangle-b(t)),$
$\displaystyle J_{a^{*},b}(x):=\\{t\in T:\langle
a^{*}(t),x\rangle-b(t))=f_{a^{*},b}(x)\\}\ \ {\rm for\ each}\ x\in\mathbb{X}.$

It is easy to verify that $J_{a^{*},b}(x)$ is a compact subset of $T$ for each
$x\in\mathbb{X}$.

Recall that $\mathcal{S}_{a^{*},b}$ admits a global error bound, if there
exists $\sigma>0$ such that

$\sigma\mathbf{d}(x,\mathcal{S}_{a^{*},b})\leq[f_{a^{*},b}(x)]_{+},\ {\rm for\
all}\ x\in\mathbb{X}.$ (4.2)

The Hoffman constant of the semi-infinite linear system (4.1), denoted by
$\sigma(a^{*},b)$, is given by

$\sigma(a^{*},b):=\sup\\{\sigma>0:\text{such that }\eqref{4-2a}\ {\rm
holds}\\}.$ (4.3)

It is easy to verify that

$\sigma(a^{*},b)=\inf_{x\not\in\mathcal{S}_{a^{*},b}}\frac{f_{a^{*},b}(x)}{\mathbf{d}(x,\mathcal{S}_{a^{*},b})}.$

Note that

$\mathcal{S}_{a^{*},b}={\rm int}(\mathcal{S}_{a^{*},b})\cup{\rm
bdry}(\mathcal{S}_{a^{*},b}).$

We begin with characterizing the interior and the boundary of
$\mathcal{S}_{a^{*},b}$ via the following proposition.

###### Proposition 3

Let be semi-infinite linear system defined by (4.1). Then

${\rm int}(\mathcal{S}_{a^{*},b})=\\{x\in\mathcal{S}_{a^{*},b}:\langle
a^{*}(t),x\rangle<b(t)\ \ {\it for\ all}\ t\in T\\}.$ (4.4)

Thus, $x\in{\rm bdry}(\mathcal{S}_{a^{*},b})$ if and only if
$x\in\mathcal{S}_{a^{*},b}$ and there is some nonempty compact subset
$J_{x}\subseteq T$ such that

$\langle a^{*}(t),x\rangle=b(t),\forall t\in J_{x}\ \ {\it and}\ \ \langle
a^{*}(t),x\rangle<b(t),\forall t\in T\backslash J_{x}.$ (4.5)

Proof. Let $\bar{x}\in{\rm int}(\mathcal{S}_{a^{*},b})$. Then there is
$\delta>0$ such that
$\mathbf{B}(\bar{x},\delta)\subseteq\mathcal{S}_{a^{*},b}$.

Suppose on the contrary that there is some $t\in T$ such that

$\langle a^{*}(t),\bar{x}\rangle=b(t).$

Take $u_{t}\in\mathbb{X}$ such that $\|u_{t}\|=1$ and $\langle
a^{*}(t),u_{t}\rangle>0$. Then for any $r\in(0,\delta)$, one has

$\langle a^{*}(t),\bar{x}+ru_{t}\rangle>b(t)$

which contradicts
$\bar{x}+ru_{t}\in\mathbf{B}(\bar{x},\delta)\subseteq\mathcal{S}_{a^{*},b}$.

On the other hand, let $\bar{x}\in\mathcal{S}_{a^{*},b}$ be such that $\langle
a^{*}(t),\bar{x}\rangle<b(t)$ for all $t\in T$. For any $t\in T$, by the
continuity of $a^{*}$ and $b$, there exists $\delta_{t}>0$ such that

$\langle a^{*}(s),x\rangle\leq b(s),\ \ \forall
x\in\mathbf{B}(\bar{x},\delta_{t})\ {\rm and}\ \forall
s\in\mathbf{B}_{\rho}(t,\delta_{t}):=\\{t^{\prime}\in
T:\rho(t^{\prime},t)<\delta_{t}\\}.$ (4.6)

Using the compactness of $T$, there exist $t_{1},\cdots,t_{m}\in T$ such that

$T\subseteq\bigcup_{i=1}^{m}\mathbf{B}_{\rho}(t_{i},\frac{\delta_{t_{i}}}{2}).$
(4.7)

Let $\delta:=\min\\{\frac{\delta_{1}}{2},\cdots,\frac{\delta_{n}}{2}\\}$. We
claim that

$\mathbf{B}(\bar{x},\delta)\subseteq\mathcal{S}_{a^{*},b}.$

Indeed, for any $x\in\mathbf{B}(\bar{x},\delta)$ and any $s\in T$, by (4.7),
there exists $j\in\\{1,\cdots,m\\}$ such that
$s\in\mathbb{B}_{\rho}(t_{j},\frac{\delta_{t_{j}}}{2})$. Note that
$x\in\mathbf{B}(\bar{x},\delta)\subseteq\mathbf{B}(\bar{x},\delta_{t_{j}})$
and it follows from (4.6) that

$\langle a^{*}(s),x\rangle\leq b(s).$

This implies that the claim holds. The proof is complete. ∎

The following theorem gives primal criteria on the Hoffman’s constant for
system (4.1) under perturbations on the problem data.

###### Theorem 4.1

Consider the following statements:

  * (a)

Suppose that

$\inf\left\\{\Big{|}\inf_{\|h\|=1}\sup_{t\in J}\langle
a^{*}(t),h\rangle\Big{|}:J\subseteq T\ {\it is\ compact}\right\\}>0.$ (4.8)

Then there exist $c,\varepsilon>0$ such that for all $\widetilde{x}\in{\rm
bdry}(\mathcal{S}_{a^{*},b})$ and
$\widetilde{u}^{*}\in\mathbb{B}_{\mathbb{X}^{*}}$, one has
$\sigma(\widetilde{a}^{*},\widetilde{b})\geq c$, where
$\widetilde{a}^{*}:T\rightarrow\mathbb{X}^{*}$ and
$\widetilde{b}:T\rightarrow\mathbb{R}$ are defined as follows:

$\widetilde{a}^{*}(t):=a^{*}(t)+\varepsilon\widetilde{u}^{*}\ \ {\it and}\ \
\widetilde{b}(t):=b(t)+\varepsilon\langle\widetilde{u}^{*},\widetilde{x}\rangle\
\ {\it for\ all}\ t\in T.$ (4.9)

Moreover, the following inequality on Hoffman’s constant $\sigma(a^{*},b)$
holds:

$\sigma(a^{*},b)\geq\inf\left\\{\Big{|}\inf_{\|h\|=1}\sup_{t\in J}\langle
a^{*}(t),h\rangle\Big{|}:J\subseteq T\ {\it is\ compact}\right\\}.$ (4.10)

  * (b)

Suppose that there exists a sequence $\\{x_{k}\\}\subseteq{\rm
bdry}(\mathcal{S}_{a^{*},b})$ such that

$\inf_{k\geq 1}\Big{|}\inf_{\|h\|=1}\sup_{t\in J_{a^{*},b}(x_{k})}\langle
a^{*}(t),h\rangle\Big{|}=0.$ (4.11)

Then for any $\varepsilon>0$ there exist $x_{\varepsilon}\in{\rm
bdry}(\mathcal{S}_{a^{*},b})$ and $u^{*}_{\varepsilon}\in\mathbb{X}^{*}$ with
$\|u^{*}_{\varepsilon}\|\leq\varepsilon$ such that
$\sigma(a_{\varepsilon}^{*},b_{\varepsilon})<\varepsilon$, where
$a_{\varepsilon}^{*}(t):=a^{*}(t)+u^{*}_{\varepsilon}$ and
$b_{\varepsilon}(t):=b(t)+\langle u^{*}_{\varepsilon},x_{\varepsilon}\rangle$
for all $t\in T$.

Proof. We denote

$\tau:=\inf\left\\{\Big{|}\inf_{\|h\|=1}\sup_{t\in J}\langle
a^{*}(t),h\rangle\Big{|}:J\subseteq T\ {\it is\ compact}\right\\}>0.$

Let $F:T\times\mathbb{X}\rightarrow\mathbb{R}$ be defined by

$F(t,x):=\langle a^{*}(t),x\rangle-b(t),\ \forall(t,x)\in T\times\mathbb{X},$

and let $f_{t}(x):=F(t,x)$ for all $(t,x)\in T\times\mathbb{X}$ and $f$ be
given in (3.3). Then $f=f_{a^{*},b}$,
$F\in\mathcal{C}(T\times\mathbb{X},\mathbb{R})$ and each $f_{t}$ is convex.

$({\rm a})$ We first show that $f_{t},f$ and $\tau$ satisfy assumptions as
given in (3.7) and (3.8) of Theorem 3.2.

Indeed, for each $x\in\mathcal{S}_{a^{*},b}$, one has that $J_{a^{*},b}(x)$ is
a compact subset of $T$ and thus

$\inf\left\\{\Big{|}\inf_{\|h\|=1}\sup_{t\in J_{a^{*},b}(x)}\langle
a^{*}(t),h\rangle\Big{|}:x\in\mathcal{S}_{a^{*},b}\right\\}\geq\tau.$

Note that

$\inf_{\|h\|=1}d^{+}f(x,h)=\inf_{\|h\|=1}\sup_{t\in
T_{f}(x)}d^{+}f_{t}(x,h)=\inf_{\|h\|=1}\sup_{t\in J_{a^{*},b}(x)}\langle
a^{*}(t),h\rangle$

and it follows from (4.8) that $f_{t},f$ and $\tau$ satisfy assumptions in
(3.7) and (3.8).

Based on the equivalence of (i) and (iii) in Theorem 3.2, one derives the
conclusion. We next prove (4.10). Using the proof of (i)$\Rightarrow$(ii) in
Theorem 3.2, for any $\varepsilon\in(0,\tau)$ sufficiently small, one has (ii)
as given in Theorem 3.2 holds for $c:=\tau-\varepsilon$, and this implies that

$\sigma(a^{*},b)\geq c=\tau-\varepsilon.$

Letting $\varepsilon\rightarrow 0^{+}$, one obtains (4.10).

$({\rm b})$ Suppose that there exists a sequence $\\{x_{k}\\}\subseteq{\rm
bdry}(\mathcal{S}_{a^{*},b})$ such that (4.11) holds. Then using the proof of
(iii)$\Rightarrow$(i) in Theorem 3.2, yields the conclusion. The proof is
complete.∎

As an application of Theorem 4.1, let us give an example where one can verify
the uniform bound of the Hoffman constant for the given linear system.

###### Example 1

Let $\mathbb{X}:=\mathbb{R}^{2}$, $T:=\\{1,2,3\\}$, let
$a^{*}:T\rightarrow\mathbb{R}^{2},b:T\rightarrow\mathbb{R}$ be defined as
follows:

$a^{*}(1):=(1,1),a^{*}(2):=(-2,1),a^{*}(3):=(1,-2),b(1):=1,b(2):=2,b(3):=2.$

Consider the following linear system:

$\langle a^{*}(t),x\rangle\leq b(t),\ \ t\in T.$ (4.12)

Take $J_{1}:=\\{1,2\\},J_{2}:=\\{2,3\\},J_{3}:=\\{1,3\\},J_{4}:=\\{1,2,3\\}$.
Consider the following four min-max optimization problems:

$OP(J_{k})\ \ \ \ \min_{\|h\|=1}\max_{t\in J_{k}}\langle
a^{*}(t),h\rangle,k=1,2,3,4.$ (4.13)

We denote by $\theta_{k}$ and $h_{k}$ the optimal value and the optimal
solution of $OP(J_{k})$ for each $k$, respectively. Solving these four
problems we obtain that

$\displaystyle\theta_{1}=-1,h_{1}=(0,-1);\theta_{2}=-\frac{\sqrt{2}}{2},h_{2}=(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2});$
$\displaystyle\theta_{3}=-1,h_{3}=(-1,0);\theta_{4}=\sqrt{2},h_{4}=(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}).$

This implies that

$\min\left\\{\Big{|}\min_{\|h\|=1}\max_{t\in J}\langle
a^{*}(t),h\rangle\Big{|}:J\subseteq T\right\\}=\frac{\sqrt{2}}{2}>0,$

and then Theorem 4.1 implies that the Hoffman constant ($\sigma(\cdot)$) for
system (4.12) is uniformly bounded under perturbations on the problem data
$(a^{*},b)$.

Finally, let us show that the uniform bound of the Hoffman constant may not be
satisfied if the condition (4.8) is violated in Theorem 4.1.

###### Example 2

Let $\mathbb{X}:=\mathbb{R}^{2}$, $T:=\\{1,2\\}$, and let
$a^{*}:T\rightarrow\mathbb{R}^{2},b:T\rightarrow\mathbb{R}$ be defined as
follows:

$a^{*}(1):=(1,1),a^{*}(2):=(-1,-1),b(1)=b(2):=0.$

Consider the following linear system:

$\langle a^{*}(t),x\rangle\leq b(t),\ \ t\in T.$ (4.14)

Then
$\mathcal{S}(a^{*},b)=\\{x=(x_{1},x_{2})\in\mathbb{R}^{2}:x_{1}+x_{2}=0\\}$.
Note that

$\min_{\|h\|=1}\max_{t\in T}\langle a^{*}(t),h\rangle=0$

and thus from Theorem 4.1, the Hoffman constant for system (4.14) is not
uniformly bounded under perturbations on the problem data $(a^{*},b)$.

Indeed, for any $\varepsilon>0$, select
$u_{\varepsilon}^{*}:=(0,\varepsilon)$, $x_{\varepsilon}:=(0,0)$ and define
$a_{\varepsilon}^{*}(t):=a^{*}(t)+u_{\varepsilon}^{*},b_{\varepsilon}(t):=b(t)$
for $t=1,2$. Then
$f_{a_{\varepsilon}^{*},b_{\varepsilon}}(x):=\max\\{x_{1}+x_{2}+\varepsilon
x_{2},-x_{1}-x_{2}+\varepsilon x_{2}\\}$ for all
$x=(x_{1},x_{2})\in\mathbb{R}^{2}$ and

$\mathcal{S}(a_{\varepsilon}^{*},b_{\varepsilon})=\\{x=(x_{1},x_{2})\in\mathbb{R}^{2}:x_{1}+(1+\varepsilon)x_{2}\leq
0,x_{1}+(1-\varepsilon)x_{2}\geq 0\\}.$

Take $u_{\varepsilon}:=(-\varepsilon,\varepsilon)$ and thus

$\frac{f_{a_{\varepsilon}^{*},b_{\varepsilon}}(u_{\varepsilon})}{\mathbf{d}(u_{\varepsilon},\mathcal{S}(a_{\varepsilon}^{*},b_{\varepsilon}))}=\frac{\varepsilon^{2}}{\sqrt{2}\varepsilon}=\frac{\varepsilon}{\sqrt{2}},$

which implies that $\sigma(a_{\varepsilon}^{*},b_{\varepsilon})<\varepsilon$.

##  5 Concluding Remarks and Perspectives

This paper is devoted to studying stability of local and global error bounds
for semi-infinite convex constraint systems in a Banach space. We establish
primal characterizations of the stability of error bounds in terms of
directional derivatives. It is proved that verifying the stability of the
error bounds for semi-infinite convex constraint systems is equivalent to
solving several minimax problems defined by directional derivatives of
component functions. Stability under perturbations of the problem data only
requires that all component functions have the same linear perturbations. When
applied to sensitivity analysis of Hoffman’s constants for semi-infinite
linear systems, these stability results lead to conditions ensuring that
Hoffman’s constants can be uniformly bounded.

In a future work, we plan to study applications of these results to
sensitivity analysis of variational problems and convergence analysis of
computational algorithms. Primal criteria of stability of error bounds for
inequality systems defined by non-convex functions are also to be considered.

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