Lipschitz modulus of linear and convex systems with the Hausdorff metric

TL;DR

This paper investigates the Lipschitz stability of feasible sets in linear and convex systems using the Hausdorff metric, providing explicit formulas and extending previous Chebyshev-based results to a Hausdorff framework.

Contribution

It introduces explicit formulas for the Lipschitz modulus of feasible set mappings in the Hausdorff metric and extends stability analysis from Chebyshev to Hausdorff perturbations for linear and convex systems.

Findings

Explicit Lipschitz modulus formulas for linear systems under Hausdorff perturbations

Extension of stability results from Chebyshev to Hausdorff metric

New insights into convex system stability via linearization techniques

Abstract

This paper analyzes the Lipschitz behavior of the feasible set in two parametric settings, associated with linear and convex systems in R^n. To start with, we deal with the parameter space of linear (finite/semi-infinite) systems identified with the corresponding sets of coefficient vectors, which are assumed to be closed subsets of R^(n+1). In this framework, where the Hausdorff distance is used to measure the size of perturbations, an explicit formula for computing the Lipschitz modulus of the feasible set mapping is provided. As direct antecedent, we appeal to its counterpart in the parameter space of all linear systems with a fixed index set, T, where the Chebyshev (pseudo) distance was considered to measure the perturbations. Indeed, the stability (and, particularly, Lipschitz properties) of linear systems in the Chebyshev framework has been widely analyzed in the literature. Here, through an appropriate indexation strategy, we take advantage of previous results to derive the new ones in the Hausdorff setting. In a second stage, the possibility of perturbing directly the set of coefficient vectors of a linear system allows us to provide new contributions on the Lipschitz behavior of convex systems via linearization techniques.

Equations

$$\{a^{\prime }x\le b,\left(a,b\right)\in U\},U\in CL\left({\mathbb{R}}^{n+1}\right),$$

$$f\left(x\right)\le 0,f\in \Gamma ,$$

$$\{a^{\prime }x\le b:b=a^{\prime }z-f\left(z\right),\left(z,a\right)\in \mathrm{gph}\partial f\},$$

$$\partial f\left(z\right):=\{a\in {\mathbb{R}}^n\mid f\left(x\right)\ge f\left(z\right)+a^{\prime }\left(x-z\right),\text{ for all }x\in {\mathbb{R}}^n\}.$$

$$\mathcal{F}\left(U\right):=\left\{x\in {\mathbb{R}}^n:a^{\prime }x\le b\text{ for all }\left(a,b\right)\in U\right\},U\in CL\left({\mathbb{R}}^{n+1}\right).$$

$$\left\{a_t^{\prime }x\le b_t,t\in T\right\},$$

$$d\left(x_1,\mathcal{M}\left(y_2\right)\right)\le \kappa d\left(y_1,y_2\right)\text{ for all }{y}_1,y_2\subset V\text{ and all }{x}_1\in \mathcal{M}\left(y_1\right)\cap W.$$

$$\mathrm{lip}\mathcal{M}\left(y_0,x_0\right)=\underset{\left(x,y\right)\to \left(x_0,y_0\right)}{\mathrm{limsup}}\frac{d\left(x,\mathcal{M}(y)\right)}{d\left(y,{\mathcal{M}}^{-1}(x)\right)},$$

$$\mathrm{clm}\mathcal{M}\left(y_0,x_0\right)=\underset{x\to x_0}{\mathrm{limsup}}\frac{d\left(x,\mathcal{M}(y_0)\right)}{d\left(y_0,{\mathcal{M}}^{-1}(x)\right)}.$$

$$\Vert \left(a,b\right)\Vert =\max \left\{{\Vert a\Vert }_{\ast },\left|b\right|\right\},\left(a,b\right)\in {\mathbb{R}}^{n+1},$$

$${\Vert a\Vert }_{\ast }=\underset{\Vert x\Vert \le 1}{\sup }{a}^{\prime }x.$$

$$d_H\left(U_1,U_2\right):=\max \{e\left(U_1,U_2\right),e\left(U_2,U_1\right)\},$$

$$e\left(U_i,U_j\right):=\inf \left\{\epsilon >0:U_i\subset U_j+\epsilon \mathbb{B}\right\}=\sup \left\{d\left(x,U_j\right):x\in U_i\right\},$$

$$d_{\infty }\left({\sigma }_1,{\sigma }_2\right):=\underset{t\in T}{\sup }\Vert {\sigma }_1\left(t\right)-{\sigma }_2\left(t\right)\Vert ,$$

$${\mathcal{F}}^T\left(\sigma \right):=\left\{x\in {\mathbb{R}}^n:a_t^{\prime }x\le b_t,t\in T\right\},\sigma ={\left(a_t,b_t\right)}_{t\in T}\in {\left({\mathbb{R}}^{n+1}\right)}^T.$$

$$\mathrm{rge}\left(\sigma \right)=U,$$

$$\mathcal{I}_{U}\left(t\right):=\left\{\begin{tabular}[]{cc}$t$&$\text{if }t\in U,$\\ $P_{U}\circ P_{U_{0}}(t)$&$\text{if }t\notin U,$\end{tabular}\right.$$

$$d_{\infty }\left({\mathcal{I}}_U,{\mathcal{I}}_{U_0}\right)=d_H\left(U,U_0\right)\text{ for all }U\in CL\left({\mathbb{R}}^{n+1}\right).$$

$$d_K\left(f_1,f_2\right)=\underset{x\in K}{\sup }\left|f_1\left(x\right)-f_2\left(x\right)\right|.$$

$$\partial f\left(x\right):=\{u\in {\mathbb{R}}^n\mid f\left(z\right)\ge f\left(x\right)+u^{\prime }\left(z-x\right),\text{ for all }z\in {\mathbb{R}}^n\}.$$

$$\partial f\left(x_0+\delta \mathbb{B}\right)\subset \partial f_0\left(x_0\right)+\epsilon \mathbb{B},$$

$$\partial f_1\left(x_0\right)\subset \partial f_2\left(x_0+\sqrt{\delta }\mathbb{B}\right)+4\sqrt{\delta }\mathbb{B}.$$

$$\partial f\left(K_0+\delta \mathbb{B}\right)\subset \partial f_0\left(K_0\right)+\epsilon \mathbb{B},$$

$$\partial f_1\left(K_0\right)\subset \partial f_2\left(K_0+\sqrt{\delta }\mathbb{B}\right)+4\sqrt{\delta }\mathbb{B}.$$

$$d_H\left(\partial f\left(K_0+\sqrt{\delta }\mathbb{B}\right),\partial f_0\left(K_0\right)\right)\le \epsilon .$$

$$\partial f_1\left(x_0\right)\subset \partial f_2\left(x_0+\sqrt{\delta }\mathbb{B}\right)+4\sqrt{\delta }\mathbb{B}\subset \partial f_2\left(K_0+\sqrt{\delta }\mathbb{B}\right)+4\sqrt{\delta }\mathbb{B}.$$

$$\partial f\left(K_0+{\delta }_1\mathbb{B}\right)\subset \partial f_0\left(K_0\right)+\epsilon \mathbb{B},\text{ provided that }{d}_K\left(f,f_0\right)\le {\delta }_1,f\in \Gamma .$$

$$0<{\delta }_0:=\min \{{\delta }_1^2,{\alpha }^2,{\left(\frac{\epsilon }{4}\right)}^2\},$$

$$\partial f\left(K_0+\sqrt{\delta }\mathbb{B}\right)\subset \partial f\left(K_0+{\delta }_1\mathbb{B}\right)\subset \partial f_0\left(K_0\right)+\epsilon \mathbb{B}.$$

$$\partial f_0\left(K_0\right)\subset \partial f\left(K_0+\sqrt{\delta }\mathbb{B}\right)+4\sqrt{\delta }\mathbb{B}\subset \partial f\left(K_0+\sqrt{\delta }\mathbb{B}\right)+\epsilon \mathbb{B},$$

Full text

**Keywords. **Lipschitz modulus · Feasible set mapping · Uncertain inequality systems · Hausdorff metric · Indexation

**Mathematics Subject Classification: **90C31, 49J53, 49K40, 90C05, 90C25, 90C34.

1 Introduction

This paper is initially focussed on the Lipschitz behavior of the *feasible set *associated with a parametric family of *linear inequality systems *of the form:

[TABLE]

where $x\in\mathbb{R}^{n}$ is the vector of variables, $CL\left(\mathbb{R}^{n+1}\right)$ is the *parameter space *of all nonempty closed subsets in $\mathbb{R}^{n+1}.$ Elements in $U\in CL\left(\mathbb{R}^{n+1}\right)$ are denoted as $\left(a,b\right),$ where $a\in\mathbb{R}^{n}$ and $b\in\mathbb{R}.$ Given $x,y\in\mathbb{R}^{n},$ $x^{\prime}y$ represents the usual inner product of $x$ and $y.$ When $U$ is an infinite set, (1) is a linear semi-infinite inequality system. Observe that, in this framework, perturbations fall on $U$ and, so, obviously, two different systems, associated with different sets $U_{1},U_{2}\in CL\left(\mathbb{R}^{n+1}\right),$ can have different cardinality. This setting includes as a particular case the parametric family of linear systems coming from linearizing convex inequalities of the form

[TABLE]

where $\Gamma$ represents the set of all finite-valued convex functions on $\mathbb{R}^{n}.$ Specifically, the feasible set of (2) does coincide with the one of the linear system,

[TABLE]

where $\mathrm{gph}\partial f$ represents the graph of the subdifferential mapping $\partial f:\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n}$ given by

[TABLE]

The main objectives of this work consist of analyzing the Lipschitzian behavior of the parametrized linear system (1) and to apply the obtained results to derive new contributions on the convex case (2) via the standard linearization (3). We emphasize the fact that previous results about stability of subdifferential (traced out from [2]) are also used in the study of this convex case.

Formally, associated with (1), we consider the feasible set mapping, $\mathcal{F}:CL\left(\mathbb{R}^{n+1}\right)\rightrightarrows\mathbb{R}^{n},$ which assigns to each $U\in CL\left(\mathbb{R}^{n+1}\right)$ the set of solutions of the corresponding system:

[TABLE]

The parameter space, $CL\left(\mathbb{R}^{n+1}\right),$ will be endowed with Hausdorff distance (see Section 2 for details). For convenience, we deal with closed sets, but the study could be carried out with general nonempty sets, since both the feasible set mapping and the Hausdorff distance (pseudo-distance in such a case) do not distinguish between nonempty sets and their closures.

The main original contributions of the present paper consists of providing a formula for computing the Lipschitz modulus of $\mathcal{F}$ at $\left(U_{0},x_{0}\right)\in\mathrm{gph}\mathcal{F}$ and, in a second stage, to derive a Lipschitzian type condition for the feasible set of the parametrized convex system (2). Roughly speaking we provide measures (or estimations) of the rate of variation of feasible points, around a nominal one $x_{0}\in\mathbb{R}^{n}$ with respect to perturbations of the nominal parameter ($U_{0}\in CL\left(\mathbb{R}^{n+1}\right)$ in the case of linear systems and $f_{0}\in\Gamma$ in the convex case).

As immediate antecedents of the present work we cite [4] (see also updated results in [3]) and [5]. The first paper computes the Lipschitz modulus of the feasible set mapping in the context of linear systems with an arbitrarily fixed index set $T$ of the form

[TABLE]

where $x\in\mathbb{R}^{n}$ is the variable and $\left(a_{t},b_{t}\right)_{t\in T}\in\left(\mathbb{R}^{n+1}\right)^{T}.$ The parameter space considered there, $\left(\mathbb{R}^{n+1}\right)^{T},$ is formed by all functions from $T$ to $\mathbb{R}^{n+1}$ and it is endowed with the (extended) Chebyshev distance. The reader is addressed to the monograph [10, Chapter 6] for a comprehensive study of such systems. The results of [4] do not apply directly to our current setting unless some appropriate connection between both parameter spaces, $CL\left(\mathbb{R}^{n+1}\right)$ and $\left(\mathbb{R}^{n+1}\right)^{T},$ was established. In relation to this point, we appeal to paper [5], which provides the motivation and background from the methodological point of view. That paper is focussed on the *calmness modulus *(see again Section 2), and takes advantage of previous results developed in the context of systems (5), to derive new contributions for the parametrized system (1). Formally, [5] introduces an appropriate *indexation mapping, *assigning to each set in $CL\left(\mathbb{R}^{n+1}\right)$ an element in $\left(\mathbb{R}^{n+1}\right)^{T}$ in such a way that the Hausdorff distance around $U_{0}\in CL\left(\mathbb{R}^{n+1}\right)$ translates into the Chebyshev distance around its image in $CL\left(\mathbb{R}^{n+1}\right)$. That indexation strategy is shown to be inappropriate for studying the Lipschitz (instead of calmness) modulus and, in relation to this fact, a new indexation strategy is introduced in Section 3.

The problem of analyzing the relationship among different parametric contexts was also addressed in [6] and [7] from a different perspective, mainly focussed on the lower semicontinuity of the feasible set mapping.

Now we summarize the structure of the paper. Section 2 gathers some definitions and key results of the background on the Lipschitz modulus in the context of systems (5), indexations, and stability of subdifferentials. Section 3 develops the study of the Lipschitz modulus of $\mathcal{F},$ including the definition of an appropriate indexation which allows us to take advantage of the background about systems (5). Finally, Section 4 applies the results of previous section to tackle the convex case.

2 Preliminaries and first results

To start with, recall that a set-valued mapping $\mathcal{M}:Y\rightrightarrows X$ between metric spaces (both distances denoted by $d$) has the Aubin property (also called pseudo-Lipschitz –cf. [11]– or Lipschitz-like –cf. [12]–) at $\left(y_{0},x_{0}\right)\in\mathrm{gph}\mathcal{M}$ if there exist a constant $\kappa\geq 0$ and neighborhoods $W$ of $x_{0}$ and $V$ of $y_{0}$ such that

[TABLE]

The infimum of constants $\kappa$ over all $\left(\kappa,W,V\right)$ satisfying (6) is called the Lipschitz modulus of $\mathcal{M}$ at $\left(y_{0},x_{0}\right),$ denoted by $\mathrm{lip}\mathcal{M}\left(y_{0},x_{0}\right),$ and it is defined as $+\infty$ when the Aubin property fails at $\left(y_{0},x_{0}\right).$ The Aubin property of $\mathcal{M}$ at $\left(y_{0},x_{0}\right)\in\mathrm{gph}\mathcal{M}$ is known to be equivalent to the metric regularity of its inverse mapping $\mathcal{M}^{-1}$ at $\left(x_{0},y_{0}\right)\in\mathrm{gph}\mathcal{M};$ moreover, $\mathrm{lip}\mathcal{M}\left(y_{0},x_{0}\right)$ is known to coincide with the modulus of metric regularity of $\mathcal{M}^{-1}$ at $\left(x_{0},y_{0}\right).\,$So, we can write

[TABLE]

under the conventions $\frac{0}{0}:=0,$ $\frac{1}{0}:=+\infty,$ and $\frac{1}{+\infty}:=0.$

The particularization of (6) to $y_{2}=y_{0}$ yields the definition of *calmness *of $\mathcal{M}$ at $\left(y_{0},x_{0}\right),$ whose associated calmness modulus, $\mathrm{clm}\mathcal{M}\left(y_{0},x_{0}\right),$ is defined analogously. It is also known that the calmness of $\mathcal{M}$ at $\left(y_{0},x_{0}\right)$ is equivalent to the metric subregularity of $\mathcal{M}^{-1}$ at $\left(x_{0},y_{0}\right)$, and that the corresponding moduli do coincide; so,

[TABLE]

Clearly $\mathrm{clm}\mathcal{M}\left(y^{0},x_{0}\right)\leq\mathrm{lip}\mathcal{M}\left(y^{0},x_{0}\right).$ For additional information about the Aubin property, calmness, and related topics of variational analysis, the reader is addressed to [9, 11, 12, 14].

2.1 Indexation strategies and calmness of linear systems

For comparative purposes and as a motivation of the results of Section 3, this subsection recalls some details about the indexation introduced in [5]. First, we fix the topologies considered in the space of variables, $\mathbb{R}^{n},$ and the parameter spaces, $CL\left(\mathbb{R}^{n+1}\right)$ and $\left(\mathbb{R}^{n+1}\right)^{T}$.

Unless otherwise stated, $\mathbb{R}^{n}$ (space of variables) is equipped with an arbitrary norm, $\left\|\cdot\right\|,$ while $\mathbb{R}^{n+1}$ (space of coefficient vectors of linear systems) is endowed with the norm

[TABLE]

where $\left\|\cdot\right\|_{\ast}$ represents the dual norm of $\left\|\cdot\right\|$ in $\mathbb{R}^{n}$, which is given by

[TABLE]

The space $CL\left(\mathbb{R}^{n+1}\right)$ is endowed with the (extended) Hausdorff distance $d_{H}:CL\left(\mathbb{R}^{n+1}\right)\times CL\left(\mathbb{R}^{n+1}\right)\rightarrow[0,+\infty]$ given by

[TABLE]

where $e\left(U_{i},U_{j}\right),$ $i,j=1,2,$ represents the excess of $U_{i}$ over $U_{j},$

[TABLE]

where $\mathbb{B}$ denotes the closed unit ball in $\mathbb{R}^{n+1}.$ See [1, Section 3.2] for details about the Hausdorff distance in general settings.

In $\left(\mathbb{R}^{n+1}\right)^{T},$ the (extended) Chebyshev (or supremum) distance, $d_{\infty}:\left(\mathbb{R}^{n+1}\right)^{T}\times\left(\mathbb{R}^{n+1}\right)^{T}\rightarrow[0,+\infty],$ given by

[TABLE]

is considered.

As commented in the introduction, paper [5] analyzes the calmness of $\mathcal{F}$ at $\left(U_{0},x_{0}\right)\in\mathrm{gph}\mathcal{F}$ via the calmness of the feasible set mapping associated with systems (5), $\mathcal{F}^{T}:\left(\mathbb{R}^{n+1}\right)^{T}\rightrightarrows\mathbb{R}^{n},$ which is given by

[TABLE]

To do this, a particular *indexation mapping *between $CL\left(\mathbb{R}^{n+1}\right)$ and $\left(\mathbb{R}^{n+1}\right)^{T}$ is introduced. Recall that $\sigma\in\left(\mathbb{R}^{n+1}\right)^{T}$ is said to be an indexation of $U\in CL\left(\mathbb{R}^{n+1}\right)$ if

[TABLE]

where $\mathrm{\,`rge}$’ means range (or image); specifically, [5] considers T:=\mathbb{R}^{n+1}\and assigns to each $U\in CL\left(\mathbb{R}^{n+1}\right)$ an indexation $\mathcal{I}_{U}\in\left(\mathbb{R}^{n+1}\right)^{\mathbb{R}^{n+1}}$ defined as

[TABLE]

where, for each $U_{1}\in$ $CL\left(\mathbb{R}^{n+1}\right),$ $P_{U_{1}}:\mathbb{R}^{n+1}\rightarrow\mathbb{R}^{n+1}$ is a particular selection of the metric projection multifunction on $U_{1};$ i.e., $P_{U_{1}}\left(t\right)$ is a best approximation of $t\in\mathbb{R}^{n+1}$ on $U_{1}\in CL\left(\mathbb{R}^{n+1}\right).$ Observe that, in particular, $\mathcal{I}_{U_{0}}=P_{U_{0}}.$ A comparative analysis with other possible indexations, and particularly one given in [8], is carried out in [5, Section 3]. Theorem 3.1 in [5] shows that

[TABLE]

Example 3.1 in the same paper shows that $U\mapsto P_{U}$ is not an adequate indexation mapping in relation to calmness, as far as Chebyshev distances between projections, $d_{\infty}\left(P_{U},P_{U_{0}}\right),$ can be much larger than Hausdorff distances between sets $d_{H}\left(U,U_{0}\right).$

Indexation mapping $\mathcal{I}$ in (10) is suitable for the study of the calmness property of $\mathcal{F},$ but it is no longer enough for the Aubin property, for which we need more, namely: $d_{\infty}\left(\sigma_{1},\sigma_{2}\right)=d_{H}\left(U_{1},U_{2}\right)$ when $\sigma_{1}$ and $\sigma_{2}$ are indexations of two sets $U_{1}$ and $U_{2}$ close enough to the nominal set $U_{0}.$ The price to pay is that the definition of $\sigma_{2}$ depends not only on $U_{0},$ but also on $U_{1}.$ Such an indexation strategy is defined in the proof of Theorem 3.2 and constitutes one of the main contributions of the this section.

2.2 On the stability of subdifferentials

This subsection gathers some stuff traced out from [2] about stability of subdifferentials of convex functions at a point $x_{0}\in\mathbb{R}^{n}$ and provides some extensions and consequences on the stability over a compact set $K_{0}\subset\mathbb{R}^{n}.$ This results will be used in Section 4.

Given any two functions $f_{1},f_{2}\in\Gamma$ and a compact subset $K\subset\mathbb{R}^{n}$ we use the notation

[TABLE]

The following theorem gathers two stability conditions for subdifferentials. The first one, which is a direct consequence of [13, Theorem 24.5], provides the Hausdorff upper semicontinuity of the multifunction which assigns to each pair $\left(f,x\right)\in\Gamma\times\mathbb{R}^{n}$ the subdifferential of $f$ at $x,$

[TABLE]

On the other hand, condition $\left(ii\right)$ expresses a certain uniform lower Hölder type property.

Theorem 2.1

Let $x_{0}\in\mathbb{R}^{n},$ $\alpha>0,$ and $K:=x_{0}+\alpha\mathbb{B}.$ One has:

$\left(i\right)$* *[2, Prop. 2.1] Given $f_{0}\in\Gamma$ and $\varepsilon>0,$ there exists $\delta>0$ such that

[TABLE]

provided that $f\in\Gamma$ satisfies $d_{K}\left(f,f_{0}\right)\leq\delta.\vskip 6.0pt plus 2.0pt minus 2.0pt$

$\left(ii\right)$*[2, Thm. 3.4] *For any $0<\delta\leq\alpha^{2},$ and any $f_{1},f_{2}\in\Gamma$ such that $d_{K}(f_{1},f_{2})\leq\delta,$ we have

[TABLE]

Corollary 2.1

Let $K_{0}\subset\mathbb{R}^{n}$ a compact set, $\alpha>0,$ and $K:=K_{0}+\alpha\mathbb{B}.$ One has:

$\left(i\right)$* Given $f_{0}\in\Gamma$ and $\varepsilon>0,$ there exists $\delta>0$ such that*

[TABLE]

provided that $f\in\Gamma$ satisfies $d_{K}\left(f,f_{0}\right)\leq\delta.$

$\left(ii\right)$* For any $0<\delta\leq\alpha^{2},$ and any $f_{1},f_{2}\in\Gamma$ such that $d_{K}(f_{1},f_{2})\leq\delta,$ we have*

[TABLE]

$\left(iii\right)$* Given $f_{0}\in\Gamma$ and $\varepsilon>0,$ there exists $\delta_{0}>0$ such that for any $0<\delta\leq\delta_{0},$ and any $f\in\Gamma,$ with $d_{K}\left(f,f_{0}\right)\leq\delta,$ one has*

[TABLE]

Proof $\left(i\right)$ follows the same argument of the proof of Theorem 2.1$\left(i\right).$ Here we present a sketch for completeness. Arguing by contradiction, assume the existence of sequences $\left\{f_{r}\right\}\subset\Gamma$ and $\{\left(x_{r},u_{r}\right)\}\subset\mathbb{R}^{n}\times\mathbb{R}^{n}$ such that $d_{K}\left(f_{r},f_{0}\right)\leq\frac{1}{r},$ $x_{r}\in K_{0}+\frac{1}{r}\mathbb{B},$ and $u_{r}\in\partial f_{r}\left(x_{r}\right)\diagdown\partial f_{0}\left(x_{0}\right)+\varepsilon\mathbb{B},$ $r=1,2,...$ In this way we reach a contradiction (see [13, Theorem 24.5]) as far as we may assume without loss of generality that $\{x_{r}\}$ converges to a certain $x_{0}\in K_{0}.$

$\left(ii\right)$ Comes straightforwardly from 2.1$\left(ii\right).$ Indeed, let $0<\delta\leq\alpha^{2},$ let $f_{1},f_{2}\in\Gamma$ be such that $d_{K}(f_{1},f_{2})\leq\delta,$ and take any x_{0}\in K_{0}.\,\We have $d_{x_{0}+\alpha\mathbb{B}}(f_{1},f_{2})\leq d_{K}(f_{1},f_{2})\leq\delta,$ which entails

[TABLE]

$\left(iii\right)$ Take $f_{0}\in\Gamma$ and $\varepsilon>0.$ From condition \left(i\right)\there exists $\delta_{1}>0$ such that

[TABLE]

We may assume $\delta_{1}\leq 1.$

Define

[TABLE]

and take $0<\delta\leq\delta_{0},$ and $f\in\Gamma$ such that $d_{K}\left(f,f_{0}\right)\leq\delta.$ Then, since $\sqrt{\delta}\leq\sqrt{\delta_{0}}\leq\delta_{1}$ and $\delta\leq\delta_{1}^{2}\leq\delta_{1}$ (which yields $d_{K}\left(f,f_{0}\right)\leq\delta_{1}),$ one has,

[TABLE]

On the other hand since $0<\delta\leq\alpha^{2},$ condition $\left(ii\right)$ yields

[TABLE]

where the last inclusion comes from $\delta\leq\left(\frac{\varepsilon}{4}\right)^{2}.$ $\square$

3 Lipschitz modulus of $\mathcal{F}$ in the Hausdorff setting

This section provides a point-based formula for $\mathrm{lip\,}\mathcal{F}\left(U_{0},x_{0}\right)$ through a previously established expression of $\mathrm{lip\,}\mathcal{F}^{T}\left(\sigma_{0},x_{0}\right),$ which is recalled in the next theorem. First, we introduce some notation: Given $X\subset\mathbb{R}^{k},$ $k\in\mathbb{N},$ we denote by conv$X$ the *convex hull *of $X,$ and $\mathrm{int}X,$ $\mathrm{cl}X$ and $\mathrm{bd}X$ stand, respectively, for the interior, the closure and the boundary of $X.$

Theorem 3.1

(see [4, Theorem 1$\left(ii\right)$])* Let $\left(\sigma_{0},x_{0}\right)\in\mathrm{gph}\mathcal{F}^{T},$ with $\sigma_{0}=\left(a_{t}^{0},b_{t}^{0}\right)_{t\in T}\in\left(\mathbb{R}^{n+1}\right)^{T}.$ Assume that $\left\{a_{t}^{0},\text{ }t\in T\right\}$ is bounded. Then*

[TABLE]

where

[TABLE]

The following lemma constitutes a key step for deriving the announced formula for $\mathrm{lip}\mathcal{F}\left(U_{0},x_{0}\right)$. In it we construct appropriate indexations of sets $U_{1},U_{2}\in CL\left(\mathbb{R}^{n+1}\right),$ denoted by $\sigma_{1},\sigma_{2},$ which preserve the distance between them; i.e., $d_{H}\left(U_{1},U_{2}\right)=d_{\infty}\left(\sigma_{1},\sigma_{2}\right),$ and and we obtain Lipschitz estimates for each $d_{\infty}\left(\sigma_{i},\sigma_{0}\right)$ in terms of $d_{H}\left(U_{j},U_{0}\right),$ $j=1,2.$ At this moment, we observe that, in general, given any* $\sigma_{1},\sigma_{2}\in\left(\mathbb{R}^{n+1}\right)^{T},$ $T$ being arbitrary, one has*

[TABLE]

For simplicity in the notation, in the lemma let us write $P_{i}\left(t\right)$ instead $P_{U_{i}}\left(t\right),$ for $t\in\mathbb{R}^{n+1}$ and $i=0,1,2.$

Lemma 3.1

Let $U_{0}\in CL\left(\mathbb{R}^{n+1}\right),$ $T:=\mathbb{R}^{n+1},$ and $\sigma_{0}:=P_{0}\in\left(\mathbb{R}^{n+1}\right)^{T}.$ Associated with each pair of subsets $\left(U_{1},U_{2}\right)\in CL\left(\mathbb{R}^{n+1}\right)\times CL\left(\mathbb{R}^{n+1}\right)$ let us define a pair of functions $\left(\sigma_{1},\sigma_{2}\right)\in\left(\mathbb{R}^{n+1}\right)^{T}\times\left(\mathbb{R}^{n+1}\right)^{T}$ as follows: for each $t\in\mathbb{R}^{n+1}$,

[TABLE]

Then we have

[TABLE]

Proof Take any $\left(U_{1},U_{2}\right)\in CL\left(\mathbb{R}^{n+1}\right)\times CL\left(\mathbb{R}^{n+1}\right),$ and the associated $\left(\sigma_{1},\sigma_{2}\right)\in\left(\mathbb{R}^{n+1}\right)^{T}\times\left(\mathbb{R}^{n+1}\right)^{T}$ as in the statement of the lemma. First, let us see that

[TABLE]

For $t\in U_{1}$ we have

[TABLE]

For $t\in U_{2}$ we have

[TABLE]

For $t\notin U_{1}\cup U_{2}$ we have

[TABLE]

In summary, (15) holds in any case.

Now, let us check that

[TABLE]

For $t\in U_{1}\cup U_{2}$ the arguments are completely analogous to those of $\sigma_{1}.$ For $t\notin U_{1}\cup U_{2}$ we have

[TABLE]

So, we have established (16).

The last step consists of checking $d_{\infty}\left(\sigma_{1},\sigma_{2}\right)=d_{H}\left(U_{1},U_{2}\right).$ On the one hand,

[TABLE]

and, analogously, $\sup_{t\in U_{2}}\left\|\sigma_{1}\left(t\right)-\sigma_{2}\left(t\right)\right\|=e\left(U_{2},U_{1}\right).$ On the other hand, for all $t\notin U_{1}\cup U_{2}$ we have

[TABLE]

$\square$

Theorem 3.2

Let $U_{0}\in CL\left(\mathbb{R}^{n+1}\right)$ be such that $\left\{a\in\mathbb{R}^{n}:\left(a,b\right)\in U_{0}\right\}$ is bounded. Then,

[TABLE]

where

[TABLE]

Proof For simplicity, let us denote by $\kappa_{0}$ the right-hand side of (17). Take the indexation of $U_{0},$ $\sigma_{0}:=P_{U_{0}},$ and observe that the pair $\left(\sigma_{0},x_{0}\right)$ satisfies all the hypotheses of Theorem 3.1. Accordingly, $\mathrm{lip\,}\mathcal{F}^{T}\left(\sigma_{0},x_{0}\right)=\kappa_{0}.$ The rest of the proof is devoted to showing that $\mathrm{lip\,}\mathcal{F}^{T}\left(\sigma_{0},x_{0}\right)=\mathrm{lip\,}\mathcal{F}\left(U_{0},x_{0}\right).$ Let $\varepsilon>0$ be arbitrarily given. By the definition of $\mathrm{lip\,}\mathcal{F}^{T}\left(\sigma_{0},x_{0}\right)$, there exists $\delta>0$ such that, for all $\sigma_{1},\sigma_{2}\in\left(\mathbb{R}^{n+1}\right)^{T}$ with $d_{\infty}\left(\sigma_{i},\sigma_{0}\right)<\delta,$ $i=1,2,$ and all $x^{1}\in\mathcal{F}^{T}\left(\sigma_{1}\right)$ with $\left\|x^{1}-x_{0}\right\|<\delta,$ one has

[TABLE]

We are going to prove that, for all $U_{1},U_{2}\in CL\left(\mathbb{R}^{n+1}\right)$ with $d_{H}\left(U_{i},U_{0}\right)<\delta/3,$ $i=1,2,$ and all $x^{1}\in\mathcal{F}\left(U_{1}\right)$ with $\left\|x^{1}-x_{0}\right\|<\delta,$ one has

[TABLE]

Once this is proved, we will have $\mathrm{lip\,}\mathcal{F}\left(U_{0},x_{0}\right)\leq\mathrm{lip\,}\mathcal{F}^{T}\left(\sigma_{0},x_{0}\right).$ Let $U_{1},$ $U_{2},$ and $x^{1}$ be given as above. Associated with the pair $\left(U_{1},U_{2}\right)$ consider the pair of indexations $\left(\sigma_{1},\sigma_{2}\right)\in\left(\mathbb{R}^{n+1}\right)^{T}\times\left(\mathbb{R}^{n+1}\right)^{T}$ as in the previous lemma. Then, we have $d_{\infty}\left(\sigma_{1},\sigma_{2}\right)=d_{H}\left(U_{1},U_{2}\right)$ and

[TABLE]

Moreover, it is clear that $\mathrm{rge\,}\sigma_{i}=U_{i},$ for $i=1,2.$ More in detail, the inclusion $\subset$ is evident since we are selecting projections on $U_{i},$ and $\supset$ comes from $\sigma_{i}\left(t\right)=t$ for $t\in U_{i}.$ Then, the aimed result will follow straightforwardly from (18). Specifically,

[TABLE]

This finishes the proof of $\mathrm{lip\,}\mathcal{F}\left(U_{0},x_{0}\right)\leq\mathrm{lip\,}\mathcal{F}^{T}\left(\sigma_{0},x_{0}\right).$

The opposite inequality follows from (14). More in detail, assume that (19) holds for all $U_{1},U_{2}\in CL\left(\mathbb{R}^{n+1}\right)$ with $d_{\infty}\left(U_{i},U_{0}\right)<\delta,$ $i=1,2,$ and all $x^{1}\in\mathcal{F}\left(U_{1}\right)$ with $\left\|x^{1}-x_{0}\right\|<\delta,$ for some $\varepsilon>0$ and some associated $\delta>0$; and, for the same $\delta$, consider any pair $\sigma_{1},\sigma_{2}\in\left(\mathbb{R}^{n+1}\right)^{T}$ with $d_{\infty}\left(\sigma_{i},\sigma_{0}\right)<\delta,$ $i=1,2,$ and any $x^{1}\in\mathcal{F}^{T}\left(\sigma_{1}\right).$ Then, appealing to (14), we conclude from (19) that

[TABLE]

$\square$

The following lemma constitutes the counterpart of [3, Thm. 1, Lem. 2] in the context of systems (1). We omit the proof since it follows straightforwardly from the original reference (for systems (5)), as far as it only involves a fix system (it is not of parametric nature). We say that $U\in CL\left(\mathbb{R}^{n+1}\right)$ satisfies the strong Slater condition (SSC, in brief) when $\widehat{x}\in\mathbb{R}^{n}$ exists such that $\sup_{\left(a,b\right)\in U}\left(a^{\prime}\widehat{x}-b\right)<0;$ in such a case $\widehat{x}$ is called a strong Slater point of $U.$

Lemma 3.2

We have that

$\left(i\right)$* $U_{0}$ satisfies the SSC if and only if $0_{n}\notin C_{U_{0}};$*

$\left(ii\right)$* Assume that $\left\{a\in\mathbb{R}^{n}:\left(a,b\right)\in U_{0}\right\}$ is bounded. Then, $x_{0}$ is a strong Slater point of $U_{0}$ if and only if $C_{U_{0}}=\emptyset.$*

As a consequence of the previous lemma, we derive the following corollary which gather two special particular cases.

Remark 3.1

Note that under the assumptions of Theorem 3.2:**

$\left(i\right)$* $\mathrm{lip\,}\mathcal{F}\left(U_{0},x_{0}\right)=+\infty$ if and only if $0_{n}\in C_{U_{0}},$ equivalently SSC is not satisfied at $U_{0}$ .*

$\left(ii\right)$* $\mathrm{lip\,}\mathcal{F}\left(U_{0},x_{0}\right)=0$ if and only if $C_{U_{0}}=\emptyset,$ *equivalently, $x_{0}$ is an SS element.

Corollary 3.1

*(see [4, Proposition 1]) *Let $U_{0}\in CL\left(\mathbb{R}^{n+1}\right)$ be such that $\left\{a\in\mathbb{R}^{n}:\left(a,b\right)\in U_{0}\right\}$ is bounded and assume that SSC holds at $U_{0}.$ Then $\mathcal{F}$ has the Aubin property at $\left(U_{0},x_{0}\right)$ for any $x_{0}\in\mathcal{F}\left(U_{0}\right)$.

4 Application to convex inequalities

This section is devoted to apply the previous results about linear systems to the convex case. Throughout this section $\mathbb{R}^{n}$ is considered to be endowed with the Euclidean norm, denoted in the same way for simplicity, $\left\|\cdot\right\|,$ and $\mathbb{B}$ is the corresponding closed unit ball.

We consider the parameterized family of convex inequalities (2) and the corresponding feasible set mapping $\mathcal{L}:\Gamma\rightrightarrows\mathbb{R}^{n}$ assigning to each convex function $f\in\Gamma$ its zero (sub)level set

[TABLE]

It is well-known that, for each $f\in\Gamma,$ $\mathcal{L}\left(f\right)\subset\mathbb{R}^{n}$ is a closed convex set and, as commented in Section 1, via a standard linearization, it can be written as the feasible set of a linear semi-infinite inequality system of the form; i.e.,

[TABLE]

First, let us see that we can reduce the index set of system (20) to a certain subset of $\mathrm{gph}\partial f$.

Lemma 4.1

Let $f\in\Gamma$ and $X\subset\mathbb{R}^{n}$ be an open set such that $\mathcal{L}\left(f\right)\subset X.$ Then,

[TABLE]

Moreover, $X$ can be replaced in (21) with any $S\supset X$.

Proof The inclusion ‘$\subset$’ is trivial. Let us prove ‘$\supset$’ reasoning by contradiction. Assume the existence of $x_{0}\in\mathbb{R}^{n}$ such that

[TABLE]

and $x_{0}\notin\mathcal{L}\left(f\right),$ which entails $f\left(x_{0}\right)>0.$ Indeed, we have $x_{0}\notin X;$ otherwise, taking $z=x_{0}$ in (22), we would have $a{\acute{}}x_{0}\leq a{\acute{}}x_{0}-f\left(x_{0}\right),$ for any $a\in\partial f\left(x_{0}\right)$, yielding the contradiction $f\left(x_{0}\right)\leq 0.$ Once we know that $x_{0}\notin X,$ pick any $x_{1}\in\mathcal{L}\left(f\right)\subset X$ and define $x^{\lambda}:=\left(1-\lambda\right)x_{1}+\lambda x_{0}$ for any $0<\lambda<1.$ Observe that for each $0<\lambda<1,$ $x^{\lambda}$ also verifies the linear inequalities of the right member in (22), i. e.,

[TABLE]

Then, arguing as in the previous paragraph, $x^{\lambda}\notin X\,,$ $0<\lambda<1$, which represents a contradiction since we can choose $x^{\lambda}$ sufficiently close to $x^{1}$ to ensure $x^{\lambda}\in X.$

Finally, from (20) and (21), it is obvious that, $X$ can be replaced by any $S\supset X$. $\square$

From now on we use the notation: $f_{0}\in\Gamma$ is our nominal convex function, $\alpha_{0}>0$ is a fixed scalar, and $E_{0}\subset\mathbb{R}^{n}$ is the $\alpha_{0}$-enlargement of the nominal feasible set $\mathcal{L(}f_{0});$ i.e.,

[TABLE]

As a particular consequence of the previous lemma, we can write

[TABLE]

Going further, the following lemma ensures that we can keep the same $E_{0}$ in the linear representation of $\mathcal{L(}f_{1})$ provided that $f_{1}\in\Gamma$ is close enough to $f_{0}$ in relation to the pseudo-distance $d_{E_{0}}$ defined in (12).

Lemma 4.2

Assume that $\mathcal{L(}f_{0})$ is bounded. There exists $\eta>0$ such that

[TABLE]

whenever $f_{1}\in\Gamma,$ with $d_{E_{0}}\left(f_{1},f_{0}\right)<\eta.$

($E_{0}$ can be replaced by any set $E\supset E_{0}$).

Proof Obviously $\mathcal{L}\left(f_{0}\right)+\frac{\mathcal{\alpha}_{0}}{2}\mathbb{B}$ is a compact convex subset of $\mathbb{R}^{n}$ and so the following minimum is attained:

[TABLE]

for some $\widetilde{x}\in\mathrm{bd}\,\left(\mathcal{L}\left(f_{0}\right)+\frac{\mathcal{\alpha}_{0}}{2}\mathbb{B}\right).$ Observe that $m>0,$ since otherwise we would have $\widetilde{x}\in\mathcal{L}\left(f_{0}\right),$ and then $\widetilde{x}\in\widetilde{x}+\frac{\mathcal{\alpha}_{0}}{2}\mathbb{B}\subset\mathcal{L}\left(f_{0}\right)+\frac{\mathcal{\alpha}_{0}}{2}\mathbb{B},$ which would yield the contradiction $\widetilde{x}\in\mathrm{int}\,\left(\mathcal{L}\left(f_{0}\right)+\frac{\mathcal{\alpha}_{0}}{2}\mathbb{B}\right).$ Take

[TABLE]

and consider any convex function $f_{1}$ such that $d_{E_{0}}\left(f_{1},f_{0}\right)<\eta.$ Let us see that

[TABLE]

For any $x\in\mathrm{bd}\,\left(\mathcal{L}\left(f_{0}\right)+\frac{\mathcal{\alpha}_{0}}{2}\mathbb{B}\right)$ we have

[TABLE]

while, for $x\in\mathcal{L}\left(f_{0}\right)$ one has

[TABLE]

Now, arguing by contradiction, assume that there exists $x_{1}\in\mathbb{R}^{n}$ such that $f_{1}\left(x_{1}\right)\leq 0$ and $x_{1}\notin\mathcal{L}\left(f_{0}\right)+\frac{\mathcal{\alpha}_{0}}{2}\mathbb{B}.$ Take any $x_{0}\in\mathcal{L}\left(f_{0}\right)$ (in particular, $x_{0}\in\mathrm{int}\,\left(\mathcal{L}\left(f_{0}\right)+\frac{\mathcal{\alpha}_{0}}{2}\mathbb{B}\right)$) and let $\lambda\in\left]0,1\right[$ such that

[TABLE]

Then, we attain the contradiction, with (24),

[TABLE]

The fact that $E_{0}$ can be replaced by any subset containing it comes from the standard linearization of the convex inequality $f_{1}\left(x\right)\leq 0$ where the whole graph, $\mathrm{gph}\partial f_{1},$ is used. $\square$

The following lemma constitutes a key tool for our purposes.

Lemma 4.3

Let $K_{1},K_{2}\subset\mathbb{R}^{n}$ be compact sets, let $f_{1},f_{2}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be convex functions, and consider

[TABLE]

Then,

[TABLE]

where $\rho:=\max\{1+\left\|x\right\|:x\in K_{1}\cup K_{2}\}.$

Proof Take $K_{i},f_{i},U_{i},$ for $i=1,2,$ and $\rho,$ as in the statement of the lemma. Let us establish the inequality

[TABLE]

which yields by symmetry the aimed inequality (25). For simplicity, in this proof we use the notation

[TABLE]

Specifically, take any $\left(a_{1},b_{1}\right)\in U_{1},$ and let us prove the existence of $\left(a_{2},b_{2}\right)\in U_{2}$ such that

[TABLE]

By definition, $\left(a_{1},b_{1}\right)\in U_{1}$ entails the existence of $z_{1}\in K_{1}$ such that

[TABLE]

Since $\partial f_{1}\left(K_{1}\right)$ and $\partial f_{2}\left(K_{1}\right)$ are compact subsets in $\mathbb{R}^{n}$ (see again [13, Theorem 24.7]), in particular we have that

[TABLE]

and, so, we can write

[TABLE]

Define

[TABLE]

and let us establish (27) for such an element $\left(a_{2},b_{2}\right)$.

On the one hand,

[TABLE]

where for the first inequality we have applied the fact that $a_{2}\in\partial f_{2}\left(z_{2}\right).$

On the other hand, since $a_{1}\in\partial f_{1}\left(z_{1}\right),$ we have

[TABLE]

So, we have established

[TABLE]

Finally, we have (recall that $\rho>1),$

[TABLE]

which yields (27) and the proof is complete. $\square$

The following theorem provides the announced results about the Lipschitzian behavior of the feasible set of convex inequalities. It appeals to the constant

[TABLE]

where

[TABLE]

Before the theorem, the next proposition says that $\kappa_{0}$ is finite when the convex inequality ‘$f_{0}\left(x\right)\leq 0$’ has a strict solution (i.e., when this convex inequality verifies the Slater condition).

Proposition 4.1

There exists $z_{0}\in\mathbb{R}^{n}$ such that $f_{0}\left(z_{0}\right)<0$ if and only if $0_{n}\notin C_{U_{0}}.$

Proof According to Lemma 3.2, we only have to prove that the existence of $z_{0}\in\mathbb{R}^{n}$ such that $f_{0}\left(z_{0}\right)<0$ is equivalent to SSC at $U_{0}.$

Take $z_{0}\in\mathbb{R}^{n}$ such that $f_{0}\left(z_{0}\right)<0$ and let us see that $z_{0}$ is a SS point of $U_{0}.$ Observe that

[TABLE]

equivalently

[TABLE]

So, $\sup_{\binom{a}{v}\in U_{0}}\left(a^{\prime}z_{0}-v\right)\leq f_{0}\left(z_{0}\right)<0.$

Reciprocally, let $z_{0}$ be a SS point of $U_{0}$, in particular, $z_{0}\in\mathcal{L(}f_{0})\subset E_{0}$ and, so, taking any $a\in\partial f_{0}\left(z_{0}\right),$ we have that

[TABLE]

which entails $f_{0}\left(z_{0}\right)<0.$ $\square$

Theorem 4.1

Let $\kappa>\kappa_{0},$ $\alpha>0,$ $E:=E_{0}+\alpha\mathbb{B},$ and $\rho:=\max\{1+\left\|x\right\|:x\in E\}.$ Then, there exist $\delta_{0}>0$ such that $0<\delta\leq\delta_{0}$ implies

[TABLE]

provided that $f_{1},f_{2}\in\Gamma$, $x_{1}\in\mathcal{L(}f_{1}),$ $d_{{}_{E}}\left(f_{i},f_{0}\right)\leq\delta,$ $i=1,2,$ and $\left\|x_{1}-x_{0}\right\|\leq\delta.$

Proof Take $\kappa>\kappa_{0}$ and fix $\varepsilon>0.$ Theorem 3.2 ensures the existence of $\delta_{1}>0$ such that $d_{H}\left(U_{i},U\right)\leq\delta_{1},$ $U_{i}\in CL\left(\mathbb{R}^{n+1}\right),$ $i=1,2,...,$ and $\left\|x_{1}-x_{0}\right\|\leq\delta,$ $x_{1}\in\mathcal{F}\left(U_{1}\right),$ imply

[TABLE]

On the other hand, according to Corollary 2.1$\left(iii\right),$ choose $\delta_{2}>0$ such that $d_{{}_{E}}\left(f,f_{0}\right)\leq\delta\leq\delta_{2}$ implies ,

[TABLE]

Let $\eta>0$ be as in Lemma 4.2, and consider

[TABLE]

Now consider $0<\delta\leq\delta_{0},$ and $f_{1},f_{2}\in\Gamma$, with $d_{{}_{E}}\left(f_{i},f_{0}\right)\leq\delta.$ Define,

[TABLE]

Appealing to Lemma 4.3, we have, for $i=1,2,$

[TABLE]

Moreover, since $d_{{}_{E}}\left(f,f_{0}\right)\leq\delta\leq\eta,$ we have

[TABLE]

Consequently, appealing to (30) in the particular case $U_{i}=U_{i}^{\delta},$ $i=1,2,$ we conclude

[TABLE]

where in the second inequality we have appealed again to Lemma 4.3. $\square$

4.1 The convex differentiable case

Throughout this subsection we assume that our nominal function $f_{0}\in\Gamma$ is differentiable, so that we write $\nabla f_{0}$ instead of $\partial f_{0}$. The following theorem provides the counterpart of Corollary 2.1$\left(iii\right)$ under differentiability of $f_{0}$.

Theorem 4.2

Let $K_{0}\subset\mathbb{R}^{n}$ a compact set, $\alpha>0,$ and $K:=K_{0}+\alpha\mathbb{B}.$ Given $\varepsilon>0,$ there exists $\delta>0$ such that for any $f\in\Gamma,$ with $d_{K}\left(f,f_{0}\right)\leq\delta,$ one has

[TABLE]

Proof Take $\varepsilon>0.$ From Theorem 2.1 \left(i\right)\there exists $\delta_{1}>0$ such that

[TABLE]

In particular, $\partial f\left(K_{0}\right)\subset\nabla f_{0}\left(K_{0}\right)+\varepsilon\mathbb{B},$ if $d_{K}\left(f,f_{0}\right)\leq\delta_{1},$ $f_{1}\in\Gamma.$

Let us prove the existence of $\delta_{2}>0\,$such that

[TABLE]

In such a case, just take $\delta:=\min\{\delta_{1},\delta_{2}\}$ to finish the proof.

Arguing by contradiction, assume the existence of a sequence of convex functions $\{f_{r}\},$ with $d_{K}\left(f_{r},f_{0}\right)\leq\frac{1}{r}$ such that

[TABLE]

For each $r,$ let $x^{r}\in K_{0}$ such that

[TABLE]

The compactness of $K_{0},$ and consequently of $\nabla f_{0}\left(K_{0}\right)$ allows us to assume that $\{x^{r}\}$ and $\{\nabla f_{0}\left(x^{r}\right)\}\,$converge to $\overline{x}\in K_{0}$ and $\nabla f_{0}\left(\overline{x}\right),$ respectively (see again [13, Theorem 24.5]). This fact, together with (31) yields the existence of $r_{0}\in\mathbb{N}$ such that

[TABLE]

On the other hand, [13, Theorem 24.5] guarantees, for $r$ large enough,

[TABLE]

which represents a contradiction. $\square$

Following the proof of Theorem 4.1, appealing to the previous theorem instead of Corollary 2.1$\left(iii\right)$ we derive the following corollary. Recall that $E_{0}$ and $k_{0}$ are defined in (23) and (29), respectively.

Corollary 4.1

Let $\kappa>\kappa_{0},$ $\alpha>0,$ $E:=E_{0}+\alpha\mathbb{B},$ and $\rho:=\max\{1+\left\|x\right\|:x\in E\}.$ There exist $\delta>0$ such that for any $f_{1},f_{2}\in\Gamma$, with $d_{{}_{E}}\left(f_{i},f_{0}\right)\leq\delta,$ $i=1,2,$ and any $x^{1}\in\mathcal{L(}f_{1}),$ with $\left\|x^{1}-x_{0}\right\|\leq\delta,$ one has

[TABLE]