Subdifferential stability and subdifferential sum rules

TL;DR

This paper investigates the stability properties of subdifferentials and strong slopes of lower semicontinuous functions under Wijsman perturbations, and relates subdifferential sum rules to these stability results.

Contribution

It provides new insights into the stability of subdifferentials under Wijsman convergence and unifies subdifferential sum rules within this stability framework.

Findings

Subdifferentials are stable under Wijsman perturbations.

Subdifferential sum rules are special cases of stability results.

The paper establishes conditions for stability of the strong slope.

Abstract

In the first part, we discuss the stability of the strong slope and of the subdifferential of a lower semicontinuous function with respect to Wijsman perturbations of the function, i.e. perturbations described via Wijsman convergence. In the second part, we show how subdifferential sum rules can be viewed as special cases of subdifferential stability results.

Equations

$$f▽g:x\mapsto (f▽g)(x):=\underset{z\in X}{\inf }(f(z)+g(x-z))$$

$$d_S(x)=d(x,S):=\inf \{\Vert x-a\Vert :a\in S\},$$

$$B_{\delta }(S):=\{x\in X:d_S(x)\le \delta \}.$$

$$\delta_{S}(x):=\left\{\begin{array}[]{ll}0&\mbox{if }\leavevmode\nobreak\ x\in S\\ \infty&\mbox{otherwise}.\end{array}\right.$$

$$\mathrm{Li}{S}_n$$

$$\mathrm{Ls}{S}_n$$

$$\mathrm{Li}{S}_n$$

$$\mathrm{Ls}{S}_n$$

$$\mathrm{D}(A,B):=\inf \{\Vert a-b\Vert :a\in A,b\in B\}=\inf \{d(a,B):a\in A\}.$$

$$\mathrm{D}(\mathrm{hypo}g,\mathrm{epi}h)$$

$$({\mathrm{i}}^{\prime })\text{leavevmode}\text{leavevmode}x\in \mathrm{dom}f\Rightarrow \exists (x_n{)}_n\subset X:x_n\to x,\underset{n}{\mathrm{limsup}}{f}_n(x_n)\le f(x).$$

$$\underset{n}{\mathrm{limsup}}d((x,\alpha ),\mathrm{epi}{f}_n)\le \underset{n}{\mathrm{limsup}}d((x,\alpha ),(x_n,\alpha +\epsilon ))=\epsilon .$$

$$\alpha +\gamma <\underset{B_{\lambda +\delta }(x)}{\inf }f.$$

$$\mathrm{D}(B_{\lambda +\delta }(x)\times \alpha ,\mathrm{epi}f)=\epsilon >0.$$

$$f(z)-\alpha =d((z,\alpha ),(z,f(z)))\ge \mathrm{D}(B_{\lambda +\delta }(x)\times \alpha ,\mathrm{epi}f)=\epsilon ,$$

$$\underset{{\delta }^{\prime }>0}{\sup }\underset{B_{\bar{\lambda }+{\delta }^{\prime }}(x)}{\inf }f\ge \underset{B_{\lambda +\delta }(x)}{\inf }f>\alpha .$$

$$\underset{\delta >0}{\sup }\underset{n}{\mathrm{liminf}}\mathrm{D}(B_{\lambda +\delta }(x)\times \alpha ,\mathrm{epi}{f}_n)>0,$$

$$r_S(f):=\underset{\delta >0}{\sup }\underset{B_{\delta }(S)}{\inf }f=\underset{\delta \to 0}{\lim }\inf \{f(x):d_S(x)\le \delta \}.$$

$$f(x):=\left\{\begin{array}[]{ll}-1/n&\mbox{if }\leavevmode\nobreak\ x=e_{i}/n+e_{1}/in,\leavevmode\nobreak\ \leavevmode\nobreak\ n=1,2,\ldots,\leavevmode\nobreak\ \leavevmode\nobreak\ i=1,2,\ldots\\ 0&\mbox{otherwise}.\end{array}\right.$$

$$r_{B_{\lambda }(0)}(f)=-1/n,\underset{B_{\lambda }(0)}{\inf }f=-1/(n+1).$$

$$r_S(f)=\underset{n\to \infty }{\lim }\inf \{f(x)+n{d}_S^p(x):x\in X\}.$$

$$\underset{X}{\inf }(f+\eta d_S^p)\le \underset{B_{\delta }(S)}{\inf }f+\eta {\delta }^p\le \underset{\delta >0}{\sup }\underset{B_{\delta }(S)}{\inf }f+\epsilon ,$$

$$\underset{B_{\delta }(S)}{\inf }f\le f(x)+\eta d_S^p(x),\forall x\in X.$$

$$\gamma <\underset{\eta >0}{\sup }\underset{X}{\inf }(f+\eta d_S^p),$$

$$r_{B_{\lambda }(x)}(f_S)\le \underset{n\to \infty }{\mathrm{liminf}}\inf \{f_n(z):z\in B_{\lambda }(x)\}=\underset{n\to \infty }{\lim }\inf \{f_{B_{\lambda }(x)}(z)+n{d}_S(z):z\in X\}.$$

$$f_n(x)\le \underset{y\in X}{\inf }(f(x+y)+n\Vert y-v\Vert )+n\Vert v\Vert =f_n(x+v)+n\Vert v\Vert ,$$

$$\underset{B_{\lambda }(x)}{\inf }{f}_n=\underset{x^{\prime }\in B_{\lambda }(x)}{\inf }\underset{z\in X}{\inf }(f(z)+n\Vert z-x^{\prime }\Vert )=\underset{z\in X}{\inf }(f(z)+n{d}_{B_{\lambda }(x)}(z)).$$

$$\underset{n\to \infty }{\lim }\underset{B_{\lambda }(x)}{\inf }{f}_n=r_{B_{\lambda }(x)}(f).$$

$$|\nabla f|(x):=\underset{\begin{array}{c}y\to x\\ y\ne x\end{array}}{\mathrm{limsup}}\frac{(f(x)-f(y){)}^{+}}{\Vert x-y\Vert },$$

$$\underset{n}{\mathrm{limsup}}|\nabla f_n|(x_n)\le |\nabla f|(x).$$

$$\left\{\begin{array}[]{l}\|\bar{x}_{n}-x_{n}\|<\varepsilon,\quad|f_{n}(\bar{x}_{n})-f_{n}(x_{n})|\leq(\sigma+2\varepsilon)\varepsilon,\par\\ |\nabla f_{n}|(\bar{x}_{n})\leq\sigma+3\varepsilon.\end{array}\right.$$

Full text

Subdifferential stability and subdifferential sum rules

March 19, 2024

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Dedicated to Alex Ioffe on the occasion of his 80th birthday

Abstract. In the first part, we discuss the stability of the strong slope and of the subdifferential of a lower semicontinuous function with respect to Wijsman perturbations of the function, i.e. perturbations described via Wijsman convergence. In the second part, we show how subdifferential sum rules can be viewed as special cases of subdifferential stability results.

Keywords: lower semicontinuity, Wijsman convergence, slope, subdifferential calculus, trustworthiness.

2010 Mathematics Subject Classification: 49J52, 49J53, 49J45, 26E15.

1 Introduction

In the first part, we study the stability of the strong slope [5] and of the subdifferential of a lower semicontinuous function with respect to variational perturbations of the function. This issue was initiated in our work [8]. There we showed that the slope of the sum $f+x^{*}$ of a lower semicontinuous function $f$ and of a continuous linear functional $x^{*}$ is stable under slice perturbations of $f$, where the notion of slice convergence for lower semicontinuous, non necessarily convex, functions was introduced in [7]. Here we show instead that the slope of $f$ is stable under Wijsman perturbations of $f$, a weaker and classical notion of convergence (see [2, 16]). Our preceding result can be recovered from the following fact (see Theorem 3.4): A sequence $(f_{n})_{n}$ is slice convergent to $f$ if and only if the sequence $(f_{n}+x^{*})_{n}$ is Wijsman convergent to $f+x^{*}$ for every $x^{*}\in X^{*}$. Applications to the stability of trustworthy subdifferentials (see [10]) under Wijsman convergence are adapted from [8].

In the second part, we show how subdifferential sum rules can be viewed as special cases of subdifferential stability results.

The results of this article were largely announced in [13].

Notation. Except where otherwise stated, $X$ stands for a real Banach space and $X^{\ast}$ for its topological dual. All functions are assumed to be extended-real-valued and lower semicontinuous (lsc); ${\rm LSC}(X)$ denotes the space of all such functions on $X$. For $f\in{\rm LSC}(X)$, we denote by ${\rm dom}\kern 1.49994ptf:=\{x\in X:f(x)<\infty\}$ the effective domain of $f$, by ${\rm graph}\kern 1.49994ptf:=\{\,(x,\alpha):f(x)=\alpha\,\}$ the *graph *of $f$, by ${\rm epi}\kern 1.49994ptf:=\{\,(x,\alpha):f(x)\leq\alpha\,\}$ the epigraph of $f$ and by ${\rm hypo}\kern 1.49994ptf:=\{\,(x,\alpha):f(x)\geq\alpha\,\}$ the hypograph of $f$. We write $x\to_{f}\bar{x}$ to say that $x\to\bar{x}$ and $f(x)\to f(\bar{x})$. For any two functions $f,g\in{\rm LSC}(X)$ we denote by

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the inf-convolution of $f$ and $g$. The closed $\varepsilon$-ball centered at point $x$ is written $B_{\varepsilon}(x)$. For a subset $S\subset X$ and a norm $\|\cdot\|$ on $X$, the distance of a point $x\in X$ to $S$ is given by

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and the closed uniform $\delta$-neighborhood of $S$ ($\delta\geq 0$) is defined by

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The diameter of $S$ is given by ${\rm diam}\kern 1.49994pt(S):=\sup\,\{\,\|x-y\|:x,y\in S\,\}\,,$ and the indicator of $S$ is the function $\delta_{S}:X\to{\mathbb{R}}\cup\{\infty\}$ defined by

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For $f:X\to{\mathbb{R}}\cup\{\infty\}$ et $S\subset X$, we write $f_{S}:=f+\delta_{S}$ for the ‘restriction’ of $f$ to $S$, and $\inf_{S}f:=\inf f(S)$.

2 Convergences of sets

We recall (see [2, Definition 5.2.1]) that the lower and upper limits of a sequence of sets $(S_{n})_{n}$ in a Hausdorff topological space $Y$ are respectively defined by

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In a metric space $Y$ these formulas reduce to

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Definitions (1b)–(2b) go back to Peano (1887, 1908), definitions (1)–(2) were popularized by Kuratowski (1948): see Dolecki-Greco [6] for historical comments.

Lower and upper limits of a sequence of sets describe two symmetric behaviors of the sequence with respect to individual points of the space: a point $y$ is in $\mathrm{Li}\kern 1.49994ptS_{n}$ if and only if every neighborhood of $y$ ‘hits’ $S_{n}$ eventually, while a point $y$ is not in $\mathrm{Ls}\kern 1.49994ptS_{n}$ if and only if some neighborhood of $y$ ‘misses’ $S_{n}$ eventually. In normed spaces, this hit-and-miss behavior is more conveniently described by using gap distances. We recall that the gap distance between two sets $A$, $B$ of $(Y,\|.\|)$ is given by

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Proposition 2.1**.**

Let $Y$ be a normed space and let $\{S_{n},S\}\subset Y$ with $S$ closed. Then:

(a)* \begin{array}[t]{lcl}\displaystyle S\subset\mathrm{Li}\kern 1.49994ptS_{n}&\Longleftrightarrow&\displaystyle\forall y\in Y,\quad\limsup_{n}\,d(y,S_{n})\leq d(y,S)\\ &\Longleftrightarrow&\displaystyle\forall S^{\prime}\subset Y,\quad\limsup_{n}\,{\rm D}(S^{\prime},S_{n})\leq{\rm D}(S^{\prime},S).\end{array}*

(b)* $\displaystyle\leavevmode\nobreak\ S\supset\mathrm{Ls}\kern 1.49994ptS_{n}\leavevmode\nobreak\ \Longleftrightarrow\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \forall y\in Y,\quad d(y,S)>0\Rightarrow\sup_{\delta>0}\liminf_{n}{\rm D}(B_{\delta}(y),S_{n})>0.$*

(c)* $\displaystyle\leavevmode\nobreak\ d(y,S)\leq\liminf_{n}d(y,S_{n})\leavevmode\nobreak\ \Longleftrightarrow\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \forall\lambda\geq 0,\quad{\rm D}(B_{\lambda}(y),S)>0\Rightarrow\sup_{\delta>0}\liminf_{n}{\rm D}(B_{\lambda+\delta}(y),S_{n})>0.$*

**Proof. **The above assertions are certainly well known, but we provide a full proof for completeness.

(a) The first property implies the second one: if $\gamma>d(y,S)$, then the open set $V:={\rm int}\kern 1.49994ptB(y,\gamma)$ contains a point $z$ in $S\subset\mathrm{Li}\kern 1.49994ptS_{n}$, and since $V\in\mathcal{N}(z)$, it follows from (1) that $S_{n}\cap V\neq\emptyset$ eventually, hence $\limsup_{n}d(y,S_{n})\leq\gamma$, as required. The second property implies the third one: if $\gamma>{\rm D}(S^{\prime},S)$, then $d(z,S)<\gamma$ for some $z\in S^{\prime}$, so $\limsup_{n}{\rm D}(S^{\prime},S_{n})\leq\limsup_{n}d(z,S_{n})\leq d(z,S)<\gamma$. The third property implies the first one: if $y\in S$, then $\limsup_{n}d(y,S_{n})=0$ by the third property, so $y\in\mathrm{Li}\kern 1.49994ptS_{n}$ by (1b).

(b) To prove ‘$\Rightarrow$’, let $d(y,S)>0$. Then, $y$ is not in $S\supset\mathrm{Ls}\kern 1.49994ptS_{n}$, so, according to the definition (2b) of the upper limit, there exist $\delta>0$ and $N\in{\mathbb{N}}$ such that $d(y,S_{n})>\delta$ for all $n\geq N$. Hence, for any $\delta^{\prime}\in(0,\delta)$, we have ${\rm D}(B_{\delta^{\prime}}(y),S_{n})\geq\delta-\delta^{\prime}>0$, showing that the property in the second half of (b) holds. To prove ‘$\Leftarrow$’, let $y\notin S$. Then, $d(y,S)>0$, so, for some $\delta>0$, ${\rm D}(B_{\delta}(y),S_{n})>0$ eventually, which implies that $S_{n}\cap B_{\delta}(y)=\emptyset$ eventually, that is, $y\notin\mathrm{Ls}\kern 1.49994ptS_{n}$.

(c) To prove ‘$\Rightarrow$’, let $\gamma:={\rm D}(B_{\lambda}(y),S)>0$. Then, $d(y,S)\geq\lambda+\gamma$: indeed, if $z\in S$, the point $y^{\prime}=y+\lambda(z-y)/\|z-y\|$ is in $B_{\lambda}(y)$, so $\|z-y\|=\lambda+\|z-y^{\prime}\|\geq\lambda+\gamma$. It follows from our assumption that $d(y,S_{n})\geq\lambda+\gamma$ eventually, hence, as easily seen, for any $\delta\in(0,\gamma)$ we have ${\rm D}(B_{\lambda+\delta}(y),S_{n})\geq\gamma-\delta>0$ eventually, showing that the second property in (c) holds. To prove ‘$\Leftarrow$’, assume $d(y,S)>\lambda$. Then, ${\rm D}(B_{\lambda}(y),S)>0$, so, ${\rm D}(B_{\lambda}(y),S_{n})>0$ eventually, hence $d(y,S_{n})>\lambda$ eventually, as required.

A sequence of sets $(S_{n})_{n}$ is declared Kuratowski convergent to $S$ (or Peano-Kuratowski convergent to $S$) provided $\mathrm{Li}\kern 1.49994ptS_{n}=\mathrm{Ls}\kern 1.49994ptS_{n}=S$ [2, Definition 5.2.3], whereas in a metric space $(Y,d)$, the sequence is declared Wijsman convergent to $S$ provided $d(y,S_{n})\to d(y,S)$ for every $y\in Y$ [2, Definition 5.2.3]. It readily follows from Proposition 2.1 that, in normed spaces, both convergences can be characterized by a hit-and-miss criterion using gap distances:

Corollary 2.1.1**.**

Let $(Y,\|.\|)$ be a normed space and let $\{S_{n},S\}\subset Y$ with $S$ closed. Then:

(a)* $(S_{n})_{n}$ is Wijsman convergent to $S$ if and only if, for every $y\in Y$ and $\lambda\geq 0$,

(W)\left\{\begin{array}[]{rl}{\rm(i)}&y\in S\Rightarrow\lim_{n}d(y,S_{n})=0;\par\\ {\rm(ii)}&{\rm D}(B_{\lambda}(y),S)>0\Rightarrow\sup_{\delta>0}\liminf_{n}{\rm D}(B_{\lambda+\delta}(y),S_{n})>0.\end{array}\right.*

(b)* $(S_{n})_{n}$ is Kuratowski convergent to $S$ if and only if ${\rm(W)}$ holds for every $y\in Y$ and $\lambda=0$.*

Since $\mathrm{Li}\kern 1.49994ptS_{n}$ and $\mathrm{Ls}\kern 1.49994ptS_{n}$ are always closed sets, the limit $S$ of a Kuratowski or Wijsman convergent sequence is always a closed set.

3 Convergences of functions

A sequence of functions $(f_{n})_{n}\subset{\rm LSC}(X)$ is declared epi-convergent (or $\Gamma$-convergent) to a function $f\in{\rm LSC}(X)$ provided the sequence of their epigraphs $({\rm epi}\kern 1.49994ptf_{n})_{n}$ is Kuratowski convergent to ${\rm epi}\kern 1.49994ptf$ in $X\times{\mathbb{R}}$ [2, Definition 5.3.1]. Likewise, the sequence $(f_{n})_{n}$ is declared Wijsman convergent to $f$ provided the sequence $({\rm epi}\kern 1.49994ptf_{n})_{n}$ is Wijsman convergent to ${\rm epi}\kern 1.49994ptf$ in $X\times{\mathbb{R}}$ supplied with the max norm $\|(x,t)\|:=\max\{\|x\|,|t|\}$.

In view of Corollary 2.1.1, epi-convergence and Wijsman convergence of functions are characterized by hit-and-miss criteria. The proof uses the following easy observation:

Fact 3.1** ([7, Lemma 2.2]).**

*For any two functions $f,g:X\to{\mathbb{R}}\cup\{\infty\}$ one has

${\rm D}({\rm hypo}\kern 1.49994ptg,{\rm epi}\kern 1.49994ptf)={\rm D}({\rm hypo}\kern 1.49994ptg,{\rm graph}\kern 1.49994ptf)={\rm D}({\rm graph}\kern 1.49994ptg,{\rm epi}\kern 1.49994ptf).$

Proposition 3.2**.**

Let $X$ be a normed space and $\{f_{n},f\}\subset{\rm LSC}(X)$. Then:

(a)* $(f_{n})_{n}$ is Wijsman convergent to $f$ if and only if for every $x\in X$, $\lambda\geq 0$ and $\alpha\in{\mathbb{R}}$,

{\rm(W_{g})}\left\{\begin{array}[]{rl}{\rm(i)}&\exists\,(x_{n})_{n}\subset X:\;x_{n}\to x,\;f_{n}(x_{n})\to f(x);\par\\ {\rm(ii)}&\displaystyle{\rm D}(B_{\lambda}(x)\times\alpha,{\rm epi}\kern 1.49994ptf)>0\Rightarrow\sup_{\delta>0}\liminf_{n}{\rm D}(B_{\lambda+\delta}(x)\times\alpha,{\rm epi}\kern 1.49994ptf_{n})>0.\end{array}\right.*

(b)* $(f_{n})_{n}$ is epi-convergent to $f$ if and only if ${\rm(W_{g})}$ holds for every $x\in X$, $\lambda=0$ and $\alpha\in{\mathbb{R}}$.*

**Proof. **We have to show that the formulas ${\rm(W_{g})}$ are equivalent to the corresponding formulas (W) in Corollary 2.1.1 with $S_{n}$, $S$ replaced by ${\rm epi}\kern 1.49994ptf_{n}$, ${\rm epi}\kern 1.49994ptf$ and $B_{\lambda}(y)$ replaced by the balls of $X\times{\mathbb{R}}$ supplied with the box norm, namely $B_{\lambda}(x)\times[\beta,\alpha]$.

We first observe that (W)(ii) and ${\rm(W_{g})}$(ii) are equivalent: this is due to the fact that for any $h$, ${\rm D}(B_{\lambda}(x)\times[\beta,\alpha],{\rm epi}\kern 1.49994pth)={\rm D}(B_{\lambda}(x)\times\alpha,{\rm epi}\kern 1.49994pth)$, as follows from Fact 3.1 with $g:=\alpha+\delta_{B_{\lambda}(x)}$:

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Next, we claim that (W)(i) is equivalent to

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Indeed, assume (W)(i) and let $x\in{\rm dom}\kern 1.49994ptf$. It follows from (W)(i) that $\lim_{n}d((x,f(x)),{\rm epi}\kern 1.49994ptf_{n})=0$, hence there exists a sequence $((x_{n},\alpha_{n}))_{n}$ in ${\rm epi}\kern 1.49994ptf_{n}$ such that $x_{n}\to x$ and $\alpha_{n}\to f(x)$. This implies that $\limsup_{n}f_{n}(x_{n})\leq\limsup_{n}\alpha_{n}=f(x)$. Conversely, assume that (i’) holds, let $(x,\alpha)\in{\rm epi}\kern 1.49994ptf$ and let $\varepsilon>0$. It follows from (i’) that for some sequence $x_{n}\to x$, $f_{n}(x_{n})<\alpha+\varepsilon$ eventually. We therefore have

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This shows that $\lim_{n}d((x,\alpha),{\rm epi}\kern 1.49994ptf_{n})=0$, as required.

To complete the proof of the proposition, it suffices to show that ${\rm(W_{g})}$(ii) implies that $f(x)\leq\liminf_{n}f_{n}(x_{n})$ for every sequence $x_{n}\to x$. Let $\alpha<f(x)$. Then, $d((x,\alpha),{\rm epi}\kern 1.49994ptf)>0$, so by ${\rm(W_{g})}$(ii) with $\lambda=0$, for some $\delta>0$ one has ${\rm D}(B_{\delta}(x)\times\alpha,{\rm epi}\kern 1.49994ptf_{n})>0$ eventually. Now, let $x_{n}\to x$. Then $x_{n}\in B_{\delta}(x)$ eventually, so $d((x_{n},\alpha),{\rm epi}\kern 1.49994ptf_{n})>0$ eventually, i.e., $\alpha<f_{n}(x_{n})$ eventually, which was to be proved.

Formulas ${\rm(W_{g})}$ suggest possible localizations of either concept of variational convergence ‘at a given point $x$’; we consider only Wijsman convergence:

Definition 1**.**

A sequence of functions $(f_{n})_{n}\subset{\rm LSC}(X)$ is Wijsman convergent to $f$ at $x$ with radius $\lambda_{x}\in(0,+\infty]$ provided ${\rm(W_{g})}$ holds at $x$ for all $\lambda\in[0,\lambda_{x})$ and all $\alpha\in{\mathbb{R}}$.

Evidently, (global) Wijsman convergence implies (local) Wijsman convergence at every point. The converse need not be true.

In [7, 8], a stronger concept of variational convergence is considered: roughly, it consists in demanding ${\rm(W_{g})}$ to hold not only for horizontal bounded slices $B_{\lambda}(x)\times\alpha$, i.e. graphs of constant maps restricted to balls, but more generally for all non-vertical bounded slices, i.e. graphs of continuous affine maps restricted to balls. The localization of this concept of convergence at an individual point reads as follows:

Definition 2**.**

A sequence of functions $(f_{n})_{n}\subset{\rm LSC}(X)$ is slice convergent to $f$ at $x\in X$ with radius $\lambda_{x}\in(0,+\infty]$ if for all functions $\varphi_{\lambda}:=\varphi+\delta_{B_{\lambda}(x)}$, with $\varphi$ affine continuous and $\lambda\in[0,\lambda_{x})$,

{\rm(s)}\left\{\!\!\begin{array}[]{rl}{\rm(i)}&\exists\,(x_{n})_{n}\subset X:\;x_{n}\to x,\;f_{n}(x_{n})\to f(x);\par\\ {\rm(ii)}&\displaystyle{\rm D}({\rm graph}\kern 1.49994pt\varphi_{\lambda},{\rm epi}\kern 1.49994ptf)>0\Rightarrow\sup_{\delta>0}\liminf_{n}{\rm D}({\rm graph}\kern 1.49994pt\varphi_{\lambda+\delta},{\rm epi}\kern 1.49994ptf_{n})>0.\end{array}\right.

The sequence $(f_{n})_{n}$ is (globally) slice convergent to $f$ if (s) holds at every point $x\in X$ with $\lambda_{x}=+\infty$.

In [7, 8], this convergence was called ball-affine convergence and it was proved that both the global and local versions of this convergence coincide with the well-known slice convergence on the space of convex lsc functions. This justifies the use in the present paper of the alternative name ‘slice’ for this convergence on the space of all lsc functions.

The precise link between Wijsman convergence and slice convergence is described in the theorem below whose proof is based on the following lemma:

Lemma 3.3**.**

Let $X$ be a normed space, $f,g:X\to{\mathbb{R}}\cup\{\infty\}$ and $x^{*}\in X^{*}$. Then, ${\rm D}({\rm graph}\kern 1.49994ptg,{\rm epi}\kern 1.49994pt(f-x^{*}))>0$ if and only if ${\rm D}({\rm graph}\kern 1.49994pt(g+x^{*}),{\rm epi}\kern 1.49994ptf)>0$.

**Proof. **It is clearly sufficient to prove that the first condition implies the second one. So, let $\varepsilon>0$ such that ${\rm D}({\rm graph}\kern 1.49994ptg,{\rm epi}\kern 1.49994pt(f-x^{*}))>\varepsilon$, and then let $\delta\in(0,\varepsilon/(1+\|x^{*}\|)$. Pick $(x,\alpha)$ in ${\rm epi}\kern 1.49994ptf$ and $(y,\beta)$ in ${\rm epi}\kern 1.49994pt(g+x^{*})$. The lemma will be proved by showing that $d((x,\alpha),(y,\beta))\geq\delta$. The case $\|y-x\|\geq\delta$ being obvious, assume $\|y-x\|<\delta<\varepsilon$. Since $(x,\alpha-\langle x^{*},x\rangle)$ is in ${\rm epi}\kern 1.49994pt(f-x^{*})$ and $(y,\beta-\langle x^{*},y\rangle)$ is in ${\rm graph}\kern 1.49994ptg$, our assumption implies that $\alpha-\langle x^{*},x\rangle>\beta-\langle x^{*},y\rangle+\varepsilon$, hence $\alpha-\beta>\varepsilon+\langle x^{*},x-y\rangle>\varepsilon-\delta\|x^{*}\|>\delta$. The proof is complete.

Theorem 3.4**.**

Let $X$ be a normed space and $\{f_{n},f\}\subset{\rm LSC}(X)$. The sequence $(f_{n})_{n}$ is slice convergent to $f$ at $x$ with radius $\lambda_{x}>0$ if and only if every sequence $(f_{n}+x^{*})_{n}$ with $x^{*}\in X^{*}$ is Wijsman convergent to $f+x^{*}$ at $x$ with radius $\lambda_{x}$.

**Proof. **It suffices to combine Definition 2, Proposition 3.2 and Lemma 3.3.

4 Wijsman convergence and uniform infimum

The following analytic characterization of Wijsman convergence of functions in terms of the lower limit of their infima on balls is an adaptation of results in [7, 8]:

Theorem 4.1**.**

*Let $X$ be a normed space and $\{f_{n},f\}\subset{\rm LSC}(X)$. Then, $(f_{n})_{n}$ is Wijsman convergent to $f$ at $x$ with radius $\lambda_{x}\in(0,+\infty]$ if and only if for every $\lambda\in[0,\lambda_{x})$,

{\rm(W_{a})}\left\{\begin{array}[]{rl}{\rm(i)}&\exists\,(x_{n})_{n}\subset X:\;x_{n}\to x,\;f_{n}(x_{n})\to f(x);\vskip 3.0pt plus 1.0pt minus 1.0pt\\ {\rm(ii)}&\displaystyle r_{B_{\lambda}(x)}(f):=\sup_{\delta>0}\inf_{B_{\lambda+\delta}(x)}f\leq\liminf_{n}\inf_{B_{\lambda}(x)}f_{n}.\end{array}\right.*

**Proof. **It suffices to show that ‘${\rm(W_{g})(ii)}$ holds for every $\lambda\in[0,\lambda_{x})$’ is equivalent to ‘${\rm(W_{a})(ii)}$ holds for every $\lambda\in[0,\lambda_{x})$’. To prove ‘$\Rightarrow$’, fix $\lambda\in[0,\lambda_{x})$, let $\alpha<\sup_{\delta>0}\inf_{B_{\lambda+\delta}(x)}f$, and take $\gamma>0$ and $\delta\in(0,\gamma)$ such that

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We have ${\rm D}(B_{\lambda}(x)\times\alpha,{\rm epi}\kern 1.49994ptf)\geq\delta>0$, because for $z\in B_{\lambda}(x)$ and $(y,\beta)\in{\rm epi}\kern 1.49994ptf$ with $\|y-z\|<\delta$, one has $\|y-x\|\leq\lambda+\delta$, so $\beta\geq f(y)\geq\alpha+\gamma\geq\alpha+\delta$, whence $d((z,\alpha),(y,\beta))\geq\beta-\alpha\geq\delta$. Applying ${\rm(W_{g})\,(ii)}$, we derive that there exists $\delta>0$ such that ${\rm D}(B_{\lambda+\delta}(x)\times\alpha,{\rm epi}\kern 1.49994ptf_{n})>0$ eventually, so $f_{n}(z)>\alpha$ for every $z$ in $B_{\lambda+\delta}(x)$ and large $n$, hence $\liminf_{n}\inf_{B_{\lambda+\delta}(x)}f_{n}\geq\alpha.$ A fortiori, $\liminf_{n}\inf_{B_{\lambda}(x)}f_{n}\geq\alpha.$ The proof that ${\rm(W_{a})\,(ii)}$ holds is complete.

To prove ‘$\Leftarrow$’, fix $\lambda\in[0,\lambda_{x})$, let $\alpha\in{\mathbb{R}}$ such that $\gamma:={\rm D}(B_{\lambda}(x)\times\alpha,{\rm epi}\kern 1.49994ptf)>0$, and take $\delta>0$ such that

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Let $z\in B_{\lambda+\delta}(x)\cap{\rm dom}\kern 1.49994ptf$. Since $(z,\alpha)\not\in{\rm epi}\kern 1.49994ptf$, one has $f(z)-\alpha>0$. Therefore,

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so $f(z)\geq\alpha+\varepsilon$ for every $z\in B_{\lambda+\delta}(x)$. Hence, $\inf_{B_{\lambda+\delta}(x)}f>\alpha$. Take $\bar{\delta}\in(0,\delta)$ such that $\bar{\lambda}:=\lambda+\bar{\delta}<\lambda_{x}$. Then,

[TABLE]

Now, applying ${\rm(W_{a})\,(ii)}$ with $\bar{\lambda}$ instead of $\lambda$, we find that $\inf_{B_{\bar{\lambda}}(x)}f_{n}=\inf_{B_{\lambda+\bar{\delta}}(x)}f_{n}>\alpha$ eventually, hence ${\rm epi}\kern 1.49994ptf_{n}\cap(B_{\lambda+\bar{\delta}}(x)\times\alpha)=\emptyset$ eventually, from which we derive that, for any $\delta\in(0,\bar{\delta})$, ${\rm D}(B_{\lambda+\delta}(x)\times\alpha,{\rm epi}\kern 1.49994ptf_{n})\geq\bar{\delta}-\delta>0$ eventually, so that

[TABLE]

as required.

The value $r_{B_{\lambda}(x)}(f):=\sup_{\delta>0}\inf_{B_{\lambda+\delta}(x)}f$ on the left of ${\rm(W_{a})\,(ii)}$ cannot be replaced by the usual infimum $\inf_{B_{\lambda}(x)}f$ when $f$ is only lsc (see Example 3 below). This is not fortuitous. In problems involving non regular functions $f:X\to{\mathbb{R}}\cup\{\infty\}$ to be minimized on a constrained set $S$, the value that naturally comes to the fore is

[TABLE]

The first explicit mention of (a variant of) this value dates back to [4]. The value as written in (3) was introduced and used in [1]. Its importance was emphasized in [12], where the concept was generalized and employed in various situations related to constrained minimization. In the process, further properties and applications have been developed in [7, 11]. The notation $r_{S}(f)$ comes from [1], slightly modifying the one in [4]. The name uniform infimum of $f$ on $S$ for $r_{S}(f)$ was proposed in [12], arguing that this value incorporates the behavior of $f$ on uniform neighborhoods of $S$. Since then, this concept has been used in the textbooks [3, 15] and in the survey [10] under different notations and names. For example, in [15], $r_{S}(f)$ is denoted $\wedge_{S}(f)$ (more or less as in [3]) and is called stabilized infimum; the usual infimum $\inf_{S}f$ is declared robust when it is equal to $r_{S}(f)$; a point $\bar{x}$ achieving this value, i.e. $f(\bar{x})=r_{S}(f)$, is called a robust minimizer (more or less as in [10]).

In general, $r_{S}(f)<\inf_{S}f$ for arbitrary lsc $f$: additional conditions (so-called qualification conditions) are required to have the equality $r_{S}(f)=\inf_{S}f$.

Example 1*.*

A lsc $f$ with $r_{B_{\lambda}(0)}(f)<\inf_{B_{\lambda}(0)}f$ for arbitrary small $\lambda$ (see also [12, Example 2], [11, Exemple 3.7]). Let $X$ be the Hilbert space $\ell^{2}({\mathbb{N}})$ and let $(e_{i})_{i\in{\mathbb{N}}}$ be its canonical basis. Define

[TABLE]

Then $f$ is lsc at every point. For $\lambda=1/n$, $n\in{\mathbb{N}}$, one has

[TABLE]

So, for every $\lambda>0$ there exists $\lambda_{n}\in(0,\lambda)$ such that $r_{B_{\lambda_{n}}(0)}(f)<\inf_{B_{\lambda_{n}}(0)}f.$

Example 2*.*

Sufficient conditions for $r_{S}(f)=\inf_{S}f$ to hold (see [1, Proposition 3.2]):

1. $S=X$,

2. $f$ is uniformly continuous on a uniform neighborhood of $S$,

3. $f$ is lsc on a neighborhood of $S$ and $S$ is compact, or $f$ is inf-compact and $S$ is closed,

4. $X={\mathbb{R}}_{+}({\rm dom}\kern 1.49994ptf-S)$, $f$ is convex lsc and $S$ is closed and convex.

Since $r_{S}(f)$ is the natural value bound to the constrained minimization problem

($\mathcal{P}$) Minimize $f(x)$ subject to $x\in S$

it is expected that $r_{S}(f)$ can be obtained as the limiting value of the unconstrained penalized problems associated with ($\mathcal{P}$). This is indeed the case. The following proposition was established in [11, Proposition 3.16] gathering earlier observations (see also [15, Proposition 1.130]). For the sake of completeness, we reproduce the proof.

Proposition 4.2**.**

Let $X$ be a normed space, $f:X\to{\mathbb{R}}\cup\{\infty\}$ bounded from below and $S\subset X$ such that $S\cap{\rm dom}\kern 1.49994ptf\neq\emptyset$. Then, for any $p>0$,

[TABLE]

**Proof. **Let $\varepsilon>0$ and $\eta>0$. Choose $\delta>0$ such that $\eta\delta^{p}<\varepsilon$. Then,

[TABLE]

showing that the first member of (4) is not smaller than the second one.

Now, let $\gamma<\sup_{\delta>0}\inf_{B_{\delta}(S)}f$. Take $\delta>0$ such that $\gamma<\inf_{B_{\delta}(S)}f$ and choose $\eta>0$ such that $\inf_{B_{\delta}(S)}f\leq\eta\delta^{p}+\inf_{X}f$ (this is possible since both $\inf_{B_{\delta}(S)}f$ and $\inf_{X}f$ are finite). We claim that

[TABLE]

This is clear if $x$ belongs to $B_{\delta}(S)$; otherwise $d_{S}(x)\geq\delta$, hence, due to our choice of $\eta$, $\inf_{B_{\delta}(S)}f\leq\inf_{X}f+\eta\delta^{p}\leq f(x)+\eta d^{p}_{S}(x)$. It follows that

[TABLE]

showing that the first member of (4) is not greater than the second one.

The next two propositions provide useful examples of Wijsman convergent sequences.

Proposition 4.3**.**

Let $X$ be a normed space, $f\in LSC(X)$ and $S\subset X$ a closed subset such that $S\cap{\rm dom}\kern 1.49994ptf\neq\emptyset$. Let $f_{n}:=f+nd_{S}^{p}$ with $p>0$. The following are equivalent:

(a)* The sequence $(f_{n})_{n}$ is Wijsman convergent to $f_{S}$ at every point $x\in X$,*

(b)* $r_{B_{\lambda}(x)}(f_{S})\leq r_{S}(f_{B_{\lambda}(x)})$ for every $x\in X$ and $\lambda>0$ small enough.*

**Proof. **We have to show that the assertions ${\rm(W_{a})}$ in Theorem 4.1 hold if and only if (b) holds. Assertion ${\rm(W_{a})(i)}$ is always satisfied at every point $x\in X$ by the constant sequence $x_{n}:=x$. Indeed, $f_{n}(x)=f_{S}(x)$ for $x\in S$ and $\lim_{n}f_{n}(x)=+\infty$ for $x\not\in S$ since $S$ is closed, so $\lim_{n}f_{n}(x)=f_{S}(x)$ for $x\not\in S$. On the other hand, ${\rm(W_{a})(ii)}$ asserts that for $\lambda>0$ small enough,

[TABLE]

Now by lower semicontinuity, $f_{B_{\lambda}(x)}$ is bounded from below for $\lambda>0$ small enough, so according to Proposition 4.2, the expression on the right hand side is equal to $r_{S}(f_{B_{\lambda}(x)})$ for $\lambda>0$ small enough. Hence, ${\rm(W_{a})(ii)}$ is equivalent to (b).

Proposition 4.4**.**

Let $X$ be a normed space and let $f\in LSC(X)$ be proper and bounded from below. Let $f_{n}:=f\triangledown n\|.\|$. Then, each $f_{n}$ is Lipschitz continuous on $X$ and the sequence $(f_{n})_{n}$ is Wijsman convergent to $f$ with $r_{B_{\lambda}(x)}(f)=\lim_{n}\inf_{B_{\lambda}(x)}f_{n}$ for every $x\in X$ and $\lambda\geq 0$.

**Proof. **We have $f_{n}(x)=\inf_{y\in X}(f(x+y)+n\|y\|)$. We first observe that $f_{n}(x)$ is finite for every $x\in X$ and $n\in{\mathbb{N}}$. Indeed, $f$ being proper, there is $y\in X$ such that $f(x+y)$ is finite, so $f_{n}(x)<+\infty$, and $f$ being bounded from below, $-\infty<\inf_{X}f\leq f_{n}(x)$. Otherwise, for any $x,v\in X$ and $n\in{\mathbb{N}}$, it holds

[TABLE]

so $|f_{n}(x)-f_{n}(x+v)|\leq n\|v\|$, proving that each $f_{n}$ is $n$-Lipschitz.

We now prove the second statement. For any $\lambda\geq 0$, one has

[TABLE]

So, taking the limit as $n\to+\infty$ on both sides, we see from Proposition 4.2 that

[TABLE]

For $\lambda=0$, (6) gives $\lim_{n\to\infty}f_{n}(x)=r_{\{x\}}f$, where $r_{\{x\}}f=\sup_{\delta>0}\inf_{B_{\delta}(x)}f=f(x)$ because $f$ is lsc, so ${\rm(W_{a})(i)}$ is satisfied at any point $x\in X$ by the constant sequence $x_{n}:=x$. On the other hand, (6) clearly implies ${\rm(W_{a})(ii)}$. This proves that the sequence $(f_{n})_{n}$ is Wijsman convergent to $f$.

Example 3*.*

Let $f$ be the lsc function defined in Example 1 and let $(f_{n})_{n}$ be the sequence given in Proposition 4.4. This sequence is Wijsman convergent to $f$ with $r_{B_{\lambda}(x)}(f)=\lim_{n}\inf_{B_{\lambda}(x)}f_{n}$ for every $x\in X$ and $\lambda\geq 0$. Since for every $\lambda_{x}>0$ there exists $\lambda\in[0,\lambda_{x})$ such that $r_{B_{\lambda}(x)}(f)<\inf_{B_{\lambda}(x)}f$, we infer that for every $\lambda_{x}>0$ there exists $\lambda\in[0,\lambda_{x})$ such that $\liminf_{n}\inf_{B_{\lambda}(x)}f_{n}<\inf_{B_{\lambda}(x)}f$. This shows that we cannot replace the value $r_{B_{\lambda}(x)}(f)$ by $\inf_{B_{\lambda}(x)}f$ in ${\rm(W_{a})(ii)}$.

5 Stability of slopes with respect to Wijsman convergence

From now on, $X$ denotes a Banach space and $f_{n},f:X\to{\mathbb{R}}\cup\{\infty\}$ denote lsc functions. The slope of $f:X\to{\mathbb{R}}\cup\{\infty\}$ at $x\in{\rm dom}\kern 1.49994ptf$, introduced in [5], is defined by

[TABLE]

where $\alpha^{+}=\max(0,\alpha)$ for $\alpha\in{\mathbb{R}}\cup\{\infty\}$.

Theorem 5.1**.**

If $(f_{n})_{n}$ is Wijsman convergent to $f$ at $x\in{\rm dom}\kern 1.49994ptf$, then there is a sequence $(x_{n})_{n}\subset X$ such that $x_{n}\to x$, $f_{n}(x_{n})\to f(x)$, and

[TABLE]

**Proof. **The proof is exactly the same as the one of [8, Theorem 3.1], where slice convergence was considered instead of Wijsman convergence, but we reproduce it for the reader’s convenience. Let $\sigma:=|\nabla f|(x)$, which we may suppose to be finite. Let $(x_{n})_{n}\subset X$ be a sequence verifying ${\rm(W_{a})\,(i)}$ and let $\lambda_{x}>0$ be such that ${\rm(W_{a})\,(ii)}$ holds for $\lambda\in[0,\lambda_{x})$. We claim that for every $\varepsilon\in(0,\lambda_{x})$ there exists $N_{\varepsilon}\in{\mathbb{N}}$ such that for each $n\geq N_{\varepsilon}$ there exists $\bar{x}_{n}\in X$ verifying

[TABLE]

Indeed, it follows from the definition of the slope of $f$ at $x$ that there exists $\lambda_{x}^{\prime}\in(0,\varepsilon)$ such that for all $\lambda\in(0,\lambda_{x}^{\prime})$,

[TABLE]

hence, for all $\lambda\in(0,\lambda_{x}^{\prime})$,

[TABLE]

Fix $\lambda\in(0,\lambda_{x}^{\prime})$. Combining the previous inequality with ${\rm(W_{a})\,(ii)}$ we get

[TABLE]

while, according to ${\rm(W_{a})\,(i)}$, for $n$ large enough one has

[TABLE]

[TABLE]

We thus derive that for each $n$ large enough one has

[TABLE]

Then, applying Ekeland’s variational principle, we get a point $\bar{x}_{n}\in X$ such that

[TABLE]

which implies that $|\nabla f_{n}|(\bar{x}_{n})\leq\sigma+3\varepsilon$. The proof of the claim is therefore complete.

Next, for each $\varepsilon=1/k<\lambda_{x}$, choose an integer $N_{k}$ and a sequence $(\bar{x}_{n,k})_{n}$ in $X$ verifying (7) for each $n\geq N_{k}$. Without loss of generality, we may assume that $N_{k+1}>N_{k}$. The desired sequence is then given by $\bar{x}_{n}:=\bar{x}_{n,k}$ for $n\in{\mathbb{N}}$ and $k$ such that $N_{k}\leq n<N_{k+1}$.

Corollary 5.1.1**.**

Assume $|\nabla f|(\bar{x})=0$. If $(f_{n})_{n}$ is Wijsman convergent to $f$ at $\bar{x}$, then there is a sequence $(x_{n})_{n}$ such that $x_{n}\to\bar{x}$, $f_{n}(x_{n})\to f(\bar{x})$ and $|\nabla f_{n}|(x_{n})\to 0$.

Corollary 5.1.2**.**

Assume $\inf_{X}f>-\infty$. If $(f_{n})_{n}$ is Wijsman convergent to $f$, then there is a sequence $(x_{n})_{n}$ such that $f_{n}(x_{n})\to\inf_{X}f$ and $|\nabla f_{n}|(x_{n})\to 0$.

**Proof. **For each positive integer $n$, let $y_{n}\in X$ such that $f(y_{n})\leq\inf_{X}f+1/n^{2}$. Apply Ekeland’s variational principle to get $z_{n}\in X$ such that $f(z_{n})\leq f(y_{n})\leq\inf_{X}f+1/n^{2}$ and $f(z_{n})\leq f(y)+(1/n)\|y-z_{n}\|$ for every $y\in X$. This implies that $|\nabla f|(z_{n})\leq 1/n$. Now, by Theorem 5.1, we can construct a sequence $(x_{n})_{n}$ such that for each $n$, $|f_{n}(x_{n})-f(z_{n})|\leq 1/n$ and $|\nabla f_{n}|(x_{n})\leq|\nabla f|(z_{n})\leq 1/n$. This sequence has the required properties.

6 Trustworthiness and stability of subdifferentials

In the sequel, we call subdifferential any operator $\partial$ that associates a set $\partial f(x)\subset X^{*}$ to any triplet $(X,f,x)$, where $X$ is Banach space, $f\in{\rm LSC}(X)$ and $x\in X$, in such a way that the following properties are satisfied:

(A1) If $f$ is convex near $x$, then $\partial f(x)=\{x^{*}\in X^{*}:\langle x^{*},y-x\rangle+f(x)\leq f(y),\leavevmode\nobreak\ \leavevmode\nobreak\ \forall y\in X\}$;

(A2) If $F(x,y)=f(x)+g(y)$, then $\partial F(x,y)\subset\partial f(x)\times\partial g(y)$;

(A3) For any $f$ and $x^{*}\in X^{*}$, $\partial(f+x^{*})(x)=\partial f(x)+x^{*}$.

There are many other basic properties shared by all interesting subdifferentials (see [10, Definition 2.1]). But in what follows we need only the three properties above.

We write $\partial f:=\{(x,x^{*})\in X\times X^{*}:x^{*}\in\partial f(x)\}$ for the graph of $\partial f$. As in [10, Definition 2.12], we say that a subdifferential $\partial$ is trustworthy on a space $X$, or that $X$ is a trustworthy space for $\partial$, if the following rule holds:

(R1) Fuzzy minimization rule. For any $f\in{\rm LSC}(X)$ and $\varphi$ convex Lipschitz, if $f+\varphi$ admits a finite local minimum at $z$, then there are sequences $((x_{n},x_{n}^{*}))_{n}\subset\partial f$ and $((y_{n},y_{n}^{*}))_{n}\subset\partial\varphi$ such that $x_{n}\to z$, $y_{n}\to z$, $f(x_{n})\to f(z)$ and $x_{n}^{*}+y_{n}^{*}\to 0$.

Example 4*.*

Main trustworthy spaces (see [12, 3, 10, 15] and the references therein):

1. $X$ is a Hilbert space, $\partial$ is the proximal subdifferential;

2. $X$ is an Asplund space, $\partial$ is the Fréchet or the limiting Fréchet subdifferential;

3. $X$ is a separable Banach space, $\partial$ is the Hadamard subdifferential;

4. $X$ is any Banach space, $\partial$ is the subdifferential of Clarke, of Michel-Penot, or of Ioffe.

The rule (R1) for trustworthiness expresses a subdifferential necessary condition for a point $z$ to be a local minimizer of the penalized function $f+\varphi$ where $\varphi$ is convex Lipschitz. In fact, trustworthiness can be characterized by various properties related to such penalized functions. For example:

(P1) Necessary condition for an approximate local minimizer. For any $f\in{\rm LSC}(X)$, $\varphi$ convex Lipschitz, $\lambda>0$ and $\sigma>0$, if

$(f+\varphi)(z)<\inf_{B_{\lambda}(z)}(f+\varphi)+\lambda\sigma,$

then there exist $(\bar{x},\bar{x}^{*})\in\partial f$ and $(\bar{y},\bar{y}^{*})\in\partial\varphi$ such that $\|\bar{x}-z\|<\lambda$, $\|\bar{y}-z\|<\lambda$, $|f(\bar{x})+\varphi(\bar{y})-(f(z)+\varphi(z)|<\lambda\sigma$ and $\|\bar{x}^{*}+\bar{y}^{*}\|<\sigma$.

(P2) Slope control. For any $f\in{\rm LSC}(X)$, $\varphi$ convex Lipschitz and $z\in{\rm dom}\kern 1.49994ptf$, there are sequences $((x_{n},x_{n}^{*}))_{n}\subset\partial f$ and $((y_{n},y_{n}^{*}))_{n}\subset\partial\varphi$ such that

$x_{n}\to_{f}z$, $y_{n}\to z$ and $\limsup_{n}\|x_{n}^{*}+y_{n}^{*}\|\leq|\nabla(f+\varphi)|(z)$.

(P3) Fréchet subdifferential control. For any $f\in{\rm LSC}(X)$, $\varphi$ convex Lipschitz and $(z,z^{*})\in\partial_{F}(f+\varphi)$, there are sequences $((x_{n},x_{n}^{*}))_{n}\subset\partial f$ and $((y_{n},y_{n}^{*}))_{n}\subset\partial\varphi$ such that

$x_{n}\to_{f}z$, $y_{n}\to z$ and $x_{n}^{*}+y_{n}^{*}\to z^{*}$.

We recall that the Fréchet subdifferential of $f$ at $\bar{x}$ is given by

[TABLE]

which, as observed in [11, Proposition 4.1], can be conveniently rewritten as

[TABLE]

Property (P1) was considered for the first time in [7] (in the special case where $\varphi$ is a continuous linear form) and in [14] for the Fréchet subdifferential and $\varphi=0$. The following proposition was established in [11, Théorème 4.2]. We briefly recall the proof for the sake of completeness.

Proposition 6.1**.**

Rule (R1) and Properties (P1)–(P3) are equivalent.

**Proof. **(R1) $\Rightarrow$ (P1) with $\varphi$ linear continuous was already observed in [7, Theorem 3.2]. Let $f\in{\rm LSC}(X)$, $\varphi$ convex Lipschitz, $\lambda>0$ and $\sigma>0$ such that $(f+\varphi)(z)<\inf_{B_{\lambda}(z)}(f+\varphi)+\lambda\sigma$. Apply Ekeland’s variational principle to $g:=f+\varphi+\delta_{B_{\lambda}(z)}$ with $0<\sigma^{\prime}<\sigma$ such that $g(z)<\inf_{X}g+\lambda\sigma^{\prime}$ and with $\lambda^{\prime}$ such that $0<\lambda^{\prime}<\lambda$. We obtain a point $\bar{z}\in B_{\lambda^{\prime}}(z)$, with $|(f+\varphi)(\bar{z})-(f+\varphi)(z)|\leq\lambda\sigma^{\prime}$, that is a local minimizer of the function $f+\varphi+\sigma^{\prime}\|\;.-\bar{z}\|$.

Now we apply (R1) to $f$ and $\psi:=\varphi+\sigma^{\prime}\|\;.-\bar{z}\|$ at point $\bar{z}$ to get $(\bar{x},\bar{x}^{*})\in\partial f$ and $(\bar{y},\bar{y}^{*})\in\partial\psi$ such that $\|\bar{x}-\bar{z}\|<\lambda-\lambda^{\prime}$, $|f(\bar{x})-f(\bar{z})|<\lambda(\sigma-\sigma^{\prime})/2$, $\varphi(\bar{y})-\varphi(z)|<\lambda(\sigma-\sigma^{\prime})/2$ and $\|\bar{x}^{*}+\bar{y}^{*}\|<\sigma-\sigma^{\prime}$. Combining the above inequalities, we infer that $\|\bar{x}-z\|<\lambda$ and $|f(\bar{x})+\varphi(\bar{y})-f(z)-\varphi(z)|<\lambda\sigma$. On the other hand, as $\bar{y}^{*}\in\partial\psi(\bar{y})$, using (A1) and standard calculus rules of convex analysis, we derive that $\bar{y}^{*}=\bar{y}^{*}_{0}+\sigma^{\prime}\xi^{*}$ where $\bar{y}^{*}_{0}\in\partial\varphi(\bar{y})$ and $\|\xi^{*}\|\leq 1$, so

[TABLE]

Thus, $(\bar{x},\bar{x}^{*})\in\partial f$ and $(\bar{y},\bar{y}^{*}_{0})\in\varphi$ satisfy the required inequalities of (P1).

(P1) $\Rightarrow$ (P2). Let $|\nabla(f+\varphi)|(z)<\sigma$. We have to show that for any $\varepsilon>0$, there exist $(x,x^{*})\in\partial f$ and $(y,y^{*})\in\partial\varphi$ such that $\|x-z\|<\varepsilon$, $\|y-z\|<\varepsilon$, $|f(x)-f(z)|<\varepsilon$ and $\|x^{*}+y^{*}\|<\sigma$. Fix $\sigma^{\prime}$ such that $|\nabla(f+\varphi)|(z)<\sigma^{\prime}<\sigma$. From the definition of the strong slope, we derive that there is $\lambda^{\prime}>0$ such that $(f+\varphi)(x)-(f+\varphi)(y)<\sigma^{\prime}\|x-y\|$ for every $y\in B_{\lambda^{\prime}}(x)$, which implies that for all $\lambda\in]0,\lambda^{\prime}]$, one has $(f+\varphi)(x)<(f+\varphi)(y)+\sigma^{\prime}\lambda$ for every $y\in B_{\lambda}(x)$. Hence, for every $\lambda\in]0,\lambda^{\prime}]$, it holds

[TABLE]

Let $\varepsilon>0$. Apply (P1) with $\lambda=\min\{\lambda^{\prime},\varepsilon,\varepsilon/\sigma\}$. We obtain $(x,x^{*})\in\partial f$ and $(y,y^{*})\in\partial\varphi$ such that $\|x-z\|<\lambda\leq\varepsilon$, $\|y-z\|<\lambda\leq\varepsilon$, $|f(x)+\varphi(y)-f(z)-\varphi(z)|<\lambda\sigma\leq\varepsilon$ and $\|x^{*}+y^{*}\|<\sigma$. Since $\varphi$ is continuous, we can manage so that the penultimate inequality induces $|f(x)-f(z)|<\varepsilon$. The proof is complete.

(P2) $\Rightarrow$ (P3). Let $z^{*}\in\partial_{F}(f+\varphi)(z)$. By (9), this amounts to $|\nabla(f+\varphi-z^{*})|(z)=0$. Applying (P2) we get sequences $((x_{n},x_{n}^{*}))_{n}\subset\partial(f-z^{*})$ and $((y_{n},y_{n}^{*}))_{n}\subset\partial\varphi$ with $x_{n}\to z$, $y_{n}\to z$, $(f-z^{*})(x_{n})\to(f-z^{*})(z)$ and $x_{n}^{*}+y_{n}^{*}\to 0$. From (A3), we derive that $\bar{x}_{n}^{*}:=x_{n}^{*}+z^{*}\in\partial f(x_{n})$. Then, the sequences $((x_{n},\bar{x}_{n}^{*}))_{n}$ and $((y_{n},y_{n}^{*}))_{n}$ satisfy the required properties.

(P3) $\Rightarrow$ (R1) is obvious.

The following theorem asserts that Property (P2) is stable with respect to Wijsman perturbations, the next one that Property (P3) is stable with respect to slice perturbations.

Theorem 6.2**.**

Assume (P2) holds on the space $X$. Let $(f_{n})_{n}$ be a sequence of lsc functions and $(\varphi_{n})_{n}$ a sequence of convex Lipschitz functions such that $(f_{n}+\varphi_{n})_{n}$ is Wijsman convergent to a function $f$ at $\bar{z}$. Then, there are elements $(x_{n},x_{n}^{*})\in\partial f_{n}$ and $(y_{n},y_{n}^{*})\in\partial\varphi_{n}$ such that $x_{n}\to\bar{z}$, $y_{n}\to\bar{z}$, $f_{n}(x_{n})+\varphi_{n}(y_{n})\to f(\bar{z})$ and $\limsup_{n}\|x^{*}_{n}+y_{n}^{*}\|\leq|\nabla f|(\bar{z})$.

**Proof. **By Theorem 5.1, there is a sequence $(z_{n})_{n}$ such that $z_{n}\to\bar{z}$, $(f_{n}+\varphi_{n})(z_{n})\to f(\bar{z})$ and $\limsup_{n}|\nabla(f_{n}+\varphi_{n})|(z_{n})\leq|\nabla f|(\bar{z})$. By (P2) applied to $f_{n}$, $\varphi_{n}$ and $z_{n}$, there are sequences $(x_{n})_{n}$ and $(y_{n})_{n}$ such that, for every $n\in{\mathbb{N}}$, $\|x_{n}-z_{n}\|\leq 1/n$, $\|y_{n}-z_{n}\|\leq 1/n$ $|f_{n}(x_{n})-f_{n}(z_{n})|\leq 1/n$ and

[TABLE]

So, for every $n\in{\mathbb{N}}$, there are elements $x_{n}^{*}\in\partial f_{n}(x_{n})$ and $y_{n}^{*}\in\partial\varphi_{n}(y_{n})$ such that

[TABLE]

Since $z_{n}\to\bar{z}$, $f_{n}(z_{n})\to f(\bar{z})$ and $\limsup_{n}|\nabla(f_{n}+\varphi_{n})|(z_{n})\leq|\nabla f|(\bar{z})$, the sequences $((x_{n},x_{n}^{*}))_{n}$ and $((y_{n},y_{n}^{*}))_{n}$ satisfy the required properties.

Theorem 6.3**.**

Assume (P2) holds on the space $X$. Let $(f_{n})_{n}$ be a sequence of lsc functions and $(\varphi_{n})_{n}$ a sequence of convex Lipschitz functions such that $(f_{n}+\varphi_{n})_{n}$ is slice convergent to a function $f$. Then, for each $(\bar{z},\bar{z}^{*})\in\partial_{F}$, there are elements $(x_{n},x_{n}^{*})\in\partial f_{n}$ and $(y_{n},y_{n}^{*})\in\partial\varphi_{n}$ such that $x_{n}\to z$, $y_{n}\to z$, $f_{n}(x_{n})+\varphi_{n}(y_{n})\to f(z)$ and $x^{*}_{n}+y_{n}^{*}\to z^{*}$.

**Proof. **Let $(\bar{z},\bar{z}^{*})\in\partial_{F}$. So $|\nabla(f-\bar{z}^{*})|(z)=0$. By Theorem 3.4, the sequence $((f_{n}-\bar{z}^{*}+\varphi_{n})_{n}$ is Wijsman convergent to $f-\bar{z}^{*}$. Applying Theorem 6.2 and (A3), we find sequences $((x_{n},x_{n}^{*}))_{n}$ and $((y_{n},y_{n}^{*}))_{n}$ such that $x_{n}\to\bar{z}$, $y_{n}\to\bar{z}$, $f_{n}(x_{n})+\varphi_{n}(y_{n})\to f(\bar{z})$ and $\|x^{*}_{n}-\bar{z}^{*}+y_{n}^{*}\|\to 0$. The conclusion follows.

Variants of Theorem 6.2 and Theorem 6.3 are considered in [7, Theorem 4.1] and [8, Theorem 5.1] and several applications are given.

7 Subdifferential Sum Rules

A family of lsc functions $\{f_{1},\ldots,f_{k}\}$ is said to be decouplable at $\bar{x}\in X$ ([11, Définition 3.5]) provided there is $\lambda_{x}>0$ such that for any $\lambda\in[0,\lambda_{x})$,

[TABLE]

It is said to be $X^{*}$-decouplable at $\bar{x}$ if $\{f_{1},\ldots,f_{k},x^{*}\}$ is decouplable at $\bar{x}$ for every $x^{*}\in X^{*}$.

Example 5*.*

Sufficient conditions for a family $\{f_{1},\ldots,f_{k}\}$ to be decouplable at $\bar{x}$:

1. All but at most one of the functions are uniformly continuous near $\bar{x}$ [12].

2. At least one of the functions has compact lower level sets near $\bar{x}$ [12].

3. The function $\sum f_{i}$ achieves a local uniform (decoupled, robust) minimum at $\bar{x}$ [12, 3, 10].

4. $k=2$ and the inf-convolution $f_{B_{\lambda}(\bar{x})}\triangledown g^{-}_{B_{\lambda}(\bar{x})}$ (with $f=f_{1}$, $g=f_{2}$) is lsc at [math] [11].

Let $F:(x_{1},\ldots,x_{k})\in X^{k}\mapsto\sum f_{i}(x_{i})$ be the decoupled sum, $B^{k}_{\lambda}(\bar{x}):=B_{\lambda}(\bar{x})\times\cdots\times B_{\lambda}(\bar{x})$ be the $\lambda$-ball of center $(\bar{x},\ldots,\bar{x})$ in $X^{k}$ with the max-morm, and let $\Delta:=\{(x,\ldots,x)\in X^{k}:x\in X\}$ be the diagonal of $X^{k}$. The two expressions in the decoupling’s condition (10) can be written as

$\displaystyle\sup_{\delta>0}\inf_{B_{\lambda+\delta}(\bar{x})}\sum f_{i}=r_{B_{\lambda}(\bar{x})}\left(\sum f_{i}\right)=r_{B^{k}_{\lambda}(\bar{x})}(F_{\Delta}),$

[TABLE]

So the decoupling’s condition (10) amounts to:

[TABLE]

Proposition 7.1**.**

The family $\{f_{1},\ldots,f_{k}\}$ is decouplable at $\bar{x}\in X$ if and only if the sequence $F_{n}:=F+nd_{\Delta}$ is Wijsman convergent to $F_{\Delta}$ at $(\bar{x},\ldots,\bar{x})$.

**Proof. **According to the above, the family $\{f_{1},\ldots,f_{k}\}$ is decouplable at $\bar{x}\in X$ if and only if (10b) holds for any $\lambda>0$ small enough, and according to Proposition 4.3, this latter condition means that the sequence $F_{n}:=F+nd_{\Delta}$ is Wijsman convergent to $F_{\Delta}$ at $(\bar{x},\ldots,\bar{x})$.

We consider generalizations of (P2) and (P3) to decouplable families of functions:

(R2) Slope control. Let $\{f_{1},\ldots,f_{k}\}\subset{\rm LSC}(X)$ be decouplable at $\bar{x}\in X$. If $|\nabla(\sum f_{i})|(\bar{x})<\infty$, then there are sequences $((x_{i,n},x_{i,n}^{*}))_{n}\subset\partial f_{i}$, $i=1,\ldots,k$, such that

$x_{i,n}\to_{f_{i}}\bar{x}$, $\limsup_{n}\|\sum x^{*}_{i,n}\|\leq\left|\nabla\left(\sum f_{i}\right)\right|(\bar{x})$ and ${\rm diam}\kern 1.49994pt(x_{1,n},\ldots,x_{k,n})\,\|x^{*}_{i,n}\|\to 0$.

(R3) Fréchet subdifferential control. Let $\{f_{1},\ldots,f_{k}\}\subset{\rm LSC}(X)$ be $X^{*}$-decouplable at $\bar{x}\in X$. For any $\bar{x}^{*}\in\partial_{F}\left(\sum f_{i}\right)(\bar{x})$, there are sequences $((x_{i,n},x_{i,n}^{*}))_{n}\subset\partial f_{i}$, $i=1,\ldots,k$, such that

$x_{i,n}\to_{f_{i}}\bar{x}$, $\sum x^{*}_{i,n}\to\bar{x}^{*}$ and ${\rm diam}\kern 1.49994pt(x_{1,n},\ldots,x_{k,n})\,\|x^{*}_{i,n}\|\to 0$.

Theorem 7.2**.**

Let $\mathcal{X}$ be a class of Banach spaces which contains Cartesian products of its elements. If (R1) holds on every space in $\mathcal{X}$, then so do (R2) and (R3).

**Proof. **Let $X$ be a space in the class $\mathcal{X}$. We show with some details that (R2) holds on $X$. By Proposition 7.1, the sequence of functions $F+nd_{\Delta}$ defined on $X^{k}$ is Wijsman convergent to the function $F_{\Delta}$ at $(\bar{x},\ldots,\bar{x})$. Since (R1) holds on $X^{k}$, we may apply Theorem 6.2 with $f_{n}:=F$, $\varphi_{n}:=nd_{\Delta}$, $f:=F_{\Delta}$ and $\bar{z}:=(\bar{x},\ldots,\bar{x})$. This produces elements $(\hat{x}_{n},\hat{x}^{*}_{n})\in\partial F$ and $(\hat{y}_{n},\hat{y}^{*}_{n})\in\partial(nd_{\Delta})$ such that

(a) $\hat{x}_{n}\to(\bar{x},\ldots,\bar{x})$, $\hat{y}_{n}\to(\bar{x},\ldots,\bar{x})$,

(b) $F(\hat{x}_{n})+nd_{\Delta}(\hat{y}_{n})\to F_{\Delta}(\bar{x},\ldots,\bar{x})=\sum f_{i}(\bar{x})$,

(c) $\limsup_{n}\|\hat{x}^{*}_{n}+\hat{y}^{*}_{n}\|\leq|\nabla F_{\Delta}|(\bar{x},\ldots,\bar{x})=\left|\nabla\left(\sum f_{i}\right)\right|(\bar{x})$.

The point $\hat{x}_{n}$ can be written as $\hat{x}_{n}=(x_{1,n},\ldots,x_{k,n})$. Since by (A2),

[TABLE]

we have $\hat{x}^{*}_{n}=(x^{*}_{1,n},\ldots,x^{*}_{k,n})$ with $x^{*}_{i,n}\in\partial f_{i}(x_{i,n})$ for every $i=1,\ldots,k$. We show that the sequences $((x_{i,n},x_{i,n}^{*}))_{n}\subset\partial f_{i}$ satisfy all the requirements.

From (a), we see that $x_{i,n}\to\bar{x}$ for each $i$. Fix $\varepsilon>0$ and $j\in\{1,\ldots,k\}$. By lower semicontinuity of the functions $f_{i}$, for $n$ sufficiently large it holds

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and by (b) above,

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Combining (11) and (12), we get that, for $n$ sufficiently large,

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hence, $f_{j}(x_{j,n})\leq f_{j}(\bar{x})+2\varepsilon$ eventually. This shows that, for every $j$, $f_{j}(x_{j,n})\to f_{j}(\bar{x})$. Finally, we have proved that $x_{i,n}\to_{f_{i}}\bar{x}$ for every $i$.

We now show that $\limsup_{n}\|\sum x^{*}_{i,n}\|\leq\left|\nabla\left(\sum f_{i}\right)\right|(\bar{x})$. Since $\hat{y}^{*}_{n}\in\partial(nd_{\Delta})(\hat{y}_{n})$, we have $\hat{y}^{*}_{n}\in\Delta^{\perp}$ and $\|\hat{y}^{*}_{n}\|\leq n$. On the other hand, $\|{\hat{x}_{n}^{*}}|\Delta\|=\|\sum x^{*}_{i,n}\|$ and (c) implies

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So, $\limsup_{n}\|\sum x^{*}_{i,n}\|\leq\left|\nabla\left(\sum f_{i}\right)\right|(\bar{x})$.

It remains to show that ${\rm diam}\kern 1.49994pt(x_{1,n},\ldots,x_{k,n})\,\|x^{*}_{i,n}\|\to 0$. Let $\varepsilon>0$ be fixed. Since for each $i=1,\ldots,n$, $f_{i}(x_{i,n})\to f_{i}(\bar{x})$, we have $|F(\hat{x}_{n})-F_{\Delta}(\bar{x},\ldots,\bar{x})|\leq\varepsilon$ eventually, so by (b) $d_{\Delta}(\hat{y}_{n})\leq\varepsilon/n$ eventually, and also $d_{\Delta}(\hat{x}_{n})\leq\varepsilon/n$ eventually since $(\hat{y}_{n})_{n}$ and $(\hat{x}_{n})_{n}$ converge to the same point. Let $\gamma=|\nabla\left(\sum f_{i}\right)|(\bar{x})$. It follows from (c) that $\|\hat{x}^{*}_{n}+\hat{y}^{*}_{n}\|\leq\gamma+\varepsilon$ eventually, so $\|\hat{x}^{*}_{n}\|\leq\gamma+\varepsilon+n$ eventually. Combining all of this, for $n$ sufficiently large it holds

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The conclusion follows directly from this last inequality. This completes the proof of (R2).

Proceeding as in Theorem 6.3, it is easy to show that (R3) follows from (R2); we omit the details.

Rules (R2) and (R3) were considered and proved to be equivalent to (R1) in [11, Théorème 5.1]. The above approach through Wijsman stability of subdifferentials is new. More equivalent properties can be found in [17, 9, 12]. See [10] for historical comments.