The maximal monotonicity of the subdifferential in locally convex spaces via upper envelopes

TL;DR

This paper investigates conditions under which the convex subdifferential is maximally monotone in locally convex spaces, providing a general theoretical framework for understanding subdifferential properties.

Contribution

It introduces equivalent conditions for maximal monotonicity of the subdifferential in the broad setting of locally convex spaces, extending existing theory.

Findings

Characterization of maximal monotonicity conditions

Extension of subdifferential theory to locally convex spaces

Theoretical framework for upper envelope representations

Abstract

Abstract Equivalent conditions that make the convex subdifferential maximal monotone are investigated in the general settings of locally convex spaces.

Equations

$${\phi }_T(x,x^{\ast }):=\sup \{⟨x-a,a^{\ast }⟩+⟨a,x^{\ast }⟩\mid (a,a^{\ast })\in \mathrm{Graph}T\},(x,x^{\ast })\in X\times X^{\ast },$$

$${\phi }_{\partial f}(x,x^{\ast })\le f^{\ast \ast }(x)+f^{\ast }(x^{\ast }).$$

$$\partial f\in \mathfrak{M}(X),g=f\mathrm{on}D(\partial f),g\ge f\Rightarrow \partial g=\partial f\in \mathfrak{M}(X).$$

$$D(\partial f)\subset \{x\in \mathrm{dom}f\mid f\mathrm{is}\mathrm{lsc}\mathrm{at}x\}\subset \{x\in \mathrm{dom}f\mid f(x)=(\mathrm{cl}\mathrm{conv}f)(x)\},$$

$$(\mathrm{dom}(\mathrm{cl}\mathrm{conv})f\setminus \mathrm{dom}f)\cap \{x\in X\mid f\mathrm{is}\mathrm{lsc}\mathrm{at}x\}=\varnothing .$$

$$f^{\cup }(x):=\sup \{⟨x-a,a^{\ast }⟩+f(a)\mid (a,a^{\ast })\in \mathrm{Graph}\partial f\}=\sup \{⟨x,a^{\ast }⟩-f^{\ast }(a^{\ast })\mid a^{\ast }\in R(\partial f)\};$$

$$f^{\#}:=f^{\cup }+{\iota }_{(\mathrm{dom}f{)}^{\#}}.$$

$$(\mathrm{Epi}f{)}^{\#}=\mathrm{Epi}(f^{\cup })\cap ((\mathrm{dom}f{)}^{\#}\times \mathbb{R})=\mathrm{Epi}(f^{\#})\ne \varnothing ;$$

$$f^{n\cup }(x):=\sup \{⟨x-a_1,a_1^{\ast }⟩+⟨a_1-a_2,a_2^{\ast }⟩+\dots$$

$$\mid (a_k,a_k^{\ast })\in \mathrm{Graph}\partial f,k=\overline{1,n}\}.$$

$$\forall a\in \mathrm{dom}f,\forall (a^{\ast },\alpha )\in N_{\mathrm{Epi}f}(a,f(a))\mathrm{with}\alpha \ne 0,⟨(x,v)-(a,f(a)),(a^{\ast },\alpha )⟩\le 0,$$

$$g^{\cup }(x^{\ast }):=\sup \{⟨x^{\ast }-a^{\ast },a⟩+g(a^{\ast })\mid (a^{\ast },a)\in \mathrm{Graph}\partial g\}=\sup \{⟨x^{\ast },a⟩-g^{\ast }(a)\mid a\in R(\partial g)\}.$$

$$f^{\ast \cup }:=(f^{\ast }{)}^{\cup },f^{\cup \ast }:=(f^{\cup }{)}^{\ast },f^{\circ }:=f^{\ast \cup \ast }=(f^{\ast \cup }{)}^{\ast }.$$

$$f^{\ast \cup }(x^{\ast })$$

$$=\sup \{⟨x^{\ast },a⟩-f(a)\mid a\in D(\partial f)\}=(f+{\iota }_{D(\partial f)}{)}^{\ast }(x^{\ast });$$

$$f^{\circ }=(f+{\iota }_{D(\partial f)}{)}^{\ast \ast }=\mathrm{cl}\mathrm{conv}(f+{\iota }_{D(\partial f)}).$$

$${\phi }_{\partial f}(x,x^{\ast })\le f^{\cup }(x)+f^{\ast \cup }(x^{\ast });$$

$${\sigma }_C^{\cup }(x^{\ast })=\sup \{⟨x^{\ast }-a^{\ast },a⟩+{\sigma }_C(a^{\ast })\mid (a,a^{\ast })\in \mathrm{Graph}{N}_C\}\ge {\sigma }_C(x^{\ast }),$$

$$f^{\cup }\le (\mathrm{cl}\mathrm{conv}f{)}^{\cup }\le \mathrm{cl}\mathrm{conv}f\le f,$$

$$\forall f\in \Gamma (X),\mathrm{int}(\mathrm{Epi}f)\ne \varnothing ,\forall x\in \mathrm{dom}f,f(x)=f^{\cup }(x).$$

$$f(x)$$

$$=\sup \{⟨x-a,a^{\ast }+n^{\ast }⟩+f(a)\mid (a,a^{\ast })\in \mathrm{Graph}\partial f,n^{\ast }\in N_{M_{f(x)}}(a),f(a)\le f(x)\}$$

$$=\sup \{⟨x-a,a^{\ast }⟩+f(a)\mid (a,a^{\ast })\in \mathrm{Graph}\partial f,f(a)\le f(x)\}$$

$$\le {\phi }_{\partial f}(x,0)+f(x).$$

$$\forall f\in \Gamma (X),\mathrm{int}(\mathrm{Epi}f)\ne \varnothing ,$$

$${\phi }_{\partial f}(x,x^{\ast })=$$

$$C^{\ast }:=x_0^{\ast }+\{y^{\ast }\in X^{\ast }\mid |⟨x-x_0,y^{\ast }⟩|\le \max \{1,|⟨x-x_0,x^{\ast }-x_0^{\ast }⟩|\}\}.$$

$$⟨x,x^{\ast }⟩\le {\phi }_{\partial (f\square {\sigma }_{C^{\ast }})}(x,x^{\ast })=\sup \{⟨x^{\ast }-a^{\ast },a⟩+⟨x,a^{\ast }⟩\mid (a,a^{\ast })\in \partial (f\square {\sigma }_{C^{\ast }})\}$$

$$=\sup \{⟨x^{\ast }-a^{\ast },b⟩+⟨x^{\ast }-a^{\ast },n⟩+⟨x,a^{\ast }⟩\mid a^{\ast }\in R(\partial f)\cap C^{\ast },a^{\ast }\in \partial f(b),n\in N_{C^{\ast }}(a^{\ast })\}$$

$$=\sup \{⟨x^{\ast }-a^{\ast },b⟩+⟨x,a^{\ast }⟩\mid a^{\ast }\in R(\partial f)\cap C^{\ast },a^{\ast }\in \partial f(b)\}\le {\phi }_{\partial f}(x,x^{\ast }).$$

$$f^{⌣}(x)$$

$$=\sup \{⟨x,a^{\ast }⟩-f^{\ast }(a^{\ast })\mid (a,a^{\ast })\in \mathrm{Graph}(\partial f),f(a)\le f(x)\},x\in X.$$

$$f_{ϵ}^{⌣}(x)$$

$$=\sup \{⟨x,a^{\ast }⟩-f^{\ast }(a^{\ast })\mid (a,a^{\ast })\in \mathrm{Graph}(\partial f),f(a)\le f(x)+ϵ\},x\in X,ϵ>0.$$

Full text

1 Preliminaries and notations

The aim of this paper is to provide equivalent conditions that allow the convex subdifferentials of all proper convex lower semicontinuous functions defined in a locally convex space be maximal monotone. Our goal will be achieved through the introduction of several types of upper envelopes for a function followed by a study of their properties in connection with the maximal monotonicity of that function subdifferential.

We will address our objective mainly via our characterization of maximal monotonicity that says that an operator is maximal monotone iff it is representable and of type NI (introduced in [10, Theorem 2.3]; see also [11, Theorem 3.4], or Theorem 10 below). That characterization is provided in terms of the Fitzpatrick function $\varphi_{T}:X\times X^{*}\rightarrow\overline{\mathbb{R}}$ of a multi-valued operator $T:X\rightrightarrows X^{*}$ which is given by (see [2])

[TABLE]

where $(X,\tau)$ is a non-trivial (that is, $X\neq\{0\}$) real Hausdorff separated locally convex space (LCS for short), $X^{\ast}$ is its topological dual usually endowed with the weak-star topology denoted by $w^{*}$, $(X^{*},w^{*})^{*}$ is identified with $X$, $\left\langle x,x^{\ast}\right\rangle:=x^{\ast}(x)=:c(x,x^{\ast})$, for $x\in X$, $x^{\ast}\in X^{\ast}$ denotes the duality product or *coupling *of $X\times X^{\ast}$, and $\operatorname*{Graph}T=\{(x,x^{*})\in X\times X^{*}\mid x^{*}\in T(x)\}$ stands for the graph of $T:X\rightrightarrows X^{*}$.

When $X$ is a Banach space, Rockafellar showed in [7, Theorem A] that if $f:X\to\overline{\mathbb{R}}$ is proper convex lower semicontinuous then its convex subdifferential $\partial f:X\rightrightarrows X^{*}$, defined by $x^{*}\in\partial f(x)$ if $f(x)$ is finite and for every $y\in X$, $f(y)\geq f(x)+\langle y-x,x^{*}\rangle$, is maximal monotone ($\partial f\in\mathfrak{M}(X)$ for short). In particular the normal cone to $C$ which is given by $N_{C}=\partial\iota_{C}\in\mathfrak{M}(X)$, whenever $C\subset X$ is closed convex. Here $\iota_{C}(x)=0$, for $x\in C$; $\iota_{C}(x)=+\infty$, for $x\in X\setminus C$ denotes the indicator function of $C$.

Due to the biconjugate formula and since the maximal monotonicity is a duality property, the same holds when $X$ is the topological dual of a Banach space endowed with the weak-star topology. Let us mention here that, when $X$ is a non-reflexive Banach space, its dual $X^{*}$ is neither a Banach space nor barreled under any topology (including its Mackey topology) compatible with the duality $(X^{*},X)$.

Part of the arguments used in [7] coming from [6, Proposition 6 (b), (c)] and [1, (A), (B)] is to show that the subdifferentiability domain of a proper convex lower semicontinuous function determines perfectly the function in the sense that the function coincide with the closed convex hull of its restriction on its subdifferentiability domain. Also, as a consequence, every proper convex lower semicontinuous function defined in a Banach space coincides with the supremum of the supporting affine functions determined by its subgradients (see again [1, Theorem 2, p. 609]).

Naturally, one may ask whether these properties are characteristic to the previously mentioned two distinct types of spaces: Banach spaces and duals of Banach spaces (endowed with the weak star topology) and, whether these properties can be extended to general LCS’s, in the sense of finding conditions such as conditions (A), (B) in [1] that allow all the convex subdifferential be maximal monotone.

It is interesting to mention here that the answer for the question stated in [1, p. 606]: “It would be interesting to know whether every space for which (A) and (B) always hold is necessarily a Banach space "in disguise" (i.e. in its Mackey topology) as the counter-examples seem to suggest.” is negative since, given a non-reflexive Banach space $X$, its topological dual $X^{*}$ has properties (A), (B) with respect to the duality $(X^{*},X)$ but no barreled (and moreover Banach space) topology on $X^{*}$ compatible with the duality $(X^{*},X)$ exists.

Our main argument is built on the introduction of several envelopes that associate naturally to a function whose subdifferential is maximal monotone.

As usual, given a LCS $(E,\mu)$ and $A\subset E$ we denote by “$A^{\#}$” the portable hull of $A$ (see [14] for more details), “$\operatorname*{conv}A$” the convex hull of $A$, “$\operatorname*{cl}_{\mu}(A)=\overline{A}^{\mu}$” the $\mu-$closure of $A$, “$\operatorname*{int}_{\mu}A$” the $\mu-$*topological interior *of $A$, “$\operatorname*{core}A$” the algebraic interior of $A$.

For $f,g:E\rightarrow\overline{\mathbb{R}}$ we set $[f\leq g]:=\{x\in E\mid f(x)\leq g(x)\}$; the sets $[f=g]$, $[f<g]$, and $[f>g]$ being defined in a similar manner. We write $f\geq g$ shorter for $f(z)\geq g(z)$, for every $z\in E$.

The class of closed convex neighborhoods of $x\in X$ in $(X,\tau)$ is denoted by $\mathscr{V}_{\tau}(x)$.

For a multi-function $T:X\rightrightarrows X^{*}$, $D(T)=\operatorname*{Pr}_{X}(\operatorname*{Graph}T)$, $R(T)=\operatorname*{Pr}_{X^{*}}(\operatorname*{Graph}T)$ stand for the domain and the range of $T$ respectively, where $\operatorname*{Pr}_{X}$, $\operatorname*{Pr}_{X^{*}}$ denote the projections of $X\times X^{*}$ onto $X$, $X^{*}$ respectively. When no confusion can occur, $T:X\rightrightarrows X^{\ast}$ will be identified with $\operatorname*{Graph}T\subset X\times X^{*}$.

We consider the following classes of functions and operators on $(X,\tau)$

$\Lambda(X)$

the class formed by proper convex functions $f:X\rightarrow\overline{\mathbb{R}}$. Recall that $f$ is proper if $\operatorname*{dom}f:=\{x\in X\mid f(x)<\infty\}$ is nonempty and $f$ does not take the value $-\infty$,

$\Gamma_{\tau}(X)$

the class of functions $f\in\Lambda(X)$ that are $\tau$–lower semi-continuous (*$\tau$–*lsc for short),

$\mathcal{M}(X)$

the class of non-empty monotone operators $T:X\rightrightarrows X^{\ast}$ ($\operatorname*{Graph}T\neq\emptyset$). Recall that $T:X\rightrightarrows X^{\ast}$ is monotone if, for all $(x_{1},x_{1}^{*}),(x_{2},x_{2}^{*})\in\operatorname*{Graph}T$, $\left\langle x_{1}-x_{2},x_{1}^{\ast}-x_{2}^{\ast}\right\rangle\geq 0$ or, equivalently, $\operatorname*{Graph}T\subset\operatorname*{Graph}(T^{+})$.

$\mathfrak{M}(X)$

the class of maximal monotone operators $T:X\rightrightarrows X^{*}$. The maximality is understood in the sense of graph inclusion as subsets of $X\times X^{*}$.

To a proper function $f:(X,\tau)\rightarrow\overline{\mathbb{R}}$ we associate the following notions:

$\operatorname*{Epi}f:=\{(x,t)\in X\times\mathbb{R}\mid f(x)\leq t\}$

is the epigraph of $f$,

$\operatorname*{conv}f:X\rightarrow\overline{\mathbb{R}}$,

the convex hull of $f$, which is the greatest convex function majorized by $f$, $(\operatorname*{conv}f)(x):=\inf\{t\in\mathbb{R}\mid(x,t)\in\operatorname*{conv}(\operatorname*{Epi}f)\}$ for $x\in X$,

$\operatorname*{cl}_{\tau}\operatorname*{conv}f:X\rightarrow\overline{\mathbb{R}}$,

the $\tau-$lsc convex hull of $f$, which is the greatest *$\tau$–*lsc convex function majorized by $f$, $(\operatorname*{cl}_{\tau}\operatorname*{conv}f)(x):=\inf\{t\in\mathbb{R}\mid(x,t)\in\operatorname*{cl}_{\tau}(\operatorname*{conv}\operatorname*{Epi}f)\}$ for $x\in X$,

$f^{\ast}:X^{\ast}\rightarrow\overline{\mathbb{R}}$

is the convex conjugate of $f:X\rightarrow\overline{\mathbb{R}}$ with respect to the dual system $(X,X^{\ast})$, $f^{\ast}(x^{\ast}):=\sup\{\left\langle x,x^{\ast}\right\rangle-f(x)\mid x\in X\}$ for $x^{\ast}\in X^{\ast}$.

Accordingly, for $C\subset X$, $\sigma_{C}(x^{*}):=\sup\{\langle x,x^{*}\rangle\mid x\in C\}=\iota_{C}^{*}(x^{*})$, for $x^{*}\in X^{*}$. Recall that $f^{**}:=(f^{*})^{*}=\operatorname*{cl}\operatorname*{conv}f$ whenever $\operatorname*{cl}\operatorname*{conv}f$ (or equivalently $f^{*}$) is proper. Here, for functions defined in $X^{*}$, all the conjugates are taken with respect to the dual system $(X^{*},X)$.

Throughout this article the conventions $\infty-\infty=\infty$, $\sup\emptyset=-\infty$, and $\inf\emptyset=\infty$ are enforced while the use of the topology notation is avoided when the topology is clearly understood.

2 Functions with maximal monotone subdifferentials

In this section we are concerned with properties of functions $f:X\to\overline{\mathbb{R}}$ for which $\partial f\in\mathfrak{M}(X)$ and try to identify the natural function framework under which we can find conditions on a function that make its subdifferential maximal monotone.

Theorem 1

Let $X$ be a LCS and let $f:X\to\overline{\mathbb{R}}$. For every $(x,x^{*})\in X\times X^{*}$,

[TABLE]

(i)* If $\operatorname*{Graph}(\partial f)\neq\emptyset$ then $f$, $\operatorname*{cl}\operatorname*{conv}f$, $f^{*}$ are proper, $f^{**}=\operatorname*{cl}\operatorname*{conv}f$, $\operatorname*{Graph}(\partial f)\subset\operatorname*{Graph}(\partial(\operatorname*{cl}\operatorname*{conv}f))$, and, for every $x\in D(\partial f)$, $\partial f(x)=\partial(\operatorname*{cl}\operatorname*{conv}f)(x)$.*

If, in addition, $\partial f\in\mathfrak{M}(X)$ then $\partial f=\partial(\operatorname*{cl}\operatorname*{conv}f)$ and $\operatorname*{conv}D(\partial f)$ is dense in $\operatorname*{conv}(\operatorname*{dom}(\operatorname*{cl}\operatorname*{conv})f)$.

(ii)* If $h:\mathbb{R}\to\overline{\mathbb{R}}$ has $\partial h\in\mathfrak{M}(\mathbb{R})$ then $h$ is proper convex. There exists a proper convex $g:\mathbb{R}\to\overline{\mathbb{R}}$ which is not lsc but $\partial g\in\mathfrak{M}(\mathbb{R})$.*

(iii)* If $X$ is at least two-dimensional then there exists $g:X\to\overline{\mathbb{R}}$ which is neither convex nor lsc such that $\partial g\in\mathfrak{M}(X)$.*

(iv)* Assume, in addition, that $X$ is a Banach space or that $f$ is continuous at some $\overline{x}\in\operatorname*{dom}f$. Then $f$ is lsc and $\partial f\in\mathfrak{M}(X)$ iff $f\in\Gamma(X)$.*

(v)* Conversely, if $\partial(\operatorname*{cl}\operatorname*{conv}f)\in\mathfrak{M}(X)$ then $f$ is proper. There exists $g:X\to\overline{\mathbb{R}}$ which is neither convex nor lsc such that $\partial(\operatorname*{cl}\operatorname*{conv}g)\in\mathfrak{M}(X)$ and $\operatorname*{Graph}(\partial g)\subsetneq\operatorname*{Graph}(\partial(\operatorname*{cl}\operatorname*{conv}g))$.*

Similarly one can show the following result.

Corollary 2

Let $X$ be a LCS and let $f:X\to\overline{\mathbb{R}}$. Assume that $X$ is a Banach space or that $f$ is continuous at some $\overline{x}\in\operatorname*{dom}f$. If $\partial f\in\mathfrak{M}(X)$ then $\operatorname*{cl}f$ is convex.

If, outside its subdifferentiability domain, we increase the values of a function whose subdifferential is maximal monotone, we still get the same maximal monotone subdifferential; but, as previously seen, this process can remove the convexity and/or the lower semicontinuity of the function given these properties are present. More precisely,

[TABLE]

Therefore, in order to be able to find conditions that make the subdifferential maximal monotone, the subdifferentiability domain of a function plays an important role.

In Banach spaces it is known that the subdifferentiability domain of a proper convex lsc function is dense in the domain of the function (see again [15, Theorem 3.1.4 (i), p. 162]). That allows to strengthen part of Theorem 1 (i), (iv).

Proposition 3

Let $X$ be a Banach space and let $f:X\to\overline{\mathbb{R}}$ be such that $\partial f\in\mathfrak{M}(X)$. Then $D(\partial f)$ is dense in $\operatorname*{dom}f$. Moreover, if $f$ is lsc at $x\in\operatorname*{dom}(\operatorname*{cl}\operatorname*{conv}f)$ then there exists $(x_{n})_{n}\subset D(\partial f)$ such that $x_{n}\to x$ in $X$ and $f(x_{n})\to f(x)=(\operatorname*{cl}\operatorname*{conv}f)(x)$. In particular

[TABLE]

[TABLE]

Based on the previous theorem we conclude that the natural function framework, under which we could identify properties of $f$ that make $\partial f$ maximal monotone, is that of proper convex lsc functions; outside this class, as the examples in this section show, that task is impossible.

3 Upper envelopes

First, given $X$ a LSC and $f:X\to\overline{\mathbb{R}}$, we define the upper envelope of the continuous affine functions defined by $\partial f$ as $f^{\cup}:X\to\overline{\mathbb{R}}$ given by

[TABLE]

and the portable hull of $f$ by

[TABLE]

Here for $C\subset X$, $C^{\#}$ denotes the portable hull of $C$ and recall that a set $C\subset X$ is portable if $C=C^{\#}$ (see [14]).

Note that, for every $C\subset X$, $\iota_{C}^{\cup}:=(\iota_{C})^{\cup}=\iota_{C^{\#}}=(\iota_{C})^{\#}=:\iota_{C}^{\#}$.

Throughout this article we use the same notation $E^{\#}$ for the portable hull of a set $E\subset X\times\mathbb{R}$. The notation $f^{\#}$ is natural via its epigraph as we will see in the next result.

Theorem 4

Let $(X,\tau)$ be a LCS and let $f:X\to\overline{\mathbb{R}}$. Then

(i)* $f^{\cup}\leq f^{\#}\leq f$ and, for every $x\in D(\partial f)$, $f^{\cup}(x)=f^{\#}(x)=f(x)$;*

(ii) $f^{\cup}$ is proper iff $\operatorname*{Graph}(\partial f)\neq\emptyset$ iff $f^{\#}$ is proper iff $\operatorname*{Graph}(\partial f)\neq\emptyset$; in this case $f^{\cup},f^{\#}\in\Gamma_{\tau}(X)$;

(iii)* $\operatorname*{Epi}(f^{\cup})\neq\emptyset$; moreover*

[TABLE]

(iv)* $\partial f^{\cup}|_{D(\partial f)}=\partial f^{\#}|_{D(\partial f)}=\partial f$, in particular $\operatorname*{Graph}\partial f\subset\operatorname*{Graph}\partial f^{\cup}\cap\operatorname*{Graph}\partial f^{\#}$;*

(v)* $f^{\cup\cup}:=(f^{\cup})^{\cup}=f^{\cup}=(f^{\cup})^{\#}=:f^{\cup\#}$, $f^{\#\#}:=(f^{\#})^{\#}=f^{\#}$;*

(vi)* $N_{\operatorname*{Epi}f}\in\mathfrak{M}(X\times\mathbb{R})$ iff $f=f^{\#}$,*

(vii)* $N_{\operatorname*{Epi}(f^{\cup})},N_{\operatorname*{Epi}(f^{\#})}\in\mathfrak{M}(X\times\mathbb{R})$ and $(\operatorname*{Epi}(f^{\cup}))^{\#}=\operatorname*{Epi}(f^{\cup})$,*

(viii)* $\partial f=\partial f^{\cup}$ iff $D(\partial f)=D(\partial f^{\cup})$; $\partial f=\partial f^{\#}$ iff $D(\partial f)=D(\partial f^{\#})$,*

(ix)* For $n\geq 2$, let $f^{n\cup}:X\to\overline{\mathbb{R}}$ be defined by*

[TABLE]

Then, for every $n\geq 2$, $f^{n\cup}=f^{\cup}$.

Let us call a function $f$ portable if $f=f^{\#}$. As previously seen

• •

For every $f$, the envelopes $f^{\cup}$, $f^{\#}$ are portable.

• •

$f$ is portable iff $\operatorname*{Epi}f$ is portable.

Remark 5

Note that $\operatorname*{Epi}(f^{\cup})$ is the intersection of all the supporting semi-spaces that contain $\operatorname*{Epi}f$ determined by non-vertical support hyperplanes to $\operatorname*{Epi}f$, that is, $\operatorname*{Epi}(f^{\cup})$ has the form of a partial portable hull of $\operatorname*{Epi}f$ and is given by $(x,v)\in\operatorname*{Epi}(f^{\cup})$ iff

[TABLE]

or $\operatorname*{Epi}(f^{\cup})=(\operatorname*{Epi}f)_{G}^{\#}$, where $G:=\{(a,\lambda;a^{*},\alpha)\in\operatorname*{Graph}(N_{\operatorname*{Epi}f})\mid\lambda=f(a),\ \alpha\neq 0\}$ (see [14]).

With respect to the dual system $(X,X^{*})$ the dual notion of the upper envelope for $g:X^{*}\to\overline{\mathbb{R}}$ is similarly defined as

[TABLE]

Naturally, for $f:X\to\overline{\mathbb{R}}$, one can consider

[TABLE]

These functions are proper iff $\operatorname*{Graph}\partial f\neq\emptyset$ in which case their expanded forms are

[TABLE]

[TABLE]

In particular $f^{*\cup}(0)=-\inf_{D(\partial f)}f$.

Remark 6

The parallelism between the normal cone and the subdifferential stops in two fundamental instances.

First, one cannot expect the Fitzpatrick function of a subdifferential to have separated variables in general. Even though, for every $f:X\to\overline{\mathbb{R}}$, $x\in X$, $x^{*}\in X^{*}$

[TABLE]

there exists examples for which the inequality in (9) can be strict. Take $f(x)=\tfrac{1}{2}x^{2}$, $x\in\mathbb{R}$ for which $\operatorname*{Graph}(\partial f)=\{(x,x)\mid x\in\mathbb{R}\}$, $\partial f\in\mathfrak{M}(\mathbb{R})$, and $\varphi_{\partial f}(x,y)=\frac{1}{4}(x+y)^{2}$, $x,y\in\mathbb{R}$, while $f^{\cup}(x)=f^{*}(x)=f(x)=\tfrac{1}{2}x^{2}$, $x\in\mathbb{R}$.

Second, even though $f^{\#\#}=f^{\#}$, $f^{\cup\cup}=f^{\cup}$ one cannot expect that $\partial f^{\cup}\in\mathfrak{M}(X)$ or $\partial f^{\#}\in\mathfrak{M}(X)$ . Note that in every LCS $X$ and for every $C\subset X$, $\sigma_{C}:(X^{*},w^{*})\to\overline{\mathbb{R}}$ has $\sigma_{C}=\sigma_{C}^{\cup}=\sigma_{C}^{\#}$ while $\partial\sigma_{C}^{(\cup)(\#)}=N_{C}{}^{-1}\in\mathfrak{M}(X^{*})$ iff $C=C^{\#}$ (see [14]).

Indeed, for every $x^{*}\in X^{*}$

[TABLE]

obtained by taking $a\in C$, $a^{*}=0$. That discrepancy comes from the fact that, in general, $\sigma_{C}^{(\cup)}\neq\sigma_{C^{\#}}$.

Notice, in passing that, for $f=\iota_{C}$ or $f=\sigma_{C}$ for $C$ closed convex, (9) becomes an identity.

Theorem 7

Let $(X,\tau)$ be a LCS and let $f:X\to\overline{\mathbb{R}}$ be such that $\operatorname*{Graph}\partial f\neq\emptyset$. Then

(i)* $f^{\cup}\leq\operatorname*{cl}\operatorname*{conv}f\leq f^{\circ}\in\Gamma_{\tau}(X)$ and, for every $x\in D(\partial f)$, $f^{\circ}(x)=f(x)$;*

(ii)* $f^{\circ\circ}=f^{\circ}$, $f^{*\circ*}=f^{\cup}$, $f^{*\cup}=f^{\circ*}$, $f^{*\circ}=f^{\cup*}$;*

(iii)* For every $x\in X$, $\partial f(x)=\partial f^{\circ}(x)\cap R(\partial f)$; in particular $\operatorname*{Graph}\partial f\subset\operatorname*{Graph}\partial f^{\circ}$;*

*(iv) $x\in D(\partial f)$ iff $\partial f^{\circ}(x)\cap R(\partial f)\neq\emptyset$; ***

(v)* $\partial f=\partial f^{\circ}$ iff $R(\partial f)=R(\partial f^{\circ})$.*

Remark 8

Given $(X,\tau)$ a LCS and $f:X\to\overline{\mathbb{R}}$ such that $\operatorname*{Graph}\partial f\neq\emptyset$ the function $g=f+\iota_{D(\partial f)}$ satisfies $\operatorname*{dom}g=D(\partial f)=D(\partial g)$, $g^{\circ}=f^{\circ}=\operatorname*{cl}\operatorname*{conv}g\leq g$ whilst $g^{\circ}\neq g$ is possible, e.g., when $D(\partial f)$ is not convex.

Also, one has

[TABLE]

with the possibility that $f^{\cup}\neq(\operatorname*{cl}\operatorname*{conv}f)^{\cup}$. For example take $f=\iota_{C}$, for which $(\operatorname*{cl}\operatorname*{conv}C)^{\#}\subsetneq C^{\#}$, e.g., $C$ is open convex in a Banach space for which $C^{\#}=X$, $\operatorname*{cl}\operatorname*{conv}C=(\operatorname*{cl}\operatorname*{conv}C)^{\#}=\overline{C}$ while $f^{\#}=f^{\cup}=\iota_{X}$, $(\operatorname*{cl}\operatorname*{conv}f)^{\#}=(\operatorname*{cl}\operatorname*{conv}f)^{\cup}=f^{\circ}=\iota_{\overline{C}}$, $\operatorname*{Graph}\partial f=C\times\{0\}$, $\operatorname*{Graph}(\partial f^{\cup})=X\times\{0\}$, $\partial f^{\circ}=N_{\overline{C}}$.

In general for $C$ non-empty closed convex $\iota_{C}^{\circ}=\iota_{C}$ and $\partial\iota_{C}^{(\circ)}=N_{C}$ which can be non-maximal monotone (see e.g. [3, 4]).

As a prelude to the remaining results of this paper we reprove a known result [12, Theorem 9, p. 282] in order to reveal a general argument for proving the maximality of the subdifferential (see Theorem 10 below) and the necessity for the introduction of another type of envelope in a natural way (see (12) below).

Theorem 9

Let $(X,\tau)$ be a LCS and let $f\in\Gamma(X)$ be continuous on $\operatorname*{int}(\operatorname*{dom}f)$. Then $\partial f\in\mathfrak{M}(X)$.

Proof. Since $\operatorname*{int}(\operatorname*{Epi}f)\neq\emptyset$ we know that $\operatorname*{Epi}f=(\operatorname*{Epi}f)^{\#}=\operatorname*{Epi}(f^{\#})$, i.e., $f=f^{\#}=f^{\cup}+\iota_{(\operatorname*{dom}f)^{\#}}$. In particular

[TABLE]

Denote the closed convex level sets of $f$ by $M_{\lambda}:=\{x\in X\mid f(x)\leq\lambda\}$, $\lambda\in\mathbb{R}$.

If $f=\iota_{C}$ for some closed convex $C\subset X$ with $\operatorname*{int}C\neq\emptyset$ then $\partial f=N_{C}\in\mathfrak{M}(X)$ (see [14, Corollary 4]) and we are done.

Assume that $f$ is non-constant on $\operatorname*{dom}f$. For every $x\in\operatorname*{dom}f$ such that $f(x)>\inf_{X}f$, $\operatorname*{int}M_{f(x)}\neq\emptyset$; in particular $\partial(f+\iota_{M_{f(x)}})=\partial f+N_{M_{f(x)}}$. Similarly, $\operatorname*{int}(\operatorname*{Epi}(f+\iota_{M_{f(x)}})\neq\emptyset$, $f+\iota_{M_{f(x)}}\in\Gamma(X)$ together with (10) and the fact that, due to $x\in M_{f(x)}$ every $n^{*}\in N_{M_{f(x)}}(a)$ has $\langle x-a,n^{*}\rangle\leq 0$, imply

[TABLE]

Therefore, for every $x\in\operatorname*{dom}f$, $\varphi_{\partial f}(x,0)\geq 0$. The same considerations applied to $f-x^{*}$ for an arbitrary $x^{*}\in X^{*}$ show in general that

[TABLE]

Fix $x_{0}\in\operatorname*{dom}f$, $x_{0}^{*}\in\operatorname*{dom}f^{*}$. For every $x\in X$, $x^{*}\in X^{*}$ take $C^{*}$ a closed convex weak-star neighborhood of $x_{0}^{*}$ that contains $x^{*}$ and $x-x_{0}\in\operatorname*{dom}\sigma_{C^{*}}$; such as

[TABLE]

Then $f\square\sigma_{C^{*}}=(f^{*}+\iota_{C^{*}})^{*}$ since $\iota_{C^{*}}$ is continuous at $x_{0}^{*}\in\operatorname*{dom}f^{*}$, $\operatorname*{dom}(f\square\sigma_{C^{*}})=\operatorname*{dom}f+\operatorname*{dom}\sigma_{C^{*}}\ni x$, $\operatorname*{int}(\operatorname*{Epi}(f\square\sigma_{C^{*}}))\neq\emptyset$, and $(a,a^{*})\in\partial(f\square\sigma_{C^{*}})$ iff $a\in\partial f^{*}(a^{*})+N_{C^{*}}(a^{*})$, $a^{*}\in R(\partial f)\cap C^{*}$ (see e.g. [15, Theorem 2.8.7, p. 126]). Then, according to (11)

[TABLE]

[TABLE]

[TABLE]

We showed that $\partial f$ is of NI-type so $\partial f\in\mathfrak{M}(X)$ since $\partial f$ is representable; a representative being given by $F(x,x^{*})=f(x)+f^{*}(x^{*})$, $x\in X$, $x^{*}\in X^{*}$; $F\in\Gamma_{\tau\times w^{*}}(X\times X^{*})$.

As we have seen in the previous proof, for the convex subdifferential the characterization of maximal monotonicity [10, Theorem 2.3] takes the following special form.

Theorem 10

Let $(X,\tau)$ be a LCS and let $f\in\Gamma_{\tau}(X)$. The following are equivalent:

(i)* $\partial f\in\mathfrak{M}(X)$,*

(ii)* under review*

(iii)* For every $(x,x^{*})\in X\times X^{*}$, $\varphi_{\partial f}(x,x^{*})\geq\langle x,x^{*}\rangle$ or $\partial f$ is of NI–type.*

Proposition 11

Let $(X,\tau)$ be a LCS and let $f\in\Gamma_{\tau}(X)$. If $\partial f\in\mathfrak{M}(X)$ then $f=f^{\cup}$ and $f^{*}=f^{*\cup}$.

Remark 12

Note again that whenever $f\in\Gamma_{\tau}(X)$ has $\operatorname*{Graph}(\partial f)\neq\emptyset$ conditions $f=f^{\cup}$ and $f^{*}=f^{*\cup}$ are equivalent to $f^{\cup}=f^{\circ}$. Indeed, if $f=f^{\cup}$ and $f^{*}=f^{*\cup}$ then, by the biconjugate formula, $f=f^{\circ}$. Conversely, if $f^{\cup}=f^{\circ}$, from $f^{\cup}\leq f\leq f^{\circ}$ we know that $f=f^{\cup}$ and $f=f^{*\cup*}$. The latter implies $f^{*}=f^{*\cup}$.**

4 Maximal monotonicity via the low-upper envelope

For $f:X\to\overline{\mathbb{R}}$ define $f^{\smallsmile}:X\to\overline{\mathbb{R}}$ by

[TABLE]

as the low-upper envelope of the continuous functions defined by $\partial f$.

Remark 13

In the case of continuity of $f$ the definition of $f^{\smallsmile}$ is suggested in the proof of Theorem 9 by the use of the implicit upper envelope $(f+\iota_{M_{f(x)}})^{\cup}(x)$ which equals $f^{\smallsmile}(x)$ due to $\partial(f+\iota_{M_{f(x)}})=\partial f+N_{M_{f(x)}}$. However in general $f^{\smallsmile}(x)\leq(f+\iota_{M_{f(x)}})(x)$.

In the case of a maximal monotone subdifferential the properness of $f^{\smallsmile}$ is ensured by Theorem 10 (ii). The properties of $f^{\smallsmile}$ are summarized in the next result.

Theorem 14

Let $(X,\tau)$ be a LCS and let $f:X\to\overline{\mathbb{R}}$. Then

(i)* $f^{\smallsmile}\leq f^{\cup}\leq f^{\#}\leq f$; for every $x\in D(\partial f)$, $f^{\smallsmile}(x)=f(x)$ and, for every $x\not\in\operatorname*{dom}f$, $f^{\smallsmile}(x)=f^{\cup}(x)$;*

(ii) $f^{\smallsmile}$ is proper iff $\operatorname*{Graph}(\partial f)\neq\emptyset$ and $f^{*}(0)=f^{*\cup}(0)$; in this case $f^{\smallsmile}\in\Gamma_{\tau}(X)$;

(iii)* For every $x\in X$, $f^{\smallsmile}(x)\leq\varphi_{\partial f}(x,0)+f(x)$.*

Corollary 15

Let $(X,\tau)$ be a LCS and let $f:X\to\overline{\mathbb{R}}$. Then, for every $x^{*}\in X^{*}$, $(f-x^{*})^{\smallsmile}$ is proper iff $\operatorname*{Graph}(\partial f)\neq\emptyset$ and $f^{*}=f^{*\cup}$.

Theorem 16

Let $(X,\tau)$ be a LCS. The following are equivalent:

(i)* For every $f\in\Gamma_{\tau}(X)$, $\partial f\in\mathfrak{M}(X)$,*

(ii)* For every $f\in\Gamma_{\tau}(X)$, $x\in\operatorname*{dom}f$, $x^{*}\in X^{*}$, $\varphi_{\partial f}(x,x^{*})\geq\langle x,x^{*}\rangle$,*

(iii)* For every $f\in\Gamma_{\tau}(X)$, $x\in\operatorname*{dom}f$, $f(x)=f^{\smallsmile}(x)$,*

(iv)* For every $f\in\Gamma_{\tau}(X)$, $f=f^{\smallsmile}$,*

(v)* For every $f\in\Gamma_{\tau}(X)$, $x\in\operatorname*{dom}f$ there is a net $\{(a_{i},a_{i}^{*})\}_{i\in I}\subset\operatorname*{Graph}(\partial f)$ such that $a_{i}\to x$, $f(a_{i})\to f(x)$, and $\langle x-a_{i},a_{i}^{*}\rangle+f(a_{i})\to f(x)$,*

(vi)* For every $f\in\Gamma_{\tau}(X)$, $x\in\operatorname*{dom}f$ there is a net $\{(a_{i},a_{i}^{*})\}_{i\in I}\subset\operatorname*{Graph}(\partial f)$ such that $\langle x-a_{i},a_{i}^{*}\rangle\to 0$,*

(vii)* For every $f\in\Gamma_{\tau}(X)$, $x\in\operatorname*{dom}f$, $\epsilon>0$ there is $(a_{\epsilon},a_{\epsilon}^{*})\in\operatorname*{Graph}(\partial f)$ such that $\langle x-a_{\epsilon},a_{\epsilon}^{*}\rangle\geq-\epsilon$.*

In this case for every $f\in\Gamma_{\tau}(X)$, $\overline{D(\partial f)}^{\tau}=\operatorname*{cl}_{\tau}(\operatorname*{dom}f)$, $\overline{R(\partial f)}^{w^{*}}=\operatorname*{cl}_{w^{*}}(\operatorname*{dom}f^{*})$ are convex.

Remark 17

For a fixed $f\in\Gamma_{\tau}(X)$, Theorem 16 none of the subpoints (ii)–(vii) imply (i). Indeed, for a closed convex $C\subset X$ we always have, for every $x\in C=\operatorname*{dom}\iota_{C}$, $x^{*}\in X^{*}$, $\varphi_{N_{C}}(x,x^{*})=\iota_{C^{\#}}(x)+\sigma_{C}(x^{*})\geq\langle x,x^{*}\rangle$, $\iota_{C}^{\smallsmile}(x)=\iota_{C}^{\cup}=\iota_{C}^{\#}(x)=\iota_{C^{\#}}(x)=\iota_{C}(x)$, when $0\in C$, $\sigma_{C}^{\smallsmile}=\sigma_{C}\in\Gamma_{w^{*}}(X^{*})$, and for subpoints (v)–(vii) we can take again $f=\iota_{C}$, $a_{i}=a_{\epsilon}=x$, $a_{i}^{*}=a_{\epsilon}^{*}=0$ while $N_{C}$ can be non-maximal monotone.**

Similar considerations can be performed for the extended notion of the low-upper envelope of $f:X\to\overline{\mathbb{R}}$ given by

[TABLE]

Theorem 18

Let $(X,\tau)$ be a LCS. The following are equivalent:

(i)* For every $f\in\Gamma_{\tau}(X)$, $\partial f\in\mathfrak{M}(X)$,*

(ii)* For every $f\in\Gamma_{\tau}(X)$, $x\in\operatorname*{dom}f$, $f(x)=\inf_{\epsilon>0}f_{\epsilon}^{\smallsmile}(x)$,*

(iii)’* For every $f\in\Gamma_{\tau}(X)$, $\epsilon>0$, $x\in\operatorname*{dom}f$, $f(x)=f_{\epsilon}^{\smallsmile}(x)$,*

(iv)’* For every $f\in\Gamma_{\tau}(X)$, $\epsilon>0$, $f=f_{\epsilon}^{\smallsmile}$.*

We conclude this paper by presenting several proofs for the maximality of the subdifferential in a Banach space. We seek minimal Banach space considerations such as a minimal impact of Bronsted-Rockafellar’s Lemma [1, Lemma, p. 608] or Ekeland’s Variational Principle.

Theorem 19

Let $(X,\|\cdot\|)$ be a Banach space. For every $f\in\Gamma(X)$, $\partial f\in\mathfrak{M}(X)$.

Proof 1. Let $f\in\Gamma(X)$. Let $x\in\operatorname*{dom}f$, $\epsilon>0$. Take $x^{*}\in\partial_{\epsilon}f(x)$ and the equivalent norm \|u\|=\left\llbracket u\right\rrbracket+|\langle u,x^{*}\rangle|, $u\in X$. According to [1, Lemma, p. 608] there exists $(a_{\epsilon},a_{\epsilon}^{*})\in\operatorname*{Graph}\partial f$ such that $\|x-a_{\epsilon}\|\leq\sqrt{\epsilon}$ and $\|x^{*}-a_{\epsilon}^{*}\|\leq\sqrt{\epsilon}$. In particular $|\langle x-a_{\epsilon},x^{*}\rangle|\leq\sqrt{\epsilon}$ and

[TABLE]

We have $|\langle x-a_{\epsilon},a_{\epsilon}^{*}\rangle|\leq|\langle x-a_{\epsilon},a_{\epsilon}^{*}-x^{*}\rangle|+|\langle x-a_{\epsilon},x^{*}\rangle|\leq\epsilon+\sqrt{\epsilon}$. Using Theorem 16 we get $\partial f\in\mathfrak{M}(X)$.

Alternatively, since $(a_{\epsilon},a_{\epsilon}^{*})\in\operatorname*{Graph}\partial f$ and $x^{*}\in\partial_{\epsilon}f(x)$ we have

[TABLE]

[TABLE]

[TABLE]

in this case Theorem 18 completes the argument.

Proof 2. Let $f\in\Gamma(X)$, $x\in\operatorname*{dom}f$, $x^{*}\in X^{*}$. According to [15, Theorems 3.1.1, 3.1.4 (i)] there is $(x_{n},x_{n}^{*})_{n\geq 1}\subset\partial f$ such that $x_{n}^{*}\in\partial_{2n^{-2}}f(x)$, $\|x_{n}-x\|\leq n^{-1}$, $|f(x_{n})-f(x)|\leq n^{-2}+n^{-1}$, for every $n\geq 1$. We have

[TABLE]

[TABLE]

Let $n\rightarrow\infty$ to get $\varphi_{\partial f}(x,x^{*})\geq\langle x,x^{*}\rangle$. According to Theorem 16, $\partial f\in\mathfrak{M}(X)$.