Maximal monotone normal cones in locally convex spaces

TL;DR

This paper investigates conditions under which the normal cone in locally convex spaces is maximal monotone, exploring implications like Bishop-Phelps and sum representability results.

Contribution

It provides new equivalent conditions for maximal monotonicity of normal cones in general locally convex spaces.

Findings

Identifies conditions for maximal monotonicity of normal cones.

Derives Bishop-Phelps and sum representability results.

Extends theory to general locally convex spaces.

Abstract

Equivalent conditions that make the normal cone maximal monotone are investigated in the general settings of locally convex spaces. Some consequences such as Bishop Phelps and sum representability results are presented in the last part.

Equations

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

$$⟨x_1-x_2,x_1^{\ast }-x_2^{\ast }⟩\ge 0$$

$$\mathrm{Supp}C=\{x\in C\mid N_C(x)\ne \{0\}\}$$

$$C^{\#}:=\{x\in X\mid \forall (a,a^{\ast })\in \mathrm{Graph}{N}_C,⟨x-a,a^{\ast }⟩\le 0\}$$

$$x\in C^{\#}\iff \forall a\in \mathrm{Supp}C,\forall a^{\ast }\in N_C(a),⟨x-a,a^{\ast }⟩\le 0,$$

$${\phi }_{N_C}(x,x^{\ast })={\iota }_{C^{\#}}(x)+{\sigma }_C(x^{\ast }).$$

$$C^{\#}:=\mathrm{Pr}{}_X(\mathrm{dom}{\phi }_{N_C})=\{x\in X\mid {\phi }_{N_C}(x,0)=(\le )0\}=D(N_C^{+}).$$

$$C_S^{\#}:=\{x\in X\mid \forall a\in C\cap S,a^{\ast }\in N_C(a),⟨x-a,a^{\ast }⟩\le 0\}.$$

$$x\in C_S^{\#}\iff \forall a\in S\cap \mathrm{Supp}C,a^{\ast }\in N_C(a),⟨x-a,a^{\ast }⟩\le 0,$$

$$N_C{|}_S=N_{C_S^{\#}}{|}_S\mathrm{iff}S\cap C=S\cap C_S^{\#}.$$

$${\psi }_{T+N_C}(x,x^{\ast })=\min \{{\psi }_{T{|}_C}(x,x^{\ast }-u^{\ast })+{\sigma }_{C_{D(T)}^{\#}}(u^{\ast })\mid u^{\ast }\in X^{\ast }\},(x,x^{\ast })\in X\times X^{\ast },$$

$$[{\psi }_{T+N_C}=c]=[{\psi }_{T{|}_C}=c]+N_C.$$

Full text

1 Preliminaries

The aim of this paper is to characterize the subsets $C$ of a locally convex space $(X,\tau)$ whose normal cone $N_{C}$ is a maximal monotone operator.

Here $(X,\tau)$ 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 $T:X\rightrightarrows X^{*}$.

Rockafellar showed in [3, Theorem A] that when $X$ is a Banach space and $f:X\to\overline{\mathbb{R}}$ 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 $C$.

Therefore it is interesting to find when a normal cone is maximal monotone outside the Banach space context.

Our main argument stems from the explicit form of the normal cone Fitzpatrick function (see Theorem 1 below) and our characterization of maximal monotone operators as representable and of type NI (see [4, Theorem 2.3] or [5, Theorem 3.4]).

Recall that the Fitzpatrick function $\varphi_{T}:X\times X^{*}\rightarrow\overline{\mathbb{R}}$ of a multi-valued operator $T:X\rightrightarrows X^{*}$ is given by (see [2])

[TABLE]

As usual, given a LCS $(E,\mu)$ and $A\subset E$ we denote by “$\operatorname*{conv}A$” the convex hull of $A$, “$\operatorname*{span}A$” the linear 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 $A$. The use of the $\mu-$notation is not enforced when the topology $\mu$ is clearly understood.

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 $z\in E$.

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 $\operatorname*{Graph}T\subset X\times X^{*}$.

The restriction of an operator $T:X\rightrightarrows X^{*}$ to $U\subset X$ is the operator $T|_{U}:X\rightrightarrows X^{*}$ defined by $\operatorname*{Graph}(T|_{U})=\operatorname*{Graph}T\cap(U\times X^{*})$.

The operator $T^{+}:X\rightrightarrows X^{\ast}$ whose graph is $\operatorname*{Graph}(T^{+}):=[\varphi_{T}\leq c]$ describes all $(x,x^{*})\in X\times X^{*}$ that are monotonically related (m.r. for short) to $T$, that is $(x,x^{*})\in[\varphi_{T}\leq c]$ iff, for every $(a,a^{*})\in\operatorname*{Graph}T$, $\langle x-a,x^{*}-a^{*}\rangle\geq 0$.

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 $(x_{1},x_{1}^{*}),(x_{2},x_{2}^{*})\in\operatorname*{Graph}T$,

[TABLE]

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 $X\times X^{*}$. It is easily seen that $T\in\mathfrak{M}(X)$ iff $T=T^{+}$.

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, $\sigma_{C}(x^{*}):=\sup\{\langle x,x^{*}\rangle\mid x\in C\}=\iota_{C}^{*}(x^{*})$, for $x^{*}\in X^{*}$. Recall that $f^{**}:=(f^{*})^{*}=\operatorname*{cl}\operatorname*{conv}f$ whenever $\operatorname*{cl}\operatorname*{conv}f$ (or equivalently $f^{*}$) is proper, where for functions defined in $X^{*}$, 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.

All the considerations and results of this paper can be done with respect to a separated dual system of vector spaces $(X,Y)$.

2 Support points and the maximality of the normal cone

Given $(X,\tau)$ a LCS and $C\subset X$, we denote by

[TABLE]

the set of *support points of *$C$ and by

[TABLE]

the *portable hull of *$C$ which is the intersection of all the supporting half-spaces that contain $C$ and are supported at points in $C$.

C^{\#}$$C$$\operatorname*{Supp}C

From their definitions, $\operatorname*{Supp}\emptyset=\operatorname*{Supp}X=\emptyset$, $\emptyset^{\#}=X$, and

[TABLE]

with the remarks that, for every $C\subset X$, one has $C^{\#}\neq\emptyset$ and $\operatorname*{cl}_{\tau}\operatorname*{conv}C\subset(\operatorname*{cl}_{\tau}\operatorname*{conv}C)^{\#}\subset C^{\#}$; while $C^{\#}=X$ when $\operatorname*{Supp}C=\emptyset$, e.g., $X^{\#}=X$.

Theorem 1

Let $X$ be a LCS. For every $C\subset X$, $(x,x^{*})\in X\times X^{*}$

[TABLE]

In particular, for every $\emptyset\neq C\subset X$, $(x,x^{*})\in X\times X^{*}$, $\varphi_{N_{C}}(x,0)=\iota_{C^{\#}}(x)\geq 0$, $\varphi_{N_{C}}(x,x^{*})\leq\iota_{C}(x)+\sigma_{C}(x^{*})$ and

[TABLE]

The last part of Theorem 1 says that any monotone extension $T$ of $N_{C}$ has $D(T)\subset C^{\#}$ or that we can extend $N_{C}$ monotonically only inside $C^{\#}\times X^{*}$.

Theorem 2

Let $X$ be a LCS and let $C\subset X$. The following are equivalent

(i)* $N_{C}\in\mathfrak{M}(X)$,*

(ii) $\varphi_{N_{C}}(x,x^{*})=\iota_{C}(x)+\sigma_{C}(x^{*})$, $(x,x^{*})\in X\times X^{*}$,

(iii)* $C^{\#}\subset(=)C$*

(iv)* $C=\{x\in X\mid\varphi_{N_{C}}(x,0)\leq 0\}$.*

Concerning the previous result note that $C$ is non-empty closed convex whenever $N_{C}\in\mathfrak{M}(X)$. Also, subpoint (iv) can be restated equivalently as

(iv)′ $C$ is non-empty and $C=(\supset)\{x\in X\mid\varphi_{N_{C}}(x,0)\leq(=)0\}$.

Theorem 2 has strong ties with the separation theorem. Assume that $C\subset X$ is closed and convex. Then for every $x\in X\setminus C$ there is $n^{*}\in X^{*}\setminus\{0\}$ such that $\langle x,n^{*}\rangle>\sigma_{C}(n^{*})=\sup\{\langle u,n^{*}\rangle\mid u\in C\}$. In the next theorem we see that the maximality of $N_{C}$ is equivalent to the possibility of picking, in the previous separation argument, of a non-zero $n^{*}$ that attains its global maximum on $C$, that is, $n^{*}\in R(N_{C})$ which is also called a support functional of $C$ (see e.g. [1]).

Theorem 3

Let $X$ be a LCS and let $C\subset X$. Then $N_{C}\in\mathfrak{M}(X)$ (or $C=C^{\#}$) iff for every $x\in X\setminus C$ there is $n^{*}\in D(\partial\sigma_{C})=R(N_{C})$ such that $\langle x,n^{*}\rangle>\sigma_{C}(n^{*})$.

Corollary 4

Let $X$ be a LCS and let $C\subset X$ be non-empty closed and convex. If $X$ is a Banach space or $\operatorname*{int}C\neq\emptyset$, or $C$ is weakly compact then $N_{C}\in\mathfrak{M}(X)$.

Note that $C^{\#}$, the portable hull of $C$, is the smallest set formed by intersecting half-spaces that contain $C$ and are supported at points in $C$, and, at the same time, the largest set on which the normal cone $N_{C}$ can be extended monotonically.

Proposition 5

Let $X$ be a LCS. For every $C\subset X$, $N_{C^{\#}}|_{C}=N_{C}$, $\operatorname*{Graph}N_{C}\subset\operatorname*{Graph}N_{C^{\#}}$, $N_{C^{\#}}\in\mathfrak{M}(X)$, and $C^{\#\#}:=(C^{\#})^{\#}=C^{\#}$.

Remark 6

Every (maximal) monotone extension of $N_{C}$ has the domain contained in $C^{\#}$, that is, $N_{C}\subset T\in\mathcal{M}(X)\Rightarrow D(T)\subset D(N_{C}^{+})=C^{\#}$. Therefore $N_{C^{\#}}$ is a maximal monotone extension of $N_{C}$ with the largest possible domain. In general, $N_{C^{\#}}$ is not the only maximal monotone extension of $N_{C}$ (even with the largest possible domain or with a normal cone structure). For example, for $C=(0,1]\subset\mathbb{R}$, $N_{C}$ admits $N_{\overline{C}}$ and $N_{C^{\#}}$ as two different maximal monotone extensions; moreover $N_{C}$ has an infinity of maximal monotone extensions with the largest possible domain $C^{\#}=(-\infty,1]$.**

The maximality of the subdifferential allows us to reprove and extend some of the Bishop-Phelps results (see [1]).

Theorem 7

Let $X$ be a LCS. If, for every closed convex $C\subset X$, $N_{C}\in\mathfrak{M}(X)$, then, for every closed convex $C\subset X$, $\operatorname*{Supp}C$ is dense in $\operatorname*{bd}C$.

Remark 8

The previous results still holds for a fixed closed convex $C\subset X$ if, for every $U\subset X$ closed convex such that $C\cap\operatorname*{int}U\neq\emptyset$, $N_{C\cap U}\in\mathfrak{M}(X)$.

Theorem 9

Let $X$ be a LCS and let $C\subset X$ be closed convex, free of lines, and finite-dimensional, that is, $\operatorname*{dim}(\operatorname*{span}C)<\infty$. Then $N_{C}\in\mathfrak{M}(X)$ and $\operatorname*{cl}_{w^{*}}R(N_{C})=\operatorname*{cl}_{w^{*}}(\operatorname*{dom}\sigma_{C})$.

If, in addition, $C$ is bounded, then $\operatorname*{cl}_{w^{*}}R(N_{C})=X^{*}$.

Proposition 10

under review

Corollary 11

Let $X$ be a LCS such that for every $f\in\Gamma(X)$, $\operatorname*{Graph}\partial f\neq\emptyset$. Then for every closed convex bounded $C\subset X$ the support functionals of $C$ are weak-star dense in $X^{*}$.

Corollary 12

Let $(X,\|\cdot\|)$ be a Banach space. Then for every closed convex bounded $C\subset X$ the support functionals of $C$ are strongly dense in $X^{*}$.

If $C\subset X$ is merely closed and convex then the support functionals of $C$ are strongly dense in the domain of $\sigma_{C}$. In other words for every $\epsilon>0$, $f\in X^{*}$ such that $\sup_{C}f<+\infty$ there is $g\in X^{*}$, $x_{0}\in C$ such that $g(x_{0})=\sup_{C}g$ (that is, $g\in N_{C}(x_{0})$) and $\|f-g\|\leq\epsilon$.

3 Partial portable hulls and sum representability

Let $(X,\tau)$ be a LCS and $C,S\subset X$. We denote by $C_{S}^{\#}$ the partial portable hull of $C$ on $S$ which is the intersection of all the (supporting) half-spaces that contain $C$ and are supported at points in $S$,

[TABLE]

Equivalently

[TABLE]

with the remarks that $C_{S}^{\#}=X$ when $S\cap\operatorname*{Supp}C=\emptyset$ and that, for every $C,S\subset X$, $C_{S}^{\#}\neq\emptyset$, $C\subset\operatorname*{cl}_{\tau}\operatorname*{conv}C\subset C^{\#}\subset C_{S}^{\#}$, $C_{S}^{\#}=C_{S\cap C}^{\#}=C_{S\cap\operatorname*{Supp}C}^{\#}$.

Note also that $C_{S}^{\#}=C^{\#}$ whenever $S\supset\operatorname*{Supp}C$.

C^{\#}_{S}$$C$$S\cap\operatorname*{Supp}C

Proposition 13

Let $X$ be a LCS and let $C,S\subset X$. Then

(i)* $\operatorname*{Graph}(N_{C}|_{S})\subset\operatorname*{Graph}N_{C_{S}^{\#}}$ and*

[TABLE]

In particular $N_{C^{\#}}|_{C}=N_{C}$.

(ii)* $(C_{S}^{\#})_{S}^{\#}=(C_{S}^{\#})^{\#}=C_{S}^{\#}$. In particular $N_{C_{S}^{\#}}\in\mathfrak{M}(X)$.*

Theorem 14

under review

Recall the following notion ( see Definition 14 in [6])

Definition 15

Let $(X,\tau)$ be a LCS. An operator $T:X\rightrightarrows X^{*}$ is representable in $C\subset X$ or $C-$representable if $C\cap D(T)\neq\emptyset$ and there is $h\in\Gamma_{\tau\times w^{*}}(X\times X^{*})$ such that $h\geq c$ and $[h=c]\cap C\times X^{*}=\operatorname*{Graph}(T|_{C})$.

Using the partial portable hull we can recover the representability of the sum between a representable operator and the normal cone (see Theorem 35 in [6])

Theorem 16

Let $X$ be a LCS, let $T:X\rightrightarrows X^{*}$, and let $C\subset X$ be closed convex such that $D(T)\cap\operatorname*{int}C\neq\emptyset$. Then $\psi_{T+N_{C}}=\psi_{T|_{C}}\square_{2}\sigma_{C_{D(T)}^{\#}\times\{0\}}$ or

[TABLE]

[TABLE]

In particular, if, in addition, $T$ is $C-$representable then $T+N_{C}$ is representable.

4 Concluding remarks

Let $(X,\tau)$ be a LCS and let us call a set $C\subset X$ portable if $C=C^{\#}$ (or $N_{C}\in\mathfrak{M}(X)$). Some of the results of this paper can be summarized as follows:

• •

$C\subset X$ is portable iff for every $x\in X\setminus C$ there is $n^{*}\in D(\partial\sigma_{C})=R(N_{C})$ such that $\langle x,n^{*}\rangle>\sigma_{C}(n^{*})$ or, equivalently, $[\sigma_{C-x}<0]\cap R(N_{C})\neq\emptyset$.

• •

If $C\subset X$ is closed convex and $X$ is a Banach space or $\operatorname*{int}C\neq\emptyset$, or $C$ is weakly compact then $C$ is portable.

• •

For every $C,S\subset X$, $C_{S}^{\#}$ is portable.

• •

If every closed convex $C\subset X$ is portable then for every closed convex $C\subset X$, $\operatorname*{Supp}C$ is dense in $\operatorname*{bd}C$.

• •

Every closed convex, free of lines, and finite-dimensional set is portable.

The results in this article are an incentive for studying the following problems:

(P1)

Find cha racterizations of all LCS $X$ with the property that, for every closed convex $C\subset X$, $N_{C}\in\mathfrak{M}(X)$.

(P2)

Find characterizations of all LCS $X$ with the property that, for every $f\in\Gamma(X)$, $\operatorname*{Graph}\partial f\neq\emptyset$.

(P3)

Find characterizations of all LCS $X$ with the property that, for every $f\in\Gamma(X)$, $\partial f\in\mathfrak{M}(X)$.