Barrier functions in the subdifferential theory

TL;DR

This paper introduces a novel approach using barrier functions to prove that monotonicity of subdifferentials implies convexity of functions, addressing technical challenges with lower semicontinuous functions.

Contribution

The paper presents a new method employing barrier functions to establish the Correa-Jofré-Thibault theorem, enhancing the theoretical toolkit for subdifferential analysis.

Findings

Barrier functions facilitate handling lower semicontinuous functions.

The new proof simplifies existing technical difficulties.

The method confirms the link between monotonicity and convexity.

Abstract

We present a new method for proving Correa-Jofr\'e-Thibault theorem that monotonicity of subdifferential implies convexity of the function. This new method is based on barrier functions. Barrier functions help overcome some of the main technical difficulties when working with lower semicontinuous functions.

Equations

$${\partial }^{c}f(x):=\{x^{\ast }\in X^{\ast }:⟨x^{\ast },y-x⟩\le f(y)-f(x),\forall y\in U\};$$

$$0\in \partial g(x)+\underset{y\to x,f(y)\to f(x)}{\mathrm{limsup}}\partial f(y)$$

$$(x_i,x_i^{\ast })\in \partial f,i=1,2\Rightarrow ⟨x_2^{\ast }-x_1^{\ast },x_2-x_1⟩\ge 0.$$

$$⟨x^{\ast },x-x_0⟩\ge 0,\forall (x,x^{\ast })\in \partial f\Rightarrow 0\in {\partial }^{c}f(x_0),$$

$$⟨x^{\ast }-x_0^{\ast },x-x_0⟩\ge 0,\forall (x,x^{\ast })\in \partial f,$$

$$\partial f\subset {\partial }^{c}f,$$

$$\mu (x)=\inf \{\lambda :\lambda >0,x\in \lambda U\}\text{mbox}forx\ne 0,\mu (0)=0.$$

$$k(x):=\frac{\mu (x)}{1-\mu (x)}=\frac{1}{1-\mu (x)}-1.$$

$$\mu (\lambda x+(1-\lambda )y)\le \lambda \mu (x)+(1-\lambda )\mu (y)\iff$$

$$1-\mu (\lambda x+(1-\lambda )y)\ge \lambda (1-\mu (x))+(1-\lambda )(1-\mu (y)).$$

$$g(\lambda x+(1-\lambda )y)\le \frac{1}{\lambda \alpha +(1-\lambda )\beta }.$$

$$\frac{1}{\lambda \alpha +(1-\lambda )\beta }\le \frac{\lambda }{\alpha }+\frac{1-\lambda }{\beta }.$$

$$0$$

$$g(\lambda x+(1-\lambda )y)\le \lambda g(x)+(1-\lambda )g(y)$$

$$\Vert x_n-y_n\Vert \to 0,$$

$$f(x_n)+g(y_n)\to \underset{U}{\inf }(f+g),$$

$$⟨x_n^{\ast },x_n-\bar{x}⟩+⟨y_n^{\ast },y_n-\bar{x}⟩\to 0.$$

$$M:=\sup \{\Vert y-\bar{x}\Vert :y\in U\}.$$

$$f(z_n)+g(z_n)<\underset{U}{\inf }(f+g)+{\epsilon }_n.$$

$$f(y_n)+g(y_n)<\underset{U}{\inf }(f+g)+{\epsilon }_n,$$

$$x\to f(x)+g(x)+\epsilon \Vert x-y_n\Vert$$

$$w^{\ast }-\lim u_k^{\ast }=u^{\ast }\in -\partial (g+{\epsilon }_n\Vert \cdot -y_n\Vert )(y_n).$$

$$\Vert u^{\ast }+y_n^{\ast }\Vert \le {\epsilon }_n.$$

$$\Vert u_k-y_n\Vert <{\epsilon }_n,|f(u_k)-f(y_n)|<{\epsilon }_n,\forall k>k_1.$$

$$|f(u_k)+g(y_n)-\underset{U}{\inf }(f+g)|\le 2{\epsilon }_n,\forall k>k_1.$$

$$⟨u_k^{\ast },u_k-\bar{x}⟩+⟨y_n^{\ast },y_n-\bar{x}⟩=⟨u^{\ast }+y_n^{\ast },y_n-\bar{x}⟩+⟨u_k^{\ast }-u^{\ast },y_n-\bar{x}⟩+⟨u_k^{\ast },u_k-y_n⟩.$$

$$|⟨u_k^{\ast },u_k-\bar{x}⟩+⟨y_n^{\ast },y_n-\bar{x}⟩|\le (M+1){\epsilon }_n,\forall k>k_2.$$

$$\bar{k}=\max \{k_1,k_2\}+1\text{mbox}and(x_n,x_n^{\ast }):=(u_{\bar{k}},u_{\bar{k}}^{\ast }).$$

$$f(\overline{x})\le \sup \{f(x)+⟨x^{\ast },\overline{x}-x⟩:(x,x^{\ast })\in \partial f\},\text{mbox}forall\overline{x}\in X.$$

$$r<f(\bar{x}).$$

Full text

This new method is based on barrier functions. Barrier functions help overcome some of the main technical difficulties when working with lower semicontinuous functions.

2010 Mathematics Subject Classification: 49J52, 47H05, 52A41.

Keywords: subdifferential, monotonicity, convex function, barrier function

1 Introduction

In 1990’s Correa, Jofré and Thibault in series of papers proved that a convex lower semicontinuous function can be characterized by monotonicity property of its subdifferential – in reflexive Banach space for the Clarke subdifferential in [2] and in any Banach space for axiomatically introduced subdifferential in [3] and for more general axiomatic presubdifferential in [4].

The main tool for proving this characterization is the Mean Value Theorem of Zagrodny [13] which holds for a lower semicontinuous function in a Banach space for any presubdifferential, see [11].

Jules and Lassonde prove a subdifferential test for optimality, see [7] of Minty type [8] involving a subdifferential satisfying certain axioms.

In setting up the axiomatic framework we follow [12], but we pick the apparently minimal set of axioms under which proofs can work. In this way our results are slightly more general. As mentioned below, adding another natural axiom can significantly simplify the presentation.

We work in a real Banach space $X$ with dual $X^{*}$.

Definition 1** (axioms for subdifferential).**

Multi-valued operator $\partial$ which associates to any function $f:X\to\mathbb{R}\cup\{+\infty\}$ and any $x\in X$ a (possibly empty) subset $\partial f(x)\subset X^{*}$ is feasible subdifferential if

1. (P1)

$\partial f(x)=\partial^{c}f(x)$* whenever $f$ is a convex and continuous function in a neighbourhood $U$ of $x$, where $\partial^{c}$ stands for the Fenchel subdifferential, i.e.*

[TABLE] 2. (P2)

For $f$ lower semicontinuous and $g$ convex and continuous in a neighbourhood of $x\in\mathrm{dom}\,f$,

[TABLE]

whenever $x$ is a local minimum point of $f+g$.

In more details (1) means that there are $y_{n}^{*}\in\partial f(y_{n})$ such that $y_{n}\to x$, $f(y_{n})\to f(x)$ and the sequence $y_{n}^{*}$ converges in the $w^{*}$ topology to some $y^{*}$ such that $-y^{*}\in\partial g(x)$.

Note that all presubdifferentials considered by Correa, Jofré and Thibault in [2, 3, 4], as well as subdifferentials considered by Jules and Lassonde in [7] are feasible subdifferentials in the sense of Definition 1.

In terms of Ioffe’s extensive classification, see [6], (P1) is called contiguity, while (P2) is weak-star form of trustworthness.

We prove the above mentioned results for feasible subdifferentials and in a different and unified way – by using barrier functions instead of (a variant of) Zagrodny Theorem.

For convenience of notation we often identify the map $\partial f:X\rightrightarrows X^{*}$ with its graph, that is, $(x,x^{*})\in\partial f$ is a shorthand for $x^{*}\in\partial f(x)$.

The main contribution of this work is a new method, based on [5], see also [10, p.569], for proving the following result of Correa, Jofré and Thibault.

Theorem 2** (Correa-Jofré-Thibault).**

Let $X$ be a Banach space and let $\partial$ be a feasible subdifferential.

Let $f:X\to\mathbb{R}\cup\left\{+\infty\right\}$ be a proper lower semicontinuous function.

If $\partial f$ is monotone, then $f$ is convex.

Recall that monotonicity of $\partial f$ means

[TABLE]

The routine way of demonstrating the above result can be sketched like this: examining the proof in [11] it is clear that Zagrodny Mean Value Theorem holds for any feasible subdifferential. Using it, one can prove a Minty test for optimality [8] of the following form

[TABLE]

where $f$ is proper and lower semicontinuous and $\partial$ is feasible subdifferential.

If – on top of (P1) and (P2) – $\partial$ also satisfies the natural axiom

1. (P3)

$\partial(f+x^{*})=\partial f+x^{*},\quad\forall x^{*}\in X^{*}$

(called in [7] stability property which is rather limited form of calculability axiom in [6]), then from Minty test immediately follows that $\partial\cup\partial^{c}$ has the somewhat surprising property (first noted by Jules and Lasonde) to be maximal with respect to monotonicity relation. That is, if $(x_{0},x_{0}^{*})$ is monotonous related to $\partial f$:

[TABLE]

then $(x_{0},x_{0}^{*})\in\partial^{c}f$. From the latter Moreau-Rockafellar Theorem about maximal monotonicity of $\partial^{c}f$ for convex, proper and lower semicontinuous $f$ follows immediately, but the surprising fact is that the above is true even if $\partial f$ is not itself monotone. (For precise statement see Theorem 6.)

So, in particular if $\partial$ is feasible and satisfies in addition (P3), then

[TABLE]

whenever $\partial f$ is monotone. Further the proof can be completed as we do in here presented proof of Theorem 2.

Note that the additional axiom (P3) is not really necessary.

Our approach is based on a different technique involving barrier functions instead of Zagrodny Theorem. We will also make the effort to obtain Theorem 2 for general feasible subdifferental.

The paper is organized as follows.

In Section 2 we construct and consider the class of barrier functions we use. In Section 3 we show some additional properties of the feasible subdifferential linked to (P2) axiom. Finally, in Section 4 we present our proof of Correa-Jofré-Thibault theorem.

2 Barrier functions

Let $U\subset X$ be a open, convex and bounded neighbourhood of [math], i.e. $0\in U$. Let $\mu$ be the Minkowski functional of $U$, that is,

[TABLE]

It is clear that $\mu(x)<1$ if $x\in U$, and $\mu(x)\geq 1$ if $x\not\in U$.

Let us first list few properties of Minkowski functional, see e.g. [1]:

• (i)

$\mu$ has values in $[0,+\infty)$;

• (ii)

$\mu$ is positively homogeneous, i.e. $\mu(tx)=t\mu(x)$ for all $x\in X$ and $t\geq 0$;

• (iii)

$\mu(x+y)\leq\mu(x)+\mu(y)$ for all $x,y\in X$ and hence $\mu$ is convex;

• (iv)

$\{x:\mu(x)\leq 1\}=\mathrm{cl}\,U\Rightarrow\{x:\mu(x)=1\}=\partial U:=\mathrm{cl}\,U\setminus U$, where $\mathrm{cl}\,U$ denotes the topological closure of $U$ and $\partial U$ denotes its boundary.

Moreover,

• (v)

There exists $c>0$ such that $\mu(x)\leq c\|x\|$ for all $x\in X$;

• (vi)

There exists $b>0$ such that $\mu(x)\geq b\|x\|$ for all $x\in X$;

• (vii)

$\mu$ is Lipschtz continuous.

Indeed, let $\delta>0$ be such that $\delta B_{X}\subset U$, where $B_{X}:=\{x\in X:\|x\|\leq 1\}$. Then $\displaystyle\frac{\delta x}{\|x\|}\in\delta B_{X}\subset U$, so $\displaystyle\mu\left(\frac{\delta x}{\|x\|}\right)\leq 1$, and $\mu(x)\leq\delta^{-1}\|x\|$, which is (v) with $c:=\delta^{-1}$.

Further, let $U\subset sB_{X}$, then $\displaystyle\frac{sx}{\|x\|}\not\in U$, so $\displaystyle\mu\left(\frac{sx}{\|x\|}\right)\geq 1$, and $\displaystyle\mu(x)\geq\frac{1}{s}\|x\|$ giving (vi) with $b:=s^{-1}$.

Finally, $\mu(x)=\mu((x-y)+y)\leq\mu(x-y)+\mu(y)$ by (ii). Hence, $\mu(x)-\mu(y)\leq\mu(x-y)$ and using (vi) we get $\mu(x)-\mu(y)\leq b\|x-y\|$ which yields $|\mu(x)-\mu(y)|\leq b\|x-y\|$ for all $x,y\in X$ and (vii) holds.

For $x\in U$, define the function

[TABLE]

The next lemma shows that $k$ is a barrier function for $U$, i.e. a continuous function whose values tend to infinity while the arguments tend to $\partial U$ (see e.g. [9]).

Lemma 3**.**

The function $k$ defined by (2) has the following properties:

• (a)

$k(0)=0$* and for some $b>0$, $k(x)\geq\mu(x)\geq b\|x\|$.*

• (b)

$\displaystyle\lim_{x\to\partial U}k(x)=+\infty$;

• (c)

$k$* is convex and Lipschitz on each level set $\{x:k(x)\leq\lambda\}$;*

• (d)

the function \displaystyle\overline{k}(x):=\left\{\begin{array}[]{ll}k(x),&x\in U\\ +\infty,&x\in X\setminus U\end{array}\right. is lower semicontinuous and convex.

Proof.

The properties (a) and (b) are clear.

In order to prove that $k$ is convex, it is enough to show that the function $\displaystyle g(x):=k(x)+1=\frac{1}{1-\mu(x)}$ is convex.

To this end, fix $x,\ y\in U$ and $\lambda\in[0,1]$. From (i)

[TABLE]

[TABLE]

Set $\alpha:=1-\mu(x)$, $\beta:=1-\mu(y)$, so $\alpha,\beta\in(0,1)$. Then the latter says

[TABLE]

We claim that

[TABLE]

Of course, (4) is equivalent to

[TABLE]

From (3) and (4) it follows that

[TABLE]

and $k$ is convex. Lipschitz continuity of $k$ on each level set is inherited by the Lipschitz continuity of $\mu$.

(d) follows from (b) and (c). ∎

3 Properties of feasible subdifferential

For the sake of clarity, we will take two technical parts out of the proof of the main result. Namely, we will show some additional properties of feasible subdifferential linked to (P2) axiom.

Lemma 4**.**

Let $\partial$ be a feasible subdifferential. Let $f:X\to\mathbb{R}\cup\{+\infty\}$ be lower semicontinuous and bounded below on the open, convex and bounded set $U\subset X$. Let, moreover, $\mathrm{dom}\,f\cap U\neq\emptyset$. Let $g$ be convex continuous and bounded below barrier function for $U$. Let $\bar{x}\in X$be fixed.

Then there exist sequences $(x_{n})_{n=1}^{\infty},(y_{n})_{n=1}^{\infty}\subset U$ and $x_{n}^{*}\in\partial f(x_{n})$, $y_{n}^{*}\in\partial^{c}g(y_{n})$ such that

[TABLE]

[TABLE]

[TABLE]

Proof.

Define

[TABLE]

Since $U$ is bounded, $M\in\mathbb{R}$.

Fix a sequence $(\varepsilon_{n})_{n=1}^{\infty}$ such that $\varepsilon_{n}>0$ and $\varepsilon_{n}\to 0$ as $n\to\infty$.

Pick $z_{n}\in U$ such that

[TABLE]

By Ekeland Variational Principle there are $y_{n}\in U$ such that

[TABLE]

and the function

[TABLE]

attains its minimum at $y_{n}$. By (P2) there are $u_{k}\to y_{n}$, as $k\to\infty$, and $u_{k}^{*}\in\partial f(u_{k})$ such that $f(u_{k})\to f(y_{n})$ and

[TABLE]

By Sum Theorem for the Fenchel subdifferential and the fact that $\partial\varepsilon_{n}\|\cdot-y_{n}\|=\varepsilon_{n}\partial\|\cdot-y_{n}\|$ it follows that there are $y_{n}^{*}\in\partial g(y_{n})$ such that

[TABLE]

Since $(u_{k},f(u_{k}))\to(y_{n},f(y_{n}))$ as $k\to\infty$, there is $k_{1}\in\mathbb{N}$ such that

[TABLE]

By (9) we have

[TABLE]

Note that

[TABLE]

Now, $|\langle u^{*}+y_{n}^{*},y_{n}-\bar{x}\rangle|\leq\|u^{*}+y_{n}^{*}\|\|y_{n}-\bar{x}\|\leq M\varepsilon_{n}$ by (8) and (10). Since $u_{k}^{*}$ weak-star converges to $u^{*}$, we have $\langle u_{k}^{*}-u^{*},y_{n}-\bar{x}\rangle\to 0$ as $k\to\infty$. Also, since by Banach-Steinhaus Theorem the sequence $(u_{k}^{*})_{k=1}^{\infty}$ is bounded, and $u_{k}\to y_{n}$ as $k\to\infty$, we have $|\langle u_{k}^{*},u_{k}-y_{n}\rangle|\leq\|u_{k}^{*}\|\|u_{k}-y_{n}\|\to 0$ as $k\to\infty$. Therefore, there is $k_{2}\in\mathbb{N}$ such that

[TABLE]

Set

[TABLE]

From (11), (12) and (13) it follows that so constructed sequences $(x_{n},x_{n}^{*})_{n=1}^{\infty}$ and $(y_{n},y_{n}^{*})_{n=1}^{\infty}$ satisfy (5), (6) and (7). ∎

Proposition 5**.**

Let $\partial$ be a feasible subdifferential. Let $f:X\to\mathbb{R}\cup\{+\infty\}$ be a proper and lower semicontinuous function. Then $\mathrm{dom}\,\partial f$ is nonempty and

[TABLE]

Proof.

Fix arbitrary $\bar{x}\in X$ and

[TABLE]

Fix some $y\in\mathrm{dom}\,f$.

Since $f$ is lower semicontinuous and the segment $[\bar{x},y]$ is compact, there is $\delta>0$ such that $f$ is bounded below on the set

[TABLE]

Let $k$ be the barrier function defined by (2) for the set $U:=C-\{\bar{x}\}$.

Let $\xi>0$ be such that $f>r$ on $\bar{x}+\xi B_{X}$. Fix $a>0$ and such that

[TABLE]

where $b>0$ and $k(x)\geq\mu(x)\geq b\|x\|$ (see Lemma 3 (a)).

Then it is immediate that

[TABLE]

Apply Lemma 4 to $f$ and the barrier function $g=ak(\cdot-\overline{x})$ to get $x_{n}^{*}\in\partial f(x_{n})$ and $y_{n}^{*}\in\partial^{c}k(\cdot-\bar{x})(y_{n})$ such that (5) and (6) are fulfilled and, moreover,

[TABLE]

Since $y_{n}^{*}\in\partial^{c}k(y_{n}-\bar{x})$, we have

[TABLE]

From the boundedness below of $f$ and (6) it follows that the sequence $(k(y_{n}-\bar{x}))_{n=1}^{\infty}$ is bounded. Since $k$ is Lipschitz on its level sets (see Lemma 3 (c)), from (5) it follows that

[TABLE]

These and (16) give $\langle x_{n}^{*},x_{n}-\bar{x}\rangle=\alpha_{n}-a\langle y_{n}^{*},y_{n}-\bar{x}\rangle\leq\alpha_{n}-ak(y_{n}-\bar{x})=-ak(x_{n}-\bar{x})+\alpha_{n}-a\beta_{n}$. So,

[TABLE]

From (15) it follows that $\langle x_{n}^{*},x_{n}-\bar{x}\rangle<f(x_{n})-r+\gamma_{n}$, or, equivalently,

[TABLE]

Therefore,

[TABLE]

Since $r<f(\bar{x})$ were arbitrary, we are done. ∎

4 Monotonicity and convexity

We start with the following extension to the case of feasible subdifferential of a result of Jules and Lassonde [7].

Theorem 6**.**

Let $\partial$ be a feasible subdifferential. Let $f:X\to\mathbb{R}\cup\{+\infty\}$ be a proper lower semicontinuous function. Let $(x_{0},x_{0}^{*})\in X\times X^{*}$ be in monotone relation to $\partial f$, that is,

[TABLE]

Then $(x_{0},x_{0}^{*})\in\partial^{c}f$.

Proof.

Let $r\in\mathbb{R}$ be such that

[TABLE]

Let $y\in\mathrm{dom}\,f$ be arbitrary.

Since $f$ is lower semicontinuous and the segment $[x_{0},y]$ is compact, there is $\delta>0$ such that $f$ is bounded below on

[TABLE]

Let $k$ be the barrier function defined by (2) for the set $U:=C-\{x_{0}\}$.

Let $a>0$ be arbitrary.

Obviously,

[TABLE]

is a convex and continuous barrier for $U$. So, we can apply Lemma 4 to $f$ and $g$. Since

[TABLE]

there are $x_{n}^{*}\in\partial f(x_{n})$ and $y_{n}^{*}\in\partial^{c}k(\cdot-x_{0})(y_{n})$ such that (5) and (6) are fulfilled and, moreover,

[TABLE]

Clearly, $\langle x_{0}^{*},x_{n}-y_{n}\rangle\to 0$, so (19) is equivalent to

[TABLE]

This and (17) give

[TABLE]

Since $y_{n}^{*}\in\partial^{c}k(\cdot-x_{0})(y_{n})$, we have

[TABLE]

because $k(0)=0$. But $k(y_{n}-x_{0})\geq b\|y_{n}-x_{0}\|$ (cf. Lemma 3 (a)), so (20) and (21) imply that

[TABLE]

From (5) it follows that $x_{n}\to x_{0}$ as well; and from the lower semicontinuity of $f$ and (6) it follows that

[TABLE]

In particular,

[TABLE]

Since $a>0$ was arbitrary,

[TABLE]

So, $x_{0}\in\mathrm{dom}\,f$ and, since $y\in\mathrm{dom}\,f$ was arbitrary, $x_{0}^{*}\in\partial^{c}f(x_{0})$. ∎

After all the above development, the proof of Correa-Jofré-Thibault is now almost immediate.

Theorem 2** (Correa-Jofré-Thibault).**

Let $X$ be a Banach space and $\partial$ be a feasible subdifferential.

Let $f:X\to\mathbb{R}\cup\left\{+\infty\right\}$ be a proper lower semicontinuous function.

If $\partial f$ is monotone, then $f$ is convex.

Proof.

Consider $g:X\to\mathbb{R}\cup\{+\infty\}$ defined as

[TABLE]

As a supremum of linear functions, $g$ is convex and lower semicontinuous.

By (14) we have that

[TABLE]

From Theorem 6 and monotonicity of $\partial f$ we have that $\partial f\subset\partial^{c}f$, which implies $f\geq g$. Therefore,

[TABLE]

∎