Maximal Monotone Inclusions and Fitzpatrick Functions

Abstract

In this paper, we study maximal monotone inclusions from the perspective of gap functions. We propose a very natural gap function for an arbitrary maximal monotone inclusion and will demonstrate how naturally this gap function arises from the Fitzpatrick function, which is a convex function, used to represent maximal monotone operators. This allows us to use the powerful strong Fitzpatrick inequality to analyse solutions of the inclusion. We also study the special cases of a variational inequality and of a generalized variational inequality problem. The associated notion of a scalar gap is also considered in some detail. Corresponding local and global error bounds are also developed for the maximal monotone inclusion.

Appendix

In this appendix, we first prove that the function

$$\begin{aligned} G(x) = \sup _{y \in C} \langle F(y), x - y \rangle , \end{aligned}$$

is a gap function when F is monotone.

We establish this directly. The function G is convex, proper and lower semi-continuous. When C is compact, G is continuous since it is finite. It is easy to see that $G(x) \ge 0$. If $G(x) = 0$, then we have, for all $y \in C$, $\langle F(y), x- y \rangle \le 0$. Consider a fixed $y \in C$, and construct the sequence $y_n = x + \frac{1}{n} ( y - x )$. Since C is convex, $y_n \in C$. Hence, we have $\langle F(y_n), x - y_n \rangle \le 0$. As $n \rightarrow \infty$, using the continuity of F , we have $\langle F(x), x - y \rangle \le 0$. Thus, x is a solution of $\hbox {VI}(F, C)$ since $y \in C$ was chosen arbitrarily.

Now assume that x is a solution of $\hbox {VI}(F, C)$. Hence, $\langle F(x), x - y \rangle \le 0$, for all $y \in C$. Then, using monotonicity of F we have $\langle F(y), x - y \rangle \le 0$, for all $y \in C$.

This shows that $G(x) \le 0$. Hence, $G(x) = 0$ and proves that G is a gap function when F is monotone.

We shall now state the Proposition 4.1 of Aussel and Dutta [8] adapted to our setting.

Proposition 8.1

Assume that the set-valued map T is compact-valued. The the function $\hat{g}$ is a gap function for the weak variational inequality.

We end the “Appendix” by stating the Sion’s minimax theorem as given in Komiya [19] but will present it only in our finite dimensional setting. We recall that a quasi-convex function is a function, whose lower level sets are always convex. If h is quasi-convex, then $- h$ is quasi-concave. Thus, every convex function is quasi-convex, and every concave function is quasi-concave. For more details on quasi-convex functions, see, for example, [21].

Theorem 8.1

(Sion’s minimax theorem) Let X be a compact convex set of ${\mathbb {R}}^n$ and Y be a convex set of ${\mathbb {R}}^m$ and let f be a real-valued function on $X \times Y$ such that

• (i) f (x, .) is quasi-concave and upper semi-continuous on Y for each $x \in X$;

• (ii) f ( ., y) is quasi-convex and lower semi-continuous on X for each $y \in Y$.

Then, we have

$$\begin{aligned} \inf _{x \in X} \sup _{y \in Y} f (x, y) = \sup _{y \in Y} \inf _{x \in X} f(x, y). \end{aligned}$$