Enlargements: a bridge between maximal monotonicity and convexity

Abstract

Perhaps the most important connection between maximally monotone operators and convex functions is the fact that the subdifferential of a convex function is maximally monotone. This connects convex functions with a proper subset of maximally monotone operators (i.e., the cyclically monotone operators). Our focus is to explore maps going in the opposite direction, namely those connecting an arbitrary maximally monotone map with convex functions. In this survey, we present results showing how enlargements of a maximally monotone operator $T$ provide this connection. Namely, we recall how the family of enlargements is in fact in a bijective correspondence with a whole family of convex functions. Moreover, each element in either of these families univocally defines $T$. We also show that enlargements are not merely theoretical artifacts, but have concrete advantages and applications, since they are, in some sense, better behaved than $T$ itself. Enlargements provide insights into existing tools linked to convex functions. A recent example is the use of enlargements for defining a distance between two point-to-set maps, one of them being maximally monotone. We recall this new distance here, and briefly illustrate its applications in characterizing solutions of variational problems.