An explicit example of a maximal 3-cyclically monotone operator with bizarre properties

Abstract

Subdifferential operators of proper convex lower semicontinuous functions and, more generally, maximal monotone operators are ubiquitous in optimization and nonsmooth analysis. In between these two classes of operators are the maximal n -cyclically monotone operators. These operators were carefully studied by Asplund, who obtained a complete characterization within the class of positive semidefinite (not necessarily symmetric) matrices, and by Voisei, who presented extension theorems à la Minty. All previous explicit examples of maximal n -cyclically monotone operators are maximal monotone; thus, they inherit the known good properties of maximal monotone operators. In this paper, we construct an explicit maximal 3-cyclically monotone operator with quite bizarre properties. This construction builds upon a recent, nonconstructive and Zorn’s Lemma-based, example. Our operator possesses two striking properties that sets it far apart from both the maximal monotone operator and the subdifferential operator case: it is not maximal monotone and its domain, which is closed, fails to be convex. Indeed, the domain is the boundary of the unit diamond in the Euclidean plane. The path leading to this operator requires some new results that are interesting in their own right.