The generic differentiability of convex-concave functions: Characterization

TL;DR

This paper demonstrates that convex-concave functions are generically differentiable on dense subsets, extending known results and highlighting differences from convex-convex functions, with applications to monotone operators.

Contribution

It shows that convex-concave functions have dense subsets where partial derivatives exist, a property not shared by convex-convex functions, and applies this to monotone operators.

Findings

Existence of dense subsets where partial derivatives exist

Differentiability on dense product sets for convex-concave functions

Recovery of generic single-valuedness of monotone operators

Abstract

As established by R T. Rockafellar, real valued convex-concave functions are generically differentiable. It this paper we shall show that for a convex-concave function defined on an open convex set $C \times D,$ there exist dense subsets ${\cal N}$ of $C$ and ${\cal M}$ of $D$ such that the partial derivative with respect to the first variable (resp. second variable) exists on ${\cal N} \times D$ (resp. $C \times {\cal M}$) and therefore the function is differentiable on ${\cal N} \times {\cal M}$. This is an interesting property of convex-concave functions and it does not hold for convex-convex functions. As an immediate application we recover the generic single-valuedness of monotone operators.