On the Regularity of Generalized Conjugate Functions

TL;DR

This paper studies the regularity properties of generalized conjugate functions and proximal mappings, providing conditions for their smoothness and differentiability using advanced variational analysis techniques.

Contribution

It offers new verifiable conditions for regularity and differentiability of generalized conjugates and proximal mappings, extending understanding beyond first-order properties.

Findings

Conditions for local single-valuedness and continuity of generalized proximal mappings.

Explicit derivative formulas for generalized conjugates.

Criteria for strict twice differentiability of generalized conjugates.

Abstract

We investigate regularity properties of generalized conjugate functions induced by a general coupling function and the associated generalized proximal mapping. Our main results provide verifiable conditions ensuring local single-valuedness, continuity, Lipschitz continuity, and differentiability of the generalized proximal mapping, and transfer these properties to generalized conjugates providing explicit derivative formulas. These results are based on a nonsmooth implicit function theorem for generalized equations, relying on graphical localizations and second-order variational tools. Beyond first-order regularity, we also derive conditions under which generalized conjugates are strictly twice differentiable.