Eckstein-Ferris-Pennanen-Robinson duality revisited: paramonotonicity, total Fenchel-Rockafellar duality, and the Chambolle-Pock operator

TL;DR

This paper revisits a classical duality framework in monotone operator theory, highlighting the role of paramonotonicity, characterizing total duality, and deriving projection formulas relevant to the Chambolle-Pock algorithm.

Contribution

It broadens the understanding of duality in monotone operator theory by linking paramonotonicity with saddle point coincidence and providing new projection formulas.

Findings

Paramonotonicity ensures saddle points coincide with primal-dual solution rectangles.

Total duality is characterized in the subdifferential setting.

Projection formulas are derived for sets related to the Chambolle-Pock algorithm.

Abstract

Finding zeros of the sum of two maximally monotone operators involving a continuous linear operator is a central problem in optimization and monotone operator theory. We revisit the duality framework proposed by Eckstein, Ferris, Pennanen, and Robinson from a quarter of a century ago. Paramonotonicity is identified as a broad condition ensuring that saddle points coincide with the closed convex rectangle formed by the primal and dual solutions. Additionally, we characterize total duality in the subdifferential setting and derive projection formulas for sets that arise in the analysis of the Chambolle-Pock algorithm within the recent framework developed by Bredies, Chenchene, Lorenz, and Naldi.