Eckstein-Ferris-Pennanen-Robinson duality revisited: paramonotonicity, total Fenchel-Rockafellar duality, and the Chambolle-Pock operator
Heinz H. Bauschke, Walaa M. Moursi, Shambhavi Singh
TL;DR
The paper analyzes the zero-finding problem $0\in Ax+L^*BLx$ with maximally monotone $A,B$ and linear $L$, revisiting EF duality and introducing traversals between primal and dual solutions, the saddle-point structure, and total duality criteria under paramonotonicity. It shows that paramonotonicity yields the rectangle relation ${\mathbf S}=Z\times K$ and that nonemptiness of $Z$ characterizes total duality in the subdifferential case, while providing projection formulas within the Chambolle–Pock/BCLN framework. The results extend to normal-cone and affine-subspace examples and to multi-operator problems via a product-space construction, linking to Attouch–Théra duality and PDHG. These findings offer structural insights and computable projections for solving split monotone inclusions and multi-operator feasibility problems.
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.