The operator splitting schemes revisited: primal-dual gap and degeneracy reduction by a unified analysis

TL;DR

This paper provides a unified analysis of operator splitting schemes, focusing on convergence properties, primal-dual gap, and degeneracy reduction, enhancing understanding of their theoretical foundations and potential simplifications.

Contribution

It offers a unified proximal point analysis of operator splitting schemes, analyzing convergence, primal-dual gap, and degeneracy reduction techniques.

Findings

Convergence analysis of generalized Bregman distance and primal-dual gap.

Unified framework for operator splitting schemes.

Reduction to simple resolvent exploiting structure and degeneracy.

Abstract

We revisit the operator splitting schemes proposed in a recent work of [Some extensions of the operator splitting schemes based on Lagrangian and primal-dual: A unified proximal point analysis, Feng Xue, Optimization, 2022, doi: 10.1080/02331934.2022.2057309], and further analyze the convergence of the generalized Bregman distance and the primal-dual gap of these algorithms within a unified proximal point framework. The possibility of reduction to a simple resolvent is also discussed by exploiting the structure and possible degeneracy of the underlying metric.