A Monotone+Skew Splitting Model for Composite Monotone Inclusions in Duality

TL;DR

This paper introduces a new primal-dual splitting algorithm for solving composite monotone inclusions and their duals, effectively handling large-scale problems by decomposing operators and linear transformations separately.

Contribution

It develops a unified algorithmic framework for composite monotone inclusions involving skew-symmetric operators, with proven convergence and practical applications.

Findings

Algorithms operate in a fully decomposed manner

Convergence results are established for the proposed methods

Applications to composite variational problems are demonstrated

Abstract

The principle underlying this paper is the basic observation that the problem of simultaneously solving a large class of composite monotone inclusions and their duals can be reduced to that of finding a zero of the sum of a maximally monotone operator and a linear skew-adjoint operator. An algorithmic framework is developed for solving this generic problem in a Hilbert space setting. New primal-dual splitting algorithms are derived from this framework for inclusions involving composite monotone operators, and convergence results are established. These algorithms draw their simplicity and efficacy from the fact that they operate in a fully decomposed fashion in the sense that the monotone operators and the linear transformations involved are activated separately at each iteration. Comparisons with existing methods are made and applications to composite variational problems are demonstrated.