On the complexity of the projective splitting and Spingarn's methods for the sum of two maximal monotone operators

TL;DR

This paper analyzes the iteration complexity of projective splitting and Spingarn's methods for solving the sum of two maximal monotone operators, providing new bounds and convergence rates for these algorithms.

Contribution

It offers the first complexity bounds for the pointwise and ergodic convergence of projective splitting methods and extends these results to inexact variants of the algorithms.

Findings

Established iteration-complexity bounds for projective splitting methods.

Derived complexity bounds for Spingarn's partial inverse method.

Presented inexact variants with proven convergence rates.

Abstract

In this work we study the pointwise and ergodic iteration-complexity of a family of projective splitting methods proposed by Eckstein and Svaiter, for finding a zero of the sum of two maximal monotone operators. As a consequence of the complexity analysis of the projective splitting methods, we obtain complexity bounds for the two-operator case of Spingarn's partial inverse method. We also present inexact variants of two specific instances of this family of algorithms, and derive corresponding convergence rate results.