An Inexact Spingarn's Partial Inverse Method with Applications to Operator Splitting and Composite Optimization

TL;DR

This paper introduces an inexact version of Spingarn's partial inverse method, analyzing its iteration complexity and applying it to operator splitting and composite convex optimization problems.

Contribution

It develops an inexact framework for Spingarn's method, linking it to the HPE framework, and extends it to new algorithms for operator splitting and convex optimization.

Findings

Established iteration complexity bounds for the inexact method.

Generalized Spingarn's splitting to a broader class of problems.

Provided complexity analysis for parallel forward-backward algorithms.

Abstract

We propose and study the iteration-complexity of an inexact version of the Spingarn's partial inverse method. Its complexity analysis is performed by viewing it in the framework of the hybrid proximal extragradient (HPE) method, for which pointwise and ergodic iteration-complexity has been established recently by Monteiro and Svaiter. As applications, we propose and analyze the iteration-complexity of an inexact operator splitting algorithm -- which generalizes the original Spingarn's splitting method -- and of a parallel forward-backward algorithm for multi-term composite convex optimization.