Regularized HPE-type methods for solving monotone inclusions with improved pointwise iteration-complexity bounds

TL;DR

This paper introduces regularized HPE-type methods for monotone inclusions, significantly improving pointwise iteration-complexity and approaching the ergodic complexity of existing methods.

Contribution

It proposes new regularized HPE-type algorithms that enhance convergence bounds for solving monotone inclusion problems.

Findings

Pointwise iteration-complexity is significantly improved.

Approaches ergodic iteration-complexity up to a logarithmic factor.

Methods are applicable to regularizations of the original MIP.

Abstract

This paper studies the iteration-complexity of new regularized hybrid proximal extragradient (HPE)-type methods for solving monotone inclusion problems (MIPs). The new (regularized HPE-type) methods essentially consist of instances of the standard HPE method applied to regularizations of the original MIP. It is shown that its pointwise iteration-complexity considerably improves the one of the HPE method while approaches (up to a logarithmic factor) the ergodic iteration-complexity of the latter method.