An Extension of Averaged-Operator-Based Algorithms

TL;DR

This paper extends averaged-operator algorithms to better exploit sparsity in minimization problems, improving convergence by integrating semismooth Newton methods within a Krasnosel'skib1--Mann framework.

Contribution

It introduces novel extensions of averaged-operator algorithms that incorporate semismooth Newton techniques, enhancing their ability to leverage sparsity for faster convergence.

Findings

Extended algorithms are shown to converge under certain conditions.

The new methods generalize existing schemes, including the Krasnosel'skib1--Mann iteration.

Theoretical analysis confirms improved convergence properties.

Abstract

Many of the algorithms used to solve minimization problems with sparsity-inducing regularizers are generic in the sense that they do not take into account the sparsity of the solution in any particular way. However, algorithms known as semismooth Newton are able to take advantage of this sparsity to accelerate their convergence. We show how to extend these algorithms in different directions, and study the convergence of the resulting algorithms by showing that they are a particular case of an extension of the well-known Krasnosel'ski\u{\i}--Mann scheme.