Projective Splitting with Forward Steps only Requires Continuity

TL;DR

This paper demonstrates that in finite-dimensional spaces, projective splitting algorithms can use only forward steps with a backtracking linesearch to converge under mere continuity, removing the need for Lipschitz continuity assumptions.

Contribution

It proves that the Lipschitz condition is unnecessary for convergence when using forward steps in finite-dimensional spaces, broadening the applicability of the method.

Findings

Convergence achieved with only continuity in finite dimensions.

Backtracking linesearch ensures convergence without Lipschitz assumption.

Extends the scope of projective splitting algorithms.

Abstract

A recent innovation in projective splitting algorithms for monotone operator inclusions has been the development of a procedure using two forward steps instead of the customary proximal steps for operators that are Lipschitz continuous. This paper shows that the Lipschitz assumption is unnecessary when the forward steps are performed in finite-dimensional spaces: a backtracking linesearch yields a convergent algorithm for operators that are merely continuous with full domain.