The forward-backward algorithm and the normal problem

TL;DR

This paper investigates the forward-backward algorithm's behavior in cases where the inclusion problem has no solution, introducing new formulas for normal solutions and the displacement map's range, supported by illustrative examples.

Contribution

It presents novel formulas for normal solutions and the displacement map range in the forward-backward algorithm when no solutions exist.

Findings

New formula for normal solutions in unsolvable cases

Range characterization of the displacement map

Illustrative examples demonstrating the theory

Abstract

The forward-backward splitting technique is a popular method for solving monotone inclusions that has applications in optimization. In this paper we explore the behaviour of the algorithm when the inclusion problem has no solution. We present a new formula to define the normal solutions using the forward-backward operator. We also provide a formula for the range of the displacement map of the forward-backward operator. Several examples illustrate our theory.