Extended Forward-Backward Algorithm

TL;DR

This paper introduces an extended forward-backward algorithm for finding zeros of maximal monotone operators, broadening the scope of convex optimization methods and ensuring convergence under standard assumptions.

Contribution

It extends the classical forward-backward algorithm to handle sums of maximal monotone operators without qualification conditions, with proven convergence.

Findings

Algorithm converges weakly in average under standard assumptions.

Applicable to convex constrained minimization problems without qualification.

Provides a new approach for splitting maximal monotone operators.

Abstract

We propose an extended forward-backward algorithm for approximating a zero of a maximal monotone operator which can be split as the extended sum of two maximal monotone operators. We establish the weak convergence in average of the sequence generated by the algorithm under assumptions similar to those used in classical forward-backward algorithms. This provides as a special case an algorithm for solving convex constrained minimization problems without qualification condition.