Generalized Forward-Backward Splitting

TL;DR

This paper presents a generalized forward-backward splitting algorithm for convex optimization that efficiently handles multiple non-smooth functions, with proven convergence and robustness, demonstrated through imaging inverse problems.

Contribution

It extends the forward-backward algorithm to multiple non-smooth functions, enabling more flexible convex optimization solutions.

Findings

Proven convergence in infinite-dimensional spaces.

Robustness to errors in proximity and gradient computations.

Improved performance in imaging inverse problems.

Abstract

This paper introduces the generalized forward-backward splitting algorithm for minimizing convex functions of the form $F + \sum_{i=1}^n G_i$, where $F$ has a Lipschitz-continuous gradient and the $G_i$'s are simple in the sense that their Moreau proximity operators are easy to compute. While the forward-backward algorithm cannot deal with more than $n = 1$ non-smooth function, our method generalizes it to the case of arbitrary $n$. Our method makes an explicit use of the regularity of $F$ in the forward step, and the proximity operators of the $G_i$'s are applied in parallel in the backward step. This allows the generalized forward backward to efficiently address an important class of convex problems. We prove its convergence in infinite dimension, and its robustness to errors on the computation of the proximity operators and of the gradient of $F$. Examples on inverse problems in imaging demonstrate the advantage of the proposed methods in comparison to other splitting algorithms.