Forward-Backward algorithms for weakly convex problems

2023/03/24 by Bednarczuk, Ewa, Bruccola, Giovanni, Scrivanti, Gabriele +1
#90C30 90C26 90C51 65K10 52A01 #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2303.14021

Abstract

We investigate the convergence properties of exact and inexact forward-backward algorithms to minimise the sum of two weakly convex functions defined on a Hilbert space, where one has a Lipschitz-continuous gradient. We show that the exact forward-backward algorithm converges strongly to a global solution, provided that the objective function satisfies a sharpness condition. For the inexact forward-backward algorithm, the same condition ensures that the distance from the iterates to the solution set approaches a positive threshold depending on the accuracy level of the proximal computations. As an application of the considered setting, we provide numerical experiments related to discrete tomography.

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