Forward-Backward algorithms for weakly convex problems

We investigate the convergence properties of exact and inexact forward-backward algorithms for the minimisation of 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 assumptions ensure 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 feasibility problems involving a sphere and a closed convex set, source localisation and discrete tomography.