We study the convergence of general abstract descent methods applied to a lower semicontinuous nonconvex function f that satisfies the Kurdyka-Lojasiewicz inequality in a Hilbert space. We prove that any precompact sequence converges to a critical point of f and obtain new convergence rates both for the values and the iterates. The analysis covers alternating versions of the forward-backward method with variable metric and relative errors. As an example, a nonsmooth and nonconvex version of the Levenberg-Marquardt algorithm is detailled.
@article{arxiv.1405.1357,
title = {Splitting methods with variable metric for KL functions},
author = {Pierre Frankel and Guillaume Garrigos and Juan Peypouquet},
journal= {arXiv preprint arXiv:1405.1357},
year = {2017}
}