We study the convergence of general abstract descent methods applied to a lower semicontinuous nonconvex function f that satises 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.
Submitted at Journal of Optimization Theory and Applications (JOTA).