  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.
