Splitting Methods with Variable Metric for Kurdyka-Aojasiewicz Functions and General Convergence Rates

We study the convergence of general descent methods applied to a lower semi-continuous and nonconvex function, which satisfies the Kurdyka-Aojasiewicz inequality in a Hilbert space. We prove that any precompact sequence converges to a critical point of the function, 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 detailed.