On the convergence of the proximal algorithm for nonsmooth functions involving analytic features

We study the convergence of the proximal algorithm applied to nonsmooth functions that satisfy the Lojasiewicz inequality around their generalized critical points. Typical examples of functions complying with these conditions are continuous semialgebraic or subanalytic functions. Following Lojasiewicz’s original idea, we prove that any bounded sequence generated by the proximal algorithm converges to some generalized critical point. We also obtain convergence rate results which are related to the flatness of the function by means of Lojasiewicz exponents. Apart from the sharp and elliptic cases which yield finite-time or geometric convergence, the decay estimates that are derived are of the type O(k), where s ∈ (0,+∞) depends on the flatness of the function.