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

Authors:
Hedy Attouch;Jérôme Bolte
Affiliations:
Université Montpellier II, I3M UMR CNRS 5149, Place Eugène Bataillon, 34095, Montpellier, France;Université Pierre et Marie Curie, Equipe Combinatoire et Optimisation (UMR 7090), Case 189, 4 Place Jussieu, 75252, Paris Cedex 05, France
Venue:
Mathematical Programming: Series A and B - Nonlinear convex optimization and variational inequalities
Year:
2008

Citing 0
Cited 2

Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality

Mathematics of Operations Research
A New Sequential Optimality Condition for Constrained Optimization and Algorithmic Consequences

SIAM Journal on Optimization

Quantified Score

Hi-index	0.00

Visualization

Abstract

We study the convergence of the proximal algorithm applied to nonsmooth functions that satisfy the Łjasiewicz inequality around their generalized critical points. Typical examples of functions complying with these conditions are continuous semialgebraic or subanalytic functions. Following Łjasiewicz’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 Łjasiewicz exponents. Apart from the sharp and elliptic cases which yield finite or geometric convergence, the decay estimates that are derived are of the type O(k −s ), where s ∈ (0, + ∞) depends on the flatness of the function.