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

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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.

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Authors and Affiliations

  1. I3M UMR CNRS 5149, Université Montpellier II, Place Eugène Bataillon, 34095, Montpellier, France

    Hedy Attouch

  2. Equipe Combinatoire et Optimisation (UMR 7090), Case 189, Université Pierre et Marie Curie, 4 Place Jussieu, 75252, Paris Cedex 05, France

    Jérôme Bolte

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  1. Hedy Attouch
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  2. Jérôme Bolte
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Attouch, H., Bolte, J. On the convergence of the proximal algorithm for nonsmooth functions involving analytic features. Math. Program. 116, 5–16 (2009). https://doi.org/10.1007/s10107-007-0133-5

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  • DOI: https://doi.org/10.1007/s10107-007-0133-5

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