The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems

Jérôme Bolte, Aris Daniilidis, Adrian Lewis

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248 Citations (Scopus)

Abstract

Given a real-analytic function f : ℝ n → ℝ and a critical point a ε ℝ n, the Łojasiewicz inequality asserts that there exists θ ε [1/2, 1) such that the function |f - f(a)| θ ||∇f|| _1 remains bounded around a. In this paper, we extend the above result to a wide class of nonsmooth functions (that possibly admit the value +∞), by establishing an analogous inequality in which the derivative ∇ f(x) can be replaced by any element x* of the subdifferential ∂f(x) of f. Like its smooth version, this result provides new insights into the convergence aspects of subgradient-type dynamical systems. Provided that the function f is sufficiently regular (for instance, convex or lower-C 2), the bounded trajectories of the corresponding subgradient dynamical system can be shown to be of finite length. Explicit estimates of the rate of convergence are also derived. © 2007 Society for Industrial and Applied Mathematics.
Original languageEnglish
Pages (from-to)1205-1223
JournalSIAM Journal on Optimization
Volume17
DOIs
Publication statusPublished - 1 Dec 2006

Keywords

  • Descent method
  • Dynamical system
  • Nonsmooth analysis
  • Subanalytic function
  • Subdifferential
  • ŁOjasiewicz inequality

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