Mathematical Programming

, Volume 137, Issue 1–2, pp 91–129 | Cite as

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

Full Length Paper Series A

Abstract

In view of the minimization of a nonsmooth nonconvex function f, we prove an abstract convergence result for descent methods satisfying a sufficient-decrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that the function f satisfies the Kurdyka–Łojasiewicz inequality. This assumption allows to cover a wide range of problems, including nonsmooth semi-algebraic (or more generally tame) minimization. The specialization of our result to different kinds of structured problems provides several new convergence results for inexact versions of the gradient method, the proximal method, the forward–backward splitting algorithm, the gradient projection and some proximal regularization of the Gauss–Seidel method in a nonconvex setting. Our results are illustrated through feasibility problems, or iterative thresholding procedures for compressive sensing.

Keywords

Nonconvex nonsmooth optimization Semi-algebraic optimization Tame optimization Kurdyka–Łojasiewicz inequality Descent methods Relative error Sufficient decrease Forward–backward splitting Alternating minimization Proximal algorithms Iterative thresholding Block-coordinate methods o-minimal structures 

Mathematics Subject Classification (2010)

34G25 47J25 47J30 47J35 49M15 49M37 65K15 90C25 90C53 

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Copyright information

© Springer and Mathematical Optimization Society 2011

Authors and Affiliations

  • Hedy Attouch
    • 1
  • Jérôme Bolte
    • 2
  • Benar Fux Svaiter
    • 3
  1. 1.I3M UMR CNRS 5149Université Montpellier IIMontpellierFrance
  2. 2.TSE (GREMAQ, Université Toulouse I), Manufacture des TabacsToulouseFrance
  3. 3.IMPARio de JaneiroBrazil

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