Computational Optimization and Applications

, Volume 52, Issue 3, pp 785–803 | Cite as

A Lyusternik–Graves theorem for the proximal point method

  • Francisco J. Aragón Artacho
  • Michaël Gaydu


We consider a generalized version of the proximal point algorithm for solving the perturbed inclusion yT(x), where y is a perturbation element near 0 and T is a set-valued mapping acting from a Banach space X to a Banach space Y which is metrically regular around some point \(({\bar{x}},0)\) in its graph. We study the behavior of the convergent iterates generated by the algorithm and we prove that they inherit the regularity properties of T, and vice versa. We analyze the cases when the mapping T is metrically regular and strongly regular.


Proximal point algorithm Generalized equations Perturbations Metric regularity Strong regularity 



The authors would like to thank two anonymous referees for their valuable suggestions and remarks that allowed us to improve the quality of the paper. Research of the first author was partially supported by Ministerio de Ciencia e Innovación (Spain), grant MTM2008-06695-C03-01 and program “Juan de la Cierva.” Research of the second author was partially supported by Contract EA4540 (France).


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Francisco J. Aragón Artacho
    • 1
  • Michaël Gaydu
    • 2
  1. 1.Department of Statistics and Operations ResearchUniversity of AlicanteAlicanteSpain
  2. 2.LAMIA, Dpt. de MathématiquesUniversité Antilles-GuyanePointe-à-PitreFrance

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