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A Lyusternik–Graves theorem for the proximal point method

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  • Francisco Aragón Artacho

  • Michaël Gaydu

Abstract

We consider a generalized version of the proximal point algorithm for solving the perturbed inclusion y∈T(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. Copyright Springer Science+Business Media, LLC 2012

Suggested Citation

  • Francisco Aragón Artacho & Michaël Gaydu, 2012. "A Lyusternik–Graves theorem for the proximal point method," Computational Optimization and Applications, Springer, vol. 52(3), pages 785-803, July.
    • Handle: RePEc:spr:coopap:v:52:y:2012:i:3:p:785-803
    DOI: 10.1007/s10589-011-9439-6
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    References listed on IDEAS

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    1. R. T. Rockafellar, 1976. "Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 97-116, May.
    2. Teemu Pennanen, 2002. "Local Convergence of the Proximal Point Algorithm and Multiplier Methods Without Monotonicity," Mathematics of Operations Research, INFORMS, vol. 27(1), pages 170-191, February.
    3. Stephen M. Robinson, 1980. "Strongly Regular Generalized Equations," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 43-62, February.
    4. Zhenhua Yang & Bingsheng He, 2005. "A Relaxed Approximate Proximal Point Algorithm," Annals of Operations Research, Springer, vol. 133(1), pages 119-125, January.
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