# A Lyusternik–Graves theorem for the proximal point method

## Author

Listed:
• 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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File URL: http://hdl.handle.net/10.1007/s10589-011-9439-6

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## References listed on IDEAS

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1. P. Hansen & D. Peeters & J.-F. Thisse, 1982. "An Algorithm for a Constrained Weber Problem," Management Science, INFORMS, vol. 28(11), pages 1285-1295, November.
2. Jian-lin Jiang & Ya Xu, 2006. "Minisum location problem with farthest Euclidean distances," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(2), pages 285-308, October.
3. J. Brimberg & G.O. Wesolowsky, 2002. "Minisum Location with Closest Euclidean Distances," Annals of Operations Research, Springer, vol. 111(1), pages 151-165, March.
4. Carrizosa, E. & Munoz-Marquez, M. & Puerto, J., 1998. "The Weber problem with regional demand," European Journal of Operational Research, Elsevier, vol. 104(2), pages 358-365, January.
5. repec:spr:compst:v:64:y:2006:i:2:p:285-308 is not listed on IDEAS
Full references (including those not matched with items on IDEAS)

### Keywords

Proximal point algorithm; Generalized equations; Perturbations; Metric regularity; Strong regularity;

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