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Coupling the Proximal Point Algorithm with Approximation Methods

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  • R. Cominetti

    (Universidad de Chile)

Abstract

We study the convergence of a diagonal process for minimizing a closed proper convex function f, in which a proximal point iteration is applied to a sequence of functions approximating f. We prove that, when the approximation is sufficiently fast, and also when it is sufficiently slow, the sequence generated by the method converges toward a minimizer of f. Comparison to previous work is provided through examples in penalty methods for linear programming and Tikhonov regularization.

Suggested Citation

  • R. Cominetti, 1997. "Coupling the Proximal Point Algorithm with Approximation Methods," Journal of Optimization Theory and Applications, Springer, vol. 95(3), pages 581-600, December.
    • Handle: RePEc:spr:joptap:v:95:y:1997:i:3:d:10.1023_a:1022621905645
    DOI: 10.1023/A:1022621905645
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    References listed on IDEAS

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    1. Correa Romar, 2014. "Mathematical Foci," Mathematical Economics Letters, De Gruyter, vol. 2(1-2), pages 5-11, August.
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    Cited by:

    1. Julian Rasch & Antonin Chambolle, 2020. "Inexact first-order primal–dual algorithms," Computational Optimization and Applications, Springer, vol. 76(2), pages 381-430, June.
    2. Hedy Attouch & Szilárd Csaba László, 2024. "Convex optimization via inertial algorithms with vanishing Tikhonov regularization: fast convergence to the minimum norm solution," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 99(3), pages 307-347, June.
    3. Felipe Alvarez & Miguel Carrasco & Karine Pichard, 2005. "Convergence of a Hybrid Projection-Proximal Point Algorithm Coupled with Approximation Methods in Convex Optimization," Mathematics of Operations Research, INFORMS, vol. 30(4), pages 966-984, November.
    4. Hedy Attouch & Alexandre Cabot & Zaki Chbani & Hassan Riahi, 2018. "Inertial Forward–Backward Algorithms with Perturbations: Application to Tikhonov Regularization," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 1-36, October.
    5. Fu-Quan Xia & Nan-Jing Huang, 2011. "A Projection-Proximal Point Algorithm for Solving Generalized Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 150(1), pages 98-117, July.
    6. Felipe Alvarez & Miguel Carrasco & Thierry Champion, 2012. "Dual Convergence for Penalty Algorithms in Convex Programming," Journal of Optimization Theory and Applications, Springer, vol. 153(2), pages 388-407, May.

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