Explicit convergence rate of the proximal point algorithm under R-continuity

Ba Khiet Le; Michel Théra

doi:10.3934/eect.2025016

Article Contents

2026, Volume 17: 93-105. Doi: 10.3934/eect.2025016

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Explicit convergence rate of the proximal point algorithm under R-continuity

Ba Khiet Le^1, and
Michel Théra^2, ,

1.
Optimization Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2.
Mathematics and Computer Science Department, University of Limoges, 123 Avenue Albert Thomas, 87060 Limoges CEDEX, France

^*Corresponding author: Michel Théra (michel.thera@unilim.fr)
^*Corresponding author: Michel Théra (michel.thera@unilim.fr)

Document Type: Research Article.
Dedicated to the memory of Professor Hedy Attouch

Received: August 2024

Early access: February 10, 2025

Published online: February 10, 2025

This research is funded by Ton Duc Thang University under grant number FOSTECT.2024.25. It benefited also from the support of the FMJH Program Gaspard Monge for optimization and operations research and their interactions with data science.

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Abstract

This paper provides a thorough comparison between R-continuity and other fundamental tools in optimization such as metric regularity, metric subregularity, and calmness. We show that R-continuity has some advantages in the convergence rate analysis of algorithms for solving optimization problems. We also present some properties of R-continuity and study the explicit convergence rate of the proximal point algorithm $ (\mathbf{P}\mathbf{P}\mathbf{A}) $ under the R-continuity.

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Mathematics Subject Classification: Primary: 49J52; Secondary: 49J53.

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[1]	B. Abbas, H. Attouch and B. F. Svaiter, Newton-like dynamics and forward-backward methods for structured monotone inclusions in Hilbert spaces, J. Optim. Theory Appl., 161 (2014), 331-360. doi: 10.1007/s10957-013-0414-5.
[2]	V. Acary, O. Bonnefon and B. Brogliato, Nonsmooth Modeling and Simulation for Switched Circuits, Lecture Notes in Electrical Engineering, 69. Springer Netherlands, 2011.
[3]	S. Adly, H. Attouch and M. H. Le, A doubly nonlinear evolution system with threshold effects associated with dry friction, J. Optim. Theory Appl., 203 (2024), 1188-1218. doi: 10.1007/s10957-024-02417-2.
[4]	S. Adly, M. G. Cojocaru and B. K. Le, State-dependent sweeping processes: Asymptotic behavior and algorithmic approaches, J. Optim. Theory Appl., 202 (2024), 932-948. doi: 10.1007/s10957-024-02485-4.
[5]	S. Adly, A. L. Dontchev and M. Théra, On one-sided Lipschitz stability of set-valued contractions, Numer. Funct. Anal. Optim., 35 (2014), 837-850. doi: 10.1080/01630563.2014.895760.
[6]	H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2014.
[7]	H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, Berlin, 2011.
[8]	H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Math. Studies 5, North-Holland American Elsevier, 1973.
[9]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer New York, NY, 2011.
[10]	A. V. Dmitruk and A. Y. Kruger, Metric regularity and systems of generalized equations, J. Math. Anal. Appl., 342 (2008), 864-873. doi: 10.1016/j.jmaa.2007.12.057.
[11]	A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings. A View from Variational Analysis, Springer Monographs in Mathematics. Springer, Dordrecht, 2009.
[12]	R. Henrion and J. V. Outrata, Calmness of constraint systems with applications, Math. Program., Ser. B, 104 (2005), 437-464. doi: 10.1007/s10107-005-0623-2.
[13]	A. J. Hoffman, On approximate solutions of systems of linear inequalities, J. Research Nat. Bureau Standards, 49 (1952), 263-265. doi: 10.6028/jres.049.027.
[14]	A. D. Ioffe, Metric regularity–A survey, Part I. Theory, J. Aust. Math. Soc., 101 (2016), 188-243. doi: 10.1017/S1446788715000701.
[15]	A. D. Ioffe and J. V. Outrata, On metric and calmness qualification conditions in subdifferential calculus, Set-Valued Anal., 16 (2008), 199-227. doi: 10.1007/s11228-008-0076-x.
[16]	A. Y. Kruger, Nonlinear metric subregularity, J. Optim. Theory Appl., 171 (2016), 820-855. doi: 10.1007/s10957-015-0807-8.
[17]	B. K. Le, R-continuity with applications to convergence analysis of Tikhonov regularization and DC programming, Journal of Convex Analysis, 31 (2024), 243-254.
[18]	B. Martinet, Regularisation d'inéquations variationelles par approximations successives, Rev. Francaise Inf. Rech. Oper., 4 (1970), 154-158. doi: 10.1051/m2an/197004R301541.
[19]	G. J. Minty, Monotone (Nonlinear) operators in Hilbert space, Duke Mathematical Journal, 29 (1962), 341-346. doi: 10.1215/S0012-7094-62-02933-2.
[20]	Ch. A. Micchelli, L. Chen and Y. Xu, Proximity algorithms for image models: Denoising, Inverse Problems, 27 (2011), 045009, 30 pp. doi: 10.1088/0266-5611/27/4/045009.
[21]	H. V. Ngai and M. Théra, Error bounds in metric spaces and application to the perturbation stability of metric regularity, SIAM J. Optim., 19 (2008), 1-20. doi: 10.1137/060675721.
[22]	J. P. Penot, Calculus without Derivatives, Springer New York, 2014.
[23]	S. M. Robinson, Regularity and stability for convex multivalued functions, Math. Oper. Res., 1 (1976), 130-143. doi: 10.1287/moor.1.2.130.
[24]	S. M. Robinson, Some continuity properties of polyhedral multifunctions, Math. Program. Study, 14 (1981), 206-214. doi: 10.1007/BFb0120929.
[25]	R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optimization, 14 (1976), 877-898. doi: 10.1137/0314056.
[26]	R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Grundlehren der Mathematischen Wissenschaften, 317. Springer-Verlag, Berlin, 1998.
[27]	A. Tanwani, B. Brogliato and C. Prieur, Well-posedness and output regulation for implicit time-varying evolution variational inequalities, SIAM J. Control Opti., 56 (2018), 751-781. doi: 10.1137/16M1083657.
[28]	L. Thibault, Unilateral Variational Analysis in Banach Spaces, World Scientific, 2023.
[29]	C. Ursescu, Multifunctions with closed convex graphs, Czechoslovak Math. J., 25 (1975), 438-411. doi: 10.21136/CMJ.1975.101337.
[30]	X. Y. Zheng and K. F. Ng, Metric subregularity and constraint qualifications for convex generalized equations in Banach spaces, SIAM J. Optim., 18 (2007), 437-460. doi: 10.1137/050648079.
[31]	X. Y. Zheng and K. F. Ng, Metric subregularity and calmness for nonconvex generalized equations in Banach spaces, SIAM J. Optim., 20 (2010), 2119-2136. doi: 10.1137/090772174.