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Strong Convergence of the Hybrid Halpern Type Proximal Point Algorithm
Applied and Computational Mathematics
Volume 9, Issue 6, December 2020, Pages: 187-194
Received: May 15, 2020; Accepted: Jun. 12, 2020; Published: Nov. 16, 2020
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Authors
Liu Liu, Department of Mathematics, Sichuan University of Arts and Science, Dazhou, P. R. China
Qing-bang Zhang, College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, P. R. China
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Abstract
Based on the proximal point algorithm, which is a widely used tool for solving a variety of convex optimization problems, there are many algorithms for finding zeros of maximally monotone operators. The algorithm works by applying successively so-called "resolvent" mappings with errors associated to the original object, and is weakly convergent in Hilbert space. In order to acquiring the strong convergence of the algorithm, in this paper, we construct a hybrid Halpern type proximal point algorithm with errors for approximating the zero of a maximal monotone operator, which is a combination of modified proximal point algorithm raised by Yao and Noor and Halpern inexact proximal point algorithm raised by Zhang, respectively. Then, we prove the strong convergence of our algorithm with weaker assumptions in Hilbert space. Finally, we present a numerical example to show the convergence and the convergence speed, which is not affected but accelerated by the projection in the algorithm. Our work improved and generalized some known results.
Keywords
Hybrid Halpern Type Proximal Point Algorithm, Maximal Monotone Operator, Strong Convergence, Hilbert Space
To cite this article
Liu Liu, Qing-bang Zhang, Strong Convergence of the Hybrid Halpern Type Proximal Point Algorithm, Applied and Computational Mathematics. Vol. 9, No. 6, 2020, pp. 187-194. doi: 10.11648/j.acm.20200906.13
Copyright
Copyright © 2020 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]
R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.
[2]
Güler, On the convergence of the proximal point algorithm for convex optimization, SIMA J. Control. Optim., 29 (1991), 403-419.
[3]
J. Eckstein, D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Programming, 55 (1992), 293-318.
[4]
B. S. He, L. Liao, Zh. Yang (2003). A new approximate proximal point algorithm for maximal monotone operator, Sci. China Ser. A, 46 (2): 200-206.
[5]
Zh. Yang, B. S. He, A Relaxed Approximate Proximal Point Algorithm, Ann. Oper. Res., 133 (2005), 119-125.
[6]
Q. B. Zhang, A modified proximal point algorithm with errors for approximating solution of the general variational inclusion, Operat. Research Lett., 40 (2012), 564-567.
[7]
Y. Liu, Weak convergence of a hybrid type method with errors for a maximal monotone mapping in Banach spaces, J. Inequal. Appl., 2015 (2015), 260.
[8]
Boikanyo, G. Morosanu, Four parameter proximal point algorithms, Nonlinear Anal., 74 (2011), 544-555.
[9]
H. Khatibzadeh, S. Ranjbar, On the Strong Convergence of Halpern Type Proximal Point Algorithm, J. Optim. Theory Appl., 158 (2) (2013), 385-396.
[10]
T. H. Kim, H. K. Xu, Strong convergence of modified mann iterations, Nonlinear. Anal., 61 (2005), 51-60.
[11]
S. Saejung, A supplement to a regularization method for the proximal point algorithm, J. Glob. Optim., 56 (2013), 121-129.
[12]
Ch. Tian, Y. Song, Strong convergence of a regularization method for Rockafellar's proximal point algorithm, J. Glob. Optim., 55 (2013), 831-837.
[13]
F. Wang, H. Cui, On the contraction proximal point algorithms with multi parameters. J. Glob. Optim., 54 (3) (2012), 1-7.
[14]
H. K. Xu, Strong convergence of an iterative method for non expansive and accretive operators, J. Math. Anal. Appl., 314 (2006), 631-643.
[15]
Y. Yao, M. A. Noor, On convergence criteria of generalized proximal point algorithms, J. Comput. Appl. Math., 217 (2008), 46-55.
[16]
H. K. Xu, A regularization method for the proximal point operators, J. Glob. Optim., 36 (2006), 115-125.
[17]
Q. N. Zhang, Y. S. Song, Halpern type proximal point algorithm of accretive operators, Nonlinear Appl., 75 (4) (2012), 1859-1868.
[18]
G. Marino, H. K. Xu, Convergence of generalized proximal point algorithm, Comm. Pure Appl. Anal., 3 (2004), 791-808.
[19]
M. A. Moor, Projection-proximate methods for general variational inequalities, J. Math. Anal. Appl., 318 (2006), 53-62.
[20]
H. K. Xu, Iterative algorithms for nonlinear operators, J. Londen Math. Soc., 66 (2002), 240-256.
[21]
P. E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912.
[22]
L. Laurentiu, N. Adriana, S. Andrei, An abstract proximal point algorithm, J. Global Optim., 72 (3) (2018), 553-577.
[23]
M. Gheorghe, P. Adrian, A proximal point algorithm revisited and extended, J. Optim. Theory Appl., 182 (3) (2019), 1120-1129.
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