Strong Convergence of the Hybrid Halpern Type Proximal Point Algorithm

Liu Liu; Qing-bang Zhang

doi:doi:10.11648/j.acm.20200906.13

| Peer-Reviewed

Strong Convergence of the Hybrid Halpern Type Proximal Point Algorithm

Liu Liu, Qing-bang Zhang

Published in Applied and Computational Mathematics (Volume 9, Issue 6)

Received: 15 May 2020 Accepted: 12 June 2020 Published: 16 November 2020

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

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.

Published in	Applied and Computational Mathematics (Volume 9, Issue 6)
DOI	10.11648/j.acm.20200906.13
Page(s)	187-194
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Hybrid Halpern Type Proximal Point Algorithm, Maximal Monotone Operator, Strong Convergence, Hilbert Space

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.

Cite This Article

Plain Text BibTeX RIS

APA Style

Liu Liu, Qing-bang Zhang. (2020). Strong Convergence of the Hybrid Halpern Type Proximal Point Algorithm. Applied and Computational Mathematics, 9(6), 187-194. https://doi.org/10.11648/j.acm.20200906.13

Copy | Download

ACS Style

Liu Liu; Qing-bang Zhang. Strong Convergence of the Hybrid Halpern Type Proximal Point Algorithm. Appl. Comput. Math. 2020, 9(6), 187-194. doi: 10.11648/j.acm.20200906.13

Copy | Download

AMA Style

Liu Liu, Qing-bang Zhang. Strong Convergence of the Hybrid Halpern Type Proximal Point Algorithm. Appl Comput Math. 2020;9(6):187-194. doi: 10.11648/j.acm.20200906.13

Copy | Download

@article{10.11648/j.acm.20200906.13,
  author = {Liu Liu and Qing-bang Zhang},
  title = {Strong Convergence of the Hybrid Halpern Type Proximal Point Algorithm},
  journal = {Applied and Computational Mathematics},
  volume = {9},
  number = {6},
  pages = {187-194},
  doi = {10.11648/j.acm.20200906.13},
  url = {https://doi.org/10.11648/j.acm.20200906.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20200906.13},
  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.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Strong Convergence of the Hybrid Halpern Type Proximal Point Algorithm
AU  - Liu Liu
AU  - Qing-bang Zhang
Y1  - 2020/11/16
PY  - 2020
N1  - https://doi.org/10.11648/j.acm.20200906.13
DO  - 10.11648/j.acm.20200906.13
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 187
EP  - 194
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20200906.13
AB  - 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.
VL  - 9
IS  - 6
ER  -

Copy | Download

Author Information

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

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Liu Liu, Qing-bang Zhang. (2020). Strong Convergence of the Hybrid Halpern Type Proximal Point Algorithm. Applied and Computational Mathematics, 9(6), 187-194. https://doi.org/10.11648/j.acm.20200906.13

Copy | Download

ACS Style

Liu Liu; Qing-bang Zhang. Strong Convergence of the Hybrid Halpern Type Proximal Point Algorithm. Appl. Comput. Math. 2020, 9(6), 187-194. doi: 10.11648/j.acm.20200906.13

Copy | Download

AMA Style

Liu Liu, Qing-bang Zhang. Strong Convergence of the Hybrid Halpern Type Proximal Point Algorithm. Appl Comput Math. 2020;9(6):187-194. doi: 10.11648/j.acm.20200906.13

Copy | Download

@article{10.11648/j.acm.20200906.13,
  author = {Liu Liu and Qing-bang Zhang},
  title = {Strong Convergence of the Hybrid Halpern Type Proximal Point Algorithm},
  journal = {Applied and Computational Mathematics},
  volume = {9},
  number = {6},
  pages = {187-194},
  doi = {10.11648/j.acm.20200906.13},
  url = {https://doi.org/10.11648/j.acm.20200906.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20200906.13},
  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.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Strong Convergence of the Hybrid Halpern Type Proximal Point Algorithm
AU  - Liu Liu
AU  - Qing-bang Zhang
Y1  - 2020/11/16
PY  - 2020
N1  - https://doi.org/10.11648/j.acm.20200906.13
DO  - 10.11648/j.acm.20200906.13
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 187
EP  - 194
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20200906.13
AB  - 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.
VL  - 9
IS  - 6
ER  -

Copy | Download