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Volume 9, Issue 6, December 2020, Page: 187-194
Strong Convergence of the Hybrid Halpern Type Proximal Point Algorithm
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
Received: May 15, 2020;       Accepted: Jun. 12, 2020;       Published: Nov. 16, 2020
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
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
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