Projection methods for solving split equilibrium problems

Dang Van Hieu

doi:10.3934/jimo.2019056

Article Contents

2020, Volume 16, Issue 5: 2331-2349. Doi: 10.3934/jimo.2019056

This issue Previous Article Differential equation approximation and enhancing control method for finding the PID gain of a quarter-car suspension model with state-dependent ODE Next Article A new concave reformulation and its application in solving DC programming globally under uncertain environment

Projection methods for solving split equilibrium problems

Dang Van Hieu^,

Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

^* Corresponding author: dangvanhieu@tdtu.edu.vn
^* Corresponding author: dangvanhieu@tdtu.edu.vn

Received: June 30, 2018

Revised: December 31, 2018

Early access: May 2019

Published: August 2020

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Abstract

The paper considers a split inverse problem involving component equilibrium problems in Hilbert spaces. This problem therefore is called the split equilibrium problem (SEP). It is known that almost solution methods for solving problem (SEP) are designed from two fundamental methods as the proximal point method and the extended extragradient method (or the two-step proximal-like method). Unlike previous results, in this paper we introduce a new algorithm, which is only based on the projection method, for finding solution approximations of problem (SEP), and then establish that the resulting algorithm is weakly convergent under mild conditions. Several of numerical results are reported to illustrate the convergence of the proposed algorithm and also to compare with others.

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Mathematics Subject Classification: Primary: 65K10, 65K15; Secondary: 90C33.

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Figure 1. Algorithm 1 for $ (m, k) = (30, 20) $ and different sequences of $ \beta_n $. The number of iterations is 360,353,339,360,355,376, respectively

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Figure 2. Algorithm 1 for (m; k) = (60; 40) and different sequences of β_n. The number of iterations is 258,333,336,326,291,293, respectively

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Figure 3. Algorithm 1 for (m; k) = (100; 50) and different sequences of β_n. The number of iterations is 215,236,283,280,321,290, respectively

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Figure 4. Algorithm 1 for $ (m, k) = (150,100) $ and different sequences of $ \beta_n $. The number of iterations is 161,188,219,209,245,264, respectively

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Figure 5. Experiment for the algorithms with $ (m, k) = (30, 20) $. The number of iterations is 334,240,379,168,130, respectively

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Figure 6. Experiment for the algorithms with (m; k) = (60; 40). The number of iterations is 326,221,292,129,108, respectively

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Figure 7. Experiment for the algorithms with (m; k) = (100; 50). The number of iterations is 308,250,356,114, 89, respectively

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Figure 8. Experiment for the algorithms with $ (m, k) = (150,100) $. The number of iterations is 254,192,271, 87, 69, respectively

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