数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (1): 184-204.doi: 10.1007/s10473-023-0112-9

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A RELAXED INERTIAL FACTOR OF THE MODIFIED SUBGRADIENT EXTRAGRADIENT METHOD FOR SOLVING PSEUDO MONOTONE VARIATIONAL INEQUALITIES IN HILBERT SPACES*

Duong Viet Thong1,†, Vu Tien Dung2   

  1. 1. Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam;
    2. Department of Mathematics, University of Science, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
  • 收稿日期:2021-07-06 修回日期:2021-11-03 发布日期:2023-03-01
  • 通讯作者: †Duong Viet THONG. E-mail: duongvietthong@tdmu.edu.vn

A RELAXED INERTIAL FACTOR OF THE MODIFIED SUBGRADIENT EXTRAGRADIENT METHOD FOR SOLVING PSEUDO MONOTONE VARIATIONAL INEQUALITIES IN HILBERT SPACES*

Duong Viet Thong1,†, Vu Tien Dung2   

  1. 1. Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam;
    2. Department of Mathematics, University of Science, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
  • Received:2021-07-06 Revised:2021-11-03 Published:2023-03-01
  • Contact: †Duong Viet THONG. E-mail: duongvietthong@tdmu.edu.vn
  • About author:Duong Viet Thong, E-mail: duzngvt@gmail.com

摘要: In this paper, we investigate pseudomonotone and Lipschitz continuous variational inequalities in real Hilbert spaces. For solving this problem, we propose a new method that combines the advantages of the subgradient extragradient method and the projection contraction method. Some very recent papers have considered different inertial algorithms which allowed the inertial factor is chosen in [0; 1]. The purpose of this work is to continue working in this direction, we propose another inertial subgradient extragradient method that the inertial factor can be chosen in a special case to be $1$. Under suitable mild conditions, we establish the weak convergence of the proposed algorithm. Moreover, linear convergence is obtained under strong pseudomonotonicity and Lipschitz continuity assumptions. Finally, some numerical illustrations are given to confirm the theoretical analysis.

关键词: subgradient extragradient method, inertial method, variational inequality problem, pseudomonotone mapping, strong convergence, convergence rate

Abstract: In this paper, we investigate pseudomonotone and Lipschitz continuous variational inequalities in real Hilbert spaces. For solving this problem, we propose a new method that combines the advantages of the subgradient extragradient method and the projection contraction method. Some very recent papers have considered different inertial algorithms which allowed the inertial factor is chosen in [0; 1]. The purpose of this work is to continue working in this direction, we propose another inertial subgradient extragradient method that the inertial factor can be chosen in a special case to be $1$. Under suitable mild conditions, we establish the weak convergence of the proposed algorithm. Moreover, linear convergence is obtained under strong pseudomonotonicity and Lipschitz continuity assumptions. Finally, some numerical illustrations are given to confirm the theoretical analysis.

Key words: subgradient extragradient method, inertial method, variational inequality problem, pseudomonotone mapping, strong convergence, convergence rate