A strongly convergent extragradient method with non-monotone self adaptive step size rule for solving pseudomonotone variational inequalities in a real Hilbert space

Document Type : Research Paper


1 Applied Mathematics Program, Faculty of Science and Technology, Valaya Alongkorn Rajabhat University under the Royal Patronage, Pathumthani 13180, Thailand

2 Faculty of Science and Technology, Rajamangala University of Technology Phra Nakhon (RMUTP), 1381 Pracharat 1 Road, Wongsawang, Bang Sue, Bangkok 10800, Thailand

3 Mathematics and Computing Science Program, Faculty of Science and Technology, Phetchabun Rajabhat University, Phetchabun 67000, Thailand


In this paper, a new algorithm is proposed to solve pseudo-monotone variational inequalities with the Lipschitz condition in a real Hilbert space. This problem is an exceptionally general mathematical problem in the sense that it consists of a number of the applied mathematical problems as a special instance, such as optimization problems, equilibrium models, fixed point problems, the saddle point problems, and Nash equilibrium point problems. The algorithm is formulated around two algorithms: the extra gradient algorithm and the inertial algorithm. The proposed algorithm uses a new step size rule based on local operator information rather than its Lipschitz constant or any other line search strategy and operates without any knowledge of the operator's Lipschitz constant. It presents the strong convergence of the algorithm. Finally, we conduct a number of numerical experiments to determine the performance and superiority of the described algorithm.


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Volume 13, Issue 2
July 2022
Pages 2651-2668
  • Receive Date: 08 January 2021
  • Revise Date: 17 February 2021
  • Accept Date: 09 March 2021