Relaxed extragradient algorithm for solving pseudomonotone variational inequalities in Hilbert spaces

Abstract

In this paper, we introduce a new algorithm for solving a variational inequality problem in a Hilbert space. The algorithm originates from an explicit discretization of a dynamical system in time. We establish the convergence of the algorithm for a class of non-monotone and Lipschitz continuous operators, provided by the sequentially weak-to-weak continuity of cost operators. The rate of convergence of the algorithm is also proved under some standard hypotheses. Moreover, the new algorithm uses variable step-sizes which are updated at each iteration by a cheap computation without linesearch. This step-size rule allows the resulting algorithm to work more easily without the prior knowledge of Lipschitz constant of operator. Also, it is particularly interesting in the case where the Lipschitz constant is unknown or difficult to approximate. Several numerical experiments are implemented to illustrate the theoretical results and also to compare with existing algorithms.

Keywords:

• Variational inequality problem
• pseudomonotone operator
• projection method
• relaxed extragradient algorithm
• Lipschitz condition

AMS Subject Classifications:

• 65Y05
• 65K15
• 68W10
• 47H05
• 47H10

Acknowledgments

The authors would like to thank the Associate Editor and anonymous referees for their valuable comments and suggestions which help us in improving the original version of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of the first author was supported by the National Foundation for Science and Technology Development (NAFOSTED) of Vietnam under grant number 101.01-2017.315. The third author was supported by the National Natural Science Foundation of China (11771067), the Applied Basic Project of Sichuan Province (2019YJ0204) and the Fundamental Research Funds for the Central Universities (ZYGX2019J095).