Modified accelerated algorithms for solving variational inequalities

ABSTRACT

In this paper, we propose two inertial algorithms with new stepsize rule for solving a monotone and Lipschitz variational inequality in a Hilbert space and prove some weak and strong convergence theorems of the proposed inertial algorithms. The algorithms use variable stepsizes which are updated at each iteration by a simple computation without any linesearch. A new stepsize rule presented in the paper has allowed the algorithms to work without the prior knowledge of Lipschitz constant of operator. Finally, we give several numerical results to demonstrate the computational performance of the new algorithms in comparison with other algorithms.

Keywords:

• Variational inequality
• monotone operator
• extragradient method
• subgradient extragradient method
• projection method

2010 AMS Subject Classifications:

• 65J15
• 65Y05
• 47H05
• 47J25
• 91B50

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 paper was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.01-2017.315. This research was also supported by the National Natural Science Foundation of China (11771067) and the Applied Basic Project of Sichuan Province (2019YJ0204) and the Fundamental Research Funds for the Central Universities (ZYGX2019J095).