A parallel subgradient method extended to variational inequalities involving nonexpansive mappings

ABSTRACT

In this paper, we propose and analyze the convergence of new iteration methods for finding a common point of the solution set of a class of pseudomonotone variational inequalities and the fixed point set of a finite system of nonexpansive mappings in a real Hilbert space. The idea of this algorithm is to combine the subgradient method with the parallel splitting-up techniques. The main iteration step in the proposed methods uses only one projection and does not require any Lipschitz continuous condition for the cost mapping. The convergent results are also extended to a pseudomonotone equilibrium problem involving a finite system of nonexpansive mappings. Finally, some numerical examples are developed to illustrate the behavior of the new algorithms with respect to existing algorithms.

COMMUNICATED BY:

• B. Mordukhovich

Keywords:

• Fixed point
• pseudomonotonicity
• variational inequality
• subgradient projection method
• nonexpansive mapping

2010 Mathematics Subject Classifications:

• 65 K10
• 90 C25
• 49 J35
• 47 J25
• 91B50

Acknowledgments

We are very grateful to anonymous referees for their really useful and constructive comments that helped us very much in improving the paper. This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number “101.02-2017.15”.

Disclosure statement

No potential conflict of interest was reported by the authors.