A modified subgradient extragradient method for solving the variational inequality problem

TL;DR

This paper introduces a modified subgradient extragradient method for variational inequality problems, improving stepsize selection, with proven convergence and demonstrated numerical advantages over existing methods.

Contribution

It proposes a new variant of the subgradient extragradient method with an improved stepsize strategy, enhancing convergence and performance.

Findings

The modified method converges under standard conditions.

Numerical experiments show improved performance.

The new stepsize enhances the method's efficiency.

Abstract

The subgradient extragradient method for solving the variational inequality (VI) problem, which is introduced by Censor et al. \cite{CGR}, replaces the second projection onto the feasible set of the VI, in the extragradient method, with a subgradient projection onto some constructible half-space. Since the method has been introduced, many authors proposed extensions and modifications with applications to various problems. In this paper, we introduce a modified subgradient extragradient method by improving the stepsize of its second step. Convergence of the proposed method is proved under standard and mild conditions and primary numerical experiments illustrate the performance and advantage of this new subgradient extragradient variant.