A hybrid method without extrapolation step for solving variational inequality problems

TL;DR

This paper presents a new hybrid method for solving variational inequality problems that avoids the extrapolation step, ensuring strong convergence without additional projections.

Contribution

It introduces a novel hybrid approach combining projection and outer approximation methods without extrapolation, with proven strong convergence.

Findings

Strong convergence of the proposed method

No extrapolation step needed

Effective for monotone Lipschitz-continuous mappings

Abstract

In this paper, we introduce a new method for solving variational inequality problems with monotone and Lipschitz-continuous mapping in Hilbert space. The iterative process is based on two well-known projection method and the hybrid (or outer approximation) method. However we do not use an extrapolation step in the projection method. The absence of one projection in our method is explained by slightly different choice of sets in hybrid method. We prove a strong convergence of the sequences generated by our method.