Strong convergence of inertial extragradient algorithms for solving variational inequalities and fixed point problems

TL;DR

This paper introduces two inertial extragradient algorithms that efficiently solve variational inequality and fixed point problems in Hilbert spaces, requiring minimal projections and no prior Lipschitz constant, with proven strong convergence.

Contribution

The paper presents novel inertial extragradient algorithms that do not need prior Lipschitz constant knowledge or line search, and guarantees their strong convergence.

Findings

Algorithms require only one projection per iteration.

They work without prior Lipschitz constant information.

Numerical experiments show improved efficiency.

Abstract

The paper investigates two inertial extragradient algorithms for seeking a common solution to a variational inequality problem involving a monotone and Lipschitz continuous mapping and a fixed point problem with a demicontractive mapping in real Hilbert spaces. Our algorithms only need to calculate the projection on the feasible set once in each iteration. Moreover, they can work well without the prior information of the Lipschitz constant of the cost operator and do not contain any line search process. The strong convergence of the algorithms is established under suitable conditions. Some experiments are presented to illustrate the numerical efficiency of the suggested algorithms and compare them with some existing ones.