A two-step inertial method with a new step-size rule for variational inequalities in hilbert spaces

TL;DR

This paper introduces a novel two-step inertial method with adaptive step sizes for solving variational inequalities in Hilbert spaces, removing the Lipschitz condition and improving convergence and applicability.

Contribution

It proposes a new inertial Tseng extragradient algorithm with self-adaptive and Armijo-like step sizes, extending applicability without Lipschitz assumptions.

Findings

Proves weak convergence of the algorithm.

Allows adaptive selection of step sizes for better performance.

Accelerates existing variational inequality algorithms.

Abstract

In this paper, a two-step inertial Tseng extragradient method involving self-adaptive and Armijo-like step sizes is introduced for solving variational inequalities with a quasimonotone cost function in the setting of a real Hilbert space. Weak convergence of the sequence generated by the proposed algorithm is proved without assuming the Lipschitz condition. An interesting feature of the proposed algorithm is its ability to select the better step size between the self-adaptive and Armijo-like options at each iteration step. Moreover, removing the requirement for the Lipschitz condition on the cost function broadens the applicability of the proposed method. Finally, the algorithm accelerates and complements several existing iterative algorithms for solving variational inequalities in Hilbert spaces.