# Golden ratio algorithms with new stepsize rules for variational inequalities

## TL;DR

This paper introduces two novel golden ratio algorithms with adaptive stepsize rules for solving variational inequalities, eliminating the need for prior knowledge of the Lipschitz constant and demonstrating effective convergence and performance.

## Contribution

The paper presents two new golden ratio algorithms with adaptive stepsize rules for variational inequalities, improving upon existing methods by removing the need for Lipschitz constant knowledge.

## Key findings

- Algorithms converge under standard conditions.
- The second algorithm's stepsizes stay separated from zero.
- Numerical results show competitive performance.

## Abstract

In this paper, we introduce two golden ratio algorithms with new stepsize rules for solving pseudomonotone and Lipschitz variational inequalities in finite dimensional Hilbert spaces. The presented stepsize rules allow the resulting algorithms to work without the prior knowledge of the Lipschitz constant of operator. The first algorithm uses a sequence of stepsizes which is previously chosen, diminishing and non-summable. While the stepsizes in the second one are updated at each iteration and by a simple computation. A special point is that the sequence of stepsizes generated by the second algorithm is separated from zero. The convergence as well as the convergence rate of the proposed algorithms are established under some standard conditions. Also, we give several numerical results to show the behavior of the algorithms in comparisons with other algorithms.