# Proximal extrapolated gradient methods with prediction and correction for monotone variational inequalities

## TL;DR

This paper introduces a proximal extrapolated gradient method with prediction and correction for monotone variational inequalities, enabling larger step sizes and improved numerical efficiency through theoretical convergence guarantees and practical experiments.

## Contribution

It extends proximal gradient methods by allowing larger step sizes via prediction and correction, with proven convergence and enhanced numerical performance.

## Key findings

- The method converges under a very weak condition.
- Larger step sizes improve numerical efficiency.
- Numerical experiments confirm theoretical advantages.

## Abstract

An efficient proximal-gradient-based method, called proximal extrapolated gradient method, is designed for solving monotone variational inequality in Hilbert space. The proposed method extends the acceptable range of parameters to obtain larger step sizes. The step size is predicted based a local information of the operator and corrected by linesearch procedures to satisfy a very weak condition, which is even weaker than the boundedness of sequence generated and always holds when the operator is the gradient of a convex function. We establish its convergence and ergodic convergence rate in theory under the larger range of parameters. Furthermore, we improve numerical efficiency by employing the proposed method with non-monotonic step size, and obtain the upper bound of the parameter relating to step size by an extremely simple example. Related numerical experiments illustrate the improvements in efficiency from the larger step size.