Symplectic Extra-gradient Type Method for Solving General Non-monotone Inclusion Problem

TL;DR

This paper introduces a symplectic accelerated extra-gradient method for solving general non-monotone inclusion problems, achieving faster convergence rates and improved practical efficiency through line search techniques.

Contribution

It presents a novel symplectic acceleration technique for extra-gradient methods, establishing quadratic convergence and incorporating line search for better practical performance.

Findings

Achieves inverse quadratic convergence rate.

Demonstrates faster convergence in numerical tests.

Proves convergence with line search technique.

Abstract

In recent years, accelerated extra-gradient methods have attracted much attention by researchers, for solving monotone inclusion problems. A limitation of most current accelerated extra-gradient methods lies in their direct utilization of the initial point, which can potentially decelerate numerical convergence rate. In this work, we present a new accelerated extra-gradient method, by utilizing the symplectic acceleration technique. We establish the inverse of quadratic convergence rate by employing the Lyapunov function technique. Also, we demonstrate a faster inverse of quadratic convergence rate alongside its weak convergence property under stronger assumptions. To improve practical efficiency, we introduce a line search technique for our symplectic extra-gradient method. Theoretically, we prove the convergence of the symplectic extra-gradient method with line search. Numerical tests show that this adaptation exhibits faster convergence rates in practice compared to several existing extra-gradient type methods.