  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.
