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

TL;DR

This paper tackles solving general non-monotone inclusion problems $0\in T(z)=F(z)+G(z)$ in real Hilbert spaces by developing the Symplectic Extra-gradient Type Method (SEG/SEG+) that leverages symplectic acceleration and Lyapunov analysis to achieve a fast $O(1/k^2)$ convergence, with an even faster $o(1/k^2)$ rate under stronger assumptions and weak convergence. A line-search framework is introduced to adapt to unknown Lipschitz and comonotone constants, maintaining convergence while enhancing practical performance. The authors provide extensive theoretical results, including sharpened rates and convergence under line search, and validate the approach with numerical experiments on matrix games and LASSO problems, where SEG+ with line search often outperforms existing EG-type methods. The work broadens accelerated methods for monotone/comonotone inclusions and suggests further directions in stochastic variants and parameter-tuning strategies for real-world applications.

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.

Figures (4)

• Figure 1: The required number of iterations Algorithm \ref{['al:sfbsm']} with $G=0$ respect to $D$. The termination condition is $\left\|F(x_k,y_k)\right\|\leqslant10^{-6}$.
• Figure 2: Evolution of duality gap and $\left\|\tilde{T}(z_k)\right\|^2$ versus running time.
• Figure 3: Comparison of SFBS and SPEG+.
• Figure 4: Evolution of $\left\|x_k-y_k\right\|$ and $F(y_k)-F^*$ versus running time.

Theorems & Definitions (38)

• Proposition 1: Lemma 2.47 in bauschke17
• Proposition 2: Proposition 20.38 (ii) in bauschke17
• Theorem 1
• proof
• Example 1
• Theorem 2
• proof
• Lemma 3: Estimation of error
• proof
• Theorem 4: Faster convergence rate