Accelerated Gradient and Skew-Symmetric Splitting Methods for a Class of Monotone Operator Equations

TL;DR

This paper introduces accelerated gradient and skew-symmetric splitting methods for solving a broad class of monotone operator equations, demonstrating linear convergence and optimal iteration complexity through theoretical analysis and numerical experiments.

Contribution

It extends acceleration techniques from convex minimization to general monotone operator equations, proposing new GSS and AGSS methods with proven convergence.

Findings

GSS and AGSS methods converge linearly

AGSS extends acceleration to broader class of problems

Numerical experiments confirm efficiency and robustness

Abstract

A class of monotone operator equations, which can be decomposed into sum of the gradient of a strongly convex function and a linear and skew-symmetric operator, is considered in this work. Based on discretization of the generalized gradient flow, gradient and skew-symmetric splitting (GSS) methods are proposed and proved to converge in linear rates. To further accelerate the convergence, an accelerated gradient flow is proposed and accelerated gradient and skew-symmetric splitting (AGSS) methods are developed, which extends the acceleration among the existing works on the convex minimization to a more general class of monotone operator equations. In particular, when applied to smooth saddle point systems with bilinear coupling, a linear convergent method with optimal lower iteration complexity is proposed. The robustness and efficiency of GSS and AGSS methods are verified via extensive numerical experiments.