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

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.