A Second Order Primal–Dual Dynamical System for a Convex–Concave Bilinear Saddle Point Problem

Abstract

The class of convex–concave bilinear saddle point problems encompasses many important convex optimization models arising in a wide array of applications. The most of existing primal–dual dynamical systems for saddle point problems are based on first order ordinary differential equations (ODEs), which only own the \({\mathcal {O}}(1/t)\) convergence rate in the convex case, and fast convergence rate analysis always requires some additional assumption such as strong convexity. In this paper, based on second order ODEs, we consider a general inertial primal–dual dynamical system, with damping, scaling and extrapolation coefficients, for a convex–concave bilinear saddle point problem. By the Lyapunov analysis approach, under appropriate assumptions, we investigate the convergence rates of the primal–dual gap and velocities, and the boundedness of the trajectories for the proposed dynamical system. With special parameters, our results can recover the Polyak’s heavy ball acceleration scheme and Nesterov’s acceleration scheme. We also provide numerical examples to support our theoretical claims.