“Second-Order Primal” + “First-Order Dual” Dynamical Systems With Time Scaling for Linear Equality Constrained Convex Optimization Problems

Abstract

Second-order dynamical systems are important tools for solving optimization problems, and most of the existing works in this field have focused on unconstrained optimization problems. In this article, we propose an inertial primal–dual dynamical system with constant viscous damping and time scaling for the linear equality constrained convex optimization problem, which consists of a second-order ordinary differential equation (ODE) for the primal variable and a first-order ODE for the dual variable. When the scaling satisfies certain conditions, we prove its convergence property without assuming strong convexity. Even the convergence rate can become exponential when the scaling grows exponentially. We also show that the obtained convergence property of the dynamical system is preserved under a small perturbation.