Convergence rates of mixed primal-dual dynamical systems with Hessian driven damping

Abstract

For a linear equality constrained convex optimization problem, we initially propose a mixed primal-dual dynamical system with Hessian driven damping. This dynamical system comprises a second-order ordinary differential equation (ODE) with damping and Hessian driven damping for the primal variable, and a first-order ODE for the dual variable. Utilizing the Lyapunov analysis approach, we prove that all the Lagrangian residual, the objective residual and the feasibility violation enjoy fast convergence under general scaling coefficients. We further analyse the convergence rate results under specific scaling coefficients. Our mixed dynamical system is extended to solve multi-block optimization problems, and we also consider the discrete case of the mixed dynamical system. This is the first study to explore primal-dual dynamical systems with Hessian driven damping for linear equality constrained problems.