Abstract:

This paper proposes a mixed primal-dual dynamical system with constant damping and Hessian-driven damping for solving linearly constrained optimization problems. The system consists of a second-order ordinary differential equation (ODE) with Hessian-driven damping for the primal variable and a first-order ordinary differential equation for the dual variable. By constructing an appropriate Lyapunov function, we analyze the convergence properties of the primal-dual gap, the feasibility measure and the objective function value, and establish exponential convergence rates under suitable scaling coefficients. Based on a time discretization of the continuous-time system, we derive an inertial primal-dual algorithm and validate the theoretical findings through numerical experiments, demonstrating the effectiveness and robustness of the proposed method. 

Key words: Mixed primal-dual dynamical system, Hessian-driven damping, Lyapunov function, Convergence rate, Linearly constrained optimization