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 $\alpha /t$ 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.

Keywords:

• Linear equality constrained convex optimization problem
• mixed primal-dual dynamical system
• Hessian driven damping
• time discretization
• convergence rate

AMS SUBJECT CLASSIFICATIONS:

• 34D05
• 37N40
• 46N10
• 90C25

Acknowledgments

The authors would like to thank the reviewers and the editor for their helpful comments and suggestions, which significantly improve the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).