Primal-dual dynamics featuring Hessian-driven damping and variable mass for convex optimization problems

TL;DR

This paper introduces a novel primal-dual dynamical system with Hessian-driven damping and variable mass, achieving strong convergence and improved rates for convex optimization with linear constraints.

Contribution

It proposes a new Tikhonov regularized primal-dual system with time-dependent parameters, providing convergence analysis and numerical validation.

Findings

Trajectory converges strongly to the minimal norm solution.

Convergence rates for primal-dual gap, objective residual, and feasibility are established.

Adjusting parameters improves convergence rates.

Abstract

This paper deals with a new Tikhonov regularized primal-dual dynamical system with variable mass and Hessian-driven damping for solving a convex optimization problem with linear equality constraints. The system features several time-dependent parameters: variable mass, slow viscous damping, extrapolation, and temporal scaling. By employing the Lyapunov analysis approach, we obtain the strong convergence of the trajectory generated by the proposed system to the minimal norm solution of the optimization problem, as well as convergence rate results for the primal-dual gap, the objective residual, and the feasibility violation. We also show that the convergence rates of the primal-dual gap, the objective residual, and the feasibility violation can be improved by appropriately adjusting these parameters. Further, we conduct numerical experiments to demonstrate the effectiveness of the theoretical results.