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

Xiangkai Sun, Feng Guo, Liang He, Xiaole Guo

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.

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

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.

Paper Structure

This paper contains 5 sections, 7 theorems, 91 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Strong convergence of trajectory to the minimal norm solution
  4. Numerical experiments
  5. Conclusion

Key Result

lemma 1

He2022 Assume that $g:[t_0,+\infty)\rightarrow\mathcal{X}$ is a continuous differentiable function, $\eta:[t_0,+\infty)\rightarrow[0,+\infty)$ is a continuoud differentiable function, $t_0>0$, and $C\geq0$. If then $\sup_{t\geq t_0}\|g(t)\|<+\infty.$

Figures (3)

  • Figure 1: Error analysis of system (\ref{['dyn1']}) under different mass functions.
  • Figure 2: Error analysis of system (\ref{['dyn1']}) under different time scaling functions.
  • Figure 3: The behaviors of the trajectory generated by system (\ref{['dyn1']}).

Theorems & Definitions (13)

  • lemma 1
  • lemma 2
  • remark 1
  • lemma 3
  • proposition 1
  • proof
  • theorem 1
  • proof
  • remark 2
  • theorem 2
  • ...and 3 more