"Second-Order Primal'' + "First-Order Dual'' Dynamical Systems with Time Scaling for Linear Equality Constrained Convex Optimization Problems

TL;DR

This paper introduces a novel inertial primal-dual dynamical system with time scaling for solving linear equality constrained convex optimization problems, achieving convergence without strong convexity and exponential rates under certain conditions.

Contribution

It proposes a new second-order primal-dual dynamical system with time scaling, extending existing methods to constrained problems and analyzing its convergence properties.

Findings

Proves convergence without strong convexity.

Achieves exponential convergence rate with exponential scaling.

Convergence is robust under small perturbations.

Abstract

Second-order dynamical systems are important tools for solving optimization problems, and most of existing works in this field have focused on unconstrained optimization problems. In this paper, we propose an inertial primal-dual dynamical system with constant viscous damping and time scaling for the linear equality constrained convex optimization problem, which consists of a second-order ODE for the primal variable and a first-order ODE for the dual variable. When the scaling satisfies certain conditions, we prove its convergence property without assuming strong convexity. Even the convergence rate can become exponential when the scaling grows exponentially. We also show that the obtained convergence property of the dynamical system is preserved under a small perturbation.