A General Mixed-Order Primal-Dual Dynamical System with Tikhonov Regularization

TL;DR

This paper introduces a flexible primal-dual dynamical system with Tikhonov regularization for convex optimization, analyzing its convergence properties and demonstrating its effectiveness through numerical experiments.

Contribution

It presents a generalized mixed-order primal-dual dynamical system with Tikhonov regularization, unifying and extending existing models with proven convergence rates and strong convergence results.

Findings

Convergence rate of O(1/(t^2β(t))) for objective error

Convergence rate of o(1/β(t)) for primal-dual gap

Numerical experiments confirm theoretical convergence

Abstract

In a Hilbert space, we propose a class of general mixed-order primal-dual dynamical systems with Tikhonov regularization for a convex optimization problem with linear equality constraints. The proposed dynamical system is characterized by three time-dependent parameters, i.e., general viscous damping, time scaling, and Tikhonov regularization coefficients, which can incorporate as special cases some existing mixed-order primal-dual dynamical systems in the literature. With some appropriate conditions on the parameters, we analyze by constructing suitable Lyapunov functions the asymptotic convergence properties of the proposed dynamical system, where a convergence rate of O(1/(t^2\beta(t))) for the objective function error and a convergence rate of o(1/\beta(t)) for the primal-dual gap are established. Moreover, we further prove the strong convergence of the trajectory generated by the proposed dynamical system. Finally, we carry out some numerical experiments to illustrate the obtained theoretical results of the proposed dynamical system.