Exponential convergence rates of a second-order dynamic system and algorithm for a linear equality constrained optimization problem

Abstract

The O(1/t²) convergence rate of second-order dynamical systems with asymptotic vanishing viscous damping is faster than the O(1/t) rate of systems with fixed viscous damping in unconstrained and linear equality constrained optimization problems. We explore whether the performance of systems with vanishing viscous damping remains superior when both use time scaling. We compare the best polynomial convergence rates of vanishing damping systems with the best exponential convergence rates of a fixed damping system with time scaling for linear equality constrained problems. We prove that the primal-dual trajectory weakly converges to an optimal solution. Additionally, we present an inertial algorithm derived from the implicit discretization of the dynamical system, establishing exponential convergence rates for the primal-dual gap, feasibility measure, and objective value without assuming strong convexity. The sequence of iterates generated by the inertial algorithm weakly con-verges to an optimal solution when the objective function is proper, convex, and lower semicontinuous. These results align with those in the continuous setting.

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Identifiers

ORCID for Phan Tu Vuong: orcid.org/0000-0002-1474-994X

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