Fast inertial dynamic algorithm with smoothing method for nonsmooth convex optimization

TL;DR

This paper introduces a fast inertial second-order dynamic algorithm with smoothing for nonsmooth convex optimization, achieving weak convergence to optimal solutions and an objective convergence rate of o(t^-2).

Contribution

It proposes a novel inertial dynamic method that approximates nonsmooth functions with smooth ones, providing convergence guarantees and stability analysis.

Findings

Weak convergence of trajectories to optimal solutions.

Convergence rate of objective function values as o(t^-2).

Algorithm stability under perturbations.

Abstract

In order to solve the minimization of a nonsmooth convex function, we design an inertial second-order dynamic algorithm, which is obtained by approximating the nonsmooth function by a class of smooth functions. By studying the asymptotic behavior of the dynamic algorithm, we prove that each trajectory of it weakly converges to an optimal solution under some appropriate conditions on the smoothing parameters, and the convergence rate of the objective function values is o(t^-2). We also show that the algorithm is stable, that is, this dynamic algorithm with a perturbation term owns the same convergence properties when the perturbation term satisfies certain conditions. Finally, we verify the theoretical results by some numerical experiments.