Inertial dynamical systems and accelerated algorithms with implicit Hessian-driven damping for nonconvex optimization

TL;DR

This paper studies inertial dynamical systems with implicit Hessian-driven damping for strongly quasiconvex nonconvex optimization, establishing exponential convergence and developing accelerated algorithms with robustness to perturbations.

Contribution

It introduces a novel inertial dynamical system with implicit Hessian damping and derives accelerated algorithms with proven convergence rates for strongly quasiconvex functions.

Findings

Exponential convergence rates without Lipschitz gradient assumption

Development of inertial accelerated algorithms via discretization

Numerical experiments confirming theoretical convergence

Abstract

This paper is devoted to the investigation of inertial dynamical systems with implicit Hessian-driven damping for strongly quasiconvex optimization which is a specific class of nonconvex optimization problems. We first establish exponential convergence rate properties for this system without requiring Lipschitz continuity of the gradient on the function. Then, we obtain an inertial accelerated algorithm for minimizing strongly quasiconvex functions through natural explicit time discretization to the dynamical system. Meanwhile, we consider an exogenous additive perturbation term to this dynamical system and obtain the corresponding algorithm. By utilizing the Lyapunov method, we establish convergence rates of iterative sequences and their function values. Furthermore, we conduct numerical experiments to illustrate the theoretical results.