Heavy Ball and Nesterov Accelerations with Hessian-Driven Damping for Nonconvex Optimization

In this work, we investigate a second-order dynamical system with Hessian-driven damping tailored for a class of nonconvex functions called strongly quasiconvex. Building upon this continuous-time model, we derive two discrete-time gradient-based algorithms through time discretizations. The first is a Heavy Ball method with Hessian correction, incorporating curvature-dependent terms that arise from discretizing the Hessian damping component. The second is a Nesterov-type accelerated method with adaptive momentum, featuring correction terms that account for local curvature. Both algorithms aim to enhance stability and convergence performance, particularly by mitigating oscillations commonly observed in classical momentum methods. Furthermore, in both cases we establish linear convergence to the optimal solution for the iterates and functions values. Our approach highlights the rich interplay between continuous-time dynamics and discrete optimization algorithms in the setting of strongly quasiconvex objectives. Numerical experiments are presented to support obtained results.