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

Abstract: In this work, we investigate a second-order dynamical system with Hessian-driven damping tailored for a class of nonconvex functions called strongly quasiconvex. Buil-ding upon this continuous-time model, we derive two discrete-time gra-dient-based algorithms through time discretizations. The first is a Heavy Ball method with Hessian correction, incorporating cur-va-tu-re-dependent terms that arise from discretizing the Hessian damping component. The second is a Nesterov-type accelerated method with adaptive momentum, fea-tu-ring correction terms that account for local curvature. Both algorithms aim to enhance stability and convergence performance, particularly by mi-ti-ga-ting oscillations commonly observed in cla-ssi-cal momentum me-thods. Furthermore, in both cases we establish li-near 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 se-tting of strongly quasiconvex objectives. Numerical experiments are presented to support obtained results.