Fast Optimization of Charged Particle Dynamics with Damping

SIAM Journal on Optimization, Volume 34, Issue 3, Page 2287-2313, September 2024.
Abstract. In this paper, the convergence analysis of accelerated second-order methods for convex optimization problems is developed from the point of view of autonomous dissipative inertial continuous dynamics in the magnetic field. Different from the classical heavy ball model with damping, we consider the motion of a charged particle in a magnetic field model involving the linear asymptotic vanishing damping. It is a coupled ordinary differential system by adding the magnetic coupled term [math] to the heavy ball system with [math]. In order to develop fast optimization methods, our first contribution is to prove the global existence and uniqueness of a smooth solution under certain regularity conditions of this system via the Banach fixed point theorem. Our second contribution is to establish the convergence rate of corresponding algorithms involving inertial features via discrete time versions of inertial dynamics under the magnetic field. Meanwhile, the connection of algorithms between the heavy ball model and the motion of a charged particle in a magnetic field model is established.