FAST CONVEX OPTIMIZATION VIA INERTIAL DYNAMICS COMBINING VISCOUS AND HESSIAN-DRIVEN DAMPING WITH TIME RESCALING

In a Hilbert setting, we develop fast methods for convex uncon-strained optimization. We rely on the asymptotic behavior of an inertial system combining geometric damping with temporal scaling. The convex function to minimize enters the dynamic via its gradient. The dynamic includes three co-efficients varying with time, one is a viscous damping coefficient, the second is attached to the Hessian-driven damping, the third is a time scaling coeffi-cient. We study the convergence rate of the values under general conditions involving the damping and the time scale coefficients. The obtained results are based on a new Lyapunov analysis and they encompass known results on the subject. We pay particular attention to the case of an asymptotically van-ishing viscous damping, which is directly related to the accelerated gradient method of Nesterov. The Hessian-driven damping significantly reduces the oscillatory aspects. We obtain an exponential rate of convergence of values without assuming the strong convexity of the objective function. The tempo-ral discretization of these dynamics opens the gate to a large class of inertial optimization algorithms.

Automated summary

This paper develops fast methods for convex unconstrained optimization in a Hilbert setting. It introduces an inertial system with geometric damping and temporal scaling, and analyzes its asymptotic behavior. The obtained results include existing research on the topic and provide convergence rate analysis under general conditions. The paper focuses on the case of asymptotically vanishing viscous damping and demonstrates exponential convergence rate. This work opens up new possibilities for the development of inertial optimization algorithms.