# First-order optimization algorithms via inertial systems with Hessian driven damping

## TL;DR

This paper introduces new first-order optimization algorithms inspired by inertial systems with Hessian-driven damping, achieving rapid convergence and extending to non-smooth convex functions with acceleration techniques.

## Contribution

The paper develops novel first-order algorithms based on inertial dynamics with Hessian-driven damping, extending them to non-smooth functions and incorporating acceleration via time scale factors.

## Key findings

- Algorithms exhibit rapid convergence towards zero gradients.
- Extension to non-smooth convex functions using Moreau envelope.
- Numerical results support theoretical convergence claims.

## Abstract

In a Hilbert space setting, for convex optimization, we analyze the convergence rate of a class of first-order algorithms involving inertial features. They can be interpreted as discrete time versions of inertial dynamics involving both viscous and Hessian-driven dampings. The geometrical damping driven by the Hessian intervenes in the dynamics in the form $\nabla^2 f (x(t)) \dot{x} (t)$. By treating this term as the time derivative of $ \nabla f (x (t)) $, this gives, in discretized form, first-order algorithms in time and space. In addition to the convergence properties attached to Nesterov-type accelerated gradient methods, the algorithms thus obtained are new and show a rapid convergence towards zero of the gradients. On the basis of a regularization technique using the Moreau envelope, we extend these methods to non-smooth convex functions with extended real values. The introduction of time scale factors makes it possible to further accelerate these algorithms. We also report numerical results on structured problems to support our theoretical findings.