# Newton-like dynamics associated to nonconvex optimization problems

## TL;DR

This paper introduces a Newton-like dynamical system for nonconvex optimization, demonstrating convergence to critical points under certain conditions and providing convergence rates based on the Kurdyka- property.

## Contribution

It proposes a novel dynamical system framework for nonconvex optimization and establishes convergence results and rates under the Kurdyka- property.

## Key findings

- Limit points are contained in the set of critical points.
- Trajectory convergence to critical points is proven under the Kurdyka- property.
- Convergence rates depend on the  exponent.

## Abstract

We consider the dynamical system \begin{equation*}\left\{ \begin{array}{ll} v(t)\in\partial\phi(x(t))\\ \lambda\dot x(t) + \dot v(t) + v(t) + \nabla \psi(x(t))=0, \end{array}\right.\end{equation*} where $\phi:\R^n\to\R\cup\{+\infty\}$ is a proper, convex and lower semicontinuous function, $\psi:\R^n\to\R$ is a (possibly nonconvex) smooth function and $\lambda>0$ is a parameter which controls the velocity. We show that the set of limit points of the trajectory $x$ is contained in the set of critical points of the objective function $\phi+\psi$, which is here seen as the set of the zeros of its limiting subdifferential. If the objective function satisfies the Kurdyka-\L{}ojasiewicz property, then we can prove convergence of the whole trajectory $x$ to a critical point. Furthermore, convergence rates for the orbits are obtained in terms of the \L{}ojasiewicz exponent of the objective function, provided the latter satisfies the \L{}ojasiewicz property.