Newton-like dynamics associated to nonconvex optimization problems

Abstract

We consider the dynamical system \begin{equation*}\left\{ \begin{array}{ll} v(t)\in\partialϕ(x(t))\\ λ\dot x(t) + \dot v(t) + v(t) + \nabla ψ(x(t))=0, \end{array}\right.\end{equation*} where $\varphi :{\mathbb{R}}^n\to \mathbb{R}\cup \{+\infty \}$ is a proper, convex and lower semicontinuous function, $\psi :{\mathbb{R}}^n\to \mathbb{R}$ 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 $\varphi +\psi$, which is here seen as the set of the zeros of its limiting subdifferential. If the objective function satisfies the Kurdyka-Łojasiewicz 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 Łojasiewicz exponent of the objective function, provided the latter satisfies the Łojasiewicz property.