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 $ϕ:\R^n\to\R\cup\{+\infty\}$ is a proper, convex and lower semicontinuous function, $ψ:\R^n\to\R$ is a (possibly nonconvex) smooth function and $λ>0$ 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 $ϕ+ψ$, 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.