Newtonlike dynamics associated to nonconvex optimization problems
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 KurdykaŁojasiewicz 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2017
 DOI:
 10.48550/arXiv.1703.01339
 arXiv:
 arXiv:1703.01339
 Bibcode:
 2017arXiv170301339I
 Keywords:

 Mathematics  Optimization and Control;
 Mathematics  Dynamical Systems;
 34G25;
 47J25;
 47H05;
 90C26;
 90C30;
 65K10