Convergence rates for an inertial algorithm of gradient type associated to a smooth nonconvex minimization

Abstract

We investigate an inertial algorithm of gradient type in connection with the minimization of a nonconvex differentiable function. The algorithm is formulated in the spirit of Nesterov's accelerated convex gradient method. We prove some abstract convergence results which applied to our numerical scheme allows us to show that the generated sequences converge to a critical point of the objective function, if a regularization of the objective function satisfies the Kurdyka-{\L}ojasiewicz property. Further, we provide convergence rates for the generated sequences and the objective function values formulated in terms of the {\L}ojasiewicz exponent. Finally, we obtain sublinear convergence rates for the objective function values in the iterates in the case when the objective function is convex.