Properties and Proximal Point Type Methods for Strongly Quasiconvex Functions in Hilbert Spaces

Abstract

We study strongly quasiconvex functions defined on Hilbert spaces and taking extended real values from both theoretical and algorithmic points of view. To this end, we verify the existence of (global) minimizers and the properties of the proximal operator for such functions, previously investigated merely on finitely dimensional spaces. Subsequently, we discuss a relaxed-inertial proximal point-type algorithm for solving optimization problems consisting in minimizing strongly quasiconvex functions over closed convex sets in Hilbert spaces. We obtained a Q-linear convergence rate for the relaxed proximal point-type method and, for the corresponding function values, an asymptotic rate of O(1/k) to the optimal value of the considered optimization problem.