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

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 establish the existence of global minimizers and investigate several properties of the proximal operator for this class of functions, extending to the Hilbert space setting results previously known mainly in finite dimensions. Subsequently, we propose a relaxed-inertial proximal point-type algorithm for solving optimization problems consisting in minimizing strongly quasiconvex functions over closed convex sets in Hilbert spaces. Under suitable assumptions, we prove the strong convergence of the generated sequences to the unique minimizers of the considered functions. We also derive convergence rate estimates for the relaxed proximal point method when minimizing such functions, including stepwise linear contraction estimates and a refined contraction factor. In the standard proximal point case, we show that both the iterates and the function values converge geometrically to the optimal values.