On Strongly Quasiconvex Functions: Existence Results and Proximal Point Algorithms

We prove that every strongly quasiconvex function is 2-supercoercive (in particular, coercive). Furthermore, we investigate the usual properties of proximal operators for strongly quasiconvex functions. In particular, we prove that the set of fixed points of the proximal operator coincides with the unique minimizer of a lower semicontinuous strongly quasiconvex function. As a consequence, we implement the proximal point algorithm for finding the unique solution of the minimization problem of a strongly quasiconvex function by using a positive sequence of parameters bounded away from 0 and, in particular, we revisit the general quasiconvex case. Moreover, a new characterization for convex functions is derived from this analysis. Finally, an application for a strongly quasiconvex function which is neither convex nor differentiable nor locally Lipschitz continuous is provided.