### Speaker: Profesor Felipe Lara Obreque

### Department of Mathematics, University of Tarapacá, Arica, Chile

### Date: October 06, 2021 at 10:00 am (Chilean-time)

**Title**: On strongly quasiconvex functions: theory and applications

**Abstract: **In this talk, we present a new existence result for the classof lsc strongly quasiconvex functions by showing 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 lsc strongly quasiconvex function. As a consequence, we implemented the proximal point algorithm for finding the unique solution of the minimization problem by using a positive sequence of parameters bounded away from 0 and, in particular, we revisited the general quasiconvex case. Moreover, a new subdifferential for nonconvex functions and a new characterization for convex functions is derived from our study. Finally, an application for a strongly quasiconvex function which is neither convex nor differentiable nor locally Lipschitz continuous, is provided.

**Venue**: Online via Google Meet: https://meet.google.com/vpx-itqq-oik

**A brief biography of the speaker**: Felipe Lara is assistant professor at the Department of Mathematics, University of Tarapacá, Arica, Chile. He obtained his Ph.D. degree at the University of Concepción in 2015. His research interests are in continuous optimization, especially in nonconvex nonsmooth optimization.

**Coordinators**: Fabián Flores-Bazán (CMM, Universidad de Concepción) and Abderrahim Hantoute (Alicante)