# Optimization and Equilibrium

## Variants of the A-HPE and large-step A-HPE algorithms for strongly convex problems with applications to accelerated high-order tensor methods.

Event Date: Nov 17, 2021 in Optimization and Equilibrium, Seminars

Abstract: For solving strongly convex optimization problems, we propose and study the global convergence of variants of the A-HPE and large-step A-HPE algorithms of Monteiro and Svaiter. We prove linear and the superlinear $\mathcal{O}\left(k^{\,-k\left(\frac{p-1}{p+1}\right)}\right)$ global rates for the proposed variants of the A-HPE and large-step A-HPE methods, respectively. The parameter $p\geq 2$ appears in the (high-order) large-step condition of the new large-step A-HPE algorithm. We apply our  results to high-order tensor methods, obtaining a new inexact (relative- error) tensor...

## El Lema de Farkas: Algunas extensiones y aplicaciones.

Event Date: Nov 03, 2021 in Optimization and Equilibrium, Seminars

Abstract:  Tras revisar la versión clásica del lema de Farkas y sus aplicaciones, se presentan algunas extensiones a sistemas con infinitas inecuaciones, con infinitas variables o ambas cosas a la vez, junto con algunas de sus respectivas aplicaciones.

## Stochastic incremental mirror descent algorithms with Nesterov smoothing.

Event Date: Oct 20, 2021 in Optimization and Equilibrium, Seminars

Abstract: We propose a stochastic incremental mirror a prox-friendly proper, convex and lower semicontinuous function. Different to the previous cdescent algorithm constructed by means of the Nesterov smoothing for minimizing a sum of finitely many proper, convex and lower semicontinuous functions over a nonempty closed convex set in an Euclidean space. The algorithm can be adapted in order to minimize (in the same setting) a sum of finitely many proper, convex and lower semicontinuous functions composed with linear operators. Another modification of the scheme leads to a stochastic...

## Constant Along Primal Rays Conjugacies and the l0 Pseudonorm.

Event Date: Oct 13, 2021 in Optimization and Equilibrium, Seminars

Abstract: he so-called l0 pseudonorm counts the number of nonzero components of a vector. It is standard in sparse optimization problems. However, as it is a discontinuous and nonconvex function, the l0 pseudonorm cannot be satisfactorily handled with the Fenchel conjugacy. In this talk, we present the Euclidean Capra-conjugacy, which is suitable for the l0 pseudonorm, as this latter is “convex” in the sense of generalized convexity (equal to its biconjugate). We immediately derive a convex factorization property (the l0 pseudonorm coincides, on the unit sphere, with a convex lsc function)...

## On strongly quasiconvex functions: theory and applications.

Event Date: Oct 06, 2021 in Optimization and Equilibrium, Seminars

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...