# Optimization and Equilibrium

## On diametrically maximal sets, maximal premonotone maps and premonotone bifunctions.

Event Date: Nov 18, 2020 in Optimization and Equilibrium, Seminars

Abstract: First, we study diametrically maximal sets in the Euclidean space (those which are not properly contained in a set with the samediameter), establishing their main properties. Then, we use these sets for exhibiting an explicit family of maximal premonotone operators. We also establish some relevant properties of maximal premonotone operators, like their local boundedness, and finally we introduce the notion of premonotone bifunctions, presenting a canonical relation between premonotone operators and bifunctions, that extends the well known one, which holds in the monotone...

## An algebraic view of the smallest strictly monotonic function.

Event Date: Nov 04, 2020 in Optimization and Equilibrium, Seminars

Abstract: The talk concerns with one of the most popular functions to derive nonconvex separation results. Complete characterizations for both its level sets and basic properties such as monotonicity and convexity are provided in terms of its parameters. Most of these characterizations work without considering any additional requirement or assumption. Finally, as an application, a vectorial form of the Ekeland variational principle is provided.

## Principal-Agent problem in insurance: from discrete-to continuous-time.

Event Date: Oct 07, 2020 in Optimization and Equilibrium, Seminars

Abstract: In this talk we present a contracting problem between an insurance buyer and the seller, subject to prevention efforts in the form of self-insurance and self-protection. We start with a static formulation, corresponding to an optimization problem with variational inequality constraint, and extend the main properties of the optimal contract to the continuous-time formulation, corresponding to a stochastic control problem in weak form under non-singular measures.

## Sigma-convex functions and Sigma-subdifferentials.

Event Date: Sep 23, 2020 in Optimization and Equilibrium, Seminars

Abstract:  In this talk we present and study the notion of $\sigma$-subdifferential of a proper function $f$ which contains theClarke-Rockafellar subdifferential of $f$ under some mild assumptions on $f$. We show that some well known properties of the convex function, namely Lipschitz property in the interior of its domain, remain valid for thelarge class of $\sigma$-convex functions.

## Generalized Newton Algorithms for Tilt-Stable Minimizers in Nonsmooth Optimization.

Event Date: Sep 02, 2020 in Optimization and Equilibrium, Seminars

Abstract: This talk aims at developing two versions of the generalized Newton method to compute local minimizers for nonsmooth problems of unconstrained and constraned optimization that satisfy an important stability property known as tilt stability. We start with unconstrained minimization of continuously differentiable cost functions having Lipschitzian gradients and suggest two second-order algorithms of the Newton type: one involving coderivatives of Lipschitzian gradient mappings, and the other based on graphical derivatives of the latter. Then we proceed with the propagation of these...