# On the construction of maximal p-cyclically monotone operators

### Date:  January 20,  2021 at 10:00 (Chilean-time)

Title:   On the construction of maximal p-cyclically monotone operators

Abstract: In this talk we deal with the construction of explicit examples of maximal p-cyclically maximal monotone operators. To the date, there is only one instance of an explicit example of a maximal 2-cyclically monotone operator that is not maximal monotone. We present several other examples, and a proposal of how such examples can be constructed.

A recorded video of the conference is …. ;  the slides can be downloaded here

A brief biography of the speaker: Orestes Bueno is an Associate Professor at Universidad del Pacífico, Lima, Perú. He obtained his PhD at the Instituto de Matemática Pura e Aplicada (IMPA), Brazil, in 2012. His main interests are: Maximal Monotone Operators, Generalized Convexity and Monotonicity, Functional Analysis.

Coordinators: Fabián Flores-Bazán (Universidad de Concepción) and Abderrahim Hantoute (CMM).

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

### Date:  November 18,  2020 at 10:00

Title: On diametrically maximal sets, maximal premonotone maps and premonotone bifunctions

Abstract: First, we study diametrically maximal sets in the Euclidean space (those which are not properly contained in a set with the same diameter), 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 case.

A recorded video of the conference is …. ;  the slides can be downloaded here

A brief biography of the speaker: Wilfredo Sosa es profesor del Programa de Graduados de Economía de la Universidad Católica de Brasilia, Brazil; Egresado de la Universidad de Ingeniería de Lima, Perú. Formado en el IMPA de Rio de Janeiro Brasil. Co-Fundador del IMCA de Lima Peru. Miembro titular de la Academia de Ciencias de Perú. Areas de interés: Optimization theory; Duality theory; Equilibrium theory; Mathematical economy.

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

# An algebraic view of the smallest strictly monotonic function

### Date:  November 04,  2020 at 10:00

Title: An algebraic view of the smallest strictly monotonic function

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.

A recorded video of the conference is here ;  the slides can be downloaded here

A brief biography of the speaker: César Gutiérrez (ORCID iD 0000-0002-8223-2088) is Professor at Universidad of Valladolid (Spain) and researcher of the Mathematics Research Institute of the University of Valladolid (IMUVA). He is author of 54 papers on several subjects related to vector and set-valued optimization. Currently, he is Associate Editor of Optimization.

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

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

### Date:  October 07,  2020 at 10:00

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

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.

A recorded video of the conference is here;  the slides can be downloaded here

Venue: Online via Google Meet here

A brief biography of the speaker: Nicolás Hernández is currently a Postdoctoral Researcher at the Center for Mathematical Modeling (CMM), at Universidad de Chile. He obtained his PhD in 2017, as a cotutelle between Université Paris-Dauphine and Universidad de Chile. His research areas of interest are Contract theory, stochastic control, mathematical finance, probability, optimization, game theory.

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

# Sigma-convex functions and Sigma-subdifferentials

### Date:  September 23, 2020 at 10:00

Title: Sigma-convex functions and Sigma-subdifferentials

Abstract: In this talk we present and study the notion of $\sigma$-subdifferential of a proper function $f$ which contains the Clarke-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 the large class of $\sigma$-convex functions.

A recorded video of the conference is here;  the slides can be downloaded here

A brief biography of the speaker: Mohammad Hossein Alizadeh is an Assistant
Professor at the Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran. He obtains his Ph.D. from the University the Aegean, Greece, in 2012. He is mainly interested in the following areas:

Monotone and generalized monotone operators, Monotone and generalized monotone
bifunctions, generalized convexity and generalized inverses.

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

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

### Date:  September 2, 2020 at 10:00

Title: Generalized Newton Algorithms for Tilt-Stable Minimizers in Nonsmooth Optimization

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 ofthe 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 algorithms to minimization of extended-real-valued prox-regular functions, while covering in this way problems of constrained optimization, by using Moreau envelopes. Employing advanced techniques of second-order variational analysis and characterizations of tilt stability allows us to establish the solvability of subproblems in both algorithms and to prove the Q-superlinear convergence of their iterations. Based on joint work with Ebrahim Sarabi (Miami University, USA).

A recorded video of the conference is here;  the slides can be downloaded here

A brief biography of the speaker: Prof. Boris Mordukhovich was born and educated in the former Soviet Union. He got his PhD from the Belarus State University (Minsk) in 1973. He is currently a Distinguished University Professor of Mathematics at Wayne State University. Mordukhovich is an expert in optimization, variational analysis, generalized differentiation, optimal control, and their applications to economics, engineering, behavioral sciences, and other fields. He is the author and a co-author of many papers and 5 monographs in these areas. Prof. Mordukhovich is an AMS Fellow, a SIAM Fellow, and a recipient of many international awards and honors including Doctor Honoris Causa degrees from 6 universities worldwide. He was the Founding Editor (2008) and a co-Editor-in-Chief (2009-2014) of Set-Valued and Variational Analysis, and is now an Associate Editor of many high-ranked journals including SIAM J. Optimization, JOTA, JOGO, etc. In 2016 he was elected to the Accademia Peloritana dei Pericolanti (Italy). Prof. Mordukhovich is in the list of Highly Cited Researchers in Mathematics.

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

# An overview of Sweeping Processes with applications

### Date:  August 26, 2020 at 10:00

Title: An overview of Sweeping Processes with applications

Abstract: The Moreau’s Sweeping Process is a first-order differential inclusion, involving the normal cone to a moving set depending on time. It was introduced and deeply studied by J.J. Moreau in the 1970s as a model for an elastoplastic mechanical system. Since then, many other applications have been given, and new variants have appeared. In this talk, we review the latest developments in the theory of sweeping processes and its variants. We highlight open questions and provide some applications.

This work has been supported by ANID-Chile under project Fondecyt de Iniciación 11180098.

A brief biography of the speaker: Prof. Emilio Vilches is Assistant Professor at Universidad de O’Higgins, Rancagua, Chile. He obtains his Ph.D. from the University of Chile and the University of Burgundy in 2017. He is mainly interested in the application of convex and variational analysis to nonsmooth dynamical systems.

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

# Epi-convergence, asymptotic analysis and stability in set optimization problems

### Date:  August 05, 2020 at 10:00

Title: Epi-convergence, asymptotic analysis and stability in set optimization problems

Abstract: We study the stability of set optimization problems with data that are not necessarily bounded. To do this, we use the well-known notion of epi-convergence coupled with asymptotic tools for set-valued maps. We derive characterizations for this notion that allows us to study the stability of vector and set type solutions by considering variations of the whole data (feasible set and objective map). We extend the notion of total epi-convergence to set-valued maps.

* This work has been supported by Conicyt-Chile under project FONDECYT 1181368

Joint work with Elvira Hérnández, Universidad Nacional de Educación a Distancia, Madrid, Spain

A brief biography of the speaker: Prof. Rubén López   is Professor at the University of Tarapacá,  Arica – Chile. He studied at Moscow State University – Mech Math (1996, Russia) and Universidad de Concepción – DIM (2005, Chile). He works on Optimization: asymptotic analysis, variational convergences, stability theory, approximate solutions and well-posedness.

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

# Satisfying Instead of Optimizing in the Nash Demand Games

### Date: July 22, 2020 at 10:00

Abstract: The Nash Demand Game (NDG) has been one of the first models (Nash 1953) that has tried to describe the process of negotiation, competition, and cooperation. This model is still subject to active research, in fact, it maintains a set of open questions regarding how agents optimally select their decisions and how they face uncertainty. However, the agents act rather guided by chance and necessity, with a Darwinian flavor. Satisfying, instead of optimising. The Viability Theory (VT) has this approach. Therefore, we investigate the NDG under this point of view. In particular, we ask ourselves two questions: if there are decisions in the NDG that ensure viability and if this set also contains Pareto and equilibrium strategies. Thus, carrying out the work, we find that the answers to both questions are not only affirmative, but that we also advance in characterising viable NDGs. In particular, we conclude that a certain type of NDGs ensures viability and equilibrium. Many interesting questions originate from this initial work. For example, is it possible to fully characterise the NDG by imposing viability conditions? Under what conditions does viability require cooperation? Is extreme polarisation viable?

A brief biography of the speaker: Prof. Sigifredo Laengle   is an Associate Professor at the University of Chile since 2007. He received his PhD in Germany working on the theoretical problem of the value of information in organisations. He has published articles that articulate phenomena of strategic interaction, and optimisation.

Coordinators: Abderrahim Hantoute and Fabián Flores-Bazán (Universidad de Concepción)

# Enlargements of the Moreau-Rockafellar Subdifferential

### Date: July 15, 2020 at 10:00

Abstract: The Moreau-Rockafellar subdifferential is a highly important notion in convex analysis and optimization theory. But there are many functions which fail to be subdifferentiable at certain points. In particular, there is a continuous convex function defined on $\ell^2(\mathbb{N})$, whose Moreau–Rockafellar subdifferential is empty at every point of its domain. This talk proposes some enlargements of the Moreau-Rockafellar subdifferential: the sup$^\star$-subdifferential, sup-subdifferential and symmetric subdifferential, all of them being nonempty for the mentioned function. These enlargements satisfy the most fundamental properties of the Moreau–Rockafellar subdifferential: convexity, weak$^*$-closedness, weak$^*$-compactness and, under some additional assumptions, possess certain calculus rules. The sup$^\star$ and sup subdifferentials coincide with the Moreau–Rockafellar subdifferential at every point at which the function attains its minimum, and if the function is upper semi-continuous, then there are some relationships for the other points. They can be used to detect minima and maxima of arbitrary functions.