# Closedness under addition for families of quasimonotone operators

### Universidad del Pacífico, Lima-Perú

Date:  May 11,  2022 at 12:00 am (Chilean-time)

Title:    Closedness under addition for families of quasimonotone operators

Abstract:  In this talk we will discuss some results about quasimonotone family of operators. For some notions that are extensions of monotoniticity but not beyond quasimonotonicity like pseudomonotonicity, semistrict quasimonotonicity, strict quasimonotonicity and proper quasimonotonicity, we will discuss systematically when the sum of two operators satisfying one of those properties, inherits the same property. Several examples showing the optimality in some sense of our results, are presented. Join work with: Fabián Flores-Bazán and Nicolas Hadjisavvas.

Venue: Online via Google Meet: https://meet.google.com/kzr-azwr-sen

A brief biography of the speaker: Yboon García es investigadora del Centro de Investigación-CIUP, profesora del Departamento Académico de Economía de la Universidad del Pacífico. Actualmente es directora de la Maestría en Economía de la Escuela de Gestión Pública. Es doctora en Matemática por la Universidad de las Antillas y de la Guayana (Francia) y el IMCA y la Universidad Nacional de Ingeniería; maestra en Ciencias por el IMCA-Universidad Nacional de Ingeniería, y licenciada en Matemática por la misma universidad. Ha realizado un postdoctorado en el Centro de modelamiento matemático de la Universidad de Chile. Se ha especializado en análisis convexo, análisis no lineal, análisis variacional, optimización. Es investigadora calificada en Ciencia y Tecnología del SINACYT (REGINA-CONCYTEC, Perú).

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

# Brezis pseudomonotone bifunctions and quasi equilibrium problems via penalization

### Department of Mathematics and Applications, University of Milano-Bicocca, Italy

Date:  December 15,  2021 at 10:00 am (Chilean-time)

Title:    Brezis pseudomonotone bifunctions and quasi equilibrium problems via penalization

Abstract:  We investigate quasi equilibrium problems in a reflexive Banach space under the assumption of Brezis pseudomonotonicity of the function involved. To establish existence results under weak coercivity conditions we replace the quasi equilibrium problem with a sequence of penalized usual equilibrium problems. To deal with the non compact framework, we apply a regularized version of the penalty method. The particular case of set-valued quasi variational inequalities is also considered (Joint work with Monica Bianchi and Gabor Kassay).

Venue: Online via Google Meet: https://meet.google.com/aer-kove-tns

A brief biography of the speaker: Rita Pini is a Professor at the Department of Mathematics and Applications of the University of Milano-Bicocca (Italy). She earned her master’s degree in Mathematics from the University of Milano in 1983, held a Researcher position at the University of Verona (1987-1992), and then was appointed as Associate Professor (until 2001) and Professor at the Universities of Milano and Milano-Bicocca. Her research interests include equilibrium problems, nonlinear optimization, variational analysis, convexity in Carnot groups

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

# Continuity and maximal quasimonotonicity of normal cone operators

### Speaker: Professor Nicolas Hadjisavvas,  University of the Aegean, Greece.

Date:  December 1st,  2021 at 10:00 am (Chilean-time)

Title:    Continuity and maximal quasimonotonicity of normal cone operators

Abstract:  In this talk we present some properties of the adjusted
normal cone operator of quasiconvex functions. In particular, we introduce a
new notion of maximal quasimotonicity for set-valued maps, different from
similar ones that appeared recently in the literature, and we show that this
operator is maximal quasimonotone in this sense. Among other results, we prove
the $s\times w^{\ast}$ cone upper semicontinuity of the normal cone operator
in the domain of $f$, in case the set of global minima is empty, or a
singleton, or has non empty interior (joint work with M. Bianchi and R. Pini).

Venue: Online via Google Meet: https://meet.google.com/cmx-ivvw-saq

A brief biography of the speaker: Nicolas Hadjisavvas is Professor Emeritus, University of the Aegean, Greece.  Among other responsabilities He is currently Associate editor of JOTA, JOGO, Optimization, and Optimization Letters; author of 66 papers from which he received 1392 citations according to WoS (without self-citations). In addition, He edited 4 books in Springer, and 5 special journal issues, besides He served as chair of the Working Group on Generalized Convexity (2003-2006, 2015-2018). He has been keynote or invited speaker in many Conferences or Summer Schools.

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

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

### Speaker: Professor Maicon Marques Alves

Department of Mathematics, Federal University of Santa Catarina in Florianópolis (Brazil)

Date:  November 17,  2021 at 10:00 am (Chilean-time)

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

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 O(k^(-k(p-1)/(p+1))) global rates for the proposed variants of the A-HPE and large-step A-HPE methods, respectively. The parameter p >= 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 method for (smooth) strongly convex optimization with iteration-complexity O(k^(-k(p-1)/(p+1))). In particular, for p=2, we obtain an inexact Newton-proximal algorithm with fast global O(k^(-k/3)) convergence rate.

Venue: Online via Google Meet: https://meet.google.com/xmu-cvfn-qkp

A brief biography of the speaker: M. Marques Alves is Associate Professor at the Department of Mathematics of the Federal University of Santa Catarina in Florianópolis (Brazil). He obtained his PhD in Mathematics from IMPA in 2009 and held a postdoctoral position at the Department of Industrial and Systems Engineering of the Georgia Institute of Technology (2014-2016). His current research interests include Algorithms for Convex Optimization, Monotone Operator Splitting Algorithms and applications to Statistical Learning and Inverse Problems.

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

# El Lema de Farkas: Algunas extensiones y aplicaciones

### Speaker: Profesor  Miguel A. Goberna

Universidad de Alicante, España

Date:  November 3,  2021 at 10:00 am (Chilean-time)

Title:   El Lema de Farkas: Algunas extensiones y aplicaciones

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.

Venue: Online via Google Meet: https://meet.google.com/otd-bmaa-pbb

Slides of the conference: PDF

A brief biography of the speaker: Miguel A. Goberna es Profesor Emérito de la Universidad de Alicante. Sus principales campos de interés son la optimización semi-infinita e infinita, la optimización robusta, y sus fundamentos, sobre los que ha publicado más de 125 artículos. Es también co-autor de los libros Linear Semi-Infinite Optimization (Wiley, 1998), Optimización Lineal (McGraw-Hill, Madrid, 2004), Post-Optimal Analysis in Linear Semi-Infinite Optimization, Springer, 2014), Nonlinear Optimization (Springer, 2019), y Even Convexity and Optimization: Handling Strict Inequalities (Springer, 2020), entre otros.

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

# Stochastic incremental mirror descent algorithms with Nesterov smoothing

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

Title:   Stochastic incremental mirror descent algorithms with Nesterov smoothing

Abstract:  We propose a stochastic incremental mirror descent method 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 a 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 incremental mirror descent Bregman-proximal scheme with Nesterov smoothing for minimizing the sum of finitely many proper, convex and lower semicontinuous functions with a prox-friendly proper, convex and lower semicontinuous function in the same framework. Different to the previous contributions from the literature on mirror descent methods for minimizing sums of functions, we do not require these to be (Lipschitz) continuous or differentiable. Applications in Logistics, Tomography and Machine Learning modelled as optimization problems illustrate the theoretical achievements. The talk is based on joint work with Sandy Bitterlich.

Venue: Online via Google Meet: https://meet.google.com/bhn-ufcj-wnw

A brief biography of the speaker: Sorin-Mihai GRAD is a Professor of Optimization at ENSTA Paris, after working at the Chemnitz University of Technology (where he got his PhD), Leipzig University and University of Vienna. His research interests include convex, vector and numerical optimization as well as convex analysis and applications that can be modeled as such.

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

# Constant Along Primal Rays Conjugacies and the l0 Pseudonorm

### Date:  October 13,  2021 at 10:00 am (Chilean-time)

Title:   Constant Along Primal Rays Conjugacies and the l0 Pseudonorm

Abstract: The 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) and variational formulations for the l0 pseudonorm. In a second part, we provide different extensions: the above properties hold true for a class of conjugacies depending on strictly-orthant monotonic norms (including the Euclidean norm); they hold true for nondecreasing functions of the support (including the l0 pseudonorm); more generally, we will show how Capra-conjugacies are suitable to provide convex lower bounds for zero-homogeneous functions; we will also point out how to tackle the rank matrix function. Finally, we present mathematical expressions of the Capra-subdifferential of the l0 pseudonorm, and graphical representations. This opens the way for possible suitable algorithms that we discuss.

Venue: Online via Google Meet: https://meet.google.com/jhs-kymj-gwa

A brief biography of the speaker: Michel De Lara graduated as an engineer at Ecole Polytechnique and at Ecole nationale des ponts et chaussées, where he is presently working at the mathematics research center CERMICS, after obtaining his PhD at Ecole nationale supérieure des mines de Paris. His research interests include stochastic optimization, game theory with information, generalized convexity, as well as different applications of mathematics (epidemics control, energy management).

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

# On strongly quasiconvex functions: theory and applications

### 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)

# Complejidad en Búsqueda Coordenada de Alto Orden en Bloques

### Date:  September 22,  2021 at 10:00 am (Chilean-time)

Title:   Complejidad en Búsqueda Coordenada de Alto Orden en Bloques

Abstract: Definimos métodos de alto orden para minimización en cajas con búsqueda coordenada por bloques. Probamos resultados de convergencia y complejidad para puntos estacionarios de alto orden adecuados. Exhibimos ejemplos.

Venue: Online via Google Meet: https://meet.google.com/qjd-ryiv-kmg

A brief biography of the speaker: José Mario Martínez es Profesor Emérito de la Universidad Estadual de Campinas, Presidente del Consejo Científico del Instituto de Estudios Avanzados de la Unicamp. Especialista en Optimización y Análisis Aplicada. Autor de más de 180 artículos y dos libros.

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

# Regularized version of the minimax probability machine

### Date:  September 08,  2021 at 09:30 am (Chilean-time)

Title:   Regularized  version of the minimax  probability machine

Abstract: In this talk we present novel second-order cone programming formulations for binary classification, by extending the Minimax Probability Machine (MPM) approach. Inspired by Support Vector Machines, a regularization term is included in the MPM and Minimum Error Minimax Probability Machine methods. This inclusion reduces the risk of obtaining ill-posed estimators, stabilizing the problem, and, therefore, improving the generalization performance. Our approaches are first derived as linear methods, and subsequently extended as kernel-based strategies for nonlinear classification. Experiments on well-known binary classification datasets demonstrate the virtues of the regularized formulations in terms of predictive performance.

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

Venue: Online via Google Meet: https://meet.google.com/gjq-cwic-udo

A brief biography of the speaker: Julio López is an Associate Professor at the Institute of Basic Sciences, University Diego Portales, Santiago, Chile. He obtained his PhD degree at the University of Chile, in 2009. His research interests include conic programming, convex analysis, algorithms, and machine learning.

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