Constant Rank Conditions for Second-Order Cone and Semidefinite Programming

Department of  Applied Mathematics, University of São Paulo, Brazil

Date:  1st June,  2022 at 11:00 am (Chilean-time)

Title:    Constant Rank Conditions for Second-Order Cone and Semidefinite Programming

Abstract:  In [R. Andreani, G. Haeser, L. M. Mito, H. Ramírez C., Weak notions of nondegeneracy in nonlinear semidefinite programming, arXiv:2012.14810, 2020] the classical notion of nondegeneracy (or transversality) and Robinson’s constraint qualification have been revisited in the context of nonlinear semidefinite programming exploiting the structure of the problem, namely, its eigendecomposition. This allows formulating the conditions equivalently in terms of (positive) linear independence of significantly smaller sets of vectors. Here we extend these ideas to the context of nonlinear second-order cone programming. For instance, for an m-dimensional second-order cone, instead of stating nondegeneracy at the vertex as the linear independence of m derivative vectors, we do it in terms of several statements of linear independence of two derivative vectors. This allows embedding the structure of the second-order cone into the formulation of nondegeneracy and, by extension, Robinson’s constraint qualification as well. This point of view is shown to be crucial in defining significantly weaker constraint qualifications such as the constant rank constraint qualification and the constant positive linear dependence condition. Also, these conditions are shown to be sufficient for guaranteeing global convergence of several algorithms, while still implying metric subregularity and without requiring boundedness of the set of Lagrange multipliers.

A brief biography of the speaker: Gabriel Haeser is an Associate Professor of Applied Mathematics at the University of São Paulo, Brazil. He obtained his PhD in 2009 from the University of Campinas, Brazil. He held a visiting scholar position at Stanford University in 2016-2017. His research interests include Algorithms and Optimality Conditions for Nonlinear Programming, with a recent focus on Conic Optimization.

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

Closedness under addition for families of quasimonotone operators

Speaker: Professor Yboon García Ramos

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.

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).

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).

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.

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

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.

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.

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.

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.

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)