Centro de Modelamiento Matemático - CMM
Universidad de Chile 2007
Our goal at CMM To establish meaningful and productive relationships
between advanced mathematics and all endeavors of modern society
Felipe Alvarez, Jorge Amaya, Rafael Correa, Fabián Flores (DIM, U. Concepción), Alejandro Jofré, Héctor Ramírez, Abderrahim Hantoute.
Maurice Queyranne (2008-2009)
We consider how much a decision-maker should pay for information to improve his optimal decision. We will search when the value of information is nonnegative and its role in explaining the discrepancies between equilibrium and social optimum in a game setting.
During this first Basal 4 year project (2008 –2012) the main researchers of our group have been: Felipe Álvarez, Jorge Amaya, Rafael Correa, Fabián Flores, Alejandro Jofré and Héctor Ramírez. In addition, Abderrahim Hantoute, who was postdoc until 2009, was incorporated to the group as CMM researcher in 2010. The group was completed with 6 postdocs and 2 more beginning now. We have advised 5 PhD theses and 6 more are in progress, and 13 Mathematical engineering “master” theses were completed and 3 more are in progress.
The OE group during this period have worked on the following topics:
Algorithms and sensitivity for cone programming and applications:We have developed proximal type algorithms with variable metric for solving particular instances of convex second order cone programming problems. These algorithms have been also coupled with bundle techniques in order to treat nonsmooth data. This has been successfully applied to the shape optimization of mechanical structures, and to the robust classification of data. From the sensitivity side, we have investigated the behavior of the primal-dual solution set of perturbed conic optimization problems. In particular, we have characterized the Aubin property and strong regularity in terms of second order optimality conditions. In a nonconvex setting we have analyzed the validity of the strong duality property, and characterized that in the case of one single inequality constraint.
Variational analysis and Inequalities, game theory and its applications to economics: Firstly, we have developed in a joint work with R.T. Rockafellar and R. Wets, a whole theory of existence, stability and approximations of economic equilibrium points by using a variational inequality approach. Secondly, we developed a suitable notion of convergence- lopsided convergence- to prove stability properties of general equilibrium points such as Nash, Walras, variational inequality solution points and inclusions in general. Thirdly, we have developed a game theory-stochastic optimization model to analyze agent behaviors in electricity markets. We proved a result showing some sensitivity properties and a new estimation for market power in this framework using variational analysis.
Minimal time optimal control problems and Lyapunov stability of differential inclusions:We have modeled different real-life applications via this type of optimization problem. For this, we have also worked with impulsional controls, slow-fast dynamics, among other mathematical tools. At the same time, we have characterized the so-called Lyapunov pairs associated to first-order differential inclusions governed by a Lipschitzian perturbation of a maximal monotone operator.
Convex relaxation, set-valued complementarity and vector optimization problems:We have established a quite complete relationship between a primal optimization problem and its lower semicontinuous convex relaxed form, extending a famous result by Benoist-Hiriart to the infinite-dimensional setting. We have introduced new mappings beyond copositivity and established existence and sensitivity results by means of their asymptotic set-valued maps. We have established optimality conditions via subdifferentials and characterized their solutions via scalarization. We also studied multivalued complementarity problems and multiobjective quasiconvex optimization, showing results on strict efficiency in set-valued optimization and for asymptotically bounded multifunctions and the MCP problems.
Variational principles and Lipschitz-like properties. A new variational principle for the existence of a minimizer on a drop was obtained without compactness on the constraint set and any lower boundedness condition of the objective function. From this result we deduced a simple proof the famous drop theorem by Danes, which is equivalent to the Ekeland’s variational principle. On the other hand, we have presented a unify approach for most of Lipschitz-like property used on nonsmooth and variational analysis during the last three decades.
Mathematical optimization in natural resource management: Traditional models for open pit mine planning problems use binary decision variables, giving rise to large-scale combinatorial and Mixed Integer Programming problems. As an alternative approach, we have introduced a new continuous variable framework based on calculus of variations in suitable functional spaces, which in addition allows for a refined imposition of slope constraints associated with geotechnical stability. Secondly, in the standard discrete variable setting, we have investigated robust open-pit mine planning under ore-grade uncertainty to get solutions with desirable risk-averse properties. Finally, we have developed control and viability tools with the aim of evaluating the sustainability of some Chilean fisheries and also providing possible exploitation strategies and recovery programs to decision-makers.
Publications. This activity has led to the publication of around 47 articles and to the acceptation of 12 papers in top optimization journals, such as Economic Theory,SIAM J. on Optimization, SIAM J. on Control and Optimization, Mathematical Programming, Mathematics of Operations Research, Journal of Optimization Theory and Applications, Convex Analysis, Optimization Methods and Software, Set-Valued and Variational Analysis, among others. Among these papers, we point out that "Interior Proximal Algorithm with Variable Metric for Second-order Cone Programming: Applications to Structural Optimization and Classification by Hyperplanes" (F. Álvarez, J. López, H. Ramírez; Opt. Method. Softw. 2010) was awarded the Charles Broyden 2010 prize, “On the Aubin Property of Critical Points to Perturbed Second-Order Programs” (J. Outrata, H. Ramírez; SIOPT 2011) was awarded with the Best Paper Award 2011 of the Institute of Information Theory and Automation of the Academy of Sciences of the Czech Republic. It has been also led to 9 publications in recognized conference proceedings.
Editorial work. The international recognition of the work described above is reflected first in our participation in editorial boards of important optimization journals, namely, Communications in Optimization Theory, International Journal of Optimization: Theory, Methods and Applications, Energy Optimization, Journal of Convex Analysis, Journal of Optimization Theory and Applications, Optimization, Optimization and Engineering, Optimization Research Letters, Set valued and Variational Analysis, The Open Mathematics Journal and TOP: The Journal of the Spanish Society of Statistics and Operations Research.
Conferences. Another important component of OE group is a significant number of conferences organized in Chile during this period. First, the “Third International Conference on Continuous Optimization” (ICCOPT; July 24-29, 2010): This conference is considered one of the most important events in Optimization. More than 200 participants attended this edition, including around 10 plenary speakers. This included a winter school during the two first days of the meeting on Optimization under Uncertainty, and Optimization in Natural Resources. Researchers of the CMM organized the latter. See http://www.iccopt2010.cmm.uchile.cl/ for more details. In addition, we organized the 6th French-Chilean Journeys on Optimization (May 19-21, 2008); A Thematic trimester on Variational Analysis (October 1st to December 20th, 2008): This activity consisted of weekly seminars, an undergraduate semester course, a school (November 6-19, 2008, http://variational-analysis.cmm.uchile.cl/) and a workshop (November 10-13, 2008, http://variational-analysis.cmm.uchile.cl/) on the subject. These events gave the opportunity to train undergraduate and Ph.D. students on particular topics in this field. It brought together around 50 participants, including 17 speakers, and approximately 25 students. Other conferences were: “VII France-Chilean Journeys on Optimization" (December 2011), "The 1rst and 2nd France-Chilean Workshop on Bioprocesses Modeling" (January 2010 and 2012, respectively), and the Third Focus-Seminar, “Optimization: Asymptotic, strong duality and relaxation” (April 2012).
Plenary talks and scientific committees. Furthermore, members of the group have been invited as plenary and semi-plenary speakers in reputed conferences and in the organization of numerous conferences on optimization and related areas such as the XXI International Symposium on Mathematical Programming (Berlin, Germany; August 2012), PRIMA Pacific Rim Mathematical Association (Shanghai, China; July 2013), Italian - Latin American Conference on Applied and Industrial Mathematics (Rosario, Argentina; December, 2012), International Mini-Conference on Set-Valued Variational Analysis and Optimization with Applications in Finance (Germany; August 16-19, 2012), XXV Jornadas de Matemática de la Zona Sur (Concepción, Chile; April 18-20, 2012), X International Symposium of Generalized Convexity and Monotonicity (Romania; August 22 - 27, 2011), International Symposium Mathematical Programming, chairman sessions on Energy Optimization (Chicago; August 2009), Mathematical Economic Congress - Getulio Vargas Foundation (Rio de Janeiro, Brazil; January 2008); CLAIO (Cartagena de Indias, Colombia; 2008), Conference COPI - Optimization in Energy (Paris, France; 2008), Invited special session PRIMA on Applied Mathematics and Finance (Sydney, Australia; 2009).
Projects. The above research has also permitted the creation of a patent (Invention disclosure #200800680, http://dbi.corp.hp.com/disclose/13KN9) on a method based on game theory for automatic determination of service level agreements. During this period the OE researchers have been strongly active on research and innovation projects such as the installation in Chile of the Excellence Research Center CIRIC (INRIA – Chile), in which two project teams are led by members of our group, one FONDEF project on mathematical education for training teacher; two “Associated units” of INRIA (France) on Energy Optimization and Bioprocesses Modeling and Optimization, several international cooperation projects such as STIC AmSud and ECOS programs.
Scientific network. A scientific network has been reinforced and enlarged during this first 4 years of the Basal project. We can mention the following collaborators: J. Frédéric Bonnans (INRIA, France), Jérôme Bolte (Toulouse School of Economics, France), Michel de Lara (U. Paris-Est, France), Paulo da Silva, (U. Campinas, Brasil), Miguel A. Goberna (U. Alicante, Spain), Andreas Griewank (DFG Research Center Matheon, Berlin, Germany), Vincent Martinet (INRA, France), Boris Mordukhovich (Wayne University, USA), Jiri Outrata (Czech Academic of Sciences), Alain Rapaport (INRIA-INRA, France), Terry Rockafellar (U. Seattle, USA), Lionel Thibault (U. Montpellier, France), Roger Wets (U.C. Davis, USA), among others. We have also hosted 4 foreign professors for long-term research stays: Térence Bayen from U. Montpellier, Therry Champion from U. Toulon, both supported by CNRS-France for one year, Thierry Champion from U. Toulon (2 months), Ulrich Horst, Humboldt University (1 month), Maurice Queyranne from U. British Columbia, Canada (1 year), Alberto Seeger from U. Avignon, France (2 months), Stephen Wright from U. Wisconsin, USA (1 month) and Darius Zagrodny from Cardinal Stefan Wyszynski University, Poland (2 months).
Placements.Several of our former Ph.D. and engineering students as well as postdoctoral positions have been hired in the academic and in the industry. For instance, we can mention that Juan Peypouquet has a permanent position at the Universidad Técnica Federico Santa María (Chile), while Julio López has a provisional position (young researcher) at the same university. Miguel Carrasco has a permanent position at the Universidad de Los Andes (Chile), and José Luis Malverde has been hired as mathematical engineer in the international bank “Banco Santander”.
Connections with labs. The OE’s researchers are leading the following laboratories at CMM: Network Economics Lab in which some of the results on variational inequalities, game theory and stochastic optimization described above are applied or motivated by real-life problems in economics, electricity and telecommunication markets, and risk analysis in mining.Two starting labs, one on Bio-Process and Natural Resourcesin whichdifferent real-life problems from these fields are naturally modeled by using optimal control theory. The second one is on Logistics and production related to difficult and large combinatorial problems. Finally, the CMM’s academic spin-off Center for Advanced Research in Education (CARE)at Universidad de Chile is led by one of the members of OE team.replica rolex watches in watch store| hermes birkin 40cm in bag store | patek philippe replica