El departamento de matemáticas invita a la comunidad universitaria el 25 y 26 de noviembre de 2010 en el auditorio ubicado en el edificio de Mecánica a las 6:00 p.m. a las conferencias a cargo del invitado internacional Anatoly Antipin, Computing Center of Russian Academy of Sciences, Moscow, Russia.
El doctor Anatoly Antipin, de la Federación Rusa, se encargará de dar dos conferencias y una charla sobre las posibilidades de pasantías, doctorados y maestrías auspiciadas por la Federación Rusa.
El tema a tratar en la conferencia es:
Models and methods of equilibrium programming
Scope
Applied mathematics and computer science problems are base for development of many scientific areas, in particular, such as the decision-making theory, operation research, mathematical modeling. In turn, development of these directions stimulates the researches of mathematical models and finding-out of their applied opportunities as a tool for mathematical modeling.
The decision-making theory represents an area of scientific researches claimed by practice. This area has basically two development trends. One of them is related to individual choice of alternatives. This trend is supported by an optimization problem with unique criterion at presence of constraints. Such classical approach has been suggested long ago. Another trend is related to a collective (or group) choice of the decision. Various parametrical systems of optimization problems act here as mathematical models. Most known of them are n-person games with Nash equilibrium, optimal control game problems with solutions in a class of program strategy, multi-criteria equilibrium optimization problems, economic equilibrium model and, in particular, well-known pure exchange model.
These problems allow to describe the fact of collective decision-making. Here each participant within a group independently makes the individual choice on the set of alternatives, but the whole group should be convinced that a set of individual choices answers to conditions of a group choice. The choice answering to these conditions has new quality – to coordinate partly inconsistent interests of all participants of group. In this case the considered problem is a model of a conflict, and its solution is a compromise, or equilibrium.
Collective choice models are rather popular in applications, but, unfortunately, have no advanced theory regarding to numerical methods for equilibrium problems. The equilibrium programming approach presents an attempt to construct such a theory. The theory is based on the idea of saddle point or bilinear function at which there are two variables describing alternatives of two participants and also coordination condition of their interests. It is a simple two-person zero-sum game. Within the framework of equilibrium programming theory the main task is to transfer and prove basic approaches of optimization problems onto equilibrium problems. Therefore, in this lecture course I propose to do the following:
1. To consider statements of matrix games with mixed strategies, to provide examples, to give various game interpretations, to compare them with problems of quadratic optimization, and to show, why the formulas of gradient method are different for problems of optimization and game problems.
2. To consider bilinear (saddle point) statement of an optimal control problem for linear differential systems and to give its economic interpretation by considering extra-gradient approaches for its solution.
3. To show the connection between convex programming problems and two-person games with Nash equilibrium in convex case using extra-gradient method for solution of game problems.
4. To give a statement of multi-criteria optimization problems, to formulate a multi-criteria equilibrium problem, and to suggest some approaches to construction of solution methods using economic interpretations.