Mini-corso Introduction to numerical optimization: Multiextremal, Bilevel, and Multiobjective methods 04-05/03/2017

DIMES, sala seminari, cubo 42C VI piano
Martedì, 4 Aprile 2017,        ore 15:00 – 18:00
Mercoledì, 5 Aprile2017,     ore 17:00 – 19:00
Professor V. Grishagin:
Lobachevsky State University of Nizhny Novgorod
Professor R. Paulavicius :
and
Professor E. Filatovas:
Vilnius University Institute of Mathematics and Informatics

Abstract:

Numerical Methods for Multiextremal Optimization

Vladimir Grishagin, Lobachevsky State University of Nizhny Novgorod

Many practical decision-making problems can be formulated as black-box global optimization problems. Their objective function is often Lipschitz (differentiable or not), multiextremal and hard to evaluate. Different approaches for solving these problems will be discussed. Classical and adaptive nested schemes of dimensionality reduction proposed for solving box-constrained global optimization problems will be also considered and possible ways of generalizing these schemes to constrained problems will be examined, as well.

Introduction to Bilevel Optimization

Remigijus Paulavicius, Vilnius University Institute of Mathematics and Informatics

Bilevel problems are often encountered in economics (Stackelberg games, taxation, policy decisions), transportation (network design, optimal pricing), management (network facility location, coordination of multi-divisional firms), engineering (optimal design, optimal chemical equilibria). In the first part of this talk, the focus will be given to highlight differences with single level mathematical optimization problems and other optimization problems with multiple decision levels and multiple decision makers. Then the formulation of the bilevel optimization problem will be formulated, and a summary of the properties and challenges will be given. In the second part, existing deterministic approaches developed in the literature for solving different type of bilevel problems will be reviewed. The particular interest is for the general nonconvex form which is the most challenging and only recently the first methods (Mitsos & Co. approach and Branch-and-Sandwich algorithm) to tackle this class of problems were introduced. In the last part, implementation details of the Branch-and-Sandwich algorithm based solver BASBL will be given, and detailed computational experimentation on a set of bilevel benchmark problems will be presented.

Multiobjective Optimization via Interactive Methods,
Evolutionary Algorithms, and Visualization

Ernestas Filatovas, Vilnius University Institute of Mathematics and Informatics

Many real-world problems are multiobjective, where several conflicting objective functions must be minimized. Usually, there is no solution which would be the best for all objectives, however, a set of optimal solutions in a multiobjective sense exists. These solutions are defined as the Pareto optimal solutions where none of the objective values can be improved without deteriorating other(s). Therefore, solving a multiobjective problem is usually understood as finding a Pareto optimal solution which is the most preferred for a Decision Maker (DM). Some popular approaches for solving multiobjective optimization problems as, for example, Multiple Criteria Decision Making and Evolutionary Multiobjective Optimization will be considered. Moreover, when solving multiobjective optimization problems, the DM must compare several different alternatives and select the most preferred one. The task of comparing multidimensional data is very demanding for the DM without any support. Some graphical visualization methods and tools used to support and help the DM in understanding similarities and differences between the alternative solutions will be also discussed.