Mathematisches Kolloquium - Prof. Dr. Andreas Hamel

Zeitraum: 27.04.2017 16:00 Uhr - 18:00 Uhr

Ort: Carl-Zeiß-Straße 3, SR 308

Donnerstag, 27. April 2017, 16:30 Uhr, Carl-Zeiß-Straße 3, SR 308

Prof. Dr. Andreas Hamel (Free University of Bozen/Italy)

Thema: An abstract convexity approach to set relations and set optimization

Abstract: Many order relations are defined or can be expressed via families of scalar functions, for example stochastic dominance orders, non-total preferences in economics and vector orders generated by closed convex cones by means of the bipolar theorem. It is shown that such an abstract convexity approach to order relations

(a) covers a vast variety of known and produces new examples,

(b) very naturally leads to new set relations as well as to

(c) intrinsic hull operators and complete lattices of sets,

(d) can be used to transform a (vector-type) minimization problem with respect to a non-lattice preorder into a set optimization problem with a complete lattice as image space.

Concepts for exact and approximate solutions of the resulting set optimization problems are introduced, and Weierstrass-type theorems are given. Several examples from economics, finance, statistics and multi-criteria decision making are discussed. The approach leads to new answers of questions which are subject of ongoing discussions in the respective communities such as the following. What is the (lower or upper) quantile of a multivariate random variable? How should utility maximization for incomplete preferences be done? What is the Legendre-Fenchel conjugate of a vector- or even set-valued convex function?

The talk starts with a brief introduction of set relations and set optimization with many examples. The main part is devoted to the abstract convexity approach to set optimization and includes the main results using tools from lattice theory and nonlinear functional analysis. The third part deals with a few examples in more detail: duality for vector/set optimization, multivariate statistics and risk measures for markets with transaction costs.

Joint work with Giovanni Crespi, Matteo Rocca, Carola Schrage.

Alle Interessenten sind herzlich eingeladen!