Department Mathematik » Applied Mathematics 2 » Working Group ‘Set and Vector Optimization’ » Research » Theory of Set and Vector Optimization
Applied Mathematics 2. Working Group ‘Set and Vector Optimization’. Research. Theory of Set and Vector Optimization. In vector optimization, one investigates optimization problems with a vector-valued objective function. Fundamental works concerning vector optimization date back to F.Y. Edgeworth (1881) und V. Pareto (1896). An overview of the historical development of this research area was presented in the below cited work by W. Stadler. In Germany, an upturn on this field of research began in the 70ies. The following photograph shows participants of a conference organized by P. Serafini in Italy in 1984. The theory of vector optimization is concerned with the development of optimization theory in partially ordered vector spaces. Specifically, one investigates developments outside of the classical areas such as optimality conditions and duality. Set optimization (i.e., optimization of set-valued maps) is an extension of vector optimization to the set-valued case. As an example for a concrete application, one can think of the navigation of autonomous transport robots. Such transport robots use ultrasonic sensors for the registration of obstacles. Here, one cannot determine the position of the obstacle but only the distance of the object in the radiation cone, that is, the position of an obstacle is not point, but set-valued. Therefore, questions in optimal control of such robots lead to set optimization. In recent years, derivation concepts such as the tangential Epiderivative, optimality conditions and duality results were derived. For the tangential Epiderivative arithmetic rules were proved and numerical investigations were carried out. It is desirable to deepen the theory in conlinear spaces and to include new optimality terms. Conlinear spaces were introduced by A. Hamel and generalize vector spaces, where the validity of the second distributive law is not required. This new space structure allows the treatment of power sets and convex cones. The modern theory of set optimization is ultimately concerned with vector optimization in conlinear spaces with a partial ordering structure. References: G. Eichfelder, Tangentielle Epiableitung mengenwertiger Abbildungen (Diploma thesis, University of Erlangen-Nuremberg, 2001). A. Hamel, Variational Principles on Metric and Uniform Spaces (Habilitation thesis, University of Halle-Wittenberg, 2005). J. Jahn, Vector Optimization – Theory, Applications, and Extensions (Springer, Berlin, 2004). W. Stadler, Initiators of Multicriteria Optimization, in: J. Jahn und W. Krabs (Eds.), Recent Advances and Historical Development of Vector Optimization (Springer, Lecture Notes in Economics and Mathematical Systems 294, Berlin, 1987), 3 – 47. In set-semidefinite optimization, problems with a vector-valued objective function and special inequality constraints in infinite-dimensional spaces are studied. These inequality constraints are based on a partial order in the space of continuous linear maps L(Y,Y*), which is given by the cone CLK of the so-called K-semidefinite maps: It is therefore required that the quadratic form belonging to the linear mapping A is not negative on a subset K of the space Y. The vector optimization problem with X, Y, Z topological vector spaces, Z partially ordered by a pointed ordering cone CZ, In the finite-dimensional case Y=Rn, we obtain for K=Rn semi-definite and for K=Rn+ co-positive optimization problems and we expand this important class of problems to the vector-valued case. For the K-semidefinite cone, rules for computing and various properties as well as the dual cone and the interior were investigated. For the vector optimization problem, optimality conditions and duality statements exist. To solve these problems, there is a penalty approach available. References: G. Eichfelder und J. Jahn, Set-semidefinite Optimization, Journal of Convex Analysis, 2008. Parts of this text were contributed by: Current team: J. Jahn, E. Köbis, N. Neukel Former research assistants: G. Eichfelder, Chr. Sommer Source.