New theoretical and algorithmic challenges in vector optimization

• Applicant: Professor Dr. Gert Wanka
• Subject Area: Mathematics
• Term: from 2013 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 240627717

Vector optimization is a continuously growing research area with many theoretical and practical applications. Basically, it consists in determining within a set the elements which fulfill a desired optimality property. Usually one has to determine different minimal elements of image sets of feasible sets expressed by means of various constraints through vector functions, and the feasible points where these values are attained. A classical example of a vector optimization problem is the one of jointly minimizing the risk and maximizing the expected return of an investment. There are different notions of optimality for sets, arising from practice or of merely theoretical importance, and most of them are defined by means of partial orderings induced by convex cones. The existing methods and results allow successful theoretical insights in dealing with differentiable, (generalized) convex and even some classes of nonsmooth vector optimization problems. However, the research on algorithms for solving vector optimization problems is significantly less developed than in scalar optimization, although several methods were recently successfully extended from scalar to vector optimization problems.With this project we intend to bring a significant contribution to the development of continuous vector optimization, by dealing with new theoretical and algorithmic challenges in this research field. We will investigate new directions, implement new methods and bring new ideas in this continuously growing research area. Moreover, we will provide improvements and extensions of classical results in vector optimization and, nevertheless, new developments in the algorithmic approaches to solve vector optimization problems. We intend to open new perspectives in dealing with vector optimization problems arising from different fields, especially for those whose objective and/or constraint functions lack differentiability, by treating them via techniques from the fields of convex and, respectively, nonsmooth optimization. We will investigate in five research fields of vector optimization, as follows. The first one concerns vector-minimization notions defined via generalized interiors of the ordering cones, for which we will provide new characterizations and insights. Nonsmooth vector optimization problems are the second theme to be treated. For them we will deliver new optimality conditions formulated via appropriate nonsmooth subdifferentials. The third field consists of generalized convex vector optimization problems, where we will focus on dealing with problems with cone-quasiconvex objective and/or constraint vector functions by extending techniques from cone-convex vector functions. The last two planned directions of research are devoted to algorithms for solving nonsmooth vector optimization problems and their convergence properties, where we will concentrate ourselves on smoothing and splitting type methods, respectively. This project was proposed with the support of professor Radu Bot, University of Vienna. It is conducted in close cooperation with him.

• DFG Programme: Research Grants