Exact Efficient Solution of Mixed Integer Programming Problems with Multiple Objective Functions

• Applicant: Professor Dr. Stefan Ruzika
• Subject Area: Mathematics
• Term: from 2014 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 258775501

Mathematical modeling and optimization is a key skill in a wide range of future-oriented disciplines like computational engineering or applied natural sciences. Meeting the demands implicated by the typically complex problems in these fields, appropriate modeling should reflect adherence of several incommensurable objectives. As a consequence, one cannot hope for a single solution optimizing all objectives simultaneously. Instead, a set of so-called Pareto solutions turns out to be optimal in this multi-objective setting. Moreover, from a structural point of view, many models have to incorporate a mixture of variables, i. e. some variables are restricted to integral numbers since they represent indivisible quantities while others are not. The resulting entity of models and theories concerning this matter constitutes the field of multiple objective mixed integer programming. Despite the practical importance, research in this area is still in its infancy. This observation is not as surprising as it seems considering the necessity of accomplishing some preliminary work in less complex but related mathematical fields as well as the recent spread of affordable powerful computers suitable for solving large instances of optimization problems. In short, the central idea of this proposal is to develop a rigorous methodology treating multiple objective mixed integer programming holistically. This task comprises: a) mathematical analysis of the occurring structures, b) coalescence of the theoretical findings and dedicated computational skills in terms of fast algorithms, c) efficient implementation of these algorithms and the dissemination to the academic community. To this end, a team consisting of French computer scientists and German mathematicians join forces and combine their complementary skills. While both partners have a strong background in multiple objective programming, the German group contributes with a distinctive knowledge in discrete optimization and polyhedral theory while the French partner accounts for the development of problem-dependent, highly efficient algorithms and design of numerical solvers. As a tangible result of this cooperation, a toolbox of algorithms named vOpt, which allows the fast and correct solution of multiple objective mixed integer programming problems, will be created, implemented, and provided to the academic public. According to our knowledge, no other research project is currently devoted to these questions. Due to the cross-disciplinary relevance of multiple objective mixed integer programming, this bi-national initiative is expected to expedite a sustainable and long-needed development with an eventually significant meaning for practical problems. Due to the central position of this topic in the field of multiple objective programming, and the principle of dissemination adopted for the results and the software, the project will enjoy a high impact on the scientific community.

• DFG Programme: Research Grants
• International Connection: France
• Participating Person: Professor Dr. Xavier Gandibleux