Algorithmic approaches to set optimization

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Set optimization problems have recently gained a lot of attraction as they appear in many important and timely applications such as finance (dynamic multivariate risk measures) and robust optimization (for instance in case decision uncertainty is taken into account). The main difficulty is that the values of the objective function are now sets and that a practical relevant optimality notion, known as set approach, requires that these sets have to be compared as a whole. This also implies that there is in general an infinite number of optimal solutions and it has to be the aim to find a representation of this set. While there is a continuously growing research in set optimization, it mostly concentrates on theoretical insights as on relations to other optimality concepts or derivative concepts for optimality conditions. So far there is only very limited research on algorithms for solving set optimization problems. With this project we intend to bring a significant contribution to the development of set optimization with the set approach by providing new theoretical insights which will be used for a new solution algorithm. We will follow a completely new direction by formulating suitable multiobjective optimization problems which can then be solved with the help of parameter dependent scalar-valued subproblems. We will allow the set optimization problems to be nonlinear, but other strong assumptions as on the smoothness will be required to allow to solve the arising scalar-valued subproblems numerically. Moreover, we will also bring new ideas to this growing research area by studying concepts like quality criteria for the evaluation of representations of the image of the optimal solution sets which are now union of sets. We also aim to contribute to the extension of classical concepts from scalar-valued optimization, as local and approximate optimal solutions, to set-valued problems which will be important for any algorithmic developments in this field. Our basic idea is to use minimal value functions for sufficient conditions for optimal solutions of the set optimization problem. Based on them new multiobjective optimization problems will be constructed where the objectives of these problems will be selected adaptively within the algorithm. The arising multiobjective problems will be solved by parameter dependent subproblems. The steering of these parameters will also be done adaptively to find representations of the optimal solution sets of the set optimization problem with a high quality and numerically efficient. The difficulties from the set optimization problem (as non-convexity) directly transfer to the difficulties of the subproblems. This limits the type of problems which can practically be solved. The theoretical results will nevertheless apply to wider classes of set optimization problems and will give a completely new direction also for further theoretical examinations as on optimality conditions.