Algorithmic approaches to set optimization
Final Report Abstract
The aim of this project was to develop an algorithmic solution procedure for set optimization problems using the set approach. We achieved this via two vectorization approaches that yield multiobjective optimization problems. The first one relies on the minimization and maximization of linear functionals on a respective image set, so-called extremal value functions. These functions are then minimized simultaneously over the feasible set of the set-valued optimization problem in the sense of multiobjective optimization. The second one avoids choosing specific functionals and hence, allows a more general approach and stronger theoretical results. However, the drawback of the second approach is the higher dimension of the multiobjective optimization problem and a very symmetric behavior in its solution set. This makes it numerically more laborious to obtain solutions. For both approaches we proved strong properties. Mainly, the set of weakly minimal solutions of the set-valued optimization problem can be approximated arbitrarily well by either of the vectorizations’ set of weakly minimal elements. In some cases the vectorizations are even equivalent to the initial set-valued problem. These properties allow to investigate the multiobjective optimization problems in order to obtain an approximation of the set of weakly minimal solutions of the set-valued problem. This is a valuable property since it allows to apply the solution techniques from this field of optimization and to solve the problem that way. Both vectorization approaches have been studied intensively, also considering sufficient conditions that, imposed on the set-valued problem, imply certain properties, like (Lipschitz-)continuity and (quasi-)convexity, for the respective vectorization. These properties are useful for solving the multiobjective optimization problems given by the vectorizations in the second step. Additionally, we introduced a quality measure that allows to compare different approximations and to decide which one is more favorable as well as to rate the quality of a given approximation.
Publications
- Optimality conditions for set optimization using a directional derivative based on generalized Steiner sets, Optimization (2020)
R. Baier, G. Eichfelder and T. Gerlach
(See online at https://doi.org/10.1080/02331934.2020.1812605) - On convexity and quasiconvexity of extremal value functions in set optimization, Appl. Set-Valued Anal. Optim. 3, (2021), 293-308
T. Gerlach, S. Rocktäschel
(See online at https://doi.org/10.23952/asvao.3.2021.3.04) - Solving set-valued optimization problems using a multiobjective approach, Optimization (2021)
G. Eichfelder, S. Rocktäschel
(See online at https://doi.org/10.1080/02331934.2021.1988596) - A Vectorization Scheme for Nonconvex Set Optimization Problems. SIAM J. on Optimization (2022)
G. Eichfelder, E. Quintana, S. Rocktäschel
(See online at https://doi.org/10.1137/21M143683X)
