Dr. Julia Fligge-Niebling
Nonconvex and mixed integer multiobjective optimization with an application to decision uncertainty
Published on: 08. January 2020, Link to pdf
Multiobjective optimization problems commonly arise in different fields like economics or engineering. In general, when dealing with several conflicting objective functions, there is an infinite number of optimal solutions which cannot usually be determined analytically.
This thesis presents new branch-and-bound-based approaches for computing the globally optimal solutions of multiobjective optimization problems of various types. New algorithms are proposed for smooth multiobjective nonconvex optimization problems with convex constraints as well as for multiobjective mixed integer convex optimization problems. Both algorithms guarantee a certain accuracy of the computed solutions, and belong to the first deterministic algorithms within their class of optimization problems. Additionally, a new approach to compute a covering of the optimal solution set of multiobjective optimization problems with decision uncertainty is presented. The three new algorithms are tested numerically. The results are evaluated in this thesis as well.
The branch-and-bound based algorithms deal with box partitions and use selection rules, discarding tests and termination criteria. The discarding tests are the most important aspect, as they give criteria whether a box can be discarded as it does not contain any optimal solution. We present discarding tests which combine techniques from global single objective optimization with outer approximation techniques from multiobjective convex optimization and with the concept of local upper bounds from multiobjective combinatorial optimization. The new discarding tests aim to find appropriate lower bounds of subsets of the image set in order to compare them with known upper bounds numerically.
Dr. Jana Thomann
A trust region approach for multi-objective heterogeneous optimization
Published on: March 13, 2019, Link to pdf
This thesis presents a trust region approach for multi-objective optimization problems with heterogeneous objective functions. One of the objective functions is an expensive black-box function, not given analytically, but for example by a simulation. Computing function values is assumed to be time-consuming and derivative information is not available with reasonable effort. The other objective functions are assumed to be given analytically and function evaluations and derivatives are easily available with low numerical effort.
A basic algorithm for such optimization problems is presented. It is an iterative approach using local model functions and a search direction which is defined in the image space. The algorithm generates a sequence of iteration points. It is proved that the accumulation point of this sequence fulfills a necessary condition for local optimality. Moreover, several modifications of the basic algorithm are presented that make more use of the heterogeneity of the objective functions and partly produce several points as output.
Numerical results for the basic algorithm and several modifications are presented and discussed. They confirm the theoretical findings and show the usefulness of the approaches. Moreover, an application-motivated optimization problem from fluid dynamics is considered and the results are presented and interpreted according to the application.