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Prinz, Sebastian; Thomann, Jana; Eichfelder, Gabriele; Boeck, Thomas; Schumacher, Jörg;
Expensive multi-objective optimization of electromagnetic mixing in a liquid metal. - In: Optimization and engineering : international multidisciplinary journal to promote optimizational theory & applications in engineering science.. - Dordrecht [u.a.] : Springer Science + Business Media B.V, ISSN 1573-2924, (2020), insges. 25 S.
- Published: 07 October 2020

This paper presents a novel trust-region method for the optimization of multiple expensive functions. We apply this method to a biobjective optimization problem in fluid mechanics, the optimal mixing of particles in a flow in a closed container. The three-dimensional time-dependent flows are driven by Lorentz forces that are generated by an oscillating permanent magnet located underneath the rectangular vessel. The rectangular magnet provides a spatially non-uniform magnetic field that is known analytically. The magnet oscillation creates a steady mean flow (steady streaming) similar to those observed from oscillating rigid bodies. In the optimization problem, randomly distributed mass-less particles are advected by the flow to achieve a homogeneous distribution (objective function 1) while keeping the work done to move the permanent magnet minimal (objective function 2). A single evaluation of these two objective functions may take more than two hours. For that reason, to save computational time, the proposed method uses interpolation models on trust-regions for finding descent directions. We show that, even for our significantly simplified model problem, the mixing patterns vary significantly with the control parameters, which justifies the use of improved optimization techniques and their further development.



https://doi.org/10.1007/s11081-020-09561-4
Fern, Florian; Füßl, Roland; Eichfelder, Gabriele; Manske, Eberhard; Kühnel, Michael;
Coordinate transformation and its uncertainty under consideration of a non orthogonal coordinate base. - In: Measurement science and technology. - Bristol : IOP Publ., ISSN 1361-6501, (2020), insges. 7 S.
- Accepted Manuscript online 8 July 2020

Nanopositioning and nanomeasuring machines are 3D coordinate measuring systems with nanometer precision at measurement volumes in the cubic centimeter range. The coordinate base is formed by an interferometer system with a common mirror corner. The orthogonality deviations of the mirror corner require a coordinate transformation of the measuring axes. The uncertainty of the coordinate transformation must be taken into account in the overall measurement uncertainty budget. Starting from a complete transformation model, the result of model simplications on the transformation behaviour is analysed and discussed.



https://doi.org/10.1088/1361-6501/aba3f5
Baier, Robert; Eichfelder, Gabriele; Gerlach, Tobias;
Optimality conditions for set optimization using a directional derivative based on generalized Steiner sets. - In: Optimization. - London [u.a.] : Taylor & Francis, ISSN 1029-4945, (2020), insges. 42 S.
- Published online: 14 Sep 2020

Set-optimization has attracted increasing interest in the last years, as for instance uncertain multiobjective optimization problems lead to such problems with a set-valued objective function. Thereby, from a practical point of view, most of all the so-called set approach is of interest. However, optimality conditions for these problems, for instance using directional derivatives, are still very limited. The key aspect for a useful directional derivative is the definition of a useful set difference for the evaluation of the numerator in the difference quotient. We present here a new set difference which avoids the use of a convex hull and which applies to arbitrary convex sets, and not to strictly convex sets only. The new set difference is based on the new concept of generalized Steiner sets. We introduce the Banach space of generalized Steiner sets as well as anembedding of convex sets in this space using Steiner points.In this Banach space we can easily define a difference and a directional derivative. We use the latter for new optimality conditions for set optimization. Numerical examples illustrate the new concepts.



https://doi.org/10.1080/02331934.2020.1812605
Hildenbrandt, Regina;
The k-server problem with parallel requests and the compound work function algorithm. - In: Baltic journal of modern computing : BJMC.. - [S.l.], ISSN 2255-8950, Bd. 8 (2020), 1, S. 1-20

In this paper the compound work function algorithm for solving the generalized k-server problem is proposed. This problem is an online k-server problem with parallel requests where several servers can also be located on one point. In 1995 Koutsoupias and Papadimitriouhave proved that the well-known work function algorithm is competitive for the (usual) k-server problem. A proof, where a potential-like function argument is included, was given by Borodinand El-Yaniv in 1998. Unfortunately, certain techniques of these proofs cannot be applied to show that a natural generalization of the work function algorithm is competitive for the problem with parallel requests. Values of work functions, which are used by the compound work function algorithm are derived from a surrogate problem, where at most one server must be moved in servicing the request in each step. We can show that the compound work function algorithm is competitive with the same bound of the ratio as in the case of the usual problem.



https://doi.org/10.22364/bjmc.2020.8.1.01
Rocktäschel, Stefan;
A branch-and-bound algorithm for multiobjective mixed-integer convex optimization. - Wiesbaden : Springer Spektrum, 2020. - VIII, 70 Seiten. . - (BestMasters) ISBN 978-3-658-29148-8

Eichfelder, Gabriele; Niebling, Julia; Rocktäschel, Stefan;
An algorithmic approach to multiobjective optimization with decision uncertainty. - In: Journal of global optimization : an international journal dealing with theoretical and computational aspects of seeking global optima and their applications in science, management and engineering.. - Dordrecht [u.a.] : Springer Science + Business Media B.V, ISSN 1573-2916, Bd. 77 (2020), 1, S. 3-25

In real life applications, optimization problems with more than one objective function are often of interest. Next to handling multiple objective functions, another challenge is to deal with uncertainties concerning the realization of the decision variables. One approach to handle these uncertainties is to consider the objectives as set-valued functions. Hence, the image of one decision variable is a whole set, which includes all possible outcomes of this decision variable. We choose a robust approach and thus these sets have to be compared using the so-called upper-type less order relation. We propose a numerical method to calculate a covering of the set of optimal solutions of such an uncertain multiobjective optimization problem. We use a branch-and-bound approach and lower and upper bound sets for being able to compare the arising sets. The calculation of these lower and upper bound sets uses techniques known from global optimization, as convex underestimators, as well as techniques used in convex multiobjective optimization as outer approximation techniques. We also give first numerical results for this algorithm.



https://doi.org/10.1007/s10898-019-00815-9
Niebling, Julia;
Nonconvex and mixed integer multiobjective optimization with an application to decision uncertainty. - Ilmenau : Universitätsbibliothek, 2019. - 1 Online-Ressource (iii, 163, XXXV Seiten).
Technische Universität Ilmenau, Dissertation 2019

Multikriterielle Optimierungprobleme sind in diversen Anwendungsgebieten wie beispielsweise in den Wirtschafts- oder Ingenieurwissenschaften zu finden. Da hierbei mehrere konkurrierende Zielfunktionen auftreten, ist die Lösungsmenge eines derartigen Optimierungsproblems im Allgemeinen unendlich groß und kann meist nicht in analytischer Form berechnet werden. In dieser Dissertation werden neue Branch-and-Bound basierte Algorithmen zur Lösung verschiedener Klassen von multikriteriellen Optimierungsproblemen entwickelt und vorgestellt. Der Branch-and-Bound Ansatz ist eine typische Methode der globalen Optimierung. Einer der neuen Algorithmen löst glatte multikriterielle nichtkonvexe Optimierungsprobleme mit konvexen Nebenbedingungen, während ein zweiter zur Lösung multikriterieller gemischt-ganzzahliger konvexer Optimierungsprobleme dient. Beide Algorithmen garantieren eine gewisse Genauigkeit der berechneten Lösungen und gehören damit zu den ersten deterministischen Algorithmen ihrer Art. Zusätzlich wird ein Algorithmus zur Berechnung einer Überdeckung der Lösungsmenge multikriterieller Optimierungsprobleme mit Entscheidungsunsicherheit vorgestellt. Alle drei Algorithmen wurden numerisch getestet. Die Ergebnisse werden ebenfalls in dieser Arbeit ausgewertet. Die neuen Algorithmen arbeiten alle mit Boxunterteilungen und nutzen Auswahlregeln, sowie Verwerfungs- und Terminierungskriterien. Dabei spielen gute Verwerfungskriterien eine zentrale Rolle. Diese entscheiden, ob eine Box verworfen werden kann, da diese sicher keine Optimallösung enthält. Die neuen Verwerfungskriterien nutzen Methoden aus der globalen skalarwertigen Optimierung, Approximationstechniken aus der multikriteriellen konvexen Optimierung sowie ein Konzept aus der kombinatorischen Optimierung. Dabei werden stets untere Schranken der Bildmengen konstruiert, die mit bisher berechneten oberen Schranken numerisch verglichen werden können.



https://www.db-thueringen.de/receive/dbt_mods_00040364
Thomann, Jana; Eichfelder, Gabriele;
Representation of the Pareto front for heterogeneous multi-objective optimization. - In: Journal of applied and numerical optimization. - [Edmonton] : Biemdas Academic Publishers, ISSN 2562-5535, Bd. 1 (2019), 3, S. 293-323

Optimization problems with multiple objectives which are expensive, i.e., where function evaluations are time consuming, are difficult to solve. Finding at least one locally optimal solution is already a difficult task. In case only one of the objective functions is expensive while the others are cheap, for instance, analytically given, this can be used in the optimization procedure. Using a trust-region approach and the Tammer-Weidner-functional for finding descent directions, in [19] an algorithm was proposed which makes use of the heterogeneity of the objective functions. In this paper, we present three heuristic approaches, which allow to find additional optimal solutions of the multiobjective optimization problem and by that representations at least of parts of the Pareto front. We present the related theoretical results as well as numerical results on some test instances.



https://doi.org/10.23952/jano.1.2019.3.08
Baier, Robert; Eichfelder, Gabriele; Gerlach, Tobias;
Optimality conditions for set optimization using a directional derivative based on generalized Steiner sets. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2019. - 1 Online-Ressource (40 Seiten). . - (Preprint. - M19,09)

Set-optimization has attracted increasing interest in the last years, as for instance uncertain multiobjective optimization problems lead to such problems with a set- valued objective function. Thereby, from a practical point of view, most of all the so-called set approach is of interest. However, optimality conditions for these problems, for instance using directional derivatives, are still very limited. The key aspect for a useful directional derivative is the definition of a useful set difference for the evaluation of the numerator in the difference quotient. We present here a new set difference which avoids the use of a convex hull and which applies to arbitrary convex sets, and not to strictly convex sets only. The new set difference is based on the new concept of generalized Steiner sets. We introduce the Banach space of generalized Steiner sets as well as an embedding of convex sets in this space using Steiner points. In this Banach space we can easily define a difference and a directional derivative. We use the latter for new optimality conditions for set optimization. Numerical examples illustrate the new concepts.



https://www.db-thueringen.de/receive/dbt_mods_00040057
Eichfelder, Gabriele; Gerlach, Tobias;
On classes of set optimization problems which are reducible to vector optimization problems and its impact on numerical test instances. - In: Variational analysis and set optimization. - Boca Raton : CRC Press, Taylor & Francis Group, (2019), S. 265-290

Set optimization with the set approach has recently gained increasing interest due to its practical relevance. In this problem class one studies optimization problems with a set-valued objective map and defines optimality based on a direct comparison of the images of the objective function, which are sets here. Meanwhile, in the literature a wide range of theoretical tools as scalarization approaches and derivative concepts as well as first numerical algorithms are available. These numerical algorithms require on the one hand test instances where the optimal solution sets are known. On the other hand, in most examples and test instances in the literature only set-valued maps with a very simple structure are used. We study in this paper such special set-valued maps and we show that some of them are such simple that they can equivalently be expressed as a vector optimization problem. Thus we try to start drawing a line between simple set-valued problems and such problems which have no representation as multiobjective problems. Those having a representation can be used for defining test instances for numerical algorithms with easy verifiable optimal solution set.