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Eichfelder, Gabriele; Krüger, Corinna; Schöbel, Anita
Decision uncertainty in multiobjective optimization. - 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 15732916, Bd. 69 (2017), 2, S. 485-510

In many real-world optimization problems, a solution cannot be realized in practice exactly as computed, e.g., it may be impossible to produce a board of exactly 3.546 mm width. Whenever computed solutions are not realized exactly but in a perturbed way, we speak of decision uncertainty. We study decision uncertainty in multiobjective optimization problems and we propose the concept of decision robust efficiency for evaluating the robustness of a solution in this case. This solution concept is defined by using the framework of set-valued maps. We prove that convexity and continuity are preserved by the resulting set-valued maps. Moreover, we obtain specific results for particular classes of objective functions that are relevant for solving the set-valued problem. We furthermore prove that decision robust efficient solutions can be found by solving a deterministic problem in case of linear objective functions. We also investigate the relationship of the proposed concept to other concepts in the literature.


https://doi.org/10.1007/s10898-017-0518-9
Boeck, Thomas; Terzijska, Dzulia; Eichfelder, Gabriele
Maximum electromagnetic drag configurations for a translating conducting cylinder with distant magnetic dipoles. - In: Journal of engineering mathematics. - Dordrecht [u.a.] : Springer Science + Business Media B.V, ISSN 15732703, (2017), first online (22. Jul.), insges. 19 S.

We report a semianalytic and numerical investigation of the maximal induced Lorentz force on an electrically conducting cylinder in translation along its axis that is caused by the presence of multiple distant magnetic dipoles. The problem is motivated by Lorentz force velocimetry, where induction creates a drag force on a magnet system placed next to a conducting flow. The magnetic field should maximize this drag force, which is usually quite small. Our approach is based on a long-wave theory developed for a single distant magnetic dipole. We determine the optimal orientations of the dipole moments providing the strongest Lorentz force for different dipole configurations using numerical optimization methods. Different constraints are considered for dipoles arranged on a concentric circle in a plane perpendicular to the cylinder axis. In this case, the quadratic form for the force in terms of the dipole moments can be obtained analytically, and it resembles the expression of the energy in a classical spin model. When all dipoles are equal and their positions on the circle are not constrained, the maximal force results when all dipoles are gathered in one point with all dipole moments pointing in radial direction. When the dipoles are equal and have equidistant spacing on the circle, we find that the optimal orientations of the dipole moments approach a limiting distribution. It differs from the so-called Halbach distribution that provides a uniform magnetic field in the cross section of the cylinder. The corresponding force is about 10% larger than that for the Halbach distribution but 60% smaller than for the unconstrained dipole positions. With the so-called spherical constraint for a classical spin model, the maximal force can be found from the eigenvalues of the coefficient matrix. It is typically 10% larger than the maximal force for equal dipoles because the constraint is weaker. We also study equal and evenly spaced dipoles along one or two lines parallel to the cylinder axis. The patterns of optimal magnetic moment orientations are fairly similar for different dipole numbers when the inter-dipole distance is within a certain interval. This behavior can be explained by reference to the magnetic field distribution of a single distant dipole on the cylinder axis.


https://doi.org/10.1007/s10665-017-9916-8
Bao, Truong Quang; Eichfelder, Gabriele; Soleimani, Behnam; Tammer, Christiane
Ekeland's variational principle for vector optimization with variable ordering structure. - In: Journal of convex analysisJournal of convex analysis : an international scientific journal. - Lemgo : Heldermann - Lemgo : Heldermann, ISSN 09446532, Bd. 24 (2017), 2, S. 393-415

There are many generalizations of Ekeland's variational principle for vector optimization problems with fixed ordering structures, i.e., ordering cones. These variational principles are useful for deriving optimality conditions, [epsilon]-Kolmogorov conditions in approximation theory, and [epsilon]-maximum principles in optimal control. Here, we present several generalizations of Ekeland's variational principle for vector optimization problems with respect to variable ordering structures. For deriving these variational principles we use nonlinear scalarization techniques. Furthermore, we derive necessary conditions for approximate solutions of vector optimization problems with respect to variable ordering structures using these variational principles and the subdifferential calculus by Mordukhovich.


Hildenbrandt, Regina
The k-server problem with parallel requests and the compound work function algorithm. - Ilmenau : Technische Universität, Institut für Mathematik. - 1 Online-Ressource (20 Seiten). - (Preprint. - M17,04)

In this paper we consider k-server problems with parallel requests where several servers can also be located on one point. We will distinguish the surplussituation where the request can be completely fulfilled by means of the k servers and and the scarcity-situation where the request cannot be completely met. First, we will give an example. It shows that the corresponding work function algorithm is not competitive in the case of the scarcity-situation. Until now, it remains an open question whether the work function algorithm is competitive or not in the case of the surplus-situation. Thats why, we will suggest the new "compound work function algorithm" in the following section and prove that this algorithm is also (2 k - 1)-competitive.


https://www.db-thueringen.de/receive/dbt_mods_00031742
Niebling, Julia; Eichfelder, Gabriele
A branch-and-bound algorithm for bi-objective problems. - In: Proceedings of the XIII Global Optimization Workshop : GOW'16, 4-8 September 2016, University of Minho, Braga, Portugal. - Braga, Portugal : University of Minho / Global Optimization Workshop ; 13 (Braga) : 2016.09.04-08., ISBN 978-989-20-6764-3, (2016), S. 57-60

An improved discarding test for a branch-and-bound algorithm for box-constrained bi-objective optimization problems is presented. The aim of the algorithm is to compute a covering of all global optimal solutions. We introduce the algorithm which uses selection, discarding and termination tests. The discarding tests are the most important aspect, because they examine in different ways whether a box can contain optimal solutions. For this, we are using the alphaBB-method from global scalar optimization and present and discuss an improved test compared to those from the literature.


http://hdl.handle.net/1822/42944
Eichfelder, Gabriele; Pilecka, Maria
Set approach for set optimization with variable ordering structures : part II: scalarization approaches. - In: Journal of optimization theory and applications. - Dordrecht [u.a.] : Springer Science + Business Media, ISSN 15732878, Bd. 171 (2016), 3, S. 947-963
http://dx.doi.org/10.1007/s10957-016-0993-z
Eichfelder, Gabriele; Pilecka, Maria
Set approach for set optimization with variable ordering structures : part I: set relations and relationship to vector approach. - In: Journal of optimization theory and applications. - Dordrecht [u.a.] : Springer Science + Business Media, ISSN 15732878, Bd. 171 (2016), 3, S. 931-946
http://dx.doi.org/10.1007/s10957-016-0992-0
Eichfelder, Gabriele; Krüger, Corinna; Schöbel, Anita
Decision uncertainty in multiobjective optimization. - Ilmenau : Technische Universität, Institut für Mathematik. - 1 Online-Ressource (27 Seiten). - (Preprint. - M16,06)

In many real-world optimization problems, a solution cannot be realized in practice exactly as computed, e.g., it may be impossible to produce a board of exactly 3.546˜mm width. Whenever computed solutions are not realized exactly but in a perturbed way, we speak of decision uncertainty. We study decision uncertainty in multiobjective optimization problems and we propose the concept decision robust efficiency for evaluating the robustness of a solution in this case. Therefore, we address decision uncertainty within the framework of set-valued maps. First, we prove that convexity and continuity are preserved by the resulting set-valued mappings. Second, we obtain specific results for particular classes of objective functions that are relevant for solving the set-valued problem. We furthermore prove that decision robust efficient solutions can be found by solving a deterministic problem in case of linear objective functions. We also investigate the relationship of the proposed concept to other concepts in the literature.


https://www.db-thueringen.de/receive/dbt_mods_00030032
Hildenbrandt, Regina
The k-server problem with parallel requests and the compound Harmonic algorithm. - In: Baltic journal of modern computing : BJMC. - [S.l.], ISSN 22558950, Bd. 4 (2016), 3, S. 607-629

In this paper the (randomized) compound Harmonic 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 2000 Bartal and Grove have proved that the well-known Harmonic algorithm is competitive for the (usual) k-server problem. Unfortunately, certain techniques of this proof cannot be used to show that a natural generalization of the Harmonic algorithm is competitive for the problem with parallel requests. The probabilities, which are used by the compound Harmonic algorithm are, finally, 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 Harmonic algorithm is competitive with the bound of the ratio as which has been proved by Bartal and Grove in the case of the usual problem.


http://nbn-resolving.de/urn:nbn:de:gbv:ilm1-2016200094
Eichfelder, Gabriele; Jahn, Johannes
Vector and set optimization. - In: , ISBN 978-1-4939-3094-4, (2016), S. 695-737

This chapter is devoted to recent developments of vector and set optimization. Based on the concept of a pre-order optimal elements are defined. In vector optimization properties of optimal elements and existence results are gained. Further, an introduction to vector optimization with a variable ordering structure is given. In set optimization basic concepts are summed up.


http://dx.doi.org/10.1007/978-1-4939-3094-4_17