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Bao, Truong Quang; Eichfelder, Gabriele; Soleimani, Behnam; Tammer, Christiane
Ekeland's variational principle for vector optimization with variable ordering structure. - In: Journal of convex analysis. - 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, 2017. - 1 Online-Ressource (20 Seiten). - (Preprint)

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 / Global Optimization Workshop ; 13 (Braga) : 2016.09.04-08. - Braga, Portugal : University of Minho (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, 2016. - 1 Online-Ressource (27 Seiten). - (Preprint)

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. - [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: Multiple criteria decision analysis. - New York : Springer (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
Eichfelder, Gabriele; Gerlach, Tobias; Sumi, Susanne
A modification of the [alpha]BB method for box-constrained optimization and an application to inverse kinematics. - In: EURO journal on computational optimization. - Berlin : Springer, ISSN 21924414, Bd. 4 (2016), 1, S. 93-121

For many practical applications it is important to determine not only a numerical approximation of one but a representation of the whole set of globally optimal solutions of a non-convex optimization problem. Then one element of this representation may be chosen based on additional information which cannot be formulated as a mathematical function or within a hierarchical problem formulation. We present such an application in the field of robotic design. This application problem can be modeled as a smooth box-constrained optimization problem. We extend the well-known alphaBB method such that it can be used to find an approximation of the set of globally optimal solutions with a predefined quality. We illustrate the properties and give a proof for the finiteness and correctness of our modified alphaBB method.


http://dx.doi.org/10.1007/s13675-015-0056-5
Brás, Carmo; Eichfelder, Gabriele; Júdice, Joaquim
Copositivity tests based on the linear complementarity problem. - In: Computational optimization and applications. - New York, NY [u.a.] : Springer Science + Business Media B.V, ISSN 15732894, Bd. 63 (2016), 2, S. 164-493

We present copositivity tests based on new necessary and sufficient conditions which require the solution of linear complementarity problems (LCP). We propose methodologies involving Lemkes method, an enumerative algorithm and a linear mixed-integer programming formulation to solve the required LCPs. Moreover, we discuss a new necessary condition for (strict) copositivity based on solving a linear program, which can be used as a preprocessing step. The algorithms with these three different variants are thoroughly applied to test matrices from the literature and to max-clique instances with matrices of order up to 496×496. We compare our procedures with three other copositivity tests from the literature as well as with a general global optimization solver. The numerical results are very promising and equally good and in many cases better than the results reported elsewhere.


http://dx.doi.org/10.1007/s10589-015-9772-2