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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, (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.



Thomann, Jana; Eichfelder, Gabriele;
A trust-region algorithm for heterogeneous multiobjective optimization. - In: SIAM journal on optimization. - Philadelphia, Pa. : SIAM, ISSN 1095-7189, Bd. 29 (2019), 2, S. 1017-1047

https://doi.org/10.1137/18M1173277
De Santis, Marianna; Eichfelder, Gabriele; Niebling, Julia; Rocktäschel, Stefan;
Solving multiobjective mixed integer convex optimization problems. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2019. - 1 Online-Ressource (26 Seiten). . - (Preprint. - M19,06)

Multiobjective mixed integer convex optimization refers to mathematical programming problems where more than one convex objective function needs to be optimized simultaneously and some of the variables are constrained to take integer values. We present a branch-and-bound method based on the use of properly defined lower bounds. We do not simply rely on convex relaxations, but we built linear outer approximations of the image set in an adaptive way. We are able to guarantee correctness in terms of detecting both the efficient and the nondominated set of multiobjective mixed integer convex problems according to a prescribed precision. As far as we know, the procedure we present is the first deterministic algorithm devised to handle this class of problems. Our numerical experiments show results on biobjective and triobjective mixed integer convex instances.



https://www.db-thueringen.de/receive/dbt_mods_00038620
Thomann, Jana; Eichfelder, Gabriele;
Numerical results for the multiobjective trust region algorithm MHT. - In: Data in Brief. - Amsterdam [u.a.] : Elsevier, ISSN 2352-3409, Volume 25 (2019), article 104103, Seite 1-18

https://doi.org/10.1016/j.dib.2019.104103
Niebling, Julia; Eichfelder, Gabriele;
A branch-and-bound-based algorithm for nonconvex multiobjective optimization. - In: SIAM journal on optimization. - Philadelphia, Pa. : SIAM, ISSN 1095-7189, Bd. 29 (2019), 1, S. 794-821

https://doi.org/10.1137/18M1169680
Thomann, Jana;
A trust region approach for multi-objective heterogeneous optimization. - Ilmenau : Universitätsbibliothek, 2019. - 1 Online-Ressource (iii, 202, XLI Seiten).
Technische Universität Ilmenau, Dissertation 2019

In dieser Arbeit wird ein "Trust-Region" Algorithmus für multikriterielle Optimierungsprobleme mit heterogenen Zielfunktionen vorgestellt. Eine der Zielfunktionen ist eine teure Black-Box-Funktion. Sie ist nicht analytisch gegeben, sondern beispielsweise durch eine Simulation. Für diese Funktion wird angenommen, dass die Berechnung von Funktionswerten zeitaufwändig ist und die Ableitungen nicht mit vertretbarem numerischen Aufwand berechnet werden können. Des Weiteren wird vorausgesetzt, dass die anderen Zielfunktionen analytisch gegeben sind und die Berechnung von Funktionswerten und Ableitungen mit geringem numerischen Aufwand verbunden ist. Es wird ein grundlegender Algorithmus für derartige Optimierungsprobleme vorgestellt. Der Ansatz ist iterativ und nutzt lokale Modellfunktionen und eine im Bildraum definierte Suchrichtung. Der Algorithmus erzeugt eine Folge von Iterationspunkten. Es wird bewiesen, dass der Häufungspunkt dieser Folge ein notwendiges lokales Optimalitätskriterium erfüllt. Darüber hinaus werden verschiedene Modifikationen dieses Algorithmus vorgestellt, welche die Heterogenität der Zielfunktionen weiter nutzen und teilweise mehr als einen Punkt als Ausgabe erzeugen. Des Weiteren werden Ergebnisse von numerischen Tests mit der Grundversion und einigen Modifikationen des Algorithmus präsentiert und diskutiert. Sie bestätigen die theoretischen Resultate und zeigen die Nützlichkeit der Verfahren. Der grundlegende Algorithmus wurde außerdem auf ein Anwendungsproblem der Fluiddynamik angewandt. Die zugehörigen Ergebnisse werden präsentiert und im Rahmen des Anwendungsproblems interpretiert.



https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2019000059
Eichfelder, Gabriele; Hotz, Thomas; Wieditz, Johannes;
An algorithm for computing Fréchet means on the sphere. - In: Optimization letters. - Berlin : Springer, ISSN 1862-4480, Bd. 13 (2019), 7, S. 1523-1533

For most optimisation methods an essential assumption is the vector space structure of the feasible set. This condition is not fulfilled if we consider optimisation problems over the sphere. We present an algorithm for solving a special global problem over the sphere, namely the determination of Fréchet means, which are points minimising the mean distance to a given set of points. The Branch and Bound method derived needs no further assumptions on the input data, but is able to cope with this objective function which is neither convex nor differentiable. The algorithms performance is tested on simulated and real data.



https://doi.org/10.1007/s11590-019-01415-y
Eichfelder, Gabriele; Klamroth, Kathrin; Niebling, Julia;
Using a B&B algorithm from multiobjective optimization to solve constrained optimization problems. - In: AIP conference proceedings. - Melville, NY : Inst, ISSN 15517616, Bd. 2070 (2019), S. 020028, insges. 4 S.

https://doi.org/10.1063/1.5089995
Eichfelder, Gabriele; Niebling, Julia; Rocktäschel, Stefan;
An algorithmic approach to multiobjective optimization with decision uncertainty. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2018. - 1 Online-Ressource (23 Seiten). . - (Preprint. - M18,11)

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 variable is a whole set, which includes all possible outcomes of this 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 branchand-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://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2018200159
Eichfelder, Gabriele; Pilecka, Maria;
Ordering structures and their applications. - In: Applications of Nonlinear Analysis. - Cham : Springer, (2018), S. 265-304

Ordering structures play a fundamental role in many mathematical areas. These include important topics in optimization theory such as vector optimization and set optimization, but also other subjects as decision theory use ordering structures as well. Due to strong connections between ordering structures and cones in the considered space, order theory is also used every time two elements of a space, which is more general than the real line, are compared with each other. Therefore, also cone programming possessing restrictions defined using cones, and especially semidefinite optimization where the variables are symmetric matrices, make use of ordering structures. These structures may, on the one hand, be independent of the considered element of a given space or, on the other hand, vary for each element of this space. In the last case, we speak of variable ordering structures, which is one of the important topics in the newest research on vector optimization.



https://doi.org/10.1007/978-3-319-89815-5_9
Hildenbrandt, Regina;
The k-server problem with parallel requests and the corresponding generalized paging problem. - In: Operations research proceedings 2017. - Cham, Switzerland : Springer, (2018), S. 205-211

In the present paper we give a frst summary of "competitive" algorithms for solving the "k-server problems with parallel requests" or the generalized paging problem.



Thomann, Jana; Eichfelder, Gabriele;
A trust region algorithm for heterogeneous multiobjective optimization. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2018. - 1 Online-Ressource (30 Seiten). . - (Preprint. - M18,04)
http://nbn-resolving.de/urn:nbn:de:gbv:ilm1-2018200043
Niebling, Julia; Eichfelder, Gabriele;
A branch-and-bound based algorithm for nonconvex multiobjective optimization. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2018. - 1 Online-Ressource (29 Seiten). . - (Preprint. - M18,03)
http://nbn-resolving.de/urn:nbn:de:gbv:ilm1-2018200024
Eichfelder, Gabriele; Gerlach, Tobias;
On classes of set optimization problems which are reducible to vector optimization problems and its impact on numerical test instances. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2018. - 1 Online-Ressource (25 Seiten). . - (Preprint. - M18,01)

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.



http://nbn-resolving.de/urn:nbn:de:gbv:ilm1-2017200588
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 1573-2703, Bd. 108 (2018), 1, S. 123-141

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
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 1573-2916, 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
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 0944-6532, 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. - 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, 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 1573-2878, 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 1573-2878, 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. - 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 2255-8950, 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 : state of the art surveys.. - 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 2192-4414, 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 : an international journal.. - New York, NY [u.a.] : Springer Science + Business Media B.V., ISSN 1573-2894, 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
Eichfelder, Gabriele; Gerlach, Tobias;
Characterization of properly optimal elements with variable ordering structures. - In: Optimization : a journal of mathematical programming and operations research.. - London [u.a.] : Taylor & Francis, ISSN 1029-4945, Bd. 65 (2016), 3, S. 571-588

In vector optimization with a variable ordering structure, the partial ordering defined by a convex cone is replaced by a whole family of convex cones, one associated with each element of the space. In recent publications, it was started to develop a comprehensive theory for these vector optimization problems. Thereby, also notions of proper efficiency were generalized to variable ordering structures. In this paper, we study the relation between several types of proper optimality. We give scalarization results based on new functionals defined by elements from the dual cones which allow complete characterizations also in the nonconvex case.



http://dx.doi.org/10.1080/02331934.2015.1040793
Eichfelder, Gabriele;
Variable ordering structures - what can be assumed?. - In: Dagstuhl Reports : Dokumentationen zu Dagstuhl-Seminaren und Dagstuhl-Perspektiven-Workshops.. - Wadern : Schloss Dagstuhl, ISSN 2192-5283, Bd. 5 (2015), 1, S. 102

https://doi.org/10.22032/dbt.42350
Eichfelder, Gabriele; Gandibleux, Xavier; Geiger, Martin Josef; Jahn, Johannes; Jaszkiewicz, Andrzej; Knowles, Joshua; Shukla, Pradyumn Kumar; Trautmann, Heike; Wessing, Simon;
Heterogeneous functions (WG3). - In: Dagstuhl Reports : Dokumentationen zu Dagstuhl-Seminaren und Dagstuhl-Perspektiven-Workshops.. - Wadern : Schloss Dagstuhl, ISSN 2192-5283, Bd. 5 (2015), 1, S. 121-129
- Aus: Understanding Complexity in Multiobjective Optimization (Dagstuhl Seminar 15031) S. 96-163

https://doi.org/10.22032/dbt.42198
Dempe, Stephan; Eichfelder, Gabriele; Fliege, Jörg
On the effects of combining objectives in multi-objective optimization. - In: Mathematical methods of operations research : ZOR.. - Berlin : Springer, ISSN 1432-5217, Bd. 82 (2015), 1, S. 1-18

In multi-objective optimization, one considers optimization problems with more than one objective function, and in general these objectives conflict each other. As the solution set of a multi-objective problem is often rather large and contains points of no interest to the decision-maker, strategies are sought that reduce the size of the solution set. One such strategy is to combine several objectives with each other, i.e. by summing them up, before employing tools to solve the resulting multi-objective optimization problem. This approach can be used to reduce the dimensionality of the objective space as well as to discard certain unwanted solutions, especially the 'extreme' ones found by minimizing just one of the objectives given in the classical sense while disregarding all others. In this paper, we discuss in detail how the strategy of combining objectives linearly influences the set of optimal, i.e. efficient solutions.



http://dx.doi.org/10.1007/s00186-015-0501-5
Eichfelder, Gabriele; Gerlach, Tobias; Sumi, Susanne
A modification of the [alpha]BB method for box-constrained optimization and an application to inverse kinematics. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2015. - Online-Ressource (PDF-Datei: 25 S., 1,58 MB). . - (Preprint. - M15,04)

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. For determining a representation of the global optimal solution set with a predefined quality we modify the well known BB method. We illustrate the properties and give a proof for the finiteness and correctness of our modified BB method.



http://www.db-thueringen.de/servlets/DocumentServlet?id=26001
Eichfelder, Gabriele;
Vector optimization in medical engineering. - In: Mathematics without boundaries. - New York [u.a.] : Springer, ISBN 978-1-4939-1123-3, (2014), S. 181-215

This chapter is on the theory and numerical procedures of vector optimization w.r.t. various ordering structures, on recent developments in this area and, most important, on their application to medical engineering. In vector optimization one considers optimization problems with a vector-valued objective map and thus one has to compare elements in a linear space. If the linear space is the finite dimensional space Rm this can be done componentwise. That corresponds to the notion of an EdgeworthPareto optimal solution of a multiobjective optimization problem. Among the multitude of applications which can be modeled by such a multiobjective optimization problem, we present an application in intensity modulated radiation therapy and its solution by a numerical procedure. In case the linear space is arbitrary, maybe infinite dimensional, one may introduce a partial ordering which defines how elements are compared. Such problems arise for instance in magnetic resonance tomography where the number of Hermitian matrices which have to be considered for a control of the maximum local specific absorption rate can be reduced by applying procedures from vector optimization. In addition to a short introduction and the application problem, we present a numerical solution method for solving such vector optimization problems. A partial ordering can be represented by a convex cone which describes the set of directions in which one assumes that the current values are deteriorated. If one assumes that this set may vary dependently on the actually considered element in the linear space, one may replace the partial ordering by a variable ordering structure. This was for instance done in an application in medical image registration. We present a possibility of how to model such variable ordering structures mathematically and how optimality can be defined in such a case. We also give a numerical solution method for the case of a finite set of alternatives.



Eichfelder, Gabriele; Pilecka, Maria
Set approach for set optimization with variable ordering structures. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2014. - Online-Ressource (PDF-Datei: 37 S., 434,7 KB). . - (Preprint. - M14,11)

This paper aims at combining variable ordering structures with set relations in set optimization, which have been dened using the constant ordering cone before. Since the purpose is to connect these two important approaches in set optimization, we do not restrict our considerations to one certain relation. Conversely, we provide the reader with many new variable set relations generalizing the relations from [16, 25] and discuss their usefulness. After analyzing the properties of the introduced relations, we dene new solution notions for set-valued optimization problems equipped with variable ordering structures and compare them with other concepts from the literature. In order to characterize the introduced solutions a nonlinear scalarization approach is used.



http://www.db-thueringen.de/servlets/DocumentServlet?id=25344
Eichfelder, Gabriele; Kasimbeyli, Refail
Properly optimal elements in vector optimization with variable ordering structures. - 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. 60 (2014), 4, S. 689-712

http://dx.doi.org/10.1007/s10898-013-0132-4
Terzijska, Dzulia; Porcelli, Margherita; Eichfelder, Gabriele
Multi-objective optimization in the Lorentz force velocimetry framework. - In: Book of digests & program. - Delft, (2014), S. 81-82

Bao, Truong Q.; Eichfelder, Gabriele; Soleimani, Behnam; Tammer, Christiane
Ekeland's variational principle for vector optimization with variable ordering structure. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2014. - Online-Ressource (PDF-Datei: 24 S., 351 KB). . - (Preprint. - M14,08)

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.



http://www.db-thueringen.de/servlets/DocumentServlet?id=24755
Hildenbrandt, Regina;
The k-server problem with parallel requests and the compound Harmonic algorithm. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2014. - Online-Ressource (PDF-Datei: 21 S., 163,8 KB). . - (Preprint. - M14,06)

In this paper we consider a generalized k-server problem with parallel requests where several servers can also be located on one point (which was initiated by an operations research problem). In section 4 the ''compound Harmonic algorithm'' for the generalized k-server problem is presented. Certain multi-step transition probabilities and absorbing probabilities are used by the compound Harmonic algorithm. For their computation one step of the generalized k-server problem is replaced by a number of steps of other (generalized) specific k-server problems. We show that this algorithm is competitive against an adaptive online adversary. In the case of unit distances the Harmonic algorithm and the compound Harmonic algorithm are identical.



http://www.db-thueringen.de/servlets/DocumentServlet?id=24668
Brás, Carmo; Eichfelder, Gabriele; Júdice, Joaquim
Copositivity tests based on the linear complementarity problem. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2014. - Online-Ressource (PDF-Datei: 28 S., 443 KB). . - (Preprint. - M14,05)

Copositivity tests are presented based on new necessary and suffcient conditions requiring the solution of linear complementarity problems (LCP). Methodologies involving Lemke's method, an enumerative algorithm and a linear mixed-integer programming formulation are proposed to solve the required LCPs. A new necessary condition for (strict) copositivity based on solving a Linear Program (LP) is also discussed, 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 up to dimension 496 x 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://www.db-thueringen.de/servlets/DocumentServlet?id=24543
Dempe, Stephan; Eichfelder, Gabriele; Fliege, Jörg
On the effects of combining objectives in multi-objective optimization. - Freiberg : TU Bergakad., 2014. - 15 Bl.. . - (Preprint. - 2014,04)

In multi-objective optimization, one considers optimization problems with more than one objective function, and in general these objectives conflict each other. As the solution set of a multiobjective problem is often rather large and contains points of no interest to the decision-maker, strategies are sought that reduce the size of the solution set. One such strategy is to combine several objectives with each other, i.e. by summing them up, before employing tools to solve the resulting multiobjective optimization problem. This approach can be used to reduce the dimensionality of the solution set as well as to discarde certain unwanted solutions, especially the 'extreme' ones found by minimizing just one of the objectives given in the classical sense while disregarding all others. In this paper, we discuss in detail how the strategy of combining objectives linearly influences the set of optimal, i.e. efficient solutions.



Klöppel, Michael;
Efficient numerical solution of chance constrained optimization problems with engineering applications, 2014. - Online-Ressource (PDF-Datei: VII, 106 S., 4,30 MB). : Ilmenau, Techn. Univ., Diss., 2014
- Parallel als Druckausg. erschienen

In der Praxis werden viele Prozesse durch Unsicherheiten beeinflusst. Die Auswirkungen dieser Unsicherheiten können dabei beträchtlich sein. Es ist daher sinnvoll diese Einflüsse bei der Prozessoptimierung zu betrachten. Ein Ansatz dazu ist die Nutzung der wahrscheinlichkeitsrestringierten Optimierung. Diese erfordert die Einhaltung der Nebenbedingungen nur mit einer gewissen Wahrscheinlichkeit und erlaubt damit einen Kompromiss zwischen Profit und Zuverlässigkeit. In Abhängigkeit des unterliegenden Prozesses sind mehrere Ansätze zur Umwandlung der Wahrscheinlichkeitsrestriktionen in deterministische Restriktionen möglich. Die meisten dieser Ansätze basieren auf der Berechnung hochdimensionaler Integrale. In dieser Arbeit werden entsprechende Methoden zur Berechnung solcher Integrale vorgestellt. Hauptaugenmerk liegt dabei immer auf einer möglichst effizienten numerischen Implementation. Hauptbestandteil der Arbeit ist dabei die Beschreibung von so genannten analytischen Approximationen, welche effizient für eine Vielzahl von Anwendungen eingesetzt werden können. Für diese Verfahren werden Methoden zur Berechnung der Gradienten entwickelt. Eine weitere Verringerung der Rechenzeit wird durch die effiziente Approximierung der unterliegenden Modellgleichungen erreicht. In Fallstudien aus dem Ingenieurbereich werden die analytischen Approximationen mit anderen Ansätzen verglichen. Dabei stellt sich heraus, dass diese Methoden als genereller Ansatz benutzt werden können, auch wenn andere Methoden zu leicht besseren Ergebnissen führen. Als größere Fallstudie wird eine Problem aus dem Bereich des optimalen Lastflusses gelöst. Hier zeigt sich, dass die vorgeschlagenen Ansätze bessere Ergebnisse liefern als die weithin benutzte Approximation mit normalverteilten Zufallsgrößen. Außerdem kann durch den Einsatz effizienter Methoden selbst dieses größere Beispiel in vernünftiger Rechenzeit gelöst werden.



http://www.db-thueringen.de/servlets/DocumentServlet?id=24239
Eichfelder, Gabriele;
Variable ordering structures in vector optimization. - Berlin : Springer, 2014. - xiii, 190 Seiten. . - (Vector optimization) - Description based upon print version of record
http://dx.doi.org/10.1007/978-3-642-54283-1

Hildenbrandt, Regina;
A k-server problem with parallel requests and unit distances. - In: Information processing letters : devoted to the rapid publication of short contributions to information processing.. - Amsterdam [u.a.] : Elsevier, ISSN 1872-6119, Bd. 114 (2014), 5, S. 239-246

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 surplus-situation where the request can be completely fulfilled by means of the k servers and the scarcity-situation where the request cannot be completely met. We use the method of the potential function by Bartal and Grove in order to prove that a corresponding Harmonic algorithm is competitive for the more general k-server problem in the case of unit distances. For this purpose we partition the set of points in relation to the online and offline servers' positions and then use detailed considerations related to sets of certain partitions.



http://dx.doi.org/10.1016/j.ipl.2013.12.011
Eichfelder, Gabriele; Gerlach, Tobias
Characterization of proper optimal elements with variable ordering structures. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2014. - Online-Ressource (PDF-Datei: 19 S., 192,7 KB). . - (Preprint. - M14,01)

In vector optimization with a variable ordering structure the partial ordering defined by a convex cone is replaced by a whole family of convex cones, one associated with each element of the space. As these vector optimization problems are not only of interest in applications but also mathematical challenging, in recent publications it was started to develop a comprehensive theory. In doing that also notions of proper efficiency where generalized to variable ordering structures. In this paper we study the relations between several types of proper optimality notions, among others based on local and global approximations of the considered sets. We give scalarization results based on new functionals defined by elements from the dual cones which allow characterizations also in the nonconvex case.



http://www.db-thueringen.de/servlets/DocumentServlet?id=23354
Eichfelder, Gabriele;
Numerical procedures in multiobjective optimization with variable ordering structures. - In: Journal of optimization theory and applications. - Dordrecht [u.a.] : Springer Science + Business Media, ISSN 1573-2878, Bd. 162 (2014), 2, S. 489-514

Multiobjective optimization problems with a variable ordering structure, instead of a partial ordering, have recently gained interest due to several applications. In the previous years, a basic theory has been developed for such problems. The binary relations of a variable ordering structure are defined by a cone-valued map that associates, with each element of the linear space R m, a pointed convex cone of dominated or preferred directions. The difficulty in the study of the variable ordering structures arises from the fact that the binary relations are in general not transitive. In this paper, we propose numerical approaches for solving such optimization problems. For continuous problems a method is presented using scalarization functionals, which allows the determination of an approximation of the infinite optimal solution set. For discrete problems the Jahn-Graef-Younes method, known from multiobjective optimization with a partial ordering, is adapted to allow the determination of all optimal elements with a reduced effort compared to a pairwise comparison.



http://dx.doi.org/10.1007/s10957-013-0267-y
Werner, Jürgen; Hillenbrand, Matthias; Zhao, Mingcheng; Sinzinger, Stefan
An optimization method for radial NURBS surfaces. - In: DGaO-Proceedings. - Erlangen-Nürnberg: Dt. Gesellschaft für angewandte Optik, ISSN 1614-8436, Bd. 114.2013, P54, insges. 2 S.

http://www.db-thueringen.de/servlets/DocumentServlet?id=23557
Hildenbrandt, Regina;
Partitions-requirements-matrices as optimal Markov kernels of special stochastic dynamic distance optimal partitioning problems. - In: International journal of pure and applied mathematics. - Sofia : Academic Publications, ISSN 13118080, Bd. 88 (2013), 2, S. 183-211

The Stochastic Dynamic Distance Optimal Partitioning problem (SDDP problem) is a complex Operations Research problem. The SDDP problem is based on an industrial problem, which contains an optimal conversion of machines. Partitions of integers as states of these stochastic dynamic programming problems involves combinatorial aspects of SDDP problems. Under the assumption of identical "basic costs" (in other words of "unit distances") and independent and identically distributed requirements we will show (in many cases) by means of combinatorial ideas that decisions for feasible states with least square sums of their parts are optimal solutions. Corresponding Markov kernels are called Partitions-Requirements-Matrices (PRMs). Optimal decisions of such problems can be used as approximate solutions of corresponding SDDP problems, in which the basic costs differ only slightly from each other or as starting decisions if corresponding SDDP problems are solved by iterative methods, such as the Howard algorithm.



Klöppel, Michaell; Gabash, Aouss; Geletu, Abebe; Li, Pu
Chance constrained optimal power flow with non-Gaussian distributed uncertain wind power generation. - In: 2013 12th International Conference on Environment and Electrical Engineering (EEEIC) : 5 - 8 May 2013, Wroclaw, Poland.. - Piscataway, NJ : IEEE, ISBN 978-1-4673-3060-2, (2013), S. 265-270

http://dx.doi.org/10.1109/EEEIC.2013.6549628
Dickinson, Peter J. C.; Eichfelder, Gabriele; Povh, Janez
Erratum to: On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets. - In: Optimization letters. - Berlin : Springer, ISSN 1862-4480, Bd. 7 (2013), 6, S. 1387-1397

In this paper, an erratum is provided to the article "On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets", published in Optim Lett, 2012. Due to precise observation of the first author, it has been found that the proof of Lemma 9 has a nontrivial gap, and consequently the main result (Theorem 10) is incorrect. In this erratum, we prove that Corollary 14 is still correct in the original setting while to fix the proof of Theorem 10 we need additional assumptions. We provide a list of different commonly used assumptions making this theorem to be true, and a new version of this theorem, which is now Theorem 17.



http://dx.doi.org/10.1007/s11590-013-0645-2
Eichfelder, Gabriele; Povh, Janez
On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets. - In: Optimization letters. - Berlin : Springer, ISSN 1862-4480, Bd. 7 (2013), 6, S. 1373-1386

In the paper we prove that any nonconvex quadratic problem over some set K R^n with additional linear and binary constraints can be rewritten as a linear problem over the cone, dual to the cone of K-semidefinite matrices. We show that when K is defined by one quadratic constraint or by one concave quadratic constraint and one linear inequality, then the resulting K-semidefinite problem is actually a semidefinite programming problem. This generalizes results obtained by Sturm and Zhang (Math Oper Res 28:246-267, 2003). Our result also generalizes thewell-known completely positive representation result from Burer (Math Program 120:479-495, 2009), which is actually a special instance of our result with K = R^n_+.



http://dx.doi.org/10.1007/s11590-012-0450-3
Hildenbrandt, Regina;
A new competitive ratio of the Harmonic algorithm for a k-server problem with parallel requests and unit distances. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2013. - Online-Ressource (PDF-Datei: 15 S., 152,8 KB). . - (Preprint. - M13,10)
http://www.db-thueringen.de/servlets/DocumentServlet?id=22488
Lorenz, Pierre; Klöppel, Michaell; Frost, Frank; Ehrhardt, Martin; Zimmer, Klaus; Li, Pu
Laser-induced circular nanostructures in fused silica assisted by a self-assembling chromium layer. - In: Applied surface science : a journal devoted to applied physics and chemistry of surfaces and interfaces.. - Amsterdam : Elsevier, Bd. 280 (2013), S. 933-939

http://dx.doi.org/10.1016/j.apsusc.2013.05.102
Eichfelder, Gabriele; Ha, Truong Xuan Duc
Optimality conditions for vector optimization problems with variable ordering structures. - In: Optimization : a journal of mathematical programming and operations research.. - London [u.a.] : Taylor & Francis, ISSN 1029-4945, Bd. 62 (2013), 5, S. 597-627

Our main concern in this article are concepts of nondominatedness w.r.t. a variable ordering structure introduced by Yu [P.L. Yu, Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives, J. Optim. Theory Appl. 14 (1974), pp. 319-377]. Our studies are motivated by some recent applications e.g. in medical image registration. Restricting ourselves to the case when the values of a cone-valued map defining the ordering structure are Bishop-Phelps cones, we obtain for the first time scalarizing functionals for nondominated elements, Fermat rule, Lagrange multiplier rule and duality results for a single- or set-valued vector optimization problem with a variable ordering structure.



http://dx.doi.org/10.1080/02331934.2011.575939
Geletu, Abebe; Klöppel, Michaell; Zhang, Hui; Li, Pu
Advances and applications of chance-constrained approaches to systems optimisation under uncertainty. - In: International journal of systems science. - London [u.a.] : Taylor & Francis, ISSN 1464-5319, Bd. 44 (2013), 7, S. 1209-1232

http://dx.doi.org/10.1080/00207721.2012.670310
Eichfelder, Gabriele; Kasimbeyli, Refail
Properly optimal elements in vector optimization with variable ordering structures. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2013. - Online-Ressource (PDF-Datei: 27 S., 311,4 KB). . - (Preprint. - M13,05)

In this paper, proper optimality concepts in vector optimization with variable ordering structures are introduced for the first time and characterization results via scalarizations are given. New type of scalarizing functionals are presented and their properties are discussed. The scalarization approach suggested in the paper does not require convexity and boundedness conditions.



http://www.db-thueringen.de/servlets/DocumentServlet?id=22044
Bomze, Immanuel M.; Eichfelder, Gabriele;
Copositivity detection by difference-of-convex decomposition and [omega]-subdivision. - In: Mathematical programming : Series A, Series B ; a publication of the Mathematical Programming Society.. - Berlin : Springer, ISSN 1436-4646, Bd. 138 (2013), 1/2, S. 365-400

We present three new copositivity tests based upon difference-of-convex (d.c.) decompositions, and combine them to a branch-and-bound algorithm of [omega]-subdivision type. The tests employ LP or convex QP techniques, but also can be used heuristically using appropriate test points. We also discuss the selection of efficient d.c. decompositions and propose some preprocessing ideas based on the spectral d.c. decomposition. We report on first numerical experience with this procedure which are very promising.



https://doi.org/10.1007/s10107-012-0543-x
Eichfelder, Gabriele;
Ordering structures in vector optimization and applications in medical engineering. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2013. - Online-Ressource (PDF-Datei: 33 S., 397,8 KB). . - (Preprint. - M13,01)

This manuscript is on the theory and numerical procedures of vector optimization w.r.t. various ordering structures, on recent developments in this area and, most important, on their application to medical engineering. In vector optimization one considers optimization problems with a vector-valued objective map and thus one has to compare elements in a linear space. If the linear space is the finite dimensional space R^m this can be done componentwise. That corresponds to the notion of an Edgeworth-Pareto-optimal solution of a multiobjective optimization problem. Among the multitude of applications which can be modeled by such a multiobjective optimization problem, we present an application in intensity modulated radiation therapy and its solution by a numerical procedure. In case the linear space is arbitrary, maybe infinite dimensional, one may introduce a partial ordering which defines how elements are compared. Such problems arise for instance in magnetic resonance tomography where the number of Hermitian matrices which have to be considered for a control of the maximum local specific absorption rate can be reduced by applying procedures from vector optimization. In addition to a short introduction and the application problem, we present a numerical solution method for solving such vector optimization problems. A partial ordering can be represented by a convex cone which describes the set of directions in which one assumes that the current values are deteriorated. If one assumes that this set may vary dependently on the actually considered element in the linear space, one may replace the partial ordering by a variable ordering structure. This was for instance done in an application in medical image registration. We present a possibility of how to model such variable ordering structures mathematically and how optimality can be defined in such a case. We also give a numerical solution method for the case of a finite set of alternatives.



http://www.db-thueringen.de/servlets/DocumentServlet?id=21535
Reinhardt, Rüdiger; , ; ,
Nichtlineare Optimierung : Theorie, Numerik und Experimente. - Berlin : Springer Spektrum, 2013. - X, 383 S.. ISBN 3-8274-2948-X
- Literaturverz. S. [371] - 375

Die Grundlagen zur nichtlinearen Optimierung insbesondere ausgewählte Verfahren der nichtlinearen Optimierung werden als Lehrbuch für Studenten dargestellt. Zu den Verfahren werden Experimente beschrieben, die anhand der von R. Reinhardt entwickelten und von A. Hoffmann erweiterten Lehrsoftware EdOptLab unter Matlab von jedem Leser nachvollzogen werden können. So kann der Leser die Verfahren der Optimierung selbst erleben. Das System EdOptlab wird kostenlos zur Verfügung gestellt.



Werner, Jürgen; Hillenbrand, Matthias; Hoffmann, Armin; Sinzinger, Stefan;
Automatic differentiation in the optimization of imaging optical systems. - In: Schedae informaticae. - Cracow : Institute of Computer Science, Jagiellonian University, ISSN 2083-8476, Bd. 21 (2012), S. 169-175

Automatic differentiation is an often superior alternative to numerical differentiation that is yet unregarded for calculating derivatives in the optimization of imaging optical systems. We show that it is between 8% and 34% faster than numerical differentiation with central difference when optimizing various optical systems.



http://dx.doi.org/10.4467/20838476SI.12.011.0821
Golembewski, René; Schäfer, Günter; Gerlach, Tobias
Efficient communication for large-scale robust image processing with smart camera devices. - In: IEEE Symposium on Computational Intelligence for Security and Defense Applications (CISDA), 2012 : 11 - 13 July 2012, Ottawa, Canada.. - Piscataway, NJ : IEEE, ISBN 978-1-4673-1417-6, (2012), insges. 8 S.

http://dx.doi.org/10.1109/CISDA.2012.6291526
Werner, Jürgen; Zhao, Mingcheng; Hillenbrand, Matthias; Sinzinger, Stefan
RBF-based optical surfaces. - In: DGaO-Proceedings. - Erlangen-Nürnberg: Dt. Gesellschaft für angewandte Optik, ISSN 1614-8436, Bd. 113.2012, P39, insges. 2 S.

Freeform optical surfaces offer additional degrees of freedom for designing imaging systems without rotational symmetry. This allows for a reduction in the number of optical elements, leading to more compact and lightweight systems, while at the same time improving the image quality. This also enables new areas of application. Commonly used representations for freeform surfaces are x-y-polynomials, Zernike polynomials and NURBS. Radial basis functions (RBF) have been used for many years e.g. in artificial neural networks and functional approximation and can also be used to describe optical surfaces. In this contribution we investigate properties specific to RBF-based optical surfaces and compare the performance of RBF-based surfaces to other representations in selected optical imaging systems. Interesting aspects include the dependency on the number of RBF that are summed to form the surface, the locality structure and its effects on optimization.



http://www.db-thueringen.de/servlets/DocumentServlet?id=21333
Eichfelder, Gabriele;
Cone-valued maps in optimization. - In: Applicable analysis : an international journal.. - London [u.a.] : Taylor & Francis, ISSN 1563-504X, Bd. 91 (2012), 10, S. 1831-1846

Cone-valued maps are special set-valued maps where the image sets are cones. Such maps play an important role in optimization, for instance in optimality conditions or in the context of Bishop-Phelps cones. In vector optimization with variable ordering structures, they have recently attracted even more interest. We show that classical concepts for set-valued maps as cone-convexity or monotonicity are not appropriate for characterizing cone-valued maps. For instance, every convex or monotone cone-valued map is a constant map. Similar results hold for cone-convexity, sublinearity, upper semicontinuity or the local Lipschitz property. Therefore, we also propose new concepts for cone-valued maps.



http://dx.doi.org/10.1080/00036811.2011.616499
Eichfelder, Gabriele; Jahn, Johannes
Vector optimization problems and their solution concepts. - In: Recent developments in vector optimization. - Berlin [u.a.] : Springer, ISBN 978-3-642-21113-3, (2012), S. 1-27

Eichfelder, Gabriele;
Variable ordering structures in vector optimization. - In: Recent developments in vector optimization. - Berlin [u.a.] : Springer, ISBN 978-3-642-21113-3, (2012), S. 95-126

Eichfelder, Gabriele;
Numerical procedures in multiobjective optimization with variable ordering structures. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2012. - Online-Ressource (PDF-Datei: 24 S., 365 KB). . - (Preprint. - M12,05)

Multiobjective optimization problems with a variable ordering structure instead of a partial ordering have recently gained interest due to several applications. In the last years a basic theory has been developed for such problems. The difficulty in their study arises from the fact that the binary relations of the variable ordering structure, which are defined by a cone-valued map which associates to each element of the image space a pointed convex cone of dominated or preferred directions, are in general not transitive. - In this paper we propose numerical approaches for solving such optimization problems. For continuous problems a method is presented using scalarization functionals which allows the determination of an approximation of the infinite optimal solution set. For discrete problems the Jahn-Graef-Younes method known from multiobjective optimization with a partial ordering is adapted to allow the determination of all optimal elements with a reduced effort compared to a pairwise comparison.



http://www.db-thueringen.de/servlets/DocumentServlet?id=20485
Wozniak, Sander; Gerlach, Tobias; Schäfer, Günter
Optimization-based secure multi-hop localization in wireless ad hoc networks. - In: 17th GI/ITG Conference on Communication in Distributed Systems (KiVS 2011) : March 8 - 11, 2011, Kiel, Germany.. - Wadern : Schloss Dagstuhl - Leibniz-Zentrum für Informatik, (2011), S. 182-187

http://dx.doi.org/10.4230/OASIcs.KiVS.2011.182
Hildenbrandt, Regina;
A k-server problem with parallel requests and unit distances. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2011. - Online-Ressource (PDF-Datei: 11 S., 207,5 KB). . - (Preprint. - M11,16)

In the paper a k-server problem with parallel requests where several servers can also be located on one point is considered. We show that a HARMONIC_p k-server algorithm is competitive against an adaptive online adversary in case of unit distances.



http://www.db-thueringen.de/servlets/DocumentServlet?id=19013
Geletu, Abebe; Hoffmann, Armin; Klöppel, Michael; Li, Pu
Monotony analysis and sparse-grid integration for nonlinear chance constrained process optimization. - In: Engineering optimization. - London : Taylor & Francis, ISSN 1029-0273, Bd. 43 (2011), 10, S. 1019-1041

http://dx.doi.org/10.1080/0305215X.2010.532552
Klöppel, Michael; Geletu, Abebe; Hoffmann, Armin; Li, Pu
Using sparse-grid methods to improve computation efficiency in solving dynamic nonlinear chance-constrained optimization problems. - In: Industrial & engineering chemistry research. - Columbus, Ohio : American Chemical Society, ISSN 1520-5045, Bd. 50 (2011), 9, S. 5693-5704

http://dx.doi.org/10.1021/ie102426w
Klöppel, Michael; Geletu, Abebe; Hoffmann, Armin; Li, Pu
Evaluating integration approaches to robust process optimization and control using chance constrained programming. - In: IEEE International Symposium on Computer-Aided Control System Design (CACSD), 2010 : Yokohama, Japan, 8 - 10 Sept. 2010 ; part of 2010 IEEE Multi-Conference on Systems and Control.. - Piscataway, NJ : IEEE, ISBN 978-1-4244-5355-9, (2010), S. 1079-1084

http://dx.doi.org/10.1109/CACSD.2010.5612699
Hildenbrandt, Regina;
DA stochastic dynamic programming, stochastic dynamic distance optimal partitioning problems and partitions-requirements-matrices
1. Aufl.. - Göttingen : Cuvillier, 2010. - 298 S.. - Literaturverz. S. 284 - 286

Wozniak, Sander; Gerlach, Tobias; Schaefer, Guenter
Secure multi-hop localization in wireless ad hoc networks. - In: Crossing borders within the ABC : automation, biomedical engineering and computer science ; proceedings ; 55. IWK Internationales Wissenschaftliches Kolloquium ; 13 - 17 September 2010.. - Ilmenau : Verl. ISLE, ISBN 978-3-938843-53-6, (2010), S. 761-768

http://www.db-thueringen.de/servlets/DocumentServlet?id=17364
Geletu, Abebe; Klöppel, Michael; Li, Pu; Hoffmann, Armin;
A moving horizon approach to a chance constrained nonlinear dynamic optimization problem. - In: IFAC Proceedings Volumes. - Amsterdam : Elsevier, ISSN 1474-6670, Bd. 42 (2009), 2, S. 103-107

In this paper, a chance constrained nonlinear dynamic optimization problem is considered, which will be investigated by using a moving horizon scheme. In each horizon, the chance constraints will be written (transformed) in terms of those (input) random variables with known probability distributions by using monotonicity relations. Some definitions and properties related to the required monotonicity properties are introduced. For the application problem considered these monotonicity properties hold automatically true. The chance constraints and their gradients are evaluated by computing multivariate normal integrals using direct numerical integration. Numerical experimentation results will also be reported.



https://doi.org/10.3182/20090506-3-SF-4003.00019
Böhme, Thomas; , ; ,
Ordered and linked chordal graphs. - In: Discussiones mathematicae. - Warsaw : De Gruyter Open, ISSN 2083-5892, Bd. 28 (2008), 2, S. 367-373

https://doi.org/10.7151/dmgt.1412
Hoffmann, Armin; , ; ,
Vektoranalysis, Integraltransformationen, Differenzialgleichungen, Stochastik - Theorie und Numerik. - Mathematik für Ingenieure ; 2. - München [u.a.] : Pearson Studium, 2006. - 828 S.. . - (Elektrotechnik) ISBN 3-8273-7114-7
- Literaturverz. S. 809 - 814

Der 2. Band behandelt in Teil I die mehrdimensionale Integralrechnung im Sinne von Riemann, Numerische Kubatur, Kurven und Oberflächenintegrale einschließlich der Integralsätze der Vektoranalysis. Nach einer kurzen Darlegung der Theorie einer komplexen Veränderlichen werden Fourierreihen und Integraltransformationen behandelt. Im Teil II werden sowohl die Theorie als auch die Numerik gewöhnlicher Differentialgleichungen ausgeführt. Teil III geht nach einer Einführung zur elementaren Theorie partieller Differentialgleichungen auf wichtige numerische Verfahren zur Lösung partieller Differentialgleichungen ein. Im Teil IV werden nach einer Darlegung der Maß- und Integrationstheorie im Sinne von Caratheodory die Grundzüge der Wahrscheinlichkeitsrechnung vorgestellt. Die Abbildungen, Algorithmen und Lösungen der Aufgaben sind auf der Companion-Website des Verlages abrufbar.



Hoffmann, Armin; Geletu W. Selassie, Abebe
On robustness of set-valued maps and marginal value functions. - In: Discussiones mathematicae. - Zielona Góra : Univ. of Zielona Góra Press, ISSN 1509-9407, Bd. 25 (2005), 1, S. 59-108

The ideas of robust sets, robust functions and robustness of general set valued maps were introduced by Chew & Zheng and further developed by Shi, Zheng, Zhuang , Phú, Hoffmann and Hichert to weaken up the semi-continuity requirements of certain global optimization algorithms. Robust analysis, along with measure theory, has well served as the basis for the integral global optimization method (IGOM) (Chew & Zheng ). Hence, we have attempted here to extend the robust analysis of Zheng et al. to that of robustness of set-valued maps with given structures and marginal value functions. We are also of strong conviction that, the results of our investigation could open a way to apply the IGOM for the numerical treatment of some class of parametric optimization problems, when global optima are required. - MSC 2000: 90C34, 49J52, 49J53, 90C31. - Keywords: robust set; robust function; robust set-valued map; marginal value function; piecewise lower (upper) semi-continuous; approximatable function; approximatable set-valued map; regularity condition; extended Mangasarian-Fromowitz constraint qualification.



Thierfelder, Jörg;
Eagle-Guide Nichtlineare Optimierung. - Leipzig : Ed. am Gutenbergplatz, 2005. - 83 S. . - (Eagle. - 21) ISBN 3937219218
- Hergestellt on demand

Hoffmann, Armin; , ; ,
Lineare Algebra, Analysis, Theorie und Numerik. - Mathematik für Ingenieure ; 1. - München [u.a.] : Pearson Studium, 2005. - 857 S.. . - (Elektrotechnik) ISBN 3-8273-7113-9
- Literaturverz. S. 841 - 845

Geletu W. Selassie, Abebe; Hoffmann, Armin
A conceptual method for solving generalized semi-infinite programming problems via global optimization by exact discontinuous penalization. - In: European journal of operational research : EJOR.. - Amsterdam [u.a.] : Elsevier, ISSN 0377-2217, Bd. 157 (2004), 1, S. 3-15

http://dx.doi.org/10.1016/j.ejor.2003.08.009
Thierfelder, Jörg;
Duality. - In: Mathematics of optimization. - Amsterdam [u.a.] : Elsevier, ISBN 978-0-444-50550-7, (2004), S. 459-501

Thierfelder, Jörg;
Nonsmooth optimization problems. - In: Mathematics of optimization. - Amsterdam [u.a.] : Elsevier, ISBN 978-0-444-50550-7, (2004), S. 359-457

Geletu W. Selassie, Abebe;
A coarse solution of generalized semi-infinite optimization problems via robust analysis of marginal functions and global optimization, 2004. - Online-Ressource (PDF-Datei: 182 S., 6978 KB). Ilmenau : Techn. Univ., Diss., 2004

Die Arbeit beschäftigt sich überwiegend mit theoretischen Untersuchungen zur Bestimmung grober Startlösungen für verallgemeinerte semi-infinite Optimierungsaufgaben (GSIP) mit Methoden der globalen Optimierung. GSIP Probleme besitzen im Gegensatz zu den gewöhnlichen semi-infiniten Optimierungsaufgaben (SIP) die Eigenschaft, dass die Indexmenge, die die Restriktionen beschreibt, natürlich überabzählbar ist, wie bei (SIP) aber darüber hinaus von den Problemvariablen abhängig ist, d.h. die Indexmenge ist eine Punkt-Menge Abbildung. Solche Probleme sind von sehr komplexer Struktur, gleichzeitig gibt es große Klassen von naturwissenschaftlich-technischen, ökonomischen Problemen, die in (GSIP) modelliert werden können. Im allgemeinem ist die zulässige Menge von einem (GSIP) weder abgeschlossen noch zusammenhängend. Die Abgeschlossenheit von der zulässigen Menge ist gesichert durch die Unterhalbstetigkeit der Index-Abbildung. Viele Autoren machen diese Voraussetzung, um numerische Verfahren für (GSIP) herzuleiten. Diese Arbeit versucht erstmals, ohne Unterhalbstetigkeit der Index-Abbildung auszukommen. Unter diese schwächeren Voraussetzungen kann die zulässige Menge nicht abgeschlossen sein und (GSIP) kann auch keine Lösung besitzen. Trotzdem kann man eine verallgemeinerte Minimalstelle oder eine Minimalfolge für (GSIP) bestimmen. Für diese Zwecke werden zwei numerische Zugänge vorgeschlagen. Im ersten Zugang wird der zulässige Bereich des (GSIP) durch eine (gewöhnliche) parametrische semi-infinite Approximationsaufgabe beschrieben. Die Marginalfunktion der parametrischen Aufgabe ist eine exakte Straffunktion des zulässigen Bereiches des (GSIP). Im zweiten Zugang werden zwei Straffunktionen vorgestellt. Eine verwendet die semi-infinite Restriktion direkt als einen "Max"-Straffterm und die zweite entsteht durch das "lower level Problem" des (GSIP). In beiden Zugänge müssen wir uns mit unstetigen Optimierungsaufgaben beschäftigen. Es wird gezeigt, dass die entstehende Straffunktionen oberrobust (i.A. nicht stetig) sind und damit auch hier stochastische globale Optimierungsmethoden prinzipiell anwendbar sind. Der Hauptbeitrag dieser Arbeit ist die Untersuchung von Robustheiteigenschaften von Marginalfunktionen und Punkt-Menkg-Abbildung mit bestimmte Strukturen. Dieser kann auch als eine Erweiterung der Theorie der Robusten Analysis von Chew & Zheng betrachtet werden. Gleichzeitig wird gezeigt, dass die für halbstetigen Abbildungen und Funktionen bekannten Aussagen bis auf wenige Ausnahmen in Bezug auf das Robustheitskonzept übertragen werden können.



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Giorgi, G.; Guerraggio, A.; Thierfelder, Jörg
Mathematics of optimization : smooth and nonsmooth case
1. ed.. - Amsterdam [u.a.] : Elsevier, 2004. - XV, 598 S.. ISBN 0444505504 = 978-0-444-50550-7
- Literaturangaben
http://www.loc.gov/catdir/toc/els051/2004040362.html

Hildenbrandt, Regina;
Stochastic dynamic programming with random disturbances. - In: Discussiones mathematicae. - Zielona Góra : Univ. of Zielona Góra Press, ISSN 15099423, Bd. 23 (2003), 1, S. 5-44

Hoffmann, Armin; Marx, Bernd; Metzger, Bob
Linear ODE's for describing sigma-delta-modulators, filter design and error estimations. - In: New trends in difference equations. - London [u.a.] : Taylor & Francis, (2002), S. 159-171

A Sigma-Delta Modulator of order 2 is approximated by a differential equation of first order. An error estimation with respect to the real time behavior of order O(1/m^2) is given, where m is the number of bits used for the filtering. A filter with this error bound related to the moving mean value of the input signal is proposed.



Ginchev, Ivan; Hoffmann, Armin
Approximation of set-valued functions by single-valued one. - In: Discussiones mathematicae. - Zielona Góra : Univ. of Zielona Góra Press, ISSN 1509-9407, Bd. 22 (2002), 1, S. 33-66

Hildenbrandt, Regina;
Partitions-requirements-matrices. - In: Operations research proceedings 2001. - Berlin : Springer, (2002), S. 303-310

Rückmann, Jan-J.; Stein, Oliver
On linear and linearized generalized semi-infinite optimization problems. - In: Annals of operations research. - Dordrecht [u.a.] : Springer Science + Business Media B.V, ISSN 1572-9338, Bd. 101 (2001), 1/4, S. 191-208

http://dx.doi.org/10.1023/A:1010972524021
Rückmann, Jan-J.; Shapiro, Alexander
Second-order optimality conditions in generalized semi-infinite programming. - In: Set-valued analysis : an international journal devoted to the theory of multifunctions and its applications.. - Dordrecht [u.a.] : Springer Science + Business Media B.V, ISSN 1572-932X, Bd. 9 (2001), 1/2, S. 169-186

http://dx.doi.org/10.1023/A:1011239607220
Ward, Doug E.; Fiacco, Anthony V.; Klatte, Diethard; Rückmann, Jan-J.
[Festschrift in honor of Professor Anthony V. Fiacco]. - Optimization with data perturbations ; 2. - Dordrecht : Kluwer Acad. Publ., 2001. - 460 S.. . - (Annals of operations research. - 101) - Literaturangaben

Cegielski, Andrzej; Hoffmann, Armin
Papers of the German-Polish Conference on Optimization - Methods and Applications, Zagan, Poland, September 13 - 17, 1999. - Zielona Góra : Techn. Univ. Press, 2000. - S. 141-278. . - (Discussiones mathematicae / Differential inclusions, control and optimization. - 20.2000,2)
Hichert, Jens; Hoffmann, Armin; Phú, Hoàng Xuân; Reinhardt, Rüdiger
A primal-dual integral method in global optimization. - In: Discussiones mathematicae. - Zielona Góra : Univ. of Zielona Góra Press, ISSN 1509-9407, Bd. 20 (2000), 2, S. 257-278

Hichert, Jens;
Methoden zur Bestimmung des wesentlichen Supremums mit Anwendung in der globalen Optimierung. - Aachen : Shaker, 2000. - X, 161 S.. . - (Berichte aus der Mathematik) Zugl.: Ilmenau : Techn. Univ., Diss., 1999
ISBN 3826582063

Hoffmann, Armin;
On the covariance of a random sequence generated by the fractional part of multiples of a uniformly distributed random variable. - In: Stochastics and stochastics reports. - Abingdon : Taylor & Francis, ISSN 1045-1129, Bd. 66 (1999), 1/2, S. 27-35

Kampe, Jürgen; Hichert, Jens
Ein high-level Dimensionierungsverfahren für analoge Systemkomponenten am Beispiel der Modulo-Funktion. - In: Analog '99. - Paderborn : FhG IZM - Advanced System Engineering, (1999), S. 291-298

Hoffmann, Armin; Marx, Bernd;
Mathematical modelling of a first order sigma-delta modulator. - In: Journal of difference equations and applications. - London [u.a.] : Taylor & Francis, ISSN 1563-5120, Bd. 4 (1998), 6, S. 533-550

In this paper an exact model for a basic, first order sigma-delta-modulator is derived by means of a difference equation with discontinuous nonlinearity. An explicit solution is given in terms of the greatest integer function under certain boundedness and initial conditions of the input signal. Assumptions are made under which the explicit formula remains a solution of the difference equation although the suppositions of the main theorem are violated.



http://dx.doi.org/10.1080/10236199808808161
Ginchev, Ivan; Hoffmann, Armin
The Hausdorff nearest circle to a convex compact set in the plane. - In: Zeitschrift für Analysis und ihre Anwendungen. - Berlin : EMS Publishing House, ISSN 0232-2064, Bd. 17 (1998), 2, S. 479-499

The problem of finding the nearest in the Hausdorff metric circle to a non-empty convex compact set T in the plane is considered from geometrical point of view. The consideration is based on the equivalence of this problem with the Chebyshevian best approximation of 2*pi-periodic functions by trigonometric polynomials of first order, whence it follows that the Hausdorff nearest circle to a convex compact set in the plane exists and is unique. It can be characterized by a geometric Chebyshevian alternance. As a consequence in the particular case of a polygon the centre of the circle is described as an intersection of a midline between some two vertices and a bisectrix of some two sides. In the general case geometrical algorithms corresponding to the one and the four point exchange Remez algorithms are described. They assure correspondingly linear and superlinear convergence. Following the idea, in the case of a polygon to get the exact solution in finite number of steps, a modified two point exchange algorithm is suggested and illustrated by a numerical example. An application is given to estimate the Hausdorff distance between an arbitrary convex set and its Hausdorff nearest circle. The considered problem arises as a practical problem by measuring and pattern recognition in the production of circular machine parts.



Stampa, Christoph; Neundorf, Werner; Hichert, Jens; Hoffmann, Armin; Brauer, Hartmut
Localization of multiple dipoles in bounded convex three dimensional domains by external magnetic measurements. - In: 43. Internationales Wissenschaftliches Kolloquium. - Ilmenau: Tech. Univ., (1998), insges. 6 S.

Hoffmann, Armin; Marx, Bernd; Metzger, Bob
Filtering of analog digital converters using an exact time approach. - In: 43. Internationales Wissenschaftliches Kolloquium. - Ilmenau: Tech. Univ., (1998), insges. 6 S.

Hildenbrandt, Regina;
Parametric properties of the transportation problem and relations to supermatroids. - In: Optimization : a journal of mathematical programming and operations research.. - London [u.a.] : Taylor & Francis, ISSN 1029-4945, Bd. 39 (1997), 2, S. 165-189

http://dx.doi.org/10.1080/02331939708844280
Hichert, Jens; Hoffmann, Armin; Phú, Hoàng Xuân
Convergence speed of an integral method for computing the essential supremum. - In: Developments in global optimization. - Dordrecht [u.a.] : Kluwer Academic, (1997), S. 153-170

We give an equivalence between the tasks of computing the essential supremum of a summable function and of finding a certain zero of a one-dimensional convex function. Interpreting the integral method as Newton-type method we show that in the case of objective functions with an essential supremum that is not spread the algorithm can work very slowly. For this reason we propose a method of accelerating the algorithm which is in some respect similar to the method of Aitken/Steffensen.



Ginchev, Ivan; Hoffmann, Armin
On the best approximation of set-valued functions. - In: Recent advances in optimization. - Berlin [u.a.] : Springer, (1997), S. 61-74

Let M be a Hausdorff compact topological space, let C(M) be the Banach space of the continuous on M functions supplied with the supremum norm and let V be a finite dimensional subspace of C(M). The problem of the Chebyshev approximation of a function f of C(M) by functions from V is generalized to two Chebyshev kind approximations of a point-to set mapping by a single valued continuous function from V using suitable distances between a point and a set. The first problem occur e.g. in curve fitting with noisy data or in approximating spatial bodies by circular cylinders with respect to a proper distance. The second problem is useful for calculating continuous selections with special uniform distance properties.



Hildenbrandt, Regina;
Methoden aus ganzzahliger Optimierung und Verbandstheorie zur Behandlung eines stochastischen dynamischen Transportproblems. - Stützerbach : [Libri Books on Demand]. - 164 S.. Zugl.: Ilmenau : Techn. Univ., Habil.-Schr., 1997
ISBN 3898118797
- Hergestellt on demand

Nehse, Reinhard;
[Rezension von: Dontchev, Asen L., 1948-, Well-posed optimization problems]. - In: OR spectrum. - Berlin : Springer, ISSN 1436-6304, 1, S. 28
, Rezension von : Dontchev, Asen L., 1948-: Well-posed optimization problems. - Berlin [u.a.] : Springer, 1993
https://doi.org/10.1007/BF01539877
Nehse, Reinhard;
[Rezension von: Sakawa, Masatoshi, 1947-, Fuzzy sets and interactive multiobjective optimization]. - In: OR spectrum. - Berlin : Springer, ISSN 1436-6304, 4, S. 230
, Rezension von : Sakawa, Masatoshi, 1947-: Fuzzy sets and interactive multiobjective optimization. - New York, NY [u.a.] : Plenum Press, 1993
https://doi.org/10.1007/BF01540161
Nehse, Reinhard;
[Rezension von: Gregory, Geoffrey, Decision analysis]. - In: OR spectrum. - Berlin : Springer, ISSN 1436-6304, 4, S. 208
, Rezension von : Gregory, Geoffrey: Decision analysis. - New York [u.a.] : Plenum Press, 1988
https://doi.org/10.1007/BF01540157
Phu, Hoang Xuan; Hoffmann, Armin
Essential supremum and supremum of summable functions. - In: Numerical functional analysis and optimization. - Philadelphia, Pa. : Taylor & Francis, ISSN 0163-0563, Bd. 17 (1996), 1/2, S. 167-180

Let D be a subset of R^n with positive finite Lebesgue measure and f on D an arbitrary real summable function. Then the function F of a, defined by the integral of f - a over all x of D with f(x) larger as or equal to a, is continuous, non-negative, non-increasing, convex, and has almost everywhere the measure of the level set as derivative F'(a). Further on, it holds that the essential supremum of f is given by the supremum of all a with F(a)>0. These properties can be used for computing the essential supremum of f. As example, two algorithms are stated. If the function f is dense, or lower semicontinuous, or if -f is robust, then supremum and the essential supremum of f coincide. In this case, the algorithms mentioned can be applied for determining the supremum of f, i.e., also the global maximum of f if it exists.



Nehse, Reinhard;
[Rezension von: Economic and financial modeling with mathematica®]. - In: OR spectrum. - Berlin : Springer, ISSN 1436-6304, 1, S. 4
, Rezension von
https://doi.org/10.1007/BF01719724
Nehse, Reinhard;
[Rezension von: Peressini, Anthony L., 1934-, The mathematics of nonlinear programming]. - In: OR spectrum. - Berlin : Springer, ISSN 1436-6304, 1, S. 54
, Rezension von : Peressini, Anthony L., 1934-: The mathematics of nonlinear programming. - New York, NY [u.a.] : Springer, 1993
https://doi.org/10.1007/BF01719735
Nehse, Reinhard;
[Rezension von: Hiriart-Urruty, Jean-Baptiste, 1949-, Convex analysis and minimization algorithms]. - In: OR spectrum. - Berlin : Springer, ISSN 1436-6304, 2/3, S. 192
, Rezension von : Hiriart-Urruty, Jean-Baptiste, 1949-: Convex analysis and minimization algorithms. 1, Fundamentals. - Berlin [u.a.] : Springer, 1993
Rezension von : Hiriart-Urruty, Jean-Baptiste, 1949-: Convex analysis and minimization algorithms. 2, Advanced theory and bundle methods. - Berlin [u.a.] : Springer, 1993
https://doi.org/10.1007/BF01719264
Hoffmann, Armin;
Sufficient conditions of optimality for optimal control problems governed by integral operators. - In: Operations research proceedings 1994. - Berlin [u.a.] : Springer, (1995), S. 209-214

We consider mixed suffucient conditions of optimality for abstract optimal control problems being local w.r.t. the state variable and global w.r.t. to the control variable. Using a special kind of augmented Lagrangian we don't need the kernel of the first derivative of the state equation (or inequality) for the formulation of the sufficient conditions. An example of its application to control problems with integral operators is given.



Bernhard, Frank; Hoffmann, Armin; Reinhardt, Rüdiger;
Optimal approximation of characteristics of standard thermocouples. - In: . - Padua : CLEUP, (1994), S. 1396-1401

We consider the approximation of the characteristics of thermocouples by polynomials of given degree on the largest possible range such that an apriori given error for the temperature is not violated. The used mathematical methods belongs to the field of semi-infinite optimization and the approach can be interpreted as a kind of reverse Chebyshev approximation (see also Preprint M08/94 from 22-6-94 of the Institute of Matematics of TU Ilmenau).



Hildenbrandt, Regina;
Eine Verbandsstruktur für Partitionen ganzer Zahlen. - Ilmenau : Techn. Univ., Inst. für Mathematik, 1994. - 9 S. = 392,4 KB. . - (Preprint. - M94,14)
http://www.db-thueringen.de/servlets/DocumentServlet?id=5664
Hoffmann, Armin;
Duality and optimality conditions for infinite dimensional optimization problems. - In: System modelling and optimization. - London [u.a.] : Springer, (1994), S. 425-436

Using a nonsymmetric duality for abstract continuous convex control problems optimality conditions are derived for calculating the primal and dual solutions in the case of linear on state depending dual operators. Functional and pointwise conditions are considered.



Hoffmann, Armin; Reinhardt, Rüdiger; Bernhard, Frank
Optimale Approximation der Kennlinien von Standard-Thermoelementen und anderen Sensoren. - In: Messen und Verarbeiten elektrischer und nichtelektrischer Größen. - Hagenburg : Network, (1994), S. 435-441

We consider the approximation of the characteristics of thermocouples by polynomials of given degree on the largest possible range such that an apriori given error for the temperature is not violated. The used mathematical methods belongs to the field of semi-infinite optiization and the approach can be interpreted as a kind of reverse Chebyshev approxiamtion (see also Preprint M08/94 from 22-6-94 of the Institute of Mathematics of TU Ilmenau)



Nehse, Reinhard;
[Rezension von: Holler, Manfred J., 1946-, Einführung in die Spieltheorie]. - In: OR spectrum. - Berlin : Springer, ISSN 1436-6304, 1, S. 62
, Rezension von : Holler, Manfred J., 1946-: Einführung in die Spieltheorie. - Berlin [u.a.] : Springer, 1991
https://doi.org/10.1007/BF01783419
Nehse, Reinhard;
[Rezension von: Vincke, Philippe, 1951-, Multicriteria decision-aid]. - In: OR spectrum. - Berlin : Springer, ISSN 1436-6304, 1, S. 61
, Rezension von : Vincke, Philippe, 1951-: Multicriteria decision-aid. - Chichester [u.a.] : Wiley, 1992
https://doi.org/10.1007/BF01783419
Nehse, Reinhard;
[Rezension von: Bachem, Achim, 1947-, Linear programming duality]. - In: OR spectrum. - Berlin : Springer, ISSN 1436-6304, 3, S. 179
, Rezension von : Bachem, Achim, 1947-: Linear programming duality. - Berlin [u.a.] : Springer, 1992
https://doi.org/10.1007/BF02733720
Hildenbrandt, Regina;
A special stochastic decision problem. - In: Optimization : a journal of mathematical programming and operations research.. - London [u.a.] : Taylor & Francis, ISSN 1029-4945, Bd. 28 (1993), 1, S. 95-110

http://dx.doi.org/10.1080/02331939308843906
Brosowski, Bruno; Ester, Jochen; Helbig, Siegfried; Nehse, Reinhard
Multicriteria decision : proceedings of the 14th meeting of the German Working Group "Mehrkriterielle Entscheidung" ; [held ... in Riezlern, Kleines Walsertal, Austria, in September 22 - 27, 1991]. - Frankfurt am Main [u.a.] : Lang, 1993. - 188 S.. . - (Approximation & [and] optimization. - 4) ISBN 3-631-45828-2
- Literaturangaben

Elster, Karl-Heinz
Modern mathematical methods of optimization. - Berlin : Akad.-Verl., 1993. - 415 S.. . - (Mathematical topics. - 1) ISBN 3055014529
- Literaturangaben

Nehse, Reinhard;
[Rezension von: Readings in multiple criteria decision aid]. - In: OR spectrum. - Berlin : Springer, ISSN 1436-6304, 3, S. 171-172
, Rezension von
https://doi.org/10.1007/BF01783520
Hoffmann, Armin;
The distance to the intersection of two convex sets expressed by the distances to each of them. - In: Mathematische Nachrichten. - [S.l.] : Wiley-VCH, ISSN 1522-2616, Bd. 157 (1992), 1, S. 81-98

Using the distances of a point x to two convex sets we obtain an upper estimation of the distance of x to the intersection of these two sets. Applications to the intersection of point-to-set mappings are given.



http://dx.doi.org/10.1002/mana.19921570108
Thierfelder, Jörg;
Subdifferentiale und Vektoroptimierung. - In: Wissenschaftliche Zeitschrift. - Ilmenau : Technische Hochschule, ISSN 0043-6917, Bd. 37 (1991), 3, S. 89-100

Nehse, Reinhard;
Zwei offene Probleme in der Vektoroptimierung. - In: Wissenschaftliche Zeitschrift. - Ilmenau : Technische Hochschule, ISSN 0043-6917, Bd. 37 (1991), 3, S. 45-53

Hennecke, Ralf;
Ein Trust-Region-Algorithmus für Optimierungsprobleme mit Gleichungs- und Ungleichungsrestriktionen. - In: Wissenschaftliche Zeitschrift. - Ilmenau : Technische Hochschule, ISSN 0043-6917, Bd. 37 (1991), 2, S. 5-15

Hoffmann, Armin;
Pointwise optimality criteria for optimal control problems governed by integral state relations II. - In: Wissenschaftliche Zeitschrift. - Ilmenau : Technische Hochschule, ISSN 0043-6917, Bd. 37 (1991), 3, S. 129-138