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Results: 166
Created on: Thu, 28 Mar 2024 23:08:58 +0100 in 0.0792 sec


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, 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, 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, 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, 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, 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, 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
Dempe, Stephan; Eichfelder, Gabriele; Eichfelder, Gabriele *1977-*; Fliege, Jörg
On the effects of combining objectives in multi-objective optimization. - Freiberg : TU Bergakademie, 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.



Eichfelder, Gabriele;
Vector optimization in medical engineering. - In: Mathematics without boundaries, (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