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A modification of the $$\alpha \hbox {BB}$$ α BB method for box-constrained optimization and an application to inverse kinematics

Author

Listed:
  • Gabriele Eichfelder

    (Technische Universität Ilmenau)

  • Tobias Gerlach

    (Technische Universität Ilmenau)

  • Susanne Sumi

    (Technische Universität Ilmenau)

Abstract

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 $$\alpha \hbox {BB}$$ α BB 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 $$\alpha \hbox {BB}$$ α BB method.

Suggested Citation

  • Gabriele Eichfelder & Tobias Gerlach & Susanne Sumi, 2016. "A modification of the $$\alpha \hbox {BB}$$ α BB method for box-constrained optimization and an application to inverse kinematics," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 4(1), pages 93-121, February.
    • Handle: RePEc:spr:eurjco:v:4:y:2016:i:1:d:10.1007_s13675-015-0056-5
    DOI: 10.1007/s13675-015-0056-5
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    References listed on IDEAS

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    1. A. Skjäl & T. Westerlund & R. Misener & C. A. Floudas, 2012. "A Generalization of the Classical αBB Convex Underestimation via Diagonal and Nondiagonal Quadratic Terms," Journal of Optimization Theory and Applications, Springer, vol. 154(2), pages 462-490, August.
    2. Moritz Schulze Darup & Martin Kastsian & Stefan Mross & Martin Mönnigmann, 2014. "Efficient computation of spectral bounds for Hessian matrices on hyperrectangles for global optimization," Journal of Global Optimization, Springer, vol. 58(4), pages 631-652, April.
    3. Faiz A. Al-Khayyal & James E. Falk, 1983. "Jointly Constrained Biconvex Programming," Mathematics of Operations Research, INFORMS, vol. 8(2), pages 273-286, May.
    4. Anders Skjäl & Tapio Westerlund, 2014. "New methods for calculating $$\alpha $$ BB-type underestimators," Journal of Global Optimization, Springer, vol. 58(3), pages 411-427, March.
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