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New methods for calculating $$\alpha $$ BB-type underestimators

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  • Anders Skjäl
  • Tapio Westerlund

Abstract

Most branch-and-bound algorithms in global optimization depend on convex underestimators to calculate lower bounds of a minimization objective function. The $$\alpha $$ BB methodology produces such underestimators for sufficiently smooth functions by analyzing interval Hessian approximations. Several methods to rigorously determine the $$\alpha $$ BB parameters have been proposed, varying in tightness and computational complexity. We present new polynomial-time methods and compare their properties to existing approaches. The new methods are based on classical eigenvalue bounds from linear algebra and a more recent result on interval matrices. We show how parameters can be optimized with respect to the average underestimation error, in addition to the maximum error commonly used in $$\alpha $$ BB methods. Numerical comparisons are made, based on test functions and a set of randomly generated interval Hessians. The paper shows the relative strengths of the methods, and proves exact results where one method dominates another. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:58:y:2014:i:3:p:411-427
    DOI: 10.1007/s10898-013-0057-y
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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.
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    Citations

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    Cited by:

    1. Milan Hladík, 2018. "Testing pseudoconvexity via interval computation," Journal of Global Optimization, Springer, vol. 71(3), pages 443-455, July.
    2. Dimitrios Nerantzis & Claire S. Adjiman, 2019. "Tighter $$\alpha $$ α BB relaxations through a refinement scheme for the scaled Gerschgorin theorem," Journal of Global Optimization, Springer, vol. 73(3), pages 467-483, March.
    3. Dimitrios Nerantzis & Claire S. Adjiman, 2017. "Enclosure of all index-1 saddle points of general nonlinear functions," Journal of Global Optimization, Springer, vol. 67(3), pages 451-474, March.
    4. N. Kazazakis & C. S. Adjiman, 2018. "Arbitrarily tight $$\alpha $$ α BB underestimators of general non-linear functions over sub-optimal domains," Journal of Global Optimization, Springer, vol. 71(4), pages 815-844, August.
    5. 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.
    6. Milan Hladík & Lubomir V. Kolev & Iwona Skalna, 2021. "Linear interval parametric approach to testing pseudoconvexity," Journal of Global Optimization, Springer, vol. 79(2), pages 351-368, February.
    7. M. M. Faruque Hasan, 2018. "An edge-concave underestimator for the global optimization of twice-differentiable nonconvex problems," Journal of Global Optimization, Springer, vol. 71(4), pages 735-752, August.

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    1. N. Kazazakis & C. S. Adjiman, 2018. "Arbitrarily tight $$\alpha $$ α BB underestimators of general non-linear functions over sub-optimal domains," Journal of Global Optimization, Springer, vol. 71(4), pages 815-844, August.
    2. Dimitrios Nerantzis & Claire S. Adjiman, 2019. "Tighter $$\alpha $$ α BB relaxations through a refinement scheme for the scaled Gerschgorin theorem," Journal of Global Optimization, Springer, vol. 73(3), pages 467-483, March.
    3. 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.
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