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Exclusion regions for optimization problems

Author

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  • Hermann Schichl
  • Mihály Markót
  • Arnold Neumaier

Abstract

Branch and bound methods for finding all solutions of a global optimization problem in a box frequently have the difficulty that subboxes containing no solution cannot be easily eliminated if they are close to the global minimum. This has the effect that near each global minimum, and in the process of solving the problem also near the currently best found local minimum, many small boxes are created by repeated splitting, whose processing often dominates the total work spent on the global search. This paper discusses the reasons for the occurrence of this so-called cluster effect, and how to reduce the cluster effect by defining exclusion regions around each local minimum found, that are guaranteed to contain no other local minimum and hence can safely be discarded. In addition, we will introduce a method for verifying the existence of a feasible point close to an approximate local minimum. These exclusion regions are constructed using uniqueness tests based on the Krawczyk operator and make use of first, second and third order information on the objective and constraint functions. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Hermann Schichl & Mihály Markót & Arnold Neumaier, 2014. "Exclusion regions for optimization problems," Journal of Global Optimization, Springer, vol. 59(2), pages 569-595, July.
    • Handle: RePEc:spr:jglopt:v:59:y:2014:i:2:p:569-595
    DOI: 10.1007/s10898-013-0137-z
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    References listed on IDEAS

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    1. Ralph Kearfott & Sowmya Muniswamy & Yi Wang & Xinyu Li & Qian Wang, 2013. "On smooth reformulations and direct non-smooth computations for minimax problems," Journal of Global Optimization, Springer, vol. 57(4), pages 1091-1111, December.
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    Cited by:

    1. Bettina Ponleitner & Hermann Schichl, 2021. "Exclusion regions for parameter-dependent systems of equations," Journal of Global Optimization, Springer, vol. 81(3), pages 621-644, November.
    2. Tiago Montanher & Arnold Neumaier & Ferenc Domes, 2018. "A computational study of global optimization solvers on two trust region subproblems," Journal of Global Optimization, Springer, vol. 71(4), pages 915-934, August.

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