Journal of Global Optimization

, Volume 59, Issue 2–3, pp 243–258 | Cite as

Rounding on the standard simplex: regular grids for global optimization

  • Immanuel M. Bomze
  • Stefan Gollowitzer
  • E. Alper Yıldırım
Article

Abstract

Given a point on the standard simplex, we calculate a proximal point on the regular grid which is closest with respect to any norm in a large class, including all \(\ell ^p\)-norms for \(p\ge 1\). We show that the minimal \(\ell ^p\)-distance to the regular grid on the standard simplex can exceed one, even for very fine mesh sizes in high dimensions. Furthermore, for \(p=1\), the maximum minimal distance approaches the \(\ell ^1\)-diameter of the standard simplex. We also put our results into perspective with respect to the literature on approximating global optimization problems over the standard simplex by means of the regular grid.

Keywords

Rounding Regular grid Approximation Maximin distance Proximal point 

Notes

Acknowledgments

The authors are grateful to two anonymous referees who pointed out additional related references, as well as to Oliver Stein for valuable remarks. E. Alper Yıldırım was supported in part by Turkish Scientific and Technological Research Council (TÜBİTAK) Grant 112M870 and by TÜBA-GEBİP (Turkish Academy of Sciences Young Scientists Award Program).

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Immanuel M. Bomze
    • 1
  • Stefan Gollowitzer
    • 1
  • E. Alper Yıldırım
    • 2
  1. 1. Department of Statistics and Operations ResearchUniversity of ViennaViennaAustria
  2. 2. Department of Industrial Engineering Koç UniversityIstanbulTurkey

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