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The complexity of optimizing over a simplex, hypercube or sphere: a short survey

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  • Etienne Klerk

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Abstract

We consider the computational complexity of optimizing various classes of continuous functions over a simplex, hypercube or sphere. These relatively simple optimization problems arise naturally from diverse applications. We review known approximation results as well as negative (inapproximability) results from the recent literature. Copyright The Author(s) 2008

Suggested Citation

  • Etienne Klerk, 2008. "The complexity of optimizing over a simplex, hypercube or sphere: a short survey," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 16(2), pages 111-125, June.
  • Handle: RePEc:spr:cejnor:v:16:y:2008:i:2:p:111-125
    DOI: 10.1007/s10100-007-0052-9
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    References listed on IDEAS

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    1. NESTEROV, Yu, 2003. "Random walk in a simplex and quadratic optimization over convex polytopes," CORE Discussion Papers 2003071, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Immanuel Bomze & Stefan Gollowitzer & E. Yıldırım, 2014. "Rounding on the standard simplex: regular grids for global optimization," Journal of Global Optimization, Springer, vol. 59(2), pages 243-258, July.
    2. Bomze, Immanuel M., 2012. "Copositive optimization – Recent developments and applications," European Journal of Operational Research, Elsevier, vol. 216(3), pages 509-520.
    3. Immanuel Bomze & Werner Schachinger & Gabriele Uchida, 2012. "Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization," Journal of Global Optimization, Springer, vol. 52(3), pages 423-445, March.
    4. repec:spr:jglopt:v:68:y:2017:i:2:d:10.1007_s10898-016-0469-6 is not listed on IDEAS
    5. Maziar Salahi, 2010. "Convex optimization approach to a single quadratically constrained quadratic minimization problem," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 18(2), pages 181-187, June.
    6. W. Ackooij & A. Frangioni & W. Oliveira, 2016. "Inexact stabilized Benders’ decomposition approaches with application to chance-constrained problems with finite support," Computational Optimization and Applications, Springer, vol. 65(3), pages 637-669, December.
    7. de Klerk, E. & Laurent, M., 2010. "Error bounds for some semidefinite programming approaches to polynomial minimization on the hypercube," Other publications TiSEM 619d9658-77df-4b5e-9868-0, Tilburg University, School of Economics and Management.
    8. B. G.-Tóth & E. M. T. Hendrix & L. G. Casado & I. García, 2016. "On refinement of the unit simplex using regular simplices," Journal of Global Optimization, Springer, vol. 64(2), pages 305-323, February.

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