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Random walk in a simplex and quadratic optimization over convex polytopes

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  • NESTEROV, Yu

Abstract

In this paper we develop probabilistic arguments for justifying thequality of an approximate solution for global quadratic minimization problem, obtained as a best point among all points of a uniform grid inside a polyhedral feasible set. Our main tool is a random walk inside the standard simplex, for which it is easy to find explicit probabilistic characteristics. For any integer k = 1 we can generate an approximate solution with relative accuracy 1k provided that the quadratic objective function is non-negative in all nodes of the feasible set. The complexity of the process is polynomial in the number of nodes and in the dimension of the space of variables. We extend some of the results to problems with polynomial objective function. We conclude the paper by showing that some related problems (maximization of cubic or quartic form over the Euclidean ball, and the matrix ellipsoid problem) are NP-hard.

Suggested Citation

  • NESTEROV, Yu, 2003. "Random walk in a simplex and quadratic optimization over convex polytopes," LIDAM Discussion Papers CORE 2003071, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:2003071
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    References listed on IDEAS

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    1. NESTEROV, Yurii, 1999. "Global quadratic optimization on the sets with simplex structure," LIDAM Discussion Papers CORE 1999015, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. de Klerk, E. & Pasechnik, D.V., 2007. "A linear programming reformulation of the standard quadratic optimization problem," Other publications TiSEM c3e74115-b343-4a85-976b-8, Tilburg University, School of Economics and Management.
    2. de Klerk, E. & Laurent, M. & Parrilo, P., 2006. "A PTAS for the minimization of polynomials of fixed degree over the simplex," Other publications TiSEM 603897c9-179e-43e4-9e83-6, Tilburg University, School of Economics and Management.
    3. de Klerk, E., 2006. "The Complexity of Optimizing over a Simplex, Hypercube or Sphere : A Short Survey," Other publications TiSEM 88640b6d-5240-472d-8669-4, Tilburg University, School of Economics and Management.
    4. de Klerk, E. & Pasechnik, D.V., 2005. "A Linear Programming Reformulation of the Standard Quadratic Optimization Problem," Other publications TiSEM f63bfe23-904e-4d7a-8677-8, Tilburg University, School of Economics and Management.
    5. Roland Hildebrand, 2022. "Semi-definite Representations for Sets of Cubics on the Two-dimensional Sphere," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 666-675, November.
    6. de Klerk, E., 2006. "The Complexity of Optimizing over a Simplex, Hypercube or Sphere : A Short Survey," Discussion Paper 2006-85, Tilburg University, Center for Economic Research.
    7. Christoph Buchheim & Marcia Fampa & Orlando Sarmiento, 2021. "Lower Bounds for Cubic Optimization over the Sphere," Journal of Optimization Theory and Applications, Springer, vol. 188(3), pages 823-846, March.
    8. de Klerk, E. & Pasechnik, D.V., 2005. "A Linear Programming Reformulation of the Standard Quadratic Optimization Problem," Discussion Paper 2005-24, Tilburg University, Center for Economic Research.
    9. Bo Jiang & Simai He & Zhening Li & Shuzhong Zhang, 2014. "Moments Tensors, Hilbert's Identity, and k -wise Uncorrelated Random Variables," Mathematics of Operations Research, INFORMS, vol. 39(3), pages 775-788, August.
    10. Ke Hou & Anthony Man-Cho So, 2014. "Hardness and Approximation Results for L p -Ball Constrained Homogeneous Polynomial Optimization Problems," Mathematics of Operations Research, INFORMS, vol. 39(4), pages 1084-1108, November.
    11. M. Locatelli, 2009. "Complexity Results for Some Global Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 140(1), pages 93-102, January.
    12. de Klerk, E., 2008. "The complexity of optimizing over a simplex, hypercube or sphere : A short survey," Other publications TiSEM 485b6860-cf1d-4cad-97b8-2, Tilburg University, School of Economics and Management.
    13. Lek-Heng Lim, 2017. "Self-concordance is NP-hard," Journal of Global Optimization, Springer, vol. 68(2), pages 357-366, June.
    14. 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.
    15. de Klerk, Etienne & Laurent, Monique, 2019. "A survey of semidefinite programming approaches to the generalized problem of moments and their error analysis," Other publications TiSEM d956492f-3e25-4dda-a5e2-e, Tilburg University, School of Economics and Management.

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