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Random Sampling Many-Dimensional Sets Arising in Control

Author

Listed:
  • Pavel Shcherbakov

    (Institute of Control Sciences, Russian Academy of Science, 117997 Moscow, Russia)

  • Mingyue Ding

    (Department of Biomedical Engineering, Huazhong University of Science and Technology, Wuhan 430074, China)

  • Ming Yuchi

    (Department of Biomedical Engineering, Huazhong University of Science and Technology, Wuhan 430074, China)

Abstract

Various Monte Carlo techniques for random point generation over sets of interest are widely used in many areas of computational mathematics, optimization, data processing, etc. Whereas for regularly shaped sets such sampling is immediate to arrange, for nontrivial, implicitly specified domains these techniques are not easy to implement. We consider the so-called Hit-and-Run algorithm, a representative of the class of Markov chain Monte Carlo methods, which became popular in recent years. To perform random sampling over a set, this method requires only the knowledge of the intersection of a line through a point inside the set with the boundary of this set. This component of the Hit-and-Run procedure, known as boundary oracle, has to be performed quickly when applied to economy point representation of many-dimensional sets within the randomized approach to data mining, image reconstruction, control, optimization, etc. In this paper, we consider several vector and matrix sets typically encountered in control and specified by linear matrix inequalities. Closed-form solutions are proposed for finding the respective points of intersection, leading to efficient boundary oracles; they are generalized to robust formulations where the system matrices contain norm-bounded uncertainty.

Suggested Citation

  • Pavel Shcherbakov & Mingyue Ding & Ming Yuchi, 2021. "Random Sampling Many-Dimensional Sets Arising in Control," Mathematics, MDPI, vol. 9(5), pages 1-16, March.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:5:p:580-:d:513530
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    References listed on IDEAS

    as
    1. Boris Polyak & Pavel Shcherbakov, 2017. "Why Does Monte Carlo Fail to Work Properly in High-Dimensional Optimization Problems?," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 612-627, May.
    2. Robert L. Smith, 1984. "Efficient Monte Carlo Procedures for Generating Points Uniformly Distributed over Bounded Regions," Operations Research, INFORMS, vol. 32(6), pages 1296-1308, December.
    3. Gryazina, Elena & Polyak, Boris, 2014. "Random sampling: Billiard Walk algorithm," European Journal of Operational Research, Elsevier, vol. 238(2), pages 497-504.
    4. Yurii Nesterov, 2018. "Lectures on Convex Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-91578-4, June.
    5. Ravindran Kannan & Hariharan Narayanan, 2012. "Random Walks on Polytopes and an Affine Interior Point Method for Linear Programming," Mathematics of Operations Research, INFORMS, vol. 37(1), pages 1-20, February.
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