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Randomized methods based on new Monte Carlo schemes for control and optimization

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  • Boris Polyak
  • Elena Gryazina

Abstract

We address randomized methods for control and optimization based on generating points uniformly distributed in a set. For control systems this sets are either stability domain in the space of feedback controllers, or quadratic stability domain, or robust stability domain, or level set for a performance specification. By generating random points in the prescribed set one can optimize some additional performance index. To implement such approach we exploit two modern Monte Carlo schemes for generating points which are approximately uniformly distributed in a given convex set. Both methods use boundary oracle to find an intersection of a ray and the set. The first method is Hit-and-Run, the second is sometimes called Shake-and-Bake. We estimate the rate of convergence for such methods and demonstrate the link with the center of gravity method. Numerical simulation results look very promising. Copyright Springer Science+Business Media, LLC 2011

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  • Boris Polyak & Elena Gryazina, 2011. "Randomized methods based on new Monte Carlo schemes for control and optimization," Annals of Operations Research, Springer, vol. 189(1), pages 343-356, September.
    • Handle: RePEc:spr:annopr:v:189:y:2011:i:1:p:343-356:10.1007/s10479-009-0585-5
    DOI: 10.1007/s10479-009-0585-5
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    References listed on IDEAS

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    1. 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.
    2. David E. Kaufman & Robert L. Smith, 1998. "Direction Choice for Accelerated Convergence in Hit-and-Run Sampling," Operations Research, INFORMS, vol. 46(1), pages 84-95, February.
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    1. Gryazina, Elena & Polyak, Boris, 2014. "Random sampling: Billiard Walk algorithm," European Journal of Operational Research, Elsevier, vol. 238(2), pages 497-504.

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