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Random gradient-free minimization of convex functions

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  • NESTEROV, Yurii

    ()
    (Université catholique de Louvain, CORE, B-1348 Louvain-la-Neuve, Belgium)

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    Abstract

    In this paper, we prove the complexity bounds for methods of Convex Optimization based only on computation of the function value. The search directions of our schemes are normally distributed random Gaussian vectors. It appears that such methods usually need at most n times more iterations than the standard gradient methods, where n is the dimension of the space of variables. This conclusion is true both for nonsmooth and smooth problems. For the later class, we present also an accelerated scheme with the expected rate of convergence O(n[ exp ]2 /k[ exp ]2), where k is the iteration counter. For Stochastic Optimization, we propose a zero-order scheme and justify its expected rate of convergence O(n/k[ exp ]1/2). We give also some bounds for the rate of convergence of the random gradient-free methods to stationary points of nonconvex functions, both for smooth and nonsmooth cases. Our theoretical results are supported by preliminary computational experiments.

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    File URL: http://uclouvain.be/cps/ucl/doc/core/documents/coredp2011_1web.pdf
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    Bibliographic Info

    Paper provided by Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) in its series CORE Discussion Papers with number 2011001.

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    Date of creation: 01 Jan 2011
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    Handle: RePEc:cor:louvco:2011001

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    Postal: Voie du Roman Pays 34, 1348 Louvain-la-Neuve (Belgium)
    Phone: 32(10)474321
    Fax: +32 10474304
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    Web page: http://www.uclouvain.be/core
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    Related research

    Keywords: convex optimization; stochastic optimization; derivative-free methods; random methods; complexity bounds;

    This paper has been announced in the following NEP Reports:

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    1. Winfried Pohlmeier & Luc Bauwens & David Veredas, 2007. "High frequency financial econometrics. Recent developments," ULB Institutional Repository 2013/136223, ULB -- Universite Libre de Bruxelles.
    2. Belleflamme,Paul & Peitz,Martin, 2010. "Industrial Organization," Cambridge Books, Cambridge University Press, number 9780521681599, October.
    3. ROMBOUTS, Jeroen V. K. & STENTOFT, Lars, 2010. "Option pricing with asymmetric heteroskedastic normal mixture models," CORE Discussion Papers 2010049, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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