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Convergence theory for nonconvex stochastic programming with an application to mixed logit

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Abstract

Monte Carlo methods have extensively been used and studied in the area of stochastic programming. Their convergence properties typically consider global minimizers or first-order critical points of the sample average approximation (SAA) problems and minimizers of the true problem, and show that the former converge to the latter for increasing sample size. However, the assumption of global minimization essentially restricts the scope of these results to convex problems. We review and extend these results in two directions: we allow for local SAA minimizers of possibly nonconvex problems and prove, under suitable conditions, almost sure convergence of local second-order solutions of the SAA problem to second-order critical points of the true problem. We also apply this new theory to the estimation of mixed logit models for discrete choice analysis. New useful convergence properties are derived in this context, both for the constrained and unconstrained cases, and associated estimates of the simulation bias and variance are proposed.

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Authors and Affiliations

  1. Department of Mathematics, University of Namur, Belgium

    Fabian Bastin

  2. Cerfacs, France

    Fabian Bastin

  3. Transportation Research Group, Department of Mathematics, University of Namur, Belgium

    Cinzia Cirillo & Philippe L. Toint

Authors
  1. Fabian Bastin
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  2. Cinzia Cirillo
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  3. Philippe L. Toint
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Correspondence to Fabian Bastin.

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Research Fellow of the Belgian National Fund for Scientific Research

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Bastin, F., Cirillo, C. & Toint, P. Convergence theory for nonconvex stochastic programming with an application to mixed logit. Math. Program. 108, 207–234 (2006). https://doi.org/10.1007/s10107-006-0708-6

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  • DOI: https://doi.org/10.1007/s10107-006-0708-6

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