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  \textbf{\textsc{STANFORD UNIVERSITY}}\\[5pt]
  \textbf{\textsc{DEPARTMENT OF STATISTICS}}\\[5pt]
  \Large{\textbf\textsc{{DEPARTMENTAL SEMINAR}}}
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  4:15 p.m., Tuesday, July 26, 2005\\
  Sequoia Hall Room 200\\
  (Cookies at 3:45 in 1st Floor Lounge)
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  \textsl{Professor Jack P.C. Kleijnen} \\
  Tilburg University (UvT) \& \\
 Wageningen University and Research Center (WUR) \\ 
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  \textbf{STATISTICAL TESTING OF OPTIMALITY CONDITIONS IN
    SIMULATION-BASED OPTIMIZATION}
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We derive a novel procedure for testing the Karush-Kuhn-Tucker (KKT)
first-order optimality conditions in random models. Such models arise
in simulation-based optimization with multiple random (multivariate)
outputs. We focus on expensive simulations, which have small sample
sizes.  We estimate the gradients (in the KKT conditions) through
second-order polynomials, fitted locally; we use Ordinary Least
Squares (OLS) and a central composite design (CCD). This enables us to
estimate the covariance matrix of the estimated gradients. Using these
estimates, we apply the bootstrap method to test the KKT conditions.
First, however, we apply the classic Student t test to check whether
the simulation outputs are feasible, and whether any constraints are
binding. Furthermore, we test for lack-of-fit of the fitted
polynomials. We apply our procedure to both a synthetic Monte Carlo
example and a dynamic inventory simulation.

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