optimizationnotes - MS&E 223 Simulation Brad Null...

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MS&E 223 Supplemental Notes Simulation Making Decisions via Simulation Brad Null Spring Quarter 2008-09 Page 1 of 10 Making Decisions via Simulation Ref : Law, Chapter 10; Handbook of Simulation , Chapters 17-21; Haas, Sec. 6.3.6. We give an introduction into some methods for selecting the “best” system design (or parameter setting for a given design), where the performance under each alternative is estimated via simulation. The methods presented are chosen because of simplicity–see the references for more details on exactly when they are applicable, and whether improved versions of the algorithms are available. Current understanding of the behavior of these algorithms is incomplete, and they should be applied with due caution. 1. Continuous Stochastic Optimization Robbins-Monro Algorithm Here we will consider the problem min f( ) θ∈Θ θ where, for a given value of θ , we are not able to evaluate f( ) θ analytically or numerically, but must obtain (noisy) estimates of f( ) θ using simulation. We will assume for now that the set of possible solutions Θ is uncountably infinite. In particular, suppose that Θ is an interval [ θ θ , ] of real numbers. One approach to solving the problem ) ( f min θ Θ θ is to estimate ) ( ' f θ using simulation and then use an iterative method to solve the equation ) ( ' f θ = 0. The most basic such method is called the Robbins-Monro Algorithm, and can be viewed as a stochastic version of Newton’s method for solving nonlinear equations. The basic iteration is (*) θ n+1 = - θ π n n Z n a , where a > 0 is a fixed parameter of the algorithm (the quantity a/n is sometimes called the “gain”), n Z is an unbiased estimator of ) ( ' f n θ , and θ θ θ θ θ θ θ θ < θ θ = θ π . if ; if ; if ) ( (The function ( ) π ⋅ projects the current parameter value onto the feasible set. Algorithms that iterate as described above are called stochastic approximation algorithms.) Denote by θ * the global minimizer of f. If the only local minimizer of f is θ *, then under very general conditions θ n θ *
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MS&E 223 Supplemental Notes Simulation Making Decisions via Simulation Brad Null Spring Quarter 2008-09 Page 2 of 10 as n with probability 1. (Otherwise the algorithm can converge to some other local minimizer.) For large n, it can be shown that θ n has approximately a normal distribution. Thus, if the procedure is run m independent times (where m = 5 to 10) with n iterations for each run, generating i.i.d. replicates θ n,1 , . .. , θ n,m , then a point estimator for θ * is m θ = m 1 n,j m j 1 = θ and confidence intervals can be formed in the usual way, based on the Student-t distribution with m – 1 degrees of freedom. Remark
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optimizationnotes - MS&amp;E 223 Simulation Brad Null...

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