Simulation-based optimization with stochastic approximation using common random numbers

The method of Common Random Numbers is a technique used to reduce the variance of difference estimates in simulation optimization problems. These differences are commonly used to estimate gradients of objective functions as part of the process of determining optimal values for parameters of a simulated system. Asymptotic results exist which show that using the Common Random Numbers method in the iterative Finite Difference Stochastic Approximation optimization algorithm (FDSA) can increase the optimal rate of convergence of the algorithm from the typical rate of k^–1/3 to the faster k^–1/2, where k is the algorithm’s iteration number. Simultaneous Perturbation Stochastic Approximation (SPSA) is a newer and often much more efficient optimization algorithm, and we will show that this algorithm, too, converges faster when the Common Random Numbers method is used. We will also provide multivariate asymptotic covariance matrices for both the SPSA and FDSA errors.