Adaptive multivariate three-timescale stochastic approximation algorithms for simulation based optimization

Abstract

We develop in this article, four adaptive three-timescale stochastic approximation algorithms for simulation optimization that estimate both the gradient and Hessian of average cost at each update epoch. These algorithms use four, three, two, and one simulation(s), respectively, and update the values of the decision variable and Hessian matrix components simultaneously, with estimates based on the simultaneous perturbation methodology. Our algorithms use coupled stochastic recursions that proceed using three different timescales or step-size schedules. We present a detailed convergence analysis of the algorithms and show numerical experiments using all the developed algorithms on a two-node network of M/G/1 queues with feedback for a 50-dimensional parameter vector. We provide comparisons of the performance of these algorithms with two recently developed two-timescale steepest descent simultaneous perturbation analogs that use randomized and deterministic perturbation sequences, respectively. We also present experiments to explore the sensitivity of the algorithms to their associated parameters. The algorithms that use four and three simulations, respectively, perform significantly better than the rest of the algorithms.