Measuring the Goodness of Orthogonal Array Discretizations for Stochastic Programming and Stochastic Dynamic Programming

Abstract

This paper describes a continuous space discretization scheme based on statistical experimental designs generated from orthogonal arrays (OAs) of strength three with index unity. Chen, Ruppert, and Shoemaker [Oper. Res., 47 (1999), pp. 38--53] employed this efficient discretization scheme in a numerical solution method for high-dimensional continuous-state stochastic dynamic programming (SDP). These OAs may be instrumental in reducing the dimensionality of event spaces, SDP state spaces, and first-stage decision spaces in two-stage stochastic programming. In particular, computationally efficient space-filling measures for these OAs are derived for evaluating how well a specific OA discretization fills the state space. Comparisons were made with two types of common measures: ones which maximize the average (or minimum) distance between discretization points within the OA and ones which minimize the average (or maximum) distance between discretization points and nondiscretization points lying on a full grid (i.e., points lying on the full grid that are not contained in the OA discretization). OAs of strength three were tested by fitting multivariate adaptive regression splines to data from an inventory-forecasting continuous-state stochastic dynamic program.

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