Stochastic root finding in system design

Huifen Chen, Purdue University

Abstract

The stochastic root-finding problem (SRFP) is to solve an equation $g(x) = \gamma,$ using only an unbiased estimate y(x) of g(x). Such problems arise in system design, where x is a controllable parameter, g(x) is the system performance, $\gamma$ is the desired system performance, and the estimate y(x) of g(x) is from a simulation experiment. Such problems are difficult to solve. Stochastic approximation, the only existing class of algorithms with guaranteed convergence, is slow, especially if algorithm parameters are poorly chosen; few guidelines exist for choosing parameter values. One example of the SRFP arises from Thiokol Corporation's need to determine, in real time, the constant x* so that if a system is built to meet the tolerance limits, this system will not fail with confidence $\gamma.$ For this special case, we (1) propose an efficient Monte Carlo sampling algorithm--which converges quickly and has no algorithm parameter--by reformulating this SRFP as a quantile-estimation problem, (2) improve the numerical computer code for the case of normal data, and (3) derive several properties of the tolerance-interval constant. Motivated by this reliability example, we seek algorithms to solve general SRFPs in real time using only a computer simulation routine that provides an estimate y(x) by mimicking the behavior of the modeled system. We evaluate algorithms based on computational efficiency, numerical stability, and robustness. We propose retrospective approximation algorithms that iteratively solve a sequence of deterministic sample-path equations, $\sum\sbsp{i=1}{m}y\sb{i}(x)/m = \gamma,$ with increasing sample sizes m. A final estimate is computed from those solutions. Converging under weak conditions, these algorithms differ in the deterministic root-finding method, sample-size sequence, error-tolerance sequence, final estimate, and stopping rule. We develop a probability model to set those algorithmic parameter values and then propose bounding retrospective approximation algorithms. These simpler algorithms solve sample-path equations by bounding the root and then returning the linear interpolation of the bounds. There are five algorithmic parameters: initial point, initial sample-size, sample-size multiplier, initial step size, and step-size multiplier. We discuss the choice of parameter values and provide guidelines. Bounding retrospective approximation lacks provable convergence but converges faster and is less sensitive to parameter values than stochastic approximation in our empirical results.

Degree

Ph.D.

Advisors

Schmeiser, Purdue University.

Subject Area

Industrial engineering

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