TY - JOUR

T1 - Gaussian Markov random fields for discrete optimization via simulation

T2 - framework and algorithms

AU - Salemi, Peter L.

AU - Song, Eunhye

AU - Nelson, Barry

AU - Staum, Jeremy

N1 - Copyright © 2019, INFORMS

PY - 2019/2/21

Y1 - 2019/2/21

N2 - We consider optimizing the expected value of some performance measure ofa dynamic stochastic simulation with a statistical guarantee for optimality when the decision variables are discrete, in particular, integer-ordered; the number of feasible solutions is large; and the model execution is too slow to simulate even a substantial fraction of them. Our goal is to create algorithms that stop searching when they can provide inference about the remaining optimality gap similar to the correct-selection guarantee of ranking and selection when it simulates all solutions. Further, our algorithm remains competitive with fixed-budget algorithms that search efficiently but do not provide such inference. To accomplish this we learn and exploit spatial relationships among the decisionvariables and objective function values using a Gaussian Markov random field (GMRF).Gaussian random fields on continuous domains are already used in deterministic and stochastic optimization because they facilitate the computation of measures, such as expected improvement, that balance exploration and exploitation. We show that GMRFs are particularly well suited to the discrete decision–variable problem, from both a modeling and a computational perspective. Specifically, GMRFs permit the definition of a sensible neighborhood structure, and they are defined by their precision matrices, which can be constructed to be sparse. Using this framework, we create both single and multiresolution algorithms, prove the asymptotic convergence of both, and evaluate their finite-timeperformance empirically.

AB - We consider optimizing the expected value of some performance measure ofa dynamic stochastic simulation with a statistical guarantee for optimality when the decision variables are discrete, in particular, integer-ordered; the number of feasible solutions is large; and the model execution is too slow to simulate even a substantial fraction of them. Our goal is to create algorithms that stop searching when they can provide inference about the remaining optimality gap similar to the correct-selection guarantee of ranking and selection when it simulates all solutions. Further, our algorithm remains competitive with fixed-budget algorithms that search efficiently but do not provide such inference. To accomplish this we learn and exploit spatial relationships among the decisionvariables and objective function values using a Gaussian Markov random field (GMRF).Gaussian random fields on continuous domains are already used in deterministic and stochastic optimization because they facilitate the computation of measures, such as expected improvement, that balance exploration and exploitation. We show that GMRFs are particularly well suited to the discrete decision–variable problem, from both a modeling and a computational perspective. Specifically, GMRFs permit the definition of a sensible neighborhood structure, and they are defined by their precision matrices, which can be constructed to be sparse. Using this framework, we create both single and multiresolution algorithms, prove the asymptotic convergence of both, and evaluate their finite-timeperformance empirically.

KW - large-scale discrete optimization via simulation

KW - inferential optimization

KW - Gaussian Markov random fields

U2 - 10.1287/opre.2018.1778

DO - 10.1287/opre.2018.1778

M3 - Journal article

VL - 67

SP - 250

EP - 266

JO - Operations Research

JF - Operations Research

SN - 0030-364X

IS - 1

ER -