Discrete optimization via simulation using Gaussian Markov random fields

We construct a discrete optimization via simulation (DOvS) procedure using discrete Gaussian Markov random fields (GMRFs). Gaussian random fields (GRFs) are used in DOvS to balance exploration and exploitation. They enable computation of the expected improvement (EI) due to running the simulation to evaluate a feasible point of the optimization problem. Existing methods use GRFs with a continuous domain, which leads to dense covariance matrices, and therefore can be ill-suited for large-scale problems due to slow and ill-conditioned numerical computations. The use of GMRFs leads to sparse precision matrices, on which several sparse matrix techniques can be applied. To allocate the simulation effort throughout the procedure, we introduce a new EI criterion that incorporates the uncertainty in stochastic simulation by treating the value at the current optimal point as a random variable.