First-order intrinsic Gaussian Markov random fields for discrete optimisation via simulation

We propose first-order intrinsic Gaussian Markov random fields (GMRFs) for Gaussian process (GP) regression of response surfaces, defined on a subset of the integer lattice, in operations research. These GMRFs have several desirable properties, including simple parameter estimation and no mean reversion. We focus on the application of GP regression with GMRFs to discrete optimisation via simulation (DOvS) using the Gaussian Markov improvement algorithm (GMIA). GMIA is a globally convergent GP-based algorithm for DOvS, which models the response surface as the realisation of a GMRF. The particular GMRF used in GMIA is a critical choice and influences the performance of the algorithm. We compare our first-order intrinsic GMRFs to the GMRF used in the original GMIA, provide details for parameter estimation, and discuss the global convergence of GMIA when our first-order intrinsic GMRFs are used. We then present numerical results showing the advantage of using GMIA with our first-order intrinsic GMRFs over the original GMRF.