Abstract. We extend the concept of the correlated knowledge-gradient policy for the ranking and selection of a finite set of alternatives to the case of continuous decision variables. We propose an approximate knowledge gradient for problems with continuous decision variables in the context of a Gaussian process regression model in a Bayesian setting, along with an algorithm to maximize the approximate knowledge gradient. In the problem class considered, we use the knowledge gradient for continuous parameters to sequentially choose where to sample an expensive noisy function in order to find the maximum quickly. We show that the knowledge gradient for continuous decisions is a generalization of the efficient global optimization algorithm proposed in [D. R. Jones, M. Schonlau, and W. J. Welch, J. Global Optim., 13 (1998) 1. Introduction. Our goal is to find the global maximum of a real valued continuous function that is expensive to compute and that can only be evaluated with uncertainty. We need an algorithm that can give satisfactory results with as few function evaluations as possible. For this reason, we are willing to spend extra time deciding where we would like to evaluate the function next. This problem arises in applications such as simulation optimization, the design of machinery, medical diagnostics, biosurveillance, and the design of business processes. We extend the concept of the knowledge-gradient policy for correlated beliefs (KGCB) described in [10] and [11] , originally developed to find the best of a finite set of alternatives, to problems where we are trying to optimize over a multidimensional set of continuous variables. The KGCB policy maximizes the marginal value of a single measurement and has produced very promising results in discrete ranking and selection problems without requiring the use of any tunable parameters. In [11] the KGCB policy is used in a simulation optimization application to tune a set of continuous parameters which must be discretized to perform the search. However, the KGCB policy becomes computationally too expensive when it is necessary to discretize over a large multidimensional vector. We extend the knowledge gradient to multidimensional continuous problems and then show that the knowledge-gradient concept is at least competitive with, or outperforms, specialized algorithms for specific problems. Although the concept for the knowledge gradient is very general, we choose to model the function to be optimized using Gaussian process regression with a squared exponential covariance function and model the noise in the observations as additive Gaussian noise. The knowledge gradient for continuous parameters (KGCP) policy that we
REF , and the final solution that we choose is not restricted to be one of the points we evaluate.
THE CORRELATED KNOWLEDGE GRADIENT FOR SIMULATION OPTIMIZATION OF CONTINUOUS PARAMETERS USING GAUSSIAN PROCESS REGRESSION *
SIAM Journal on Optimization
SIAM Journal on Optimization
Computer Science Mathematics