Computer experiments and global optimization

Abstract

A complex mathematical model that produces output values from input values is now commonly called a computer model. This thesis considers the problem of finding the global optimum of the response with few function evaluations. A small number of function evaluations is desirable since the computer model is often expensive (time consuming) to evaluate. The function to be optimized is modeled as a stochastic process from initial function evaluations. Points are sampled sequentially according to a criterion that combines promising prediction values with prediction uncertainty. Some graphical tools are given that allow early assessment about whether the modeling strategy will work well. The approach is generalized by introducing a parameter that controls how global versus local the search strategy is. Strategies to conduct the optimization in stages and for optimization subject to constraints on additional response variables are presented. Special consideration is given to the stopping criterion of the global optimization algorithm. The problem of achieving a tolerance on the global minimum can be represented by determining whether the first order statistic of N dependent variables is greater than a certain value. An algorithm is developed that quickly determines bounds on the probability of this event. A strategy to explore high-dimensional data informally through effect plots is presented. The interpretation of the plots is guided by pointwise standard errors of the effects which are developed. When used in the context of global optimization, the graphical analysis sheds light on the number and location of local optima.

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