Convex Estimators for Optimization of Kriging Model Problems

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This paper presents a framework for identification of the global optimum of Kriging models. The framework is based on a branch and bound scheme for sub-division of the search space into hypercubes while constructing convex under-estimators of the Kriging models. The convex under-estimators, which are a key development in this paper, provide a relaxation of the original problem. The relaxed problem has two key features: i) convex optimization algorithms such as sequential quadratic programming (SQP) are guaranteed to find the global optimum of the relaxed problem, and ii) objective value of the relaxed problem is a lower bound on the best attainable solution within a hypercube for the original (Kriging model) problem. The convex under-estimators improve in accuracy as the size of a hypercube gets smaller via the branching search. Termination of a hypercube branch is done when either: i) solution of the relaxed problem within the hypercube is no better than current best solution of the original problem, or ii) best solution of the original problem and that of the relaxed problem are within tolerance limits. To assess the significance of the proposed framework, comparison studies against genetic algorithm (GA) are conducted using Kriging models that approximate standard nonlinear test functions, as well as application problems of water desalination and vehicle crashworthiness. Results of the studies show the proposed framework deterministically providing a solution within tolerance limits from the global optimum, while GA is observed to not reliably discover the best solutions in problems with larger number of design variables.