Nested Bayesian Optimization for Computer Experiments

Computer experiments can emulate the physical systems, help computational investigations, and yield analytic solutions. They have been widely employed with many engineering applications (e.g., aerospace, automotive, energy systems. Conventional Bayesian optimization did not incorporate the nested structures in computer experiments. This paper proposes a novel nested Bayesian optimization for complex computer experiments with multi-step or hierarchical characteristics. We prove the theoretical properties of nested outputs given two cases: Gaussian or non-Gaussian. The closed forms of nested expected improvement are derived. We also propose the computational algorithms for nested Bayesian optimization. Three numerical studies show that the proposed nested Bayesian optimization outperforms the five benchmark Bayesian optimization methods ignoring the intermediate outputs of the inner computer code. The case study shows that the nested Bayesian optimization can efficiently minimize the residual stress during composite structures assembly and avoid convergence to the local optimum.