Optimization under Uncertainty: Adaptive Variance Reduction, Adaptive Metamodeling, and Investigation of Robustness Measures

This dissertation offers computational and theoretical advances for optimization under uncertainty problems that utilize a probabilistic framework for addressing such uncertainties, and adopt a probabilistic performance as objective function. Emphasis is placed on applications that involve potentially complex numerical and probability models. A generalized approach is adopted, treating the system model as a 'black-box' and relying on stochastic simulation for evaluating the probabilistic performance. This approach can impose, though, an elevated computational cost, and two of the advances offered in this dissertation aim at decreasing the computational burden associated with stochastic simulation when integrated with optimization applications.

The first one develops an adaptive implementation of importance sampling (a popular variance reduction technique) by sharing information across the iterations of the numerical optimization algorithm. The system model evaluations from the current iteration are utilized to formulate importance sampling densities for subsequent iterations with only a small additional computational effort. The characteristics of these densities as well as the specific model parameters these densities span are explicitly optimized. The second advancement focuses on adaptive tuning of a kriging metamodel to replace the computationally intensive system model. A novel implementation is considered, establishing a metamodel with respect to both the uncertain model parameters as well as the design variables, offering significant computational savings. Additionally, the adaptive selection of certain characteristics of the metamodel, such as support points or order of basis functions, is considered by utilizing readily available information from the previous iteration of the optimization algorithm.

The third advancement extends to a different application and considers the assessment of the appropriateness of different candidate robust designs. A novel robustness measure is introduced, the probability of dominance, defined as the likelihood that a given design will outperform its competing designs. This new measure ultimately provides a rational approach to quantify the preference towards each candidate design. The existence of a model prediction error is also addressed within the definition of this measure.