Single- and Multi-Objective Feasibility Robust Optimization under Interval Uncertainty with Surrogate Modeling

Abstract

This dissertation presents new methods for solving single- and multi-objective optimization problems when there are uncertain parameter values. The uncertainty in these problems is considered to come from sources with no known or assumed probability distribution, bounded only by an interval. The goal is to obtain a single solution (for single-objective optimization problems) or multiple solutions (for multi-objective optimization problems) that are optimal and “feasibly robust”. A feasibly robust solution is one that remains feasible for all values of uncertain parameters within the uncertainty interval. Obtaining such a solution can become computationally costly and require many function calls. To reduce the computational cost, the presented methods use surrogate modeling to approximate the functions in the optimization problem.This dissertation aims at addressing several key research questions. The first Research Question (RQ1) is: How can the computational cost for solving single-objective robust optimization problems be reduced with surrogate modelling when compared to previous work? RQ2 is: How can the computational cost of solving bi-objective robust optimization problems be improved by using surrogates in concert with a Bayesian optimization technique when compared to previous work? And RQ3 is: How can surrogate modeling be leveraged to make multi-objective robust optimization computationally less expensive when compared to previous work? In addressing RQ1, a new single-objective robust optimization method has been developed with improvements over an existing method from the literature. This method uses a deterministic, local solver, paired with a surrogate modelling technique for finding worst-case scenario of parameter configurations. Using this single-objective robust optimization method, improved large-scale performance and robust feasibility were demonstrated. The second method presented solves bi-objective robust optimization problems under interval uncertainty by introducing a relaxation technique to facilitate combining iterative robust optimization and Bayesian optimization techniques. This method showed improved feasibility robustness and performance at larger problem sizes over existing methods. The third method presented in this dissertation extends the current literature by considering multiple (beyond two) competing objectives for surrogate robust optimization. Increasing the number of objectives adds more dimensions and complexity to the search for solutions and can greatly increase the computational costs. In the third method, the robust optimization strategy from the bi-objective second method was combined with a new Monte Carlo approximated method. The key contributions in this dissertation are 1) a new single-objective robust optimization method combining a local optimization solver and surrogate modelling for robustness, 2) a bi-objective robust optimization method that employs iterative Bayesian optimization technique in tandem with iterative robust optimization techniques, and 3) a new acquisition function for robust optimization in problems of more than two objectives.