A Distributionally Robust Approach to Black-Box Optimization

Abstract

Deciding how to represent and manage uncertainty is a vital part of designing complex systems. Widely used is a probabilistic approach—assigning a probability distribution to each uncertain variable. However, this presents the designer with the task of assuming or estimating these probability distributions from data; a task which is inevitably prone to error. This paper addresses this challenge by formulating a distributionally robust design optimization problem, and presents computationally ecient algorithms for solving the problem. In distributionally robust optimization (DRO) methods, the designer acknowledges that they are unable to exactly specify a probability distribution for the uncertain variables, and instead specifies a so-called ambiguity set of possible distributions. This paper uses an acoustic horn design problem to explore how the error incurred in estimating a probability distribution from data a↵ects the realized performance of designs found using a traditional multi-objective optimization under uncertainty. It is found that placing some importance on a risk reduction objective results in designs that are more robust to these errors, and thus have a better mean performance realized under the true distribution than if the designer were to focus all e↵orts on optimizing for mean performance alone. In contrast, the DRO approach is able to uncover designs that are not attainable using the multi-objective approach when given the same data. These DRO designs in some cases significantly outperform those designs found using the multi-objective approach.