Robust Optimization with Moment-Dispersion Ambiguity

Abstract

Robust optimization presents a compelling methodology for optimization under uncertainty, providing a practical, ambiguity-averse evaluation of risk when the probability distribution is encapsulated by an ambiguity set. We introduce the moment-dispersion ambiguity set, an improvement on the moment-based set, enabling separate characterization of a random variable's location, dispersion, and support. To describe dispersion, we define the dispersion characteristic function, capturing complex attributes like sub-Gaussian and asymmetric dispersion, and its associated dispersion characteristic set, which serves as the input format for representing dispersion ambiguity in algebraic modeling tools. We devise a process for constructing and integrating ambiguity sets, showcasing their modeling flexibility. In particular, we introduce the independence propensity hyper-parameter to foster joint ambiguity set creation for multiple random variables, enhancing our model's real-world applicability and facilitating varying interdependence characterization without needing a correlation matrix. For ambiguous risk assessment over moment-dispersion ambiguity sets, we develop safe tractable approximations for assessing entropic risks linked with affine and convex piecewise affine cost functions, accommodating varying risk tolerances. Lastly, we demonstrate the superior numerical performance of our model over other robust optimization models by adjusting the independence propensity hyper-parameter when limited to marginal information. In data-driven experiments, we find that the moment-dispersion ambiguity sets yield less conservative decisions than classic moment-based sets and more robust decisions than Wasserstein ambiguity sets in data-limited scenarios.

Keywords: robust optimization, ambiguity set, exponential conic optimization History : May 29, 2024

JEL Classification: C61