Robust Optimization with Moment-Dispersion Ambiguity

Robust optimization models present 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 independent characterization of a random variable's central location, dispersion, and support. We propose the dispersion characteristic function to capture complex attributes like sub-Gaussian and asymmetric dispersion. We also propose the 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. We also 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 independence characterization without needing a correlation matrix. Consequently, we develop safe tractable approximations for assessing risks linked with affine and convex piecewise affine cost functions over moment-dispersion ambiguity sets, accommodating varying risk tolerances. We substantiate our approach with two numerical case studies involving appointment scheduling and portfolio optimization. When an empirical distribution is accessible, our approach demonstrates superior performance compared to empirical optimization and moment-based robust optimization methods. By adjusting the independence propensity hyper-parameter, we illustrate how our model can deliver robust yet less conservative solutions, which is particularly useful when only marginal information is accessible to the decision-maker