Optimization under moment, robust, and data-driven models of uncertainty

Abstract

We study the problem of moments and present two diverse applications that apply both the hierarchy of moment relaxation and the moment duality theory. We then propose a moment-based uncertainty model for stochastic optimization problems, which addresses the ambiguity of probability distributions of random parameters with a minimax decision rule. We establish the model tractability and are able to construct explicitly the extremal distributions. The quality of minimax solutions is compared with that of solutions obtained from other approaches such as data-driven and robust optimization approach. Our approach shows that minimax solutions hedge against worst-case distributions and usually provide low cost variability. We also extend the moment-based framework for multi-stage stochastic optimization problems, which yields a tractable model for exogenous random parameters and affine decision rules. Finally, we investigate the application of data-driven approach with risk aversion and robust optimization approach to solve staffing and routing problem for large-scale call centers. Computational results with real data of a call center show that a simple robust optimization approach can be more efficient than the data-driven approach with risk aversion.