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Many real-life decision problems in science and engineering involve uncertainty, typically modeled as random variables that follow some prescribed probability distributions. In healthcare, for example, demands for various healthcare services, the duration of service, and the availability of resources are highly uncertain. Accurate prediction of these parameters is difficult, which complicates real-world decision problems. Over the past few decades, numerous modeling paradigms have been proposed, each built on a distinct perspective on uncertainty, to address stochastic optimization problems. For example, in stochastic programming, it is often assumed that the underlying distribution of uncertainty is known, which is generally not true in many application domains. Although historical data may be available, the underlying data-generating distribution remains unobservable to the decision-maker. Robust and distributionally robust optimization address this fundamental challenge of distributional uncertainty, aiming to find robust solutions. While each approach has its benefits and drawbacks, the performance of these approaches and the trade-off between them remains unclear in application domains.
This dissertation advances the theoretical and computational frontier of stochastic optimization methodologies and their applications. In the first part of the dissertation, we propose new stochastic optimization methodologies, including new approaches for modeling uncertainty and solution methods. We first introduce a new data-driven trade-off (TRO) approach for modeling uncertainty that serves as a middle ground between the optimistic approach, which adopts a distributional belief, and the pessimistic distributionally robust optimization approach, which hedges against distributional ambiguity. We analyze several theoretical properties of this TRO approach, including conservatism, finite-sample guarantees, and asymptotic properties. From the algorithmic perspective, we propose a Spectrum Search Algorithm to identify the full spectrum of optimal solutions to the TRO model. Additionally, we propose a new inexact column-and-constraint generation (C&CG) method for solving challenging robust optimization problems. In contrast to the original C&CG method, the i-C&CG method allows solutions to the master problems to be inexact, which is desirable when solving large-scale and/or challenging problems. It is equipped with a backtracking routine that controls the trade-off between bound improvement and inexactness. Numerical experiments demonstrate the computational advantages of our i-C&CG method over state-of-the-art C&CG methods. We then build on our i-C&CG algorithm to introduce a new hybrid (primal-dual) C&CG (hC&CG) algorithm that includes two sets of optimality cuts: primal- and dual-based cuts. Our results demonstrate the superior computational performance of the hC&CG algorithm over the state-of-the-art decomposition algorithms. In the second part of the dissertation, we propose new stochastic optimization methodologies for addressing emerging real-world optimization problems. Specifically, we propose new stochastic programming and distributionally robust optimization models for a home service routing and scheduling problem, and an operating room and anesthesiologist scheduling problem.
