Doctor of Philosophy, The Ohio State University, 2024, Industrial and Systems Engineering

Decision-making problems in real life often involve uncertainties. One way to address such problems is to use stochastic optimization, where quantifying a probability distribution to represent the underlying uncertainty is critical. However, most often, only partial information about the uncertainty is available through a series of historical data and expert knowledge. This limitation becomes particularly significant if the decision maker is risk averse and needs to consider rare but high-impact events, for which the probability distribution cannot be accurately determined even with the available historical data. Distributionally Robust Stochastic Optimization (DRO) is an alternative approach that assumes that the underlying distribution is unknown but instead lies in an ambiguity set of distributions that is consistent with the available data. DRO then tries to optimize the worst-case expectation among all distributions in the ambiguity set. This dissertation focuses on effective scenarios in DROs defined using a finite number of realizations (also called scenarios) of the uncertain parameters. Effective scenarios are the critical scenarios in DRO in the sense that their removal alters the optimal objective function value. Ineffective scenarios, on the other hand, can be removed safely without changing the optimal value. In this dissertation, we investigate both the theoretical and computational aspects of effective scenarios. The first contribution of this dissertation links the effectiveness of a scenario to its worst-case distribution being always positive or uniquely zero under a general ambiguity set with finite support. We then narrow down our focus to DROs with ambiguity sets formed via the Cressie-Read power divergence family (DRO-CR) and the Wasserstein distance (DRO-W). This class of problems constitutes some of the most widely used DROs in the literature. We provide easy-to-check sufficient conditions to identify the effectiveness of scenarios fo (open full item for complete abstract)

Committee: Guzin Bayraksan (Advisor); Sam Davanloo (Committee Member); Cathy Xia (Committee Member) Subjects: Industrial Engineering; Operations Research

Doctor of Philosophy, The Ohio State University, 2018, Industrial and Systems Engineering

Many decision-making problems arising in science, engineering, and business involve uncertainties. One way to address these problems is to use stochastic optimization. A crucial task when building stochastic optimization models is quantifying a probability distribution to represent the uncertainty. Most often, partial information about the uncertainty is available through a series of historical data. In such circumstances, classical stochastic optimization models rely on approximating the underlying probability distribution. However, in many real-world applications, the underlying probability distribution cannot be accurately determined, even when historical data are available. This distributional ambiguity might lead to highly suboptimal decisions. An alternative approach to handle such an issue is to use distributionally robust stochastic optimization (DRSO for short), which assumes the underlying probability distribution is unknown but lies in an ambiguity set of distributions. Many existing studies on DRSO focus on how to construct the ambiguity set and how to transform the resulting DRSO into equivalent (well-studied) models such as mixed-integer programming and semidefinite programming. This dissertation, however, addresses more fundamental questions, in a different manner than the literature. An overarching question that motivates most of this dissertation is which scenarios/uncertainties are critical to a stochastic optimization problem? A major contribution of this dissertation is a precise mathematical definition of what is meant by a critical scenario and investigation on how to identify them for DRSO. As has never been done before for DRSO (to the best of our knowledge), we introduce the notion of effective and ineffective scenarios for DRSO. This dissertation considers DRSOs for which the ambiguity set contains all probability distributions that are not far---in the sense of the so-called total variation distance---from a (open full item for complete abstract)

Committee: Guzin Bayraksan PhD (Advisor); Antonio Conejo PhD (Committee Member); David Sivakoff PhD (Committee Member) Subjects: Industrial Engineering; Operations Research