Distributionally Robust Optimization: A Review

The concepts of risk-aversion, chance-constrained optimization, and robust optimization have developed significantly over the last decade. Statistical learning community has also witnessed a rapid theoretical and applied growth by relying on these concepts. A modeling framework, called distributionally robust optimization (DRO), has recently received significant attention in both the operations research and statistical learning communities. This paper surveys main concepts and contributions to DRO, and its relationships with robust optimization, risk-aversion, chance-constrained optimization, and function regularization

Distributionally Robust Hydrogen Optimization with Ensured Security and Multi-Energy Couplings

Power-to-gas (P2G) can convert excessive renewable energy into hydrogen via electrolysis, which can then be transported by natural gas systems to bypass constrained electricity systems. However, the injection of hydrogen could impact gas quality since gas composition fundamentally changes, adversely effecting the combustion, safety and lifespan of appliances. This paper develops a new gas quality management scheme for hydrogen injection into natural gas systems produced from excessive wind power. It introduces four gas quality indices for the integrated electricity and gas system (IEGS) measuring gas quality, considering the coordinated operation of tightly coupled infrastructures. To maintain gas quality under an acceptable range, the gas mixture of nitrogen and liquid petroleum gas with hydrogen is adopted to address the gas quality violation caused by hydrogen injection. A distributionally robust optimization (DRO) modelled by Kullback-Leibler (KL) divergence-based ambiguity set is applied to flexibly control the robustness to capture wind uncertainty. Case studies demonstrate that wind power is maximally utilized and gas mixture is effectively managed, thus improving both gas quality and performance of IEGS. The work can benefit system operators with i) ensured gas quality under hydrogen injection ii) low system operation cost and iii) high renewable energy penetratio

Convex Nonlinear and Integer Programming Approaches for Distributionally Robust Optimization of Complex Systems

The primary focus of the dissertation is to develop distributionally robust optimization (DRO) models and related solution approaches for decision making in energy and healthcare service systems with uncertainties, which often involves nonlinear constraints and discrete decision variables. Without assuming specific distributions, DRO techniques solve for solutions against the worst-case distribution of system uncertainties. In the DRO framework, we consider both risk-neutral (e.g., expectation) and risk-averse (e.g., chance constraint and Conditional Value-at-Risk (CVaR)) measures. The aim is twofold: i) developing efficient solution algorithms for DRO models with integer and/or binary variables, sometimes nonlinear structures and ii) revealing managerial insights of DRO models for specific applications. We mainly focus on DRO models of power system operations, appointment scheduling, and resource allocation in healthcare. Specifically, we first study stochastic optimal power flow (OPF), where (uncertain) renewable integration and load control are implemented to balance supply and (uncertain) demand in power grids. We propose a chance-constrained OPF (CC-OPF) model and investigate its DRO variant which is reformulated as a semidefinite programming (SDP) problem. We compare the DRO model with two benchmark models, in the IEEE 9-bus, 39-bus, and 118-bus systems with different flow congestion levels. The DRO approach yields a higher probability of satisfying the chance constraints and shorter solution time. It also better utilizes reserves at both generators and loads when the system has congested flows. Then we consider appointment scheduling under random service durations with given (fixed) appointment arrival order. We propose a DRO formulation and derive a conservative SDP reformulation. Furthermore, we study a scheduling variant under random no-shows of appointments and derive tractable reformulations for certain beliefs of no-show patterns. One preceding problem of appointment scheduling in the healthcare service operations is the surgery block allocation problem that assigns surgeries to operating rooms. We derive an equivalent 0-1 SDP reformulation and a less conservative 0-1 second-order cone programming (SOCP) reformulation for its DRO model. Finally, we study distributionally robust chance-constrained binary programs (DCBP) for limiting the probability of undesirable events, under mean-covariance information. We reformulate DCBPs as equivalent 0-1 SOCP formulations under two moment-based ambiguity sets. We further exploit the submodularity of the 0-1 SOCP reformulations under diagonal and non-diagonal matrices. We derive extended polymatroid inequalities via submodularity and lifting, which are incorporated into a branch-and-cut algorithm incorporated for efficiently solving DCBPs. We demonstrate the computational efficacy and solution performance with diverse instances of a chance-constrained bin packing problem.PHDIndustrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/149946/1/zyiling_1.pd

Addressing Nonlinearity and Uncertainty via Mixed Integer Programming Approaches

The main focus of the dissertation is to develop decision-making support tools that address nonlinearity and uncertainty appearing in real-world applications via mixed integer programming (MIP) approaches. When making decisions under uncertainty, knowing the accurate probability distributions of the uncertain parameters can help us predict their future realization, which in turn helps to make better decisions. In practice, however, it is oftentimes hard to estimate such a distribution precisely. As a consequence, if the estimated distribution is biased, the decisions thus made can end up with disappointing outcomes. To address this issue, we model uncertainty in the decision-making process through a distributionally robust optimization (DRO) approach, which aims to find a solution that hedges against the worst-case distribution within a pre-defined ambiguity set, i.e., a collection of probability distributions that share some distributional and/or statistical characteristics in common. The role of the ambiguity set is crucial as it affects both solution quality and computational tractability of the DRO model. In this dissertation, we tailor the ambiguity sets based on the available historical data in healthcare operations and energy systems, and derive efficient solution approaches for the DRO models via MIP approaches. Through extensive numerical studies, we show that the DRO solutions yield better out-of-sample performance, and the computational performance of the proposed MIP approaches is encouraging. Finally, we provide some managerial insights into the operations of healthcare and energy systems.PHDIndustrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155030/1/msryu_1.pd