K-Adaptability in Two-Stage Distributionally Robust Binary Programming

We propose to approximate two-stage distributionally robust programs with binary recourse decisions by their associated K-adaptability problems, which pre-select K candidate secondstage policies here-and-now and implement the best of these policies once the uncertain parameters have been observed. We analyze the approximation quality and the computational complexity of the K-adaptability problem, and we derive explicit mixed-integer linear programming reformulations. We also provide efficient procedures for bounding the probabilities with which each of the K second-stage policies is selected

A study of distributionally robust mixed-integer programming with Wasserstein metric: on the value of incomplete data

This study addresses a class of linear mixed-integer programming (MILP) problems that involve uncertainty in the objective function parameters. The parameters are assumed to form a random vector, whose probability distribution can only be observed through a finite training data set. Unlike most of the related studies in the literature, we also consider uncertainty in the underlying data set. The data uncertainty is described by a set of linear constraints for each random sample, and the uncertainty in the distribution (for a fixed realization of data) is defined using a type-1 Wasserstein ball centered at the empirical distribution of the data. The overall problem is formulated as a three-level distributionally robust optimization (DRO) problem. First, we prove that the three-level problem admits a single-level MILP reformulation, if the class of loss functions is restricted to biaffine functions. Secondly, it turns out that for several particular forms of data uncertainty, the outlined problem can be solved reasonably fast by leveraging the nominal MILP problem. Finally, we conduct a computational study, where the out-of-sample performance of our model and computational complexity of the proposed MILP reformulation are explored numerically for several application domains

Theory and Applications of Robust Optimization

In this paper we survey the primary research, both theoretical and applied, in the area of Robust Optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.Comment: 50 page

Unit Commitment Predictor With a Performance Guarantee: A Support Vector Machine Classifier

The system operators usually need to solve large-scale unit commitment problems within limited time frame for computation. This paper provides a pragmatic solution, showing how by learning and predicting the on/off commitment decisions of conventional units, there is a potential for system operators to warm start their solver and speed up their computation significantly. For the prediction, we train linear and kernelized support vector machine classifiers, providing an out-of-sample performance guarantee if properly regularized, converting to distributionally robust classifiers. For the unit commitment problem, we solve a mixed-integer second-order cone problem. Our results based on the IEEE 6-bus and 118-bus test systems show that the kernelized SVM with proper regularization outperforms other classifiers, reducing the computational time by a factor of 1.7. In addition, if there is a tight computational limit, while the unit commitment problem without warm start is far away from the optimal solution, its warmly started version can be solved to optimality within the time limit

Hydrogen Supply Infrastructure Network Planning Approach towards Chicken-egg Conundrum

In the early commercialization stage of hydrogen fuel cell vehicles (HFCVs), reasonable hydrogen supply infrastructure (HSI) planning decisions is a premise for promoting the popularization of HFCVs. However, there is a strong causality between HFCVs and hydrogen refueling stations (HRSs): the planning decisions of HRSs could affect the hydrogen refueling demand of HFCVs, and the growth of demand would in turn stimulate the further investment in HRSs, which is also known as the ``chicken and egg'' conundrum. Meanwhile, the hydrogen demand is uncertain with insufficient prior knowledge, and thus there is a decision-dependent uncertainty (DDU) in the planning issue. This poses great challenges to solving the optimization problem. To this end, this work establishes a multi-network HSI planning model coordinating hydrogen, power, and transportation networks. Then, to reflect the causal relationship between HFCVs and HRSs effectively without sufficient historical data, a distributionally robust optimization framework with decision-dependent uncertainty is developed. The uncertainty of hydrogen demand is modeled as a Wasserstein ambiguity set with a decision-dependent empirical probability distribution. Subsequently, to reduce the computational complexity caused by the introduction of a large number of scenarios and high-dimensional nonlinear constraints, we developed an improved distribution shaping method and techniques of scenario and variable reduction to derive the solvable form with less computing burden. Finally, the simulation results demonstrate that this method can reduce costs by at least 10.4% compared with traditional methods and will be more effective in large-scale HSI planning issues. Further, we put forward effective suggestions for the policymakers and investors to formulate relevant policies and decisions

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