Residuals-based distributionally robust optimization with covariate information

We consider data-driven approaches that integrate a machine learning prediction model within distributionally robust optimization (DRO) given limited joint observations of uncertain parameters and covariates. Our framework is flexible in the sense that it can accommodate a variety of learning setups and DRO ambiguity sets. We investigate the asymptotic and finite sample properties of solutions obtained using Wasserstein, sample robust optimization, and phi-divergence-based ambiguity sets within our DRO formulations, and explore cross-validation approaches for sizing these ambiguity sets. Through numerical experiments, we validate our theoretical results, study the effectiveness of our approaches for sizing ambiguity sets, and illustrate the benefits of our DRO formulations in the limited data regime even when the prediction model is misspecified

Optimization Methods in Electric Power Systems: Global Solutions for Optimal Power Flow and Algorithms for Resilient Design under Geomagnetic Disturbances

An electric power system is a network of various components that generates and delivers power to end users. Since 1881, U.S. electric utilities have supplied power to billions of industrial, commercial, public, and residential customers continuously. Given the rapid growth of power utilities, power system optimization has evolved with developments in computing and optimization theory. In this dissertation, we focus on two optimization problems associated with power system planning: the AC optimal power flow (ACOPF) problem and the optimal transmission line switching (OTS) problem under geomagnetic disturbances (GMDs). The former problem is formulated as a nonlinear, non-convex network optimization problem, while the latter is the network design version of the ACOPF problem that allows topology reconfiguration and considers space weather-induced effects on power systems. Overall, the goal of this research includes: (1) developing computationally efficient approaches for the ACOPF problem in order to improve power dispatch efficiency and (2) identifying an optimal topology configuration to help ISO operate power systems reliably and efficiently under geomagnetic disturbances. Chapter 1 introduces the problems we are studying and motivates the proposed research. We present the ACOPF problem and the state-of-the-art solution methods developed in recent years. Next, we introduce geomagnetic disturbances and describe how they can impact electrical power systems. In Chapter 2, we revisit the polar power-voltage formulation of the ACOPF problem and focus on convex relaxation methods to develop lower bounds on the problem objective. Based on these approaches, we propose an adaptive, multivariate partitioning algorithm with bound tightening and heuristic branching strategies that progressively improves these relaxations and, given sufficient time, converges to the globally optimal solution. Computational results show that our methodology provides a computationally tractable approach to obtain tight relaxation bounds for hard ACOPF cases from the literature. In Chapter 3, we focus on the impact that extreme GMD events could potentially have on the ability of a power system to deliver power reliably. We develop a mixed-integer, nonlinear model which captures and mitigates GMD effects through line switching, generator dispatch, and load shedding. In addition, we present a heuristic algorithm that provides high-quality solutions quickly. Our work demonstrates that line switching is an effective way to mitigate GIC impacts. In Chapter 4, we extend the preliminary study presented in Chapter 3 and further consider the uncertain nature of GMD events. We propose a two-stage distributionally robust (DR) optimization model that captures geo-electric fields induced by uncertain GMDs. Additionally, we present a reformulation of a two-stage DRO that creates a decomposition framework for solving our problem. Computational results show that our DRO approach provides solutions that are robust to errors in GMD event predictions. Finally, in Chapter 5, we summarize the research contributions of our work and provide directions for future research

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

Decomposition Methods in Column Generation and Data-Driven Stochastic Optimization

In this thesis, we are focused on tackling large-scale problems arising in two-stage stochastic optimization and the related Dantzig-Wolfe decomposition. We start with a deterministic setting, where we consider linear programs with a block-structure, but data cannot be stored centrally due to privacy concerns or decentralized storage of large datasets. The larger portion of the thesis is dedicated to the stochastic setting, where we study two-stage distributionally robust optimization under the Wasserstein ambiguity set to tackle problems with limited data. In Chapter 2, joint work with Shabbir Ahmed, we propose a fully distributed Dantzig-Wolfe decomposition (DWD) algorithm using the Alternating Direction Method of Multipliers (ADMM) method. DWD is a classical algorithm used to solve large-scale linear programs whose constraint matrix is a set of independent blocks coupled with a set of linking rows but requires to solve a master problem centrally, which can be undesirable or infeasible in certain cases due to privacy concerns or decentralized storage of data. To this end, we develop a consensus-based Dantzig-Wolfe decomposition algorithm where the master problem is solved in a distributed fashion. We detail the computational and algorithmic challenges of our method, provide bounds on the optimality gap and feasibility violation, and perform extensive computational experiments on instances of the cutting stock problem and synthetic instances using a Message Passing Interface (MPI) implementation, where we obtain high-quality solutions in reasonable time. In Chapter 3 and 4, we turn our focus to stochastic optimization, specifically applications where data is scarce and the underlying probability distribution is difficult to estimate. Chapter 3 is joint work with Anirudh Subramanyam and Kibaek Kim. Here, we consider two-stage conic DRO under the Wasserstein ambiguity set with zero-one uncertainties. We are motivated by problems arising in network optimization, where binary random variables represent failures of network components. We are interested in applications where such failures are rare and have a high impact, making it difficult to estimate failure probabilities. By using ideas from bilinear programming and penalty methods, we provide tractable approximations of our two-stage DRO model which can be iteratively improved using lift-and-project techniques. We illustrate the computational and out-of-sample performance of our method on the optimal power flow problem with random transmission line failures and a multi-commodity network design problem with random node failures. In Chapter 4, joint work with Alejandro Toriello and George Nemhauser, we study a two-stage model which arises in natural disaster management applications, where the first stage is a facility location problem, deciding where to open facilities and pre-allocate resources, and the second stage is a fixed-charge transportation problem, routing resources to affected areas after a disaster. We solve a two-stage DRO model under the Wasserstein set to deal with the lack of available data. The presence of binary variables in the second stage significantly complicates the problem. We develop an efficient column-and-constraint generation algorithm by leveraging the structure of our support set and second-stage value function, and show our results extend to the case where the second stage is a fixed-charge network flow problem. We provide a detailed discussion on our implementation, and end the chapter with computational experiments on synthetic instances and a case study of hurricane threats on the coastal states of the United States. We end the thesis with concluding remarks and potential directions for future research.Ph.D