Decomposition Methods in Column Generation and Data-Driven Stochastic Optimization
El Tonbari, Mohamed Ali El Moghazi
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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.