We address high-dimensional zero-one random parameters in two-stage convex conic optimization problems. Such parameters typically represent failures of network elements and constitute rare, high-impact random events in several applications. Given a sparse training dataset of the parameters, we motivate and study a distributionally robust formulation of the problem using a Wasserstein ambiguity set centered at the empirical distribution. We present a simple, tractable, and conservative approximation of this problem that can be efficiently computed and iteratively improved. Our method relies on a reformulation that optimizes over the convex hull of a mixed-integer conic programming representable set, followed by an approximation of this convex hull using lift-and-project techniques. We illustrate the practical viability and strong out-of-sample performance of our method on nonlinear optimal power flow problems affected by random contingencies, and we report improvements of up to 20% over existing methods. MotivationThis work is motivated by optimization problems arising in applications that are affected by an extremely large, yet finite, number of rare, high-impact random events. In particular, we are motivated by applications in which the decision-relevant random events consist of high-dimensional binary outcomes. Such applications are ubiquitous in network optimization, where the uncertain 1
Dantzig-Wolfe decomposition (DWD) is a classical algorithm for solving large-scale linear programs whose constraint matrix involves a set of independent blocks coupled with a set of linking rows. The algorithm decomposes such a problem into a master problem and a set of independent subproblems that can be solved in a distributed manner. In a typical implementation, the master problem is solved centrally. In certain settings, solving the master problem centrally is undesirable or infeasible. For example, in the case of decentralized storage of data, or when independent agents who are responsible for the subproblems desire privacy of information. In this paper, we propose a fully distributed DWD algorithm that relies on solving the master problem using a distributed consensus-based Alternating Direction Method of Multipliers (ADMM) algorithm. We derive error bounds on the optimality gap and feasibility violation of the proposed approach. We provide preliminary computational results for our algorithm using a Message Passing Interface (MPI) implementation on randomly generated instances where we obtain high quality solutions.
Problem definition: Although physical (or “social”) distancing is an important public health intervention during airborne pandemics, physical distancing dramatically reduces the effective capacity of classrooms. Academic/practical relevance: During the COVID-19 pandemic, this presented a unique problem to campus planners who hoped to deliver a meaningful amount of in-person instruction in a way that respected physical distancing. This process involved (1) assigning a mode to each offered class as remote, residential (in-person), or hybrid and (2) reassigning classrooms under severely reduced capacities to the non-remote classes. These decisions need to be made quickly and under several constraints and competing priorities, such as restrictions on changes to the timetable of classes, trade-offs between classroom density and educational benefits of in-person versus online instruction, and administrative preferences for course modes and classrooms reassignments. Methodology: We solve a flexible integer program and use hierarchical optimization to handle the multiple criteria according to priorities. We apply our methods using actual Georgia Institute of Technology (GT) student registration data, COVID-19–adjusted classroom and laboratory capacities, and departmental course mode delivery preferences. We generate optimal classroom assignments for all GT classes at the Atlanta campus and quantify the trade-offs among the competing priorities. Results: When classroom capacities decreased to 20%–25% of their normal seating capacities, optimization afforded students 15.5% more in-person contact hours compared with no room reassignments (NRRs). Among sections with an in-person preference, our model satisfies 87% of mode preferences, whereas only 47% are satisfied under NRR. Additionally, in a scenario in which all classes are preferred to be delivered in person, our model can satisfy 90% of mode preferences compared with 37% under NRR. Managerial implications: Multiobjective optimization is well suited for classroom assignment problems that campus planners usually manage sequentially and manually. Our models are computationally efficient and flexible, with the ability to handle multiple objectives with different priorities and build a new class-classrooms assignment or optimize an existing one, and they can apply under normal or sudden capacity scarcity constraints.
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