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Abstract A Low-rank Spectral Optimization Problem (LSOP) minimizes a linear objective function subject to multiple two-sided linear inequalities intersected with a low-rank and spectral constrained domain. Although solving LSOP is generally NP-hard, its partial convexification (i.e., replacing the domain with its convex hull), termed “LSOP-R, is often tractable and yields a high-quality solution. This motivates us to study the strength of LSOP-R. Specifically, we derive rank bounds for any extreme point of LSOP-R in different matrix spaces and prove their tightness. The proposed rank bounds recover two well-known results in the literature from a fresh angle and allow us to derive sufficient conditions under which the relaxation LSOP-R is equivalent to LSOP. To effectively solve LSOP-R, we develop a column generation algorithm with a vector-based convex pricing oracle and a rank-reduction algorithm, which ensures that the output solution always satisfies the theoretical rank bound. Finally, we numerically verify the strength of LSOP-R and the efficacy of the proposed algorithms. 

Abstract Urban air mobility (UAM) is an emerging air transportation mode to alleviate the ground traffic burden and achieve zero direct aviation emissions. Due to the potential economic scaling effects, the UAM traffic flow is expected to increase dramatically once implemented, and its market can be substantially large. To be prepared for the era of UAM, we study the fair and risk‐averse urban air mobility resource allocation model (FairUAM) under passenger demand and airspace capacity uncertainties for fair, safe, and efficient aircraft operations. FairUAM is a two‐stage model, where the first stage is the aircraft resource allocation, and the second stage is to fairly and efficiently assign the ground and airspace delays to each aircraft provided the realization of random airspace capacities and passenger demand. We show that FairUAM is NP‐hard even when there is no delay assignment decision or no aircraft allocation decision. Thus, we recast FairUAM as a mixed‐integer linear program (MILP) and explore model properties and strengthen the model formulation by developing multiple families of valid inequalities. The stronger formulation allows us to develop a customized exact decomposition algorithm with both benders and L‐shaped cuts, which significantly outperforms the off‐the‐shelf solvers. Finally, we numerically demonstrate the effectiveness of the proposed method and draw managerial insights when applying FairUAM to a real‐world network. 

D-optimal experimental design is a classical statistical problem in which one chooses a collection of data vectors, from some available large pool, in order to maximize a measure of predictive quality. In the classical formulation, the only constraint is on the cardinality of the collection, that is, the number of vectors chosen. We study a more general budget-constrained variant in which vectors have heterogeneous costs, and develop four new algorithms (two deterministic and two randomized) with approximation guarantees. Our methods handle heterogeneous costs using a novel exchange rule that interchanges packs of data vectors whose total costs are similar (up to some controlled amount of rounding error). The algorithms outperform the only existing method for this problem from both theoretical and empirical standpoints. Funding: The first and third authors gratefully acknowledge support from the National Science Foundation (NSF) Division of Civil, Mechanical and Manufacturing Innovation [Grant CMMI-2112828]. The second author gratefully acknowledges support from the NSF Division of Computing and Communication Foundations [Grant CCF-2246417] and Office of Naval Research [Grant N00014-24-1-2066]. 

Routing a Vehicle to Collect Data After an Earthquake In the immediate aftermath of a major earthquake, it is crucial to quickly and accurately assess structural damage throughout the region. It is especially important to identify buildings that have become unsafe in order to prioritize evacuation efforts. Only a very small number of building inspections can be feasibly performed in a narrow time frame; however, their results can then be combined with other data sources to predict damage at other locations that were not inspected. In “D-Optimal Orienteering for Postearthquake Reconnaissance Planning,” Wang, Xie, Ryzhov, Marković, and Ou present a novel nonlinear integer program that combines vehicle routing with a statistical objective, the goal being to maximize data quality. An exact method based on row and column generation is developed to solve problems with up to 200 buildings. The approach is validated in a realistic case study using real-world building data obtained from a state-of-the-art earthquake simulator. 

Sparse principal component analysis (SPCA) is designed to enhance the interpretability of traditional principal component analysis by optimally selecting a subset of features that comprise the first principal component. Given the NP-hard nature of SPCA, most current approaches resort to approximate solutions, typically achieved through tractable semidefinite programs or heuristic methods. To solve SPCA to optimality, we propose two exact mixed-integer semidefinite programs (MISDPs) and an arbitrarily equivalent mixed-integer linear program. The MISDPs allow us to design an effective branch-and-cut algorithm with closed-form cuts that do not need to solve dual problems. For the proposed mixed-integer formulations, we further derive the theoretical optimality gaps of their continuous relaxations. Besides, we apply the greedy and local search algorithms to solving SPCA and derive their first-known approximation ratios. Our numerical experiments reveal that the exact methods we developed can efficiently find optimal solutions for data sets containing hundreds of features. Furthermore, our approximation algorithms demonstrate both scalability and near-optimal performance when benchmarked on larger data sets, specifically those with thousands of features. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This research was supported in part by the Division of Civil, Mechanical and Manufacturing Innovation [Grant 224614], the Division of Computing and Communication Foundations [Grant 2246417], and the Office of Naval Research [Grant N00014-24-1-2066]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0372 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0372 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .