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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 Distributionally Favorable Optimization (DFO) is a framework for decision-making under uncertainty, with applications spanning various fields, including reinforcement learning, online learning, robust statistics, chance-constrained programming, and two-stage stochastic optimization without complete recourse. In contrast to the traditional Distributionally Robust Optimization (DRO) paradigm, DFO presents a unique challenge– the application of the inner infimum operator often fails to retain the convexity. In light of this challenge, we study the tractability and complexity of DFO. We establish sufficient and necessary conditions for determining when DFO problems are tractable (i.e., solvable in polynomial time) or intractable (i.e., not solvable in polynomial time). Despite the typical nonconvex nature of DFO problems, our results show that they are mixed-integer convex programming representable (MICP-R), thereby enabling solutions via standard optimization solvers. Finally, we numerically validate the efficacy of our MICP-R formulations. 

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]. 

In modeling battery energy storage systems (BESS) in power systems, binary variables are used to represent the complementary nature of charging and discharging. A conventional approach for these BESS optimization problems is to relax binary variables and convert the problem into a linear program. However, such linear programming relaxation models can yield unrealistic fractional solutions, such as simultaneous charging and discharging. In this paper, we develop a regularized mixed-integer programming (MIP) model for the optimal power flow (OPF) problem with BESS. We prove that, under mild conditions, the proposed regularized model admits a zero integrality gap with its linear programming relaxation; hence, it can be solved efficiently. By studying the properties of the regularized MIP model, we show that its optimal solution is also near optimal to the original OPF problem with BESS, thereby providing a valid and tight upper bound for the OPF problem with BESS. The use of the regularized MIP model allows us to solve a trilevel [Formula: see text]-[Formula: see text]-[Formula: see text] network contingency problem, which is otherwise intractable to solve. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: N. Jiang (as a graduate student at the Georgia Institute of Technology) and W. Xie were supported in part by the National Science Foundation [Grant 2246414] 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.2024.0771 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0771 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ . 

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. 

The soaring drug overdose crisis in the United States has claimed more than half a million lives in the past decade and remains a major public health threat. The ability to predict drug overdose deaths at the county level can help local communities develop action plans in response to emerging changes. Applying off-the-shelf machine learning algorithms for prediction can be challenging due to the heterogeneous risk profiles of the counties and suppressed data in common publicly available data sources. To fill these gaps, we develop a cluster-aware supervised learning (CASL) framework to enhance the prediction of county-level drug overdose deaths. This CASL model simultaneously clusters counties into groups based on geographical and socioeconomic characteristics and minimizes the loss function that accounts for suppressed values and cluster-specific regularization. Our computational study uses real-world data from 2010 to 2021, focusing on the ten states most severely impacted by the drug overdose crisis. The results demonstrate that our proposed CASL framework significantly outperforms state-of-the-art methods by achieving a superior balance in prediction accuracy for both unsuppressed and suppressed observations. The proposed model also identifies different clusters of counties, capturing heterogeneous patterns of overdose mortality among counties of diverse characteristics. 

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/ .