Quantum computing has the potential to speed up some optimization methods. One can use quantum computers to solve linear systems via quantum linear system algorithms (QLSAs). QLSAs can be used as a subroutine for algorithms that require solving linear systems, such as the dual logarithmic barrier method (DLBM) for solving linear optimization (LO) problems. In this paper, we use a QLSA to solve the linear systems arising in each iteration of the DLBM. To use the QLSA in a hybrid setting, we read out quantum states via a tomography procedure that introduces considerable error and noise. Thus, this paper first proposes an inexact feasible variant of DLBM for LO problems and then extends it to a quantum version. Our quantum approach has quadratic convergence toward the central path with inexact directions, and we show that this method has the best known [Formula: see text] iteration complexity, where n is the number of variables, [Formula: see text] is the initial duality gap, and [Formula: see text] is the accuracy. We further use iterative refinement to improve the complexity dependence on accuracy and condition number. For LO problems with quadratically more constraints than variables, the quantum complexity of our method has a sublinear dependence on dimension. Funding: This research was supported by Defense Advanced Research Projects Agency [Grant W911NF2010022] and the National Science Foundation [Grant 2128527].

In this work, we propose Wasserstein distributionally robust shallow convex neural networks (WaDiRo-SCNNs) to provide reliable nonlinear predictions when subject to adverse and corrupted data sets. Our approach is based on the reformulation of a new convex training program for rectified linear unit–based shallow neural networks, and this allows us to cast the problem into the order-1 Wasserstein distributionally robust optimization framework. Our training procedure is conservative, has low stochasticity, is solvable with open-source solvers, and is scalable to large industrial deployments. We provide out-of-sample performance guarantees, show that hard convex physical constraints can be enforced in the training program, and propose a mixed-integer convex posttraining verification program to evaluate model stability. WaDiRo-SCNN aims to make neural networks safer for critical applications, such as in the energy sector. Finally, we numerically demonstrate our model’s performance through both a synthetic experiment and a real-world power system application, namely, the prediction of hourly energy consumption in nonresidential buildings within the context of virtual power plants, and evaluate its stability across standard regression benchmark data sets. The experimental results are convincing and showcase the strengths of the proposed model. Funding: This work was possible thanks to funding from the Fonds de recherche du Québec, the Natural Sciences and Engineering Research Council of Canada, Mitacs Accelerate, and Hilo by Hydro-Québec [Grants RGPIN-2023-04235, IT35303, and IT38517; Scholarships CGRS-M and B1X].

States in the United States redraw their electoral district boundaries every 10 years. This redistricting process can be contentious and has long-lasting consequences for political representation. To reduce bias in the redistricting process, some states require bipartisan commissions; however, bipartisan commissions can still involve partisan tension if the political parties cannot compromise. We propose an optimization framework to facilitate compromise between two redistricting stakeholders. This framework seeks a midpoint between two stakeholder plans with respect to a distance metric. A midpoint can help the stakeholders visualize a potential compromise that incorporates district structure common to both of their proposed plans. First, we consider multiple distance metrics and evaluate whether midpoints with respect to these metrics are achievable and align with redistricting requirements. Then we formulate a mixed-integer linear program to find a midpoint (or any fractional point) between two given plans with respect to the transfer distance. This formulation incorporates district structure from both given plans by fixing variables; consequently, it is possible to solve some realistically sized instances exactly in reasonable amounts of time. We present experiments on grid instances and Missouri’s congressional redistricting instance to demonstrate how this method can quickly generate compromise options that align with redistricting requirements. Funding: This material is based on work supported by the National Science Foundation Graduate Research Fellowship Program [Grant DGE-1746047]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoo.2023.0029 .

The matrix sensing problem is an important low-rank optimization problem that has found a wide range of applications, such as matrix completion, phase synchornization/retrieval, robust principal component analysis (PCA), and power system state estimation. In this work, we focus on the general matrix sensing problem with linear measurements that are corrupted by random noise. We investigate the scenario where the search rank r is equal to the true rank [Formula: see text] of the unknown ground truth (the exact parametrized case), as well as the scenario where r is greater than [Formula: see text] (the overparametrized case). We quantify the role of the restricted isometry property (RIP) in shaping the landscape of the nonconvex factorized formulation and assisting with the success of local search algorithms. First, we develop a global guarantee on the maximum distance between an arbitrary local minimizer of the nonconvex problem and the ground truth under the assumption that the RIP constant is smaller than [Formula: see text]. We then present a local guarantee for problems with an arbitrary RIP constant, which states that any local minimizer is either considerably close to the ground truth or far away from it. More importantly, we prove that this noisy, overparametrized problem exhibits the strict saddle property, which leads to the global convergence of perturbed gradient descent algorithm in polynomial time. The results of this work provide a comprehensive understanding of the geometric landscape of the matrix sensing problem in the noisy and overparametrized regime. Funding: This work was supported by grants from the National Science Foundation, Office of Naval Research, Air Force Office of Scientific Research, and Army Research Office.

Distributionally robust chance-constrained programs (DR-CCPs) over Wasserstein ambiguity sets exhibit attractive out-of-sample performance and admit big-M–based mixed-integer programming reformulations with conic constraints. However, the resulting formulations often suffer from scalability issues as problem size increases. To address this shortcoming, we derive stronger formulations that scale well with respect to the problem size. Our focus is on ambiguity sets under the so-called left-hand side uncertainty, where the uncertain parameters affect the coefficients of the decision variables in the linear inequalities defining the safety sets. The interaction between the uncertain parameters and the variable coefficients in the safety set definition causes challenges in strengthening the original big-M formulations. By exploiting the connection between nominal chance-constrained programs and DR-CCP, we obtain strong formulations with significant enhancements. In particular, through this connection, we derive a linear number of valid inequalities, which can be immediately added to the formulations to obtain improved formulations in the original space of variables. In addition, we suggest a quantile-based strengthening procedure that allows us to reduce the big-M coefficients drastically. Furthermore, based on this procedure, we propose an exponential class of inequalities that can be separated efficiently within a branch-and-cut framework. The quantile-based strengthening procedure can be expensive. Therefore, for the special case of covering and packing type problems, we identify an efficient scheme to carry out this procedure. We demonstrate the computational efficacy of our proposed formulations on two classes of problems, namely stochastic portfolio optimization and resource planning. Funding: This work was supported by the DARPA Lagrange Program [Grant N660011824020], the National Science Foundation [Grant 1740707], the Office of Naval Research [Grant N00014-19-1-2321], and the Institute for Basic Science [Grants IBS-R029-C1 and IBS-R029-Y2].

Column generation is an iterative method used to solve a variety of optimization problems. It decomposes the problem into two parts: a master problem and one or more pricing problems (PP). The total computing time taken by the method is divided between these two parts. In routing or scheduling applications, the problems are mostly defined on a network, and the PP is usually an NP-hard shortest path problem with resource constraints. In this work, we propose a new heuristic pricing algorithm based on machine learning. By taking advantage of the data collected during previous executions, the objective is to reduce the size of the network and accelerate the PP, keeping only the arcs that have a high chance to be part of the linear relaxation solution. The method has been applied to two specific problems: the vehicle and crew scheduling problem in public transit and the vehicle routing problem with time windows. Reductions in computational time of up to 40% can be obtained. Funding: The authors thank Giro, Inc., and the Natural Sciences and Engineering Research Council of Canada for financial support [Grant CRDPJ 520349-17].

Gradient sampling (GS) methods for the minimization of objective functions that may be nonconvex and/or nonsmooth are proposed, analyzed, and tested. One of the most computationally expensive components of contemporary GS methods is the need to solve a convex quadratic subproblem in each iteration. By contrast, the methods proposed in this paper allow the use of inexact solutions of these subproblems, which, as proved in the paper, can be incorporated without the loss of theoretical convergence guarantees. Numerical experiments show that, by exploiting inexact subproblem solutions, one can consistently reduce the computational effort required by a GS method. Additionally, a strategy is proposed for aggregating gradient information after a subproblem is solved (potentially inexactly) as has been exploited in bundle methods for nonsmooth optimization. It is proved that the aggregation scheme can be introduced without the loss of theoretical convergence guarantees. Numerical experiments show that incorporating this gradient aggregation approach can also reduce the computational effort required by a GS method.

We study a convex quadratic nonseparable resource allocation problem that arises in the area of decentralized energy management (DEM), where unbalance in electricity networks has to be minimized. In this problem, the given resource is allocated over a set of activities that is divided into subsets, and a cost is assigned to the overall allocated amount of resources to activities within the same subset. We derive two efficient algorithms with [Formula: see text] worst-case time complexity to solve this problem. For the special case where all subsets have the same size, one of these algorithms even runs in linear time given the subset size. Both algorithms are inspired by well-studied breakpoint search methods for separable convex resource allocation problems. Numerical evaluations on both real and synthetic data confirm the theoretical efficiency of both algorithms and demonstrate their suitability for integration in DEM systems.

We study the assignment problem with chance constraints (CAP) and its distributionally robust counterpart DR-CAP. We present a technique for estimating big-M in such a formulation that takes advantage of the ambiguity set. We consider a 0-1 bilinear knapsack set to develop valid inequalities for CAP and DR-CAP. This is generalized to the joint chance constraint problem. A probability cut framework is also developed to solve DR-CAP. A computational study on problem instances obtained from using real hospital surgery data shows that the developed techniques allow us to solve certain model instances and reduce the computational time for others. The use of Wasserstein ambiguity set in the DR-CAP model improves the out-of-sample performance of satisfying the chance constraints more significantly than the one possible by increasing the sample size in the sample average approximation technique. The solution time for DR-CAP model instances is of the same order as that for solving the CAP instances. This finding is important because chance constrained optimization models are very difficult to solve when the coefficients in the constraints are random.