We consider linear combinatorial optimization problems under uncertain disruptions that increase the cost coefficients of the objective function. A decision maker, or planner, can invest resources to probe the components (i.e., the coefficients) in order to learn their disruption status. In the proposed probing optimization problem, the planner, knowing just the disruptions’ probabilities, selects which components to probe subject to a probing budget in a first decision stage. Then, the uncertainty realizes, and the planner observes the disruption status of the probed components, after which the planner solves the combinatorial problem in the second stage. In contrast to standard two-stage stochastic optimization, the planner does not have access to the full uncertainty realization in the second stage. Consequently, the planner cannot directly optimize the second-stage objective function, which is given by the actual cost after disruptions, and the decisions have to be made based on an estimate of the cost. By assuming that the estimate is given by the conditional expected cost given the information revealed by probing, we reformulate the probing optimization problem as a bilevel problem with multiple followers and propose an exact algorithm based on a value function reformulation and three heuristic algorithms. We derive theoretical results that bound the value of information and the price of not having full information and a bound on the required probing budget that attains the same performance as full information. Our extensive computational experiments suggest that probing a fraction of the components is sufficient to yield large improvements in the optimal value, that our exact algorithm is competitive for small- to medium-scale instances, and that the proposed heuristics find high-quality solutions in large-scale instances. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by the Air Force Office of Scientific Research [Grant FA9550-22-1-0236] and the Division of Civil, Mechanical and Manufacturing Innovation [Grant CMMI 2145553]. 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.2024.0629 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0629 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

There is a recent proliferation of research on the integration of machine learning and optimization. One expansive area within this research stream is optimization over pre-trained predictive models, which proposes the use of pre-trained predictive models as surrogates for uncertain or highly complex objective functions. In this setting, features of the predictive models become decision variables in the optimization problem. Despite a recent surge in publications in this area, only a few papers note the importance of incorporating trust-region considerations in this decision-making pipeline, that is, enforcing solutions to be similar to the data used to train the predictive models. Without such constraints, the evaluation of the predictive model at solutions obtained from optimization cannot be trusted and the practicality of the solutions may be unreasonable. In this paper, we provide an overview of the approaches appearing in the literature to construct a trust region and propose three alternative approaches. Our numerical evaluation highlights that trust-region constraints learned through our newly proposed approaches compare favorably with previously suggested approaches, both in terms of solution quality and computational time. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. 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.2022.0312 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0312 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

We study a class of adversarial minimum-cost flow problems where the arcs are subject to multiple ripple effect disruptions that increase their usage cost. The locations of the disruptions’ epicenters are uncertain, and the decision maker seeks a flow that minimizes cost assuming the worst-case realization of the disruptions. We evaluate the damage to each arc using a linear model, where the damage is the cumulative damage of all disruptions affecting the arc; and a maximum model, where the damage is given by the most destructive disruption affecting the arc. For both models, the arcs’ costs after disruptions are represented with a mixed-integer feasible region, resulting in a robust optimization problem with a mixed-integer uncertainty set. The main challenge to solve the problem comes from a subproblem that evaluates the worst-case cost for a given flow plan. We show that for the linear model the uncertainty set can be decomposed into a series of single disruption problems, which leads to a polynomial time algorithm for the subproblem. The uncertainty set of the maximum model, however, cannot be decomposed, and we show that the subproblem under this model is NP-hard. For this case, we further present a big-M free binary reformulation of the uncertainty set based on conflict constraints that results in a significantly smaller formulation with tighter linear programming relaxations. We extend the models by considering a less conservative approach where only a subset of the disruptions can occur and show that the properties of the linear and maximum models also hold in this case. We test our proposed approaches over real road networks and synthetics instances and show that our methods achieve orders of magnitude improvements over a standard approach from the literature. History: Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by the Air Force Office of Scientific Research [Grant FA9550-22-1-0236] and the Office of Naval Research [Grant N00014-19-1-2329]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.1243 .

We study stochastic optimization problems with objective function given by the expectation of the maximum of two linear functions defined on the component random variables of a multivariate Gaussian distribution. We consider random variables that are arbitrarily correlated, and we show that the problem is NP-hard even if the space of feasible solutions is unconstrained. We exploit a closed-form expression for the objective function from the literature to construct a cutting-plane algorithm for a highly nonlinear function, which includes the evaluation of the cumulative distribution function and probability density function of a standard normal random variable with decision variables as part of the arguments. To exhibit the model’s applicability, we consider two featured applications. The first is daily fantasy sports, where the algorithm identifies entries with positive returns during the 2018–2019 National Football League season. The second is a special case of makespan minimization for two parallel machines and jobs with uncertain processing times; for the special case where the jobs are uncorrelated, we prove the equivalence between its deterministic and stochastic versions and show that our algorithm can deliver a constant-factor approximation guarantee for the problem. The results of our computational evaluation involving synthetic and real-world data suggest that our discretization and upper bounding techniques lead to significant computational improvements and that the proposed algorithm outperforms suboptimal solutions approaches. History: Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.1259 .

We study optimization problems where the objective function is modeled through feedforward neural networks with rectified linear unit (ReLU) activation. Recent literature has explored the use of a single neural network to model either uncertain or complex elements within an objective function. However, it is well known that ensembles of neural networks produce more stable predictions and have better generalizability than models with single neural networks, which motivates the investigation of ensembles of neural networks rather than single neural networks in decision-making pipelines. We study how to incorporate a neural network ensemble as the objective function of an optimization model and explore computational approaches for the ensuing problem. We present a mixed-integer linear program based on existing popular big-M formulations for optimizing over a single neural network. We develop a two-phase approach for our model that combines preprocessing procedures to tighten bounds for critical neurons in the neural networks with a Lagrangian relaxation-based branch-and-bound approach. Experimental evaluations of our solution methods suggest that using ensembles of neural networks yields more stable and higher quality solutions, compared with single neural networks, and that our optimization algorithm outperforms (the adaption of) a state-of-the-art approach in terms of computational time and optimality gaps. History: Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete.

In recent years the use of decision diagrams within the context of discrete optimization has proliferated. This paper continues this expansion by proposing the use of decision diagrams for modeling and solving binary optimization problems with quadratic constraints. The model proposes the use of multiple decision diagrams to decompose a quadratic matrix so that each individual diagram has provably limited size. The decision diagrams are then linked through channeling constraints to ensure that the solution represented is consistent across the decision diagrams and that the original quadratic constraints are satisfied. The resulting family of decision diagrams are optimized over by a dedicated cutting-plane algorithm akin to Benders decomposition. The approach is general, in that commercial integer programming solvers can readily apply the technique. A thorough experimental evaluation on both benchmark and synthetic instances exhibits that the proposed decision diagram reformulation provides significant improvements over current methods for quadratic constraints in state-of-the-art solvers.

We study a class of sequential defender-attacker optimization problems where the defender’s objective is uncertain and depends on the operations of the attacker, which are represented by a mixed-integer uncertainty set. The defender seeks to hedge against the worst possible data realization, resulting in a robust optimization problem with a mixed-integer uncertainty set, which requires the solution of a challenging mixed-integer problem, which can be seen as a saddle-point problem over a nonconvex domain. We study two exact solution algorithms and present two feature applications for which the uncertainty is naturally modeled as a mixed-integer set. Our computational experiments show that the considered algorithms greatly outperform standard algorithms both in terms of computational time and solution quality. Moreover, our results show that modeling uncertainty with mixed-integer sets, instead of approximating the data using convex sets, results in less conservative solutions, which translates to a lower cost for the defender to protect from uncertainty. Summary of Contribution: We consider a class of defender-attacker problems where the defender has to make operational decisions that depend on uncertain actions from an adversarial attacker. Due to the type of information available to the defender, neither probabilistic modeling, nor robust optimization methods with convex uncertainty sets, are well suited to address the defender’s decision-making problem. Consequently, we frame the defender’s problem as a class of robust optimization problems with a mixed-integer uncertainty sets, and devise two exact algorithms that solve this class of problems. A comprehensive computational study shows that for the considered applications, our algorithms improves the performance of existing robust optimization approaches that can be adapted to solve this class of problems. Moreover, we show how mixed-integer uncertainty sets can reduce the level of over-conservatism that is a known issue of robust optimization approaches.

Abstract A constrained shortest path is a minimum‐cost sequence of arcs on a directed network that satisfies knapsack‐type constraints on the resource consumption over the arcs. We propose an exact method based on a recursive depth‐first search procedure known as the pulse algorithm (PA). One of the key contributions of the proposal lies in a bidirectional search strategy leveraged on parallelism. In addition, we developed a pulse‐based heuristic that quickly finds near‐optimal solutions and shows great potential for column generation (CG) schemes. We present computational experiments over large real‐road networks with up to 6 million nodes and 15 million arcs. We illustrate the use of the bidirectional PA in a CG scheme to solve a multi‐activity shift scheduling problem, where the pricing problem is modeled as a constrained‐shortest path with multiple resource constraints.

This paper examines a class of three-stage sequential defender-attacker-defender problems. In these problems the defender first selects a subset of assets to protect, the attacker next damages a subset of unprotected assets in the “interdiction” stage, after which the defender optimizes a “recourse” problem over the surviving assets. These problems are notoriously difficult to optimize, and almost always require the recourse problem to be a convex optimization problem. Our contribution is a new approach to solving defender-attacker-defender problems. We require all variables in the first two stages to be binary-valued, but allow the recourse problem to take any form. The proposed framework focuses on solving the interdiction problem by restricting the defender to select a recourse decision from a sample of feasible vectors. The algorithm then iteratively refines the sample to force finite convergence to an optimal solution. We demonstrate that our algorithm not only solves interdiction problems involving NP-hard recourse problems within reasonable computational limits, but it also solves shortest path fortification and interdiction problems more efficiently than state-of-the-art algorithms tailored for that problem, finding optimal solutions to real-road networks having up to 300,000 nodes and over 1,000,000 arcs.