Most recommender systems optimize individual item preferences rather than session-level business metrics, misaligning algorithmic objectives with platform goals. We propose a two-stage framework that directly optimizes session-level click-through rates (CTR) using counterfactual learning and trust-region constraints. Stage one trains models to predict positive session outcomes using collaborative filtering features. Stage two optimizes over these models with trust-region regularization to find alternative sessions that maximize expected CTR while ensuring prediction reliability. Using NetEase Cloud Music sessions, our LightGBM-based framework delivers substantial CTR gains across session sizes while staying within validated domains. It enables direct session-level optimization, integrates robust feedback, and applies trust regions, providing practical, business-aligned recommendations.

Linear programming (LP) relaxations are widely employed in exact solution methods for multilinear programs (MLPs). These relaxations can be obtained by using recursive McCormick linearizations (RMLs), by which an MLP is linearized by iteratively substituting bilinear products with artificial variables and constraints. This article introduces a systematic approach to identifying RMLs. We focus on identifying RMLs with a small number of artificial variables and strong LP bounds. We present a novel mechanism for representing all the possible RMLs, which we use to design an exact mixed-integer programming (MIP) formulation to identify minimum-size RMLs; this problem is NP-hard in general, but we show that it is fixed-parameter tractable if each monomial is composed of at most three variables. Moreover, we explore the structural properties of our formulation to derive an exact MIP model that identifies RMLs of a given size with the best-possible LP relaxation bound. We test our algorithms by conducting numerical experiments on a large collection of MLPs. Numerical results indicate that the RMLs obtained with our algorithms can be significantly smaller than those derived from heuristic or greedy approaches, leading, in many cases, to tighter LP relaxation bounds. Moreover, our linearization strategies can be used to reformulate MLPs as quadratically constrained programs (QCPs), which can then be efficiently solved using state-of-the-art solvers for QCPs. This QCP-based solution approach is highly beneficial for hard MLP instances. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete.

Ride-hailing services have expanded the role of shared mobility in passenger transportation systems, creating new markets and creative planning solutions for major urban centers. In this paper, we consider their use for the first-mile or last-mile passenger transportation in coordination with a mass transit service to provide a seamless multimodal transportation experience for the user. A system that provides passengers with predictable information on travel and waiting times in their commutes is immensely valuable. We envision that the passengers will inform the system of their desired travel and arrival windows so that the system can jointly optimize the schedules of passengers. The problem we study balances minimizing travel time and the number of trips taken by the last-mile vehicles, so that long-term planning, maintenance, and environmental impact are all taken into account. We focus on the case where the last-mile service aggregates passengers by destination. We show that this problem is NP-hard, and we propose a decision diagram–based branch-and-price decomposition model that can solve instances of real-world size (10,000 passengers spread over an hour, 50 last-mile destinations, 600 last-mile vehicles) in computational time (∼1 minute) that is orders of magnitude faster than the solution times of other methods appearing in the literature. Our experiments also indicate that aggregating passengers by destination on the last-mile service provides high-quality solutions to more general settings. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods and 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.2019.0163 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2019.0163 ). 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 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.

This paper provides a novel framework for solving multiobjective discrete optimization problems with an arbitrary number of objectives. Our framework represents these problems as network models, in that enumerating the Pareto frontier amounts to solving a multicriteria shortest-path problem in an auxiliary network. We design techniques for exploiting network models in order to accelerate the identification of the Pareto frontier, most notably a number of operations to simplify the network by removing nodes and arcs while preserving the set of nondominated solutions. We show that the proposed framework yields orders-of-magnitude performance improvements over existing state-of-the-art algorithms on five problem classes containing both linear and nonlinear objective functions. Summary of Contribution: Multiobjective optimization has a long history of research with applications in several domains. Our paper provides an alternative modeling and solution approach for multiobjective discrete optimization problems by leveraging graphical structures. Specifically, we encode the decision space of a problem as a layered network and propose graph reduction operators to preserve only solutions whose image are part of the Pareto frontier. The nondominated solutions can then be extracted through shortest-path algorithms on such a network. Numerical results comparing our method with state-of-the-art approaches on several problem classes, including the knapsack, set covering, and the traveling salesperson problem (TSP), suggest orders-of-magnitude runtime speed-ups for exactly enumerating the Pareto frontier, especially when the number of objective functions grows.

Business research practice is witnessing a surge in the integration of predictive modeling and prescriptive analysis. We describe a modeling framework JANOS that seamlessly integrates the two streams of analytics, allowing researchers and practitioners to embed machine learning models in an end-to-end optimization framework. JANOS allows for specifying a prescriptive model using standard optimization modeling elements such as constraints and variables. The key novelty lies in providing modeling constructs that enable the specification of commonly used predictive models within an optimization model, have the features of the predictive model as variables in the optimization model, and incorporate the output of the predictive models as part of the objective. The framework considers two sets of decision variables: regular and predicted. The relationship between the regular and the predicted variables is specified by the user as pretrained predictive models. JANOS currently supports linear regression, logistic regression, and neural network with rectified linear activation functions. In this paper, we demonstrate the flexibility of the framework through an example on scholarship allocation in a student enrollment problem and provide a numeric performance evaluation. Summary of Contribution. This paper describes a new software tool, JANOS, that integrates predictive modeling and discrete optimization to assist decision making. Specifically, the proposed solver takes as input user-specified pretrained predictive models and formulates optimization models directly over those predictive models by embedding them within an optimization model through linear transformations.

Prescriptive analytics provides organizations with scalable solutions for large-scale, automated decision making. At the core of prescriptive analytics methodology is optimization, a field devoted to the study of algorithms that solve complex decision-making problems. Optimization algorithms rely heavily on generic methods for identifying tight bounds, which provide both solutions to problems and optimality guarantees. In the last decade, decision diagrams (DDs) have demonstrated significant advantages in obtaining bounds compared with the standard linear relaxation commonly used by commercial solvers. However, the quality of the bounds computed by DDs depends heavily on the variable ordering chosen for the construction. Besides, the problem of finding an ordering that optimizes a given metric is generally NP-hard. This paper studies how machine learning, specifically deep reinforcement learning (DRL), can be used to improve bounds provided by DDs, in particular through learning a good variable ordering. The introduced DRL models improve primal and dual bounds, even over standard linear programming relaxations, and are integrated in a full-fledged branch-and-bound algorithm. This paper, therefore, provides a novel mechanism for utilizing machine learning to tighten bounds, adding to recent research on using machine learning to obtain high-quality heuristic solutions and, for the first time, using machine learning to improve relaxation bounds through a generic bounding method. We apply the methods on a classic optimization problem, the maximum independent set, and demonstrate through computational testing that optimization bounds can be significantly improved through DRL. We provide the code to replicate the results obtained on the maximum independent set. Summary of Contribution: This paper studies the use of reinforcement learning to compute a variable ordering of decision diagram-based approximations for discrete optimization problems. This is among the first works to propose the use of machine learning to improve upon generic bounding methods for discrete optimization problems, thereby establishing a critical bridge between optimization and learning.

In the bin-packing problem with minimum color fragmentation (BPPMCF), we are given a fixed number of bins and a collection of items, each associated with a size and a color, and the goal is to avoid color fragmentation by packing items with the same color within as few bins as possible. This problem emerges in areas as diverse as surgical scheduling and group event seating. We present several optimization models for the BPPMCF, including baseline integer programming formulations, alternative integer programming formulations based on two recursive decomposition strategies that utilize decision diagrams, and a branch-and-price algorithm. Using the results from an extensive computational evaluation on synthetic instances, we train a decision tree model that predicts which algorithm should be chosen to solve a given instance of the problem based on a collection of derived features. Our insights are validated through experiments on the aforementioned applications on real-world data. Summary of Contribution: In this paper, we investigate a colored variant of the bin-packing problem. We present and evaluate several exact mixed-integer programming formulations to solve the problem, including models that explore recursive decomposition strategies based on decision diagrams and a set partitioning model that we solve using branch and price. Our results show that the computational performance of the algorithms depends on features of the input data, such as the average number of items per bin. Our algorithms and featured applications suggest that the problem is of practical relevance and that instances of reasonable size can be solved efficiently.

Quantum annealing (QA) can be used to quickly obtain near-optimal solutions for quadratic unconstrained binary optimization (QUBO) problems. In QA hardware, each decision variable of a QUBO should be mapped to one or more adjacent qubits in such a way that pairs of variables defining a quadratic term in the objective function are mapped to some pair of adjacent qubits. However, qubits have limited connectivity in existing QA hardware. This has spurred work on preprocessing algorithms for embedding the graph representing problem variables with quadratic terms into the hardware graph representing qubits adjacencies, such as the Chimera graph in hardware produced by D-Wave Systems. In this paper, we use integer linear programming to search for an embedding of the problem graph into certain classes of minors of the Chimera graph, which we call template embeddings. One of these classes corresponds to complete bipartite graphs, for which we show the limitation of the existing approach based on minimum odd cycle transversals (OCTs). One of the formulations presented is exact and thus can be used to certify the absence of a minor embedding using that template. On an extensive test set consisting of random graphs from five different classes of varying size and sparsity, we can embed more graphs than a state-of-the-art OCT-based approach, our approach scales better with the hardware size, and the runtime is generally orders of magnitude smaller. Summary of Contribution: Our work combines classical and quantum computing for operations research by showing that integer linear programming can be successfully used as a preprocessing step for adiabatic quantum optimization. We use it to determine how a quadratic unconstrained binary optimization problem can be solved by a quantum annealer in which the qubits are coupled as in a Chimera graph, such as in the quantum annealers currently produced by D-Wave Systems. The paper also provides a timely introduction to adiabatic quantum computing and related work on minor embeddings.

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.

Finding tight bounds on the optimal solution is a critical element of practical solution methods for discrete optimization problems. In the last decade, decision diagrams (DDs) have brought a new perspective on obtaining upper and lower bounds that can be significantly better than classical bounding mechanisms, such as linear relaxations. It is well known that the quality of the bounds achieved through this flexible bounding method is highly reliant on the ordering of variables chosen for building the diagram, and finding an ordering that optimizes standard metrics is an NP-hard problem. In this paper, we propose an innovative and generic approach based on deep reinforcement learning for obtaining an ordering for tightening the bounds obtained with relaxed and restricted DDs. We apply the approach to both the Maximum Independent Set Problem and the Maximum Cut Problem. Experimental results on synthetic instances show that the deep reinforcement learning approach, by achieving tighter objective function bounds, generally outperforms ordering methods commonly used in the literature when the distribution of instances is known. To the best knowledge of the authors, this is the first paper to apply machine learning to directly improve relaxation bounds obtained by general-purpose bounding mechanisms for combinatorial optimization problems.

Knapsack problems play a pivotal role in the operations research literature, with various generalizations proposed and studied over the last century. Of recent interest is the quadratic multiknapsack problem (QMKP). Despite a plethora of heuristics, no exact methods for the QMKP have been published in the literature. This paper presents an exact branch-and-price algorithm for the QMKP. Experimental results indicate that the proposed algorithm is far superior, both in terms of solution times and objective function bounds, to state-of-the-art optimization technology solving a standard encoding of the problem. In addition to the algorithmic contribution, this paper studies the optimization problem of seating attendees at events, an operational challenge faced by event organizers. An optimization model for table event seating is shown to be closely related to the QMKP, and computational testing indicates that the proposed algorithm is particularly well suited for this application.

We propose a general branch-and-bound algorithm for discrete optimization in which binary decision diagrams (BDDs) play the role of the traditional linear programming relaxation. In particular, relaxed BDD representations of the problem provide bounds and guidance for branching, and restricted BDDs supply a primal heuristic. Each problem is given a dynamic programming model that allows one to exploit recursive structure, even though the problem is not solved by dynamic programming. A novel search scheme branches within relaxed BDDs rather than on values of variables. Preliminary testing shows that a rudimentary BDD-based solver is competitive with or superior to a leading commercial integer programming solver for the maximum stable set problem, the maximum cut problem on a graph, and the maximum 2-satisfiability problem. Specific to the maximum cut problem, we tested the BDD-based solver on a classical benchmark set and identified tighter relaxation bounds than have ever been found by any technique, nearly closing the entire optimality gap on four large-scale instances.

We explore the idea of obtaining bounds on the value of an optimization problem from a discrete relaxation based on binary decision diagrams (BDDs). We show how to construct a BDD that represents a relaxation of a 0-1 optimization problem, and how to obtain a bound for a separable objective function by solving a shortest (or longest) path problem in the BDD. As a test case we apply the method to the maximum independent set problem on a graph. We find that for most problem instances, it delivers tighter bounds in less computation time, than state-of-the-art integer programming software obtains by solving a continuous relaxation augmented with cutting planes.

We study propagation for the Sequence constraint in the context of constraint programming based on limited-width MDDs. Our first contribution is proving that establishing MDD-consistency for Sequence is NP-hard. Yet, we also show that this task is fixed parameter tractable with respect to the length of the sub-sequences. In addition, we propose a partial filtering algorithm that relies on a specific decomposition of the constraint and a novel extension of MDD filtering to node domains. We experimentally evaluate the performance of our proposed filtering algorithm, and demonstrate that the strength of the MDD propagation increases as the maximum width is increased. In particular, MDD propagation can outperform conventional domain propagation for Sequence by reducing the search tree size and solving time by several orders of magnitude. Similar improvements are observed with respect to the current best MDD approach that applies the decomposition of Sequence into Among constraints.