Constraint Learning to Define Trust Regions in Optimization over Pre-Trained Predictive Models

Multiobjective simulation optimization (MOSO) problems are optimization problems with multiple conflicting objectives, where evaluation of at least one of the objectives depends on a black-box numerical code or real-world experiment, which we refer to as a simulation. Whereas an extensive body of research is dedicated to developing new algorithms and methods for solving these and related problems, it is challenging and time-consuming to integrate these techniques into real-world production-ready solvers. This is partly because of the diversity and complexity of modern state-of-the-art MOSO algorithms and methods and partly because of the complexity and specificity of many real-world problems and their corresponding computing environments. The complexity of this problem is only compounded when introducing potentially complex and/or domain-specific surrogate-modeling techniques, problem formulations, design spaces, and data acquisition functions. This paper carefully surveys the current state of the art in MOSO algorithms, techniques, and solvers, as well as problem types and computational environments where MOSO is commonly applied. We then present several key challenges in the design of a parallel multiobjective simulation optimization framework (ParMOO) and how they have been addressed. Finally, we provide two case studies demonstrating how customized ParMOO solvers can be quickly built and deployed to solve real-world MOSO problems. History: Accepted by Alice Smith and Ted Ralphs, Area Editors for Software Tools. Funding: This work was supported by US DOE, Office of Science, Advanced Scientific Computing Research, SciDAC Program [Grants DE-AC02-05CH11231 and DE-AC02-06CH11357]. 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.2023.0250 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0250 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

In this paper, we present an efficient scenario reduction method for optimization problems whose objective function is the higher moment coherent risk (HMCR) measures. Compared with existing approaches, our method places greater emphasis on the characteristics of optimization problems. Because the value of HMCR measures only depends on a small subset of scenarios that correspond to high cost or loss, our approach is based mostly on the concept of ineffective scenarios previously proposed in the literature, which entails identifying the scenarios whose removal from the problem results in no change of the optimal value. We test our method on a simple portfolio optimization problem with only nine risky assets and a realistic one with 50 risky assets and cardinality constraints. Results show that our scenario reduction method can yield a more accurate optimal solution and optimal value, along with a smaller reduced-scenario set. Even at the same reduction level, our method continues to outperform the existing scenario reduction methods. Interestingly, the portfolio produced by our method is also more diversified than others. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was supported by the National Natural Science Foundation of China [Grants 71720107002 and U1901223]. 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.2022.0375 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0375 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Robust optimization (RO) is one of the key paradigms for solving optimization problems affected by uncertainty. Two principal approaches for RO, the robust counterpart method and the adversarial approach, potentially lead to excessively large optimization problems. For that reason, first-order approaches, based on online convex optimization, have been proposed as alternatives for the case of large-scale problems. However, existing first-order methods are either stochastic in nature or involve a binary search for the optimal value. We show that this problem can also be solved with deterministic first-order algorithms based on a saddle-point Lagrangian reformulation that avoids both of these issues. Our approach recovers the other approaches’ [Formula: see text] convergence rate in the general case and offers an improved [Formula: see text] rate for problems with constraints that are affine both in the decision and in the uncertainty. Experiment involving robust quadratic optimization demonstrates the numerical benefits of our approach. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. Funding: This work was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant VI.Veni.191E.035] and the Israel Science Foundation [Grant 1460/19]. 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.2022.0200 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0200 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

The redundancy allocation problem (RAP) aims to find an allocation of redundant components that maximizes system reliability under resource constraints. In this paper, we consider the RAP for complex systems with heterogeneous components, which may have bridges or interconnecting subsystems and are generally more complicated in structure than commonly studied series-parallel systems. A major challenge of the RAP is that the reliability function is nonlinear. To tackle this, we develop six exact mixed integer linear programming (MILP) formulations, which, to our knowledge, are the first of such formulations for the problem. The key idea is to use binary variables and big-M constraints to successively linearize each product term in the reliability function. In addition, we propose a novel tree representation of the reliability function to establish the interconnections between different product terms, which is then used to reduce the number of decision variables and constraints needed in the linearization. We also propose an alternative form of the reliability function that only involves the failure probabilities of subsystems, and leverage this to derive better variable bounds, smaller big-M constants, and simpler formulations. We compare the six MILP formulations by solving them directly using an MILP solver based on three sets of test instances of varying complexity. Results indicate that all formulations achieve optimal solutions quickly for the first set and within reasonable computation times for the second set. For the third set, two formulations can always be solved within a given time limit. Overall, the MILP formulation derived based on the alternative form of the reliability function and a specific type of tree representation scheme performs the best. We also demonstrate that our approach is more robust and scalable than a benchmark branch-and-bound method. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was supported by National Natural Science Foundation of China [Grants 72310107003, 72471189, 71901176]. 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.2024.0842 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0842 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Mobile technologies and AI enable health data collection from devices, allowing effective monitoring. Traditional methods often compromise privacy, but federated learning (FL) offers a potential solution. However, current FL approaches face two issues: they don’t identify key health features for clinical intervention, and their aggregation overlooks patient multidimensional heterogeneity. This study seeks to develop a new FL method to tackle these challenges and enhance privacy in mobile health monitoring. This study proposes a novel FL method combining (1) a spatial and temporal attention-based prediction model (STA-Pred) that uses attention to identify key spatial and temporal features, and (2) a multidimensional heterogeneity-based aggregation protocol (MDH-Aggr), which aggregates components based on their heterogeneity to handle multidimensional differences. Experiments on three data sets show that our method outperforms existing methods in several patient-monitoring contexts. This study enhances understanding of how to leverage mobile technologies and AI to enable privacy-preserving health monitoring that promotes the social good. Additionally, it advances FL research through two innovative designs (STA-Pred and MDH-Aggr). History: This paper has been accepted by Kaushik Dutta for the Special Issue on Responsible AI and Data Science for Social Good. Funding: Y. Chai, H. Liu, and Y. Liu are supported by the National Natural Science Foundation of China [Grants 72342011, 72322019, 72188101, and 72402001]. Dr. L. Wang’s work was in part supported by the National Natural Science Foundation of China [Grant 72271027], Hainan Provincial Natural Science Foundation of China [Grant 726MS0458], and Beijing Institute of Technology Research Fund Program for Young Scholars [Grant XSQD-202216004]. X. Liu is not supported by any funds or associated with any of the abovementioned funds. Supplemental Materials: 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.2023.0521 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0521 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/.

We propose a new form of predictive models, Partially Interpretable Estimators (PIE), which jointly train an interpretable model and a black-box model to achieve partial model transparency while maintaining high predictive performance. Our design is motivated by prior research showing that interpretability does not require exposing all model details. Therefore, our objective is to explain the main components of the prediction, withholding complicated calculations that may not be necessary for users. PIE is designed to attribute a prediction to the contribution from individual features via a sparse linear additive model to achieve interpretability while complementing the prediction with a black-box model to boost the predictive performance. As such, the linear additive model captures the primary feature contributions, while the black-box component augments PIE’s predictive power by capturing the “nuances” of feature interactions as a refinement. Moreover, we include a sparsity constraint, allowing users to adjust the model to meet domain-specific needs of interpretability. To optimize predictive performance, we propose a coordinated training algorithm that jointly trains the two components of PIE. Experimental results show that PIE achieves accuracy comparable to state-of-the-art black-box models, with human assessments confirming that its interpretability is nearly equivalent to linear models. History: Accepted by Ram Ramesh, Data Science & Machine Learning. 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.2022.0098 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0098 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

This paper develops a cost-effective sequential route recommender system to provide real-time routing recommendations for vacant taxis searching for the next passenger. We propose a prediction-and-optimization framework to recommend the searching route that maximizes the expected profit of the next successful passenger pickup based on the dynamic taxi demand-supply distribution. Specifically, this system features a deep learning-based predictor that dynamically predicts the passenger pickup probability on a road segment and a recursive searching algorithm that recommends the optimal searching route. The predictor integrates a graph convolution network (GCN) to capture the spatial distribution and a long short-term memory (LSTM) to capture the temporal dynamics of taxi demand and supply. The GCN-LSTM model can accurately predict the pickup probability on a road segment with the consideration of potential taxi oversupply. Then, the dynamic distribution of pickup probability is fed into the route optimization algorithm to recommend the optimal searching routes sequentially as route inquiries emerge in the system. The recursion tree-based route optimization algorithm can significantly reduce the computational time and provide the optimal routes within seconds for real-time implementation. Finally, extensive experiments using Beijing Taxi GPS data demonstrate the effectiveness and efficiency of the proposed recommender system. History: Accepted by Ram Ramesh, Area Editor for Data Science and Machine Learning. Funding: This work was partially supported by the Hong Kong Research Grants Council [Grants CityU 21500220, CityU 11504322] and the National Natural Science Foundation of China [Grant 72201222]. 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.2021.0112 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2021.0112 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

As an emerging technology, human-robot collaboration (HRC) has been implemented to enhance the performance of assembly lines and improve the safety of human workers. By integrating the advantages of human workers and collaborative robots (cobots), HRC enables production systems to process tasks consecutively, concurrently, or collaboratively. However, the introduction of cobots also makes the corresponding human-robot collaborative assembly line balancing problem more complex and difficult to solve. To solve this problem, we first propose an enhanced mixed integer program (EMIP) with various enhancement techniques and tighter bounds, and then, we develop an improved combinatorial Benders decomposition algorithm (Algorithm ICBD) with new local search strategies, Benders cuts, and acceleration procedures. To verify the effectiveness of our proposed model and algorithms, we conduct extensive computational experiments, and the results show that our proposed EMIP model is significantly better than the existing mixed integer program model; the percentages of instances that can obtain feasible and optimal solutions are increased from 82.42% to 100% and from 29.17% to 43.5%, respectively, whereas the average gap is decreased from 19.81% to 5.64%. In addition, our proposed Algorithm ICBD can get 100% of feasible solutions and 65.92% of optimal solutions for all of the test instances, and the average gap is only 1.49%. Moreover, compared with existing Benders decomposition methods for this problem, our approach yields comparatively better solutions in notably shorter average computational time when run in the same computational environment. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This research was supported by the National Natural Science Foundation Council of China [Grants 72401214, 92167206, 7221101377, 72471169, and 72231005], the Ministry of Education of China [Grant 24YJC630078], and Computation and Analytics of Complex Management Systems (Tianjin University). This research was also supported by the Tianjin Natural Science Foundation Project [Grant 23JCQNJC01900] and the Tianjin Philosophy and Social Science Planning Project [Grant TJGL21-016]. 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.2023.0279 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0279 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Do you remember your first video game console? We remember ours. Decades ago, they provided hours of entertainment. Now, we have repurposed them to solve dynamic and stochastic optimization problems. With deep reinforcement learning methods posting superhuman performance on a wide range of Atari games, we consider the task of representing a classic logistics problem as a game. Then, we train agents to play it. We consider several game designs for the vehicle routing problem with stochastic requests. We show how various design features impact agents’ performance, including perspective, field of view, and minimaps. With the right game design, general purpose Atari agents outperform optimization-based benchmarks, especially as the problem size grows. Our work points to the representation of dynamic and stochastic optimization problems via games as a promising research direction. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This research was enabled in part by support from Calcul Québec, the Digital Research Alliance of Canada, HEC Montreal, and the Institute for Data Valorization (IVADO). 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.2024.0838 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0838 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

In this article we propose an iterative algorithm to address the optimization problem of distributing a set of multiple highly perishable commodities in a healthcare network. In the biomedical sample transportation problem, numerous commodities with short lifespans presume multiple transportation requests at the same facility in a day and restrict the maximum time to reach their destination. These two characteristics create an interdependency between the routing and the pickup decisions in time that is highly complex. To address these timing issues, we model this problem as a service network design problem over a time-expanded network. Our solution method aggregates the network at two levels. First, the commodities are aggregated and artificially consolidated, reducing the symmetry arising when multiple transportation requests are solicited within a short period of time. Second, the space-time nodes in the network are constructed dynamically, thus reducing the size of the mathematical model to be solved at each iteration. Moreover, the method creates auxiliary networks to calculate good-quality primal bounds to the problem. Our algorithm proves to be efficient to solve a set of real-life instances from the Quebec laboratory network under the management of the Ministère de la Santé et des Services sociaux (Ministry of Health and Social Services) with a detailed network of up to 2,377 periods and 277 transportation requests. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada [Grants 2018-04609, 2020-06311]. 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.2023.0061 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0061 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

We present a new difference of convex functions algorithm (DCA) for solving linear and nonlinear mixed complementarity problems (MCPs). The approach is based on the reformulation of bilinear complementarity constraints as difference of convex (DC) functions, more specifically, the difference of scalar, convex quadratic terms. This reformulation gives rise to a DC program, which is solved via sequential linear approximations of the concave term using standard DCA techniques. The reformulation is based on a generalization of earlier results on recasting bilinearities and leads to a novel algorithmic framework for MCPs. Through extensive numerical experimentation, the proposed approach, referred to as DCA-BL, proves to be an efficient heuristic for complementarity problems. For linear complementarity problems (LCPs), we test the approach on a number of randomly generated instances by varying the size, the density, and the eigenvalue distribution of the LCP matrix, providing insights into the numerical properties of DCA-BL. In addition, we apply the framework to a market equilibrium problem and find that DCA-BL scales well on realistic LCP instances. Also, through experimentation, we find that that DCA-BL performs particularly well compared with other DC-based complementarity approaches in the literature if the LCP is highly dense, asymmetric, or indefinite. Lastly, the method is successfully applied to a set of equilibrium problems with second-order cone constraints, which give rise to nonlinear complementarity problems, with applications to stochastic equilibrium problems in water infrastructure and finance. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. Funding: This work was supported by the Division of Civil, Mechanical and Manufacturing Innovation [Grant 2113891], the Division of Electrical, Communications and Cyber Systems [Grant 2114100], the Office of Naval Research Global [Grant 00014-22-1-2649], Petrobras [Grant 4324713], the Division of Mathematical Sciences [Grant 2318519], and the Danmarks Frie Forskningsfond [Grant 0217-00009B]. 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.2024.0822 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0822 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Infodemic is a significant concern for society, and numerous studies have explored artificial intelligence (AI)-based solutions to manage it. However, existing methods fail to make sense of the complex situation of infodemic and fail to manage infodemic in a proactive manner. To tackle this challenge, we formalize infodemic topic prediction (ITP) as a distinct research problem. ITP aims to identify emerging topics to provide a global view of the situation, as well as to predict the probability of a topic becoming an infodemic to enable proactive management. Although some relevant methods exist, they are insufficient in modeling the intertopic relationships where each topic is closely connected to all others in a group of topics (called multilateral intertopic relationships). This study proposes a new hypergraph-based ITP framework that models the multilateral intertopic relationship with a novel hypergraph. Specifically, we introduce a novel temporal topic hypergraph (TTH) where topics are treated as nodes, and the multilateral intertopic relationships in both semantic and temporal perspectives are modeled by hyperedges. The main novelties are twofold. First, our TTH is a novel hypergraph that can cope with newly emerged topics through the proposed directed and undirected hyperedges. Second, we propose a novel similarity-based transformation method (STM) that reduces the complexity of hypergraph transformation from [Formula: see text] to [Formula: see text], making it scalable for social media data. Evaluations on infodemics during the COVID-19 and the Mpox pandemic demonstrate the effectiveness of our framework. This study contributes to responsible infodemic management by formalizing the task of ITP and introducing a novel framework for it, which enables public health organizations, social media platforms, and policymakers to proactively make sense of and effectively respond to infodemic. History: This paper has been accepted by Kaushik Dutta for the Special Issue on the Responsible AI and Data Science for Social Good. Funding: L. Zhao, S. Ding, Y. Chai and S. Yang are supported by the National Natural Science Foundation of China [Grants 72293581, 72188101, 72342011, and 72322019]. J. Xie and X. Fang are not supported by any funds and are not associated with any of the above mentioned funds. 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.2024.0660 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0660 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Kidney exchange problems with immunosuppressants extend the classical kidney exchange problem by allowing a limited number of arcs that do not belong to the compatibility graph to be included in the solution. Among these, the kidney exchange problem with reserve arcs (KEP-RA) assumes that any arc can be made compatible, whereas the kidney exchange problem with half-compatible arcs (KEP-HCA) assumes that this is the case only for a subset of arcs. Starting with KEP-RA, we first show that existing integer linear programming formulations for the kidney exchange problem can easily be extended to include reserve arcs. We then demonstrate that there always exists an optimal KEP-RA solution in which every cyclic set of exchanges contains at most one reserve arc, and we use this property to develop effective variable reduction procedures and new ad hoc modeling structures. We also extend these findings to the case where nondirected donors are included. Experiments show that trivial model extensions are not able to cope with medium-sized instances, whereas our enhanced models can solve instances with up to a thousand recipient-donor pairs. We also evaluate the number of transplants enabled by each reserve arc in various settings, demonstrating that although reserve arcs tend to have a diminishing return, there are instances for which this is not the case. Finally, we demonstrate how our techniques can be integrated into a variable and constraint generation algorithm to solve KEP-HCA and show that its performance complements that of a trivial model extension. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: D. Manlove was supported by the Engineering and Physical Sciences Research Council [Grant EP/X013618/1]. This work used the Dutch national e-infrastructure with the support of the SURF Cooperative [Grants EINF-8220 and EINF-12272]. 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.2024.1071 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.1071 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

We present a new flow-based formulation for identical parallel machine scheduling with a regular objective function and without idle time. The formulation is constructed with the help of a decision diagram that represents all job sequences that respect specific ordering rules. These rules rely on a partition of the planning horizon into, generally nonuniform, periods and do not exclude all optimal solutions, but they constrain solutions to adhere to a canonical form. The new formulation has numerous variables and constraints, and hence we apply a Dantzig-Wolfe decomposition to compute the linear programming relaxation in reasonable time; the resulting lower bound is stronger than the bound from the classical time-indexed formulation. We develop a branch-and-price framework that solves three instances from the literature for the first time. We compare the new formulation with the time-indexed and arc time–indexed formulation by means of a series of computational experiments. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This work was partially funded by the European Union’s Horizon 2020 research and innovation program under [Marie Skłodowska-Curie Grant 754462]. 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.2022.0301 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0301 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

We study a robust auction design problem with a minimax regret objective, in which a seller seeks a mechanism for selling multiple items to multiple bidders with additive values. The seller knows that the bidders’ values range over a box uncertainty set but has no information on their probability distribution. The robust auction design model we study requires no distributional information except for upper bounds on the bidders’ values for each item. This model is relevant if there is no trustworthy distributional information or if any distributional information is costly or time-consuming to acquire. We propose a mechanism that sells each item separately via a second price auction with a random reserve price and prove that this mechanism is optimal using duality techniques from robust optimization. We then interpret the auction design problem as a zero-sum game between the seller, who chooses a mechanism, and a fictitious adversary or “nature,” who chooses the bidders’ values from within the uncertainty set with the aim to maximize the seller’s regret. We characterize the Nash equilibrium of this game analytically when the bidders are symmetric. The Nash strategy of the seller coincides with the optimal separable second price auction, whereas the Nash strategy of nature is mixed and constitutes a probability distribution on the uncertainty set under which each bidder’s values for the items are comonotonic. We also study a restricted auction design problem over deterministic mechanisms. In this setting, we characterize the suboptimality of a separable second price auction with deterministic reserve prices and show that this auction becomes optimal if the bidders are symmetric. The optimal mechanism is derived in closed form and can easily be implemented by practitioners. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This research was funded by the Swiss National Science Foundation [Grant BSCGI0_157733] and by the Ministry of Education, Singapore, under its Academic Research Fund Tier 2 [Grant MOE-T2EP20222-0003]. 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.2022.0275 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0275 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .