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Optimal scenario reduction framework based on distance of uncertainty distribution and output performance: I. Single reduction via mixed integer linear optimization

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COMPUTERS & CHEMICAL ENGINEERING 2014年第0期70卷 50-66页

作者： Li, Zukui Floudas, Christodoulos A. Univ Alberta Dept Chem & Mat Engn Edmonton AB T6G 2V4 Canada Princeton Univ Dept Chem & Biol Engn Princeton NJ 08544 USA

Realistic decision making involves consideration of uncertainty in various parameters. While large number of scenarios brings significant challenge to computations, the scenario reduction aims at selecting a small number of representative scenarios that can capture the wide range of possible scenarios. A novel scenario reduction algorithm is proposed in this paper to incorporate the consideration of both input data and output performance of decision making. The proposed optimal scenario reduction algorithm, OSCAR, is formulated as a mixed integer linear optimization problem. It minimizes not only the probabilistic distance between the original and reduced input scenario distribution, but also minimizes the differences between the best, worst and expected performances of the output measure of the original and the reduced scenario distributions. The proposed method leads to reduced distribution not only closer to the original distribution in terms of the transportation distance, but also captures the performance of the output. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： Optimal scenario reduction Probabilistic distance mixed integer linear optimization

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On modeling the air traffic control coordination in the collision avoidance problem by mixed integer linear optimization

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ANNALS OF OPERATIONS RESEARCH 2014年第1期222卷 89-105页

作者： Alonso-Ayuso, A. Escudero, L. F. Martin-Campo, F. J. Univ Rey Juan Carlos Dept Estadist & Invest Operat Mostoles 28933 Madrid Spain Univ Complutense Madrid Dept Estadist & Invest Operat 2 Pozuelo De Alarcon 28223 Madrid Spain

A mixed integer linear optimization model is presented for providing a cooperative system between Air Traffic Control Officers who manage the airspace for aircraft conflict detection and resolution. Elsewhere we have introduced the model for dealing with velocity and altitude changes in a given air sector. In this work, we extend the model to cover the important problem of coordinating the decisions of the Air Traffic Control Officers of different air sectors. The new model allows the aircraft to ascend or descend one or more altitude levels. It is so tight that a state-of-the-art mixed integer linear optimization solver provides the solution in a very affordable computing time even for large-scale instances. It is worth pointing out that only in very few pilot instances of the testbed, the software engine needs to use the branch-and-cut phase of the solver.

关键词： Air Traffic Management Collision avoidance mixed integer linear optimization

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The Discrete Dantzig Selector: Estimating Sparse linear Models via mixed integer linear optimization

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IEEE TRANSACTIONS ON INFORMATION THEORY 2017年第5期63卷 3053-3075页

作者： Mazumder, Rahul Radchenko, Peter MIT Sloan Sch Management Operat Res Ctr Cambridge MA 02139 USA MIT Sloan Sch Management Ctr Stat Cambridge MA 02139 USA Univ Sydney Business Sch Darlington NSW 2006 Australia Univ Southern Calif Los Angeles CA 90089 USA

We propose a novel high-dimensional linear regression estimator: the Discrete Dantzig Selector, which minimizes the number of nonzero regression coefficients subject to a budget on the maximal absolute correlation between the features and residuals. Motivated by the significant advances in integer optimization over the past 10-15 years, we present a mixed integer linear optimization (MILO) approach to obtain certifiably optimal global solutions to this nonconvex optimization problem. The current state of algorithmics in integer optimization makes our proposal substantially more computationally attractive than the least squares subset selection framework based on integer quadratic optimization, recently proposed by Bertsimas et al. and the continuous nonconvex quadratic optimization framework of Liu et al.. We propose new discrete first-order methods, which when paired with the state-of-the-art MILO solvers, lead to good solutions for the Discrete Dantzig Selector problem for a given computational budget. We illustrate that our integrated approach provides globally optimal solutions in significantly shorter computation times, when compared to off-the-shelf MILO solvers. We demonstrate both theoretically and empirically that in a wide range of regimes the statistical properties of the Discrete Dantzig Selector are superior to those of popular l(1)-based approaches. We illustrate that our approach can handle problem instances with p = 10,000 features with certifiable optimality making it a highly scalable combinatorial variable selection approach in sparse linear modeling.

关键词： Sparse linear regression Dantzig selector l(0)-minimization mixed integer linear optimization mathematical optimization nonconvex optimization

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ON THE COMPLEXITY OF INVERSE mixed integer linear optimization

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SIAM JOURNAL ON optimization 2021年第4期31卷 3014-3043页

作者： Bulut, Aykut Ralphs, Ted K. Lehigh Univ Dept Ind & Syst Engn Bethlehem PA 18015 USA

Inverse optimization is the problem of determining the values of missing input parameters for an associated forward problem that are closest to given estimates and that will make a given target vector optimal. This study is concerned with the relationship of a particular inverse mixed integer linear optimization problem (MILP) to both the forward problem and the separation problem associated with its feasible region. We show that a decision version of the inverse MILP in which a primal bound is verified is coNP-complete, whereas primal bound verification for the associated forward problem is NP-complete, and that the optimal value verification problems for both the inverse problem and the associated forward problem are complete for the complexity class DP. We also describe a cutting-plane algorithm for solving inverse MILPs that illustrates the close relationship between the separation problem for the convex hull of solutions to a given MILP and the associated inverse problem. The inverse problem is shown to be equivalent to the separation problem for the radial cone defined by all inequalities that are both valid for the convex hull of solutions to the forward problem and binding at the target vector. Thus, the inverse, forward, and separation problems can be said to be equivalent.

关键词： inverse optimization mixed integer linear optimization computational complexity polynomial hierarchy

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IDOL: A web application for mixed integer linear multiobjective optimization

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SOFTWAREX 2022年 19卷

作者： Kaliszewski, Ignacy Karelkina, Olga Polish Acad Sci Syst Res Inst Ul Newelska 6 PL-01447 Warsaw Poland Warsaw Sch Informat Technol Ul Newelska 6 PL-01447 Warsaw Poland

Practical decision-making very often results in complex multiobjective optimization problems that can be solved only by advanced software. At present, it is available for singleobjective optimization and much less for multiobjective optimization. Therefore, the most common approach to multiobjective optimization is to reduce (scalarize) to its singleobjective optimization equivalent. To this aim, the objectives are aggregated into a weighted (parametrized) sum of objectives. However, this form of scalarization does not guarantee that each Pareto optimal solution can be derived. We present a free web-based application to automate scalarizing multiobjective mixed integer optimization problems by a specific form of scalarization, the process which can be tedious for large-scale instances. This form of scalarization guarantees that every Pareto optimal solution is derivable. (c) 2022 The Author(s). Published by Elsevier B.V.

关键词： Multiobjective optimization mixed integer linear optimization Chebyshev scalarization Web application

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On enhancing the explainability and fairness of tree ensembles

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2025年第1期323卷 599-608页

作者： Carrizosa, Emilio Kurishchenko, Kseniia Morales, Dolores Romero Univ Seville Inst Matemat Seville Spain Copenhagen Business Sch Dept Econ Frederiksberg Denmark

Tree ensembles are one of the most powerful methodologies in Machine Learning. In this paper, we investigate how to make tree ensembles more flexible to incorporate explainability and fairness in the training process, possibly at the expense of a decrease inaccuracy. While explainability helps the user understand the key features that play a role in the classification task, with fairness we ensure that the ensemble does not discriminate against a group of observations that share a sensitive attribute. We propose a mixed integer linear optimization formulation to train an ensemble of trees that, apart from minimizing the misclassification cost, controls for sparsity as well as the accuracy in the sensitive group. Our formulation is scalable in the number of observations since its number of binary decision variables is independent of the number of observations. In our numerical results, we show that for standard datasets used in the fairness literature, we can dramatically enhance the fairness of the benchmark, namely the popular Random Forest, while using only a few features, all without damaging the misclassification cost.

关键词： (R) Machine learning Tree ensembles Explainability Fairness mixed integer linear optimization

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Multi-criteria optimization Scheduling of Surgical Units: A Case Study at Policlinico Umberto I of Rome

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Operations Research Forum 2025年第2期6卷 1-26页

作者： Croella, Anna Livia Tronci, Edoardo Maria Nastasi, Alberto Palagi, Laura Petitti, Federico Romano, Ferdinando Department of Computer Control and Management Engineering Sapienza University of Rome Via Aroisto Rome 00185 Italy Accenture S.p.A. Rome Italy CNR-IASI Rome Italy AOU-Policlinico Umberto I Rome Italy

Nowadays, managing Operating Rooms (ORs) in healthcare systems presents a significant challenge, with underutilization of surgical facilities being a key concern. Several decision factors must be considered, including the number of ORs, operating hours, target utilization rates, and the possibility of patient transfers between surgical units (SUs). This study presents a multi-criteria mixed-integer linear optimization model designed for the specific context of Policlinico Umberto I in Rome. The model uses an open scheduling approach to determine the optimal day, time, and OR allocation for weekly surgeries. Using real data from two SUs and four ORs over twelve weeks, we assess the impact of different operating time policies on hospital Key Performance Indicators. Additionally, we introduce randomly generated data to simulate surgery priorities and further validate our model. Results demonstrate the effectiveness of mathematical optimization in improving OR utilization and overall healthcare efficiency. © The Author(s) 2025.

关键词： Health care management mixed integer linear optimization Multi-criteria optimization Operating room scheduling

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Optimal scenario reduction framework based on distance of uncertainty distribution and output performance: II. Sequential reduction

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COMPUTERS & CHEMICAL ENGINEERING 2016年 84卷 599-610页

作者： Li, Zukui Floudas, Christodoulos A. Univ Alberta Dept Chem & Mat Engn Edmonton AB T6G 2V4 Canada Texas A&M Univ Texas A&M Energy Inst College Stn TX 77843 USA

In this paper, a novel sequential scenario reduction framework for general optimization problem is proposed. The proposed method extends the previous work (Li and Floudas, 2014) and aims to tackle optimization problems with a large number of uncertain parameters and a huge number of scenarios generated from the factorial combination. The proposed method first ranks the uncertain parameters based on their effects on the optimal objective using global sensitivity analysis. Then, the parameters are sequentially considered in generating uncertainty scenarios. This method can essentially reduce the computational efforts needed for evaluating the objective values of all scenarios, which is often impractical for a huge number of scenarios. Criteria for quantifying the quality of scenario reduction are also proposed based on robust optimization and scenario optimization. Case studies are presented to illustrate the sequential scenario reduction framework and the results verify the efficiency of the proposed approach. (C) 2015 Elsevier Ltd. All rights reserved.

关键词： Optimal scenario reduction Sequential framework Uncertainty mixed integer linear optimization

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Applications of logic constrained equilibria to traffic networks and to power systems with storage

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JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY 2019年第2期70卷 310-325页

作者： Fomeni, Franklin Djeumou Gabriel, Steven A. Anjos, Miguel F. Univ Lancaster Ctr Transport & Logist CENTRAL Dept Management Sci Management Sch Lancaster England Univ Maryland Dept Mech Engn College Pk MD 20742 USA Gerad College Pk MD USA Polytech Montreal Dept Math & Ind Engn Montreal PQ Canada Gerad Montreal PQ Canada

We study equilibria in traffic networks and in power system networks with storage in the presence of logic constraints. These constraints consist of binary variables that are added to complementarity-based equilibrium models. Although these models have been thoroughly studied, the addition of logic constraints can provide additional benefits for practical applications. The main contribution of this work is to demonstrate, using two specific examples of applications, that logic constraints can render classical equilibrium models more realistic by allowing the inclusion of useful features such as equity in network flows or threshold events. Specifically, for the traffic equilibrium problem, we show how logic constraints can introduce some equity in the assignment of traffic when more than one equilibrium exists. For power system networks, we show that the presence of a storage operator acting as a service provider will not only support the operation of a power grid, but will also help stabilise the price of electricity and avoid the well-documented price-shifting effect. Unlike previous works, our model considers the storage operator as a service provider rather than a competitor to the producers. We also consider the minimum power output of production. We present results illustrating the expanded capabilities and insights provided by these new paradigms.

关键词： mixed linear complementarity problems mixed integer linear optimization traffic equilibrium energy storage power markets

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Two decomposition algorithms for solving a minimum weight maximum clique model for the air conflict resolution problem

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2017年第3期256卷 696-712页

作者： Lehouillier, Thibault Omer, Jeremy Soumis, Francois Desaulniers, Guy HEC Montreal Grp Res Decis Anal 3000 Chemin Cote St Catherine Montreal PQ H3T 2A7 Canada Ecole Polytech Montreal 2900 Blvd Edouard Montpetit Montreal PQ H3T 1J4 Canada Inst Rech Math Rennes Rennes France INSA Rennes Rennes France

In this paper, we tackle the conflict resolution problem using a new variant of the minimum-weight maximum-clique model. The problem involves identifying maneuvers that maintain the required separation distance between all pairs of a set of aircraft while minimizing fuel costs. We design a graph in which the vertices correspond to a finite set of maneuvers and the edges connect conflict-free maneuvers. A maximum clique of minimal weight yields a conflict-free situation that involves all the aircraft and minimizes the costs induced. The model uses a different cost structure compared to classical clique search problems: the costs of the vertices cannot be determined a priori, since they depend on the vertices in the clique. We formulate the problem as a mixed integer linear program. Since the modeling of the aircraft dynamics and the computation of trajectories is separated from the solution process, our mathematical framework is valid for any hypotheses on the aircraft dynamics and any choice of the available maneuvers. In particular, the aircraft can perform dynamic velocity, heading, and flight-level changes. To solve instances involving a large number of aircraft spread over several flight levels, we introduce two decomposition algorithms. The first is a sequential mixed integer linear programming procedure that iteratively refines the discretization of the maneuvers to yield a trade-off between computational time and cost. The second is a large neighborhood search heuristic that uses the first procedure as a subroutine. The best solutions for the available set of maneuvers are obtained in less than ten seconds for instances with up to 250 aircraft randomly allocated to bisten flight levels. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Air traffic control Conflict resolution mixed integer linear optimization Minimum weight maximum clique Decomposition methods

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