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mixed-integer optimization with Constraint Learning

OPERATIONS RESEARCH 2025年第2期73卷 iii-viii, 583-1150, C2-C3页

作者： Maragno, Donato Wiberg, Holly Bertsimas, Dimitris Birbil, S. . Ilker den Hertog, Dick Fajemisin, Adejuyigbe O. Univ Amsterdam Amsterdam Business Sch NL-1018 TV Amsterdam Netherlands Carnegie Mellon Univ Heinz Coll Informat Syst & Publ Policy Pittsburgh PA 15213 USA MIT Sloan Sch Management Cambridge MA 02139 USA

We establish a broad methodological foundation for mixed-integer optimization with learned constraints. We propose an end-to-end pipeline for data-driven decision making in which constraints and objectives are directly learned from data using machine learning, and the trained models are embedded in an optimization formulation. We exploit the mixed-integer optimization representability of many machine learning methods, including linear models, decision trees, ensembles, and multilayer perceptrons, which allows us to capture various underlying relationships between decisions, contextual variables, and outcomes. We also introduce two approaches for handling the inherent uncertainty of learning from data. First, we characterize a decision trust region using the convex hull of the observations to ensure credible recommendations and avoid extrapolation. We efficiently incorporate this representation using column generation and propose a more flexible formulation to deal with low-density regions and high-dimensional data sets. Then, we propose an ensemble learning approach that enforces constraint satisfaction over multiple bootstrapped estimators or multiple algorithms. In combination with domain-driven components, the embedded models and trust region define a mixed-integer optimization problem for prescription generation. We implement this framework as a Python package (OptiCL) for practitioners. We demonstrate the method in both World Food Programme planning and chemotherapy optimization. The case studies illustrate the framework's ability to generate high-quality prescriptions and the value added by the trust region, the use of ensembles to control model robustness, the consideration of multiple machine learning methods, and the inclusion of multiple learned constraints.

关键词： mixed-integer optimization machine learning constraint learning prescriptive analytics

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Feature and functional form selection in additive models via mixed-integer optimization

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COMPUTERS & OPERATIONS RESEARCH 2025年 176卷

作者： Navarro-Garcia, Manuel Guerrero, Vanesa Durban, Maria del Cerro, Arturo Univ Carlos III Madrid Dept Stat Calle Madrid 126 Getafe 28903 Spain Komorebi AI Technol Ave Gen Peron 26Planta 4 Madrid 28020 Spain

Feature selection is a recurrent research topic in modern regression analysis, which strives to build interpretable models, using sparsity as a proxy, without sacrificing predictive power. The best subset selection problem is central to this statistical task: it has the goal of identifying the subset of covariates of a given size that provides the best fit in terms of an empirical loss function. In this work, we address the problem of feature and functional form selection in additive regression models under a mathematical optimization lens. Penalized splines (P-splines) are used to estimate the smooth functions involved in the regression equation, which allow us to state the feature selection problem as a cardinality-constrained mixed-integer quadratic program (MIQP) in terms of both linear and non-linear covariates. To strengthen this MIQP formulation, we develop tight bounds for the regression coefficients. A matheuristic approach, which encompasses the use of a preprocessing step, the construction of a warm-start solution, the MIQP formulation and the large neighborhood search metaheuristic paradigm, is proposed to handle larger instances of the feature and functional form selection problem. The performance of the exact and the matheuristic approaches are compared in simulated data. Furthermore, our matheuristic is compared to other methodologies in the literature that have publicly available implementations, using both simulated and real-world data. We show that the stated approach is competitive in terms of predictive power and in the selection of the correct subset of covariates with the appropriate functional form. A public Python library is available with all the implementations of the methodologies developed in this paper.

关键词： Penalized splines Feature selection Functional form selection Matheuristic mixed-integer optimization

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Information complexity of mixed-integer convex optimization

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MATHEMATICAL PROGRAMMING 2025年第1-2期210卷 3-45页

作者： Basu, Amitabh Jiang, Hongyi Kerger, Phillip Molinaro, Marco Johns Hopkins Univ Dept Appl Math & Stat Baltimore MD USA Cornell Univ Sch Civil & Environm Engn Ithaca NY USA Microsoft Res Redmond Redmond WA 98052 USA Pontificia Univ Catolica Rio de Janeiro Comp Sci Dept Rio de Janeiro Brazil

We investigate the information complexity of mixed-integer convex optimization under different types of oracles. We establish new lower bounds for the standard first-order oracle, improving upon the previous best known lower bound. This leaves only a lower order linear term (in the dimension) as the gap between the lower and upper bounds. This is derived as a corollary of a more fundamental "transfer" result that shows how lower bounds on information complexity of continuous convex optimization under different oracles can be transferred to the mixed-integer setting in a black-box manner. Further, we (to the best of our knowledge) initiate the study of, and obtain the first set of results on, information complexity under oracles that only reveal partial first-order information, e.g., where one can only make a binary query over the function value or subgradient at a given point. We give algorithms for (mixed-integer) convex optimization that work under these less informative oracles. We also give lower bounds showing that, for some of these oracles, every algorithm requires more iterations to achieve a target error compared to when complete first-order information is available. That is, these oracles are provably less informative than full first-order oracles for the purpose of optimization.

关键词： mixed-integer optimization Convex optimization Information complexity Lower bounds

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Compressive-Sensing-Assisted mixed integer optimization for Dynamical System Discovery With Highly Noisy Data

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NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS 2025年第1期41卷

作者： Shi, Tony Ma, Mason Tran, Hoang Zhang, Guannan Univ Tennessee Knoxville Mfg Intelligence & Dynam Syst Lab Knoxville TN USA Oak Ridge Natl Lab Comp Sci & Math Div Oak Ridge TN 37830 USA

The identification of governing equations for dynamical systems is an everlasting challenge for the fundamental research in science and engineering. Machine learning has exhibited great success in learn and predicting dynamical systems from data. However, the fundamental challenges still exist: discovering the exact governing equations from highly noisy data. In the present work, we propose a compressive sensing-assisted mixed integer optimization (CS-MIO) method to make a step forward from a modern discrete optimization lens. In particular, we first formulate the problem into a mixed integer optimization model. The discrete optimization nature of the model leads to exact variable selection by means of cardinality constraint, and thereby powerful capability of exact discovery of governing equations from noisy data. Such capability is further enhanced by incorporating compressive sensing and regularization techniques for highly noisy data and high-dimensional problems. The case studies on classical dynamical systems have shown that CS-MIO can discover the exact governing equations from large-noise data, with up to two orders of magnitude larger noise compared with state-of-the-art methods. We also show its effectiveness for high-dimensional dynamical system identification through the chaotic Lorenz 96 system.

关键词： compressive sensing dynamical systems machine learning mixed-integer optimization model discovery

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mixed-integer optimization approach to learning association rules for unplanned ICU transfer

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ARTIFICIAL INTELLIGENCE IN MEDICINE 2020年第0期103卷 101806-000页

作者： Chou, Chun-An Cao, Qingtao Weng, Shao-Jen Tsai, Che-Hung Northeastern Univ Dept Mech & Ind Engn Boston MA 02115 USA Tunghai Univ Dept Ind Engn & Enterprise Informat Taichung Taiwan Taichung Vet Gen Hosp Puli Branch Dept Emergency Med Taichung Taiwan

After admission to emergency department (ED), patients with critical illnesses are transferred to intensive care unit (ICU) due to unexpected clinical deterioration occurrence. Identifying such unplanned ICU transfers is urgently needed for medical physicians to achieve two-fold goals: improving critical care quality and preventing mortality. A priority task is to understand the crucial rationale behind diagnosis results of individual patients during stay in ED, which helps prepare for an early transfer to ICU. Most existing prediction studies were based on univariate analysis or multiple logistic regression to provide one-size-fit-all results. However, patient condition varying from case to case may not be accurately examined by such a simplistic judgment. In this study, we present a new decision tool using a mathematical optimization approach aiming to automatically discover rules associating diagnostic features with high-risk outcome (i.e., unplanned transfers) in different deterioration scenarios. We consider four mutually exclusive patient subgroups based on the principal reasons of ED visits: infections, cardiovascular/respiratory diseases, gastrointestinal diseases, and neurological/other diseases at a suburban teaching hospital. The analysis results demonstrate significant rules associated with unplanned transfer outcome for each subgroups and also show comparable prediction accuracy (>70%) compared to state-of-the-art machine learning methods while providing easy-to-interpret symptom-outcome information.

关键词： Emergency department Critical care Unplanned ICU transfer Association rule mixed-integer optimization

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mixed-integer optimization Problems with Applications to Manufacturing Scheduling and Distributed Energy System Operation

Mixed-Integer Optimization Problems with Applications to Man...

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作者： Yan, Bing University of Connecticut

学位级别：Ph.D., Doctor of Philosophy

Over the past decades, mixed-integer optimization problems have attracted a lot attentions. Two popular topics are the scheduling problems of semiconductor manufacturing, and operation problems of distributed energy systems. On the one hand, the increasing pressure to meet demand is forcing semiconductor manufacturers to seek efficient scheduling methods. On the other hand, with world’s increasing energy demand and growing environmental concerns, efficient utilization of energy is essential. Lithography, with a limited number of expensive resources, is a major bottleneck in memory chip manufacturing. Because of its complex characteristics and large sizes of practical problems, developing effective scheduling approaches is challenging. In this thesis, a mixed-integer linear formulation is established for high-volume and low-variety manufacturing through novel resource-based modeling instead of traditional lot-based. To solve this problem efficiently by branch-and-cut, a two-phase approach is established based on convex hull *** solution methodology for litho machine scheduling can also be used for other mixed-integer linear problems such as distributed energy system (DES) operation. Energy demands and energy supplied by different devices are characterized by different levels of quality, which is measured by exergy in thermodynamics. Exergy is destroyed in various processes, with limited amount of exergy in fossil fuels, it is therefore important to match demand and supply in quantity and quality to avoid exergy waste. Flexible DESs provide a desirable infrastructure. An exergy-based optimization approach is therefore developed for DES operation to reduce energy costs and the exergy losses by considering the whole energy supply chain from energy resources to user demands. To capture the complicated interactions among energy devices and capture the exergy loss of each energy device, exergy networks are established with detailed device and water network models.

关键词： mixed-integer optimization semiconductor manufacturing scheduling exergy loss distributed energy system microgrid operation design branch-and-cut convex hull Surrogate Lagrangian relaxation

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Binary decision rules for multistage adaptive mixed-integer optimization

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MATHEMATICAL PROGRAMMING 2018年第2期167卷 395-433页

作者： Bertsimas, Dimitris Georghiou, Angelos MIT Sloan Sch Management 77 Massachusetts Ave Cambridge MA 02139 USA MIT Ctr Operat Res 77 Massachusetts Ave Cambridge MA 02139 USA McGill Univ Desautels Fac Management Montreal PQ H3A 1G5 Canada

Decision rules provide a flexible toolbox for solving computationally demanding, multistage adaptive optimization problems. There is a plethora of real-valued decision rules that are highly scalable and achieve good quality solutions. On the other hand, existing binary decision rule structures tend to produce good quality solutions at the expense of limited scalability and are typically confined to worst-case optimization problems. To address these issues, we first propose a linearly parameterised binary decision rule structure and derive the exact reformulation of the decision rule problem. In the cases where the resulting optimization problem grows exponentially with respect to the problem data, we provide a systematic methodology that trades-off scalability and optimality, resulting to practical binary decision rules. We also apply the proposed binary decision rules to the class of problems with random-recourse and show that they share similar complexity as the fixed-recourse problems. Our numerical results demonstrate the effectiveness of the proposed binary decision rules and show that they are (i) highly scalable and (ii) provide high quality solutions.

关键词： Adaptive optimization Binary decision rules mixed-integer optimization

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Data-Driven Identification of Characteristic Real-Driving Cycles Based on k-Means Clustering and mixed-integer optimization

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 2020年第3期69卷 2398-2410页

作者： Foerster, Daniel Inderka, Robert B. Gauterin, Frank Daimler AG D-71059 Sindelfingen Germany Karlsruhe Inst Technol Inst Vehicle Syst Technol D-76131 Karlsruhe Germany

Virtual powertrain analysis is widely applied in the automotive industry to cope with the increasing complexity and variance of future vehicle propulsion technologies. Since the vehicle-usage behavior has a strong impact on component loads, realistic computation results require accurate assumptions about these boundaries. In this context, driving cycles (DCs) are used to represent the system boundaries in vehicle operation. The aim of this article is to identify multiple characteristic driving cycles (CDCs) from extensive vehicle measurement data which represent the full variety of possible real-driving scenarios. Vehicle measurements are segmented and consumption-relevant features are extracted from each segment. These features are then used to apply clustering and classification techniques to identify characteristic groups that are consequently assigned to different driving environments and driving styles. In order to obtain even more realistic driving scenarios, a data-fusion approach is used to incorporate a road slope signal from a NASA digital elevation model for each segment. Lastly, a genetic mixed-integer optimization algorithm is proposed to efficiently generate representative DCs for each characteristic group of driving segments. The main contribution of this article is a data-driven identification of the parameter space of real-driving scenarios from extensive vehicle measurements including the implementation of road slope information. The scenarios are represented via a constrained number of compact CDCs which enables comprehensive investigations of new powertrain technologies under average as well as extreme real-driving conditions to develop efficiency-robust powertrain systems.

关键词： Mechanical power transmission Acceleration Roads Europe Databases optimization Markov processes Driving Cycle Generation Real-Driving Cycles Driving Style Driving Environment Road Slope Information k-Means Clustering Markov Chains Genetic Algorithm mixed-integer optimization

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Modeling with discrete-time recurrent fuzzy systems via mixed-integer optimization

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FUZZY SETS AND SYSTEMS 2012年 203卷 1-16页

作者： Schwung, Andreas Adamy, Juergen Tech Univ Darmstadt Inst Automat Control D-64283 Darmstadt Germany

In this paper, we present a new approach for modeling of dynamical systems with discrete-time recurrent fuzzy systems (DTRFS). The modeling problem is divided into two subproblems. The first subproblem considers the identification of the rule base of the DTRFS which can be interpreted as structural modeling. This problem is recast into integer and mixed-integer optimization problems respectively dependent on the chosen cost function and system structure. The second subproblem considers the optimization of the parameters of the system, i.e., quantitative modeling. As a result, we obtain a linguistically interpretable model of the dynamical system which additionally can be interpreted as a linguistic automaton. The applicability of the approach is shown by modeling a thermofluidic process. (C) 2012 Elsevier B.V. All rights reserved.

关键词： Discrete-time recurrent fuzzy systems Nonlinear dynamic systems Linguistic automaton Structural modeling mixed-integer optimization

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Towards simulation based mixed-integer optimization with differential equations

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NETWORKS 2018年第1期72卷 60-83页

作者： Gugat, Martin Leugering, Guenter Martin, Alexander Schmidt, Martin Sirvent, Mathias Wintergerst, David Friedrich Alexander Univ Erlangen Nurnberg FAU Lehrstuhl Angew Math 2 D-91058 Erlangen Germany Friedrich Alexander Univ Erlangen Nurnberg FAU Discrete Optimizat D-91058 Erlangen Germany Energie Campus Nurnberg D-90429 Nurnberg Germany

We propose a decomposition based method for solving mixed-integer nonlinear optimization problems with black-box nonlinearities, where the latter, for example, may arise due to differential equations or expensive simulation runs. The method alternatingly solves a mixed-integer linear master problem and a separation problem for iteratively refining the mixed-integer linear relaxation of the nonlinear equalities. The latter yield nonconvex feasible sets for the optimization model but we have to restrict ourselves to convex and monotone constraint functions. Under these assumptions, we prove that our algorithm finitely terminates with a global optimal solution of the mixed-integer nonlinear problem. Additionally, we show the applicability of our approach for three applications from optimal control with integer variables, from the field of pressurized flows in pipes with elastic walls, and from steady-state gas transport. For the latter we also present promising numerical results of our method applied to real-world instances that particularly show the effectiveness of our method for problems defined on networks.

关键词： decomposition method gas transport networks mixed-integer optimization optimization with differential equations simulation based optimization

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