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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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A test instance generator for multiobjective mixed-integer optimization

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MATHEMATICAL METHODS OF OPERATIONS RESEARCH 2024年第1期100卷 385-410页

作者： Eichfelder, Gabriele Gerlach, Tobias Warnow, Leo Tech Univ Ilmenau Inst Math PO 10 05 65 D-98684 Ilmenau Germany

Application problems can often not be solved adequately by numerical algorithms as several difficulties might arise at the same time. When developing and improving algorithms which hopefully allow to handle those difficulties in the future, good test instances are required. These can then be used to detect the strengths and weaknesses of different algorithmic approaches. In this paper we present a generator for test instances to evaluate solvers for multiobjective mixed-integer linear and nonlinear optimization problems. Based on test instances for purely continuous and purely integer problems with known efficient solutions and known nondominated points, suitable multiobjective mixed-integer test instances can be generated. The special structure allows to construct instances scalable in the number of variables and objective functions. Moreover, it allows to control the resulting efficient and nondominated sets as well as the number of efficient integer assignments.

关键词： Multiobjective optimization mixed-integer optimization Test instances Nonconvex optimization

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A genetic mixed-integer optimization of neural network hyper-parameters

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JOURNAL OF SUPERCOMPUTING 2022年第12期78卷 14680-14702页

作者： Spurlock, Kyle Elgazzar, Heba Morehead State Univ Sch Engn & Comp Sci 150 Univ Blvd Morehead KY 40351 USA

Neural networks (NN) have become immensely popular for their effectiveness and flexibility in learning complicated patterns. Despite this success, they are often considered difficult to design because of the wide variety of parameters they require. Determining the most optimal selection of parameters has become tedious and costly, and neural architecture search (NAS) methods have been employed to try and take the guesswork out of the process. A common NAS approach is the genetic algorithm (GA);however, its usage is often exclusively tied to either the learnable parameters, or the meta-parameters that augment the learning. This work proposes an experimental approach for optimizing both real-valued weights and discrete meta-parameters simultaneously. Experimental results have shown that the current approach evolves both parameter sets effectively for simple problems like Iris, but still struggles in finding an optimal model for more rigorous problems.

关键词： Genetic algorithm Deep learning mixed-integer optimization Neural architecture search

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Evolutionary mixed-integer optimization with Explicit Constraints 23

Evolutionary Mixed-Integer Optimization with Explicit Constr...

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Genetic and Evolutionary Computation Conference (GECCO)

作者： Hong, Yuan Arnold, Dirk V. Dalhousie Univ Fac Comp Sci Halifax NS Canada

ISBN: (纸本)9798400701191

Determining the feasibility of a candidate solution to a constrained black-box optimization problem may either be similarly expensive as the process of determining its quality, or it may be much cheaper. Constraints that allow obtaining degrees of feasibility or constraint violation without incurring significant computational costs are referred to as explicit. We present an evolutionary algorithm for solving mixed-integer black-box optimization problems with explicit constraints. The algorithm wraps active-set evolution strategies, an algorithm for solving continuous black-box problems with explicit constraints, in a branching mechanism that allows enforcing integrality constraints. In computer experiments we demonstrate that the algorithm solves a set of mixed-integer problems with significantly fewer objective function evaluations than several algorithms that do not exploit the explicitness of the constraints.

关键词： Constrained black-box optimization mixed-integer optimization evolution strategy

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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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Failure Mitigation and Restoration in Interdependent Networks via mixed-integer optimization

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IEEE TRANSACTIONS ON NETWORK SCIENCE AND ENGINEERING 2021年第2期8卷 1293-1304页

作者： Chen, Cheng-Lung Zheng, Qipeng P. Veremyev, Alexander Pasiliao, Eduardo L. Boginski, Vladimir Univ Cent Florida Dept Ind Engn Management Syst Orlando FL 32816 USA Air Force Res Lab Eglin AFB FL 32542 USA

We propose a new optimization model for determining optimal mitigation and restoration strategies for coupled interdependent networks in the context of preserving and/or restoring the maximum flow through the entire networked system, subject to cascading node failures that may be caused by disruptions of a subset of "seed nodes" at an initial time step. Previous related studies mainly focused on "static" strategies to mitigate cascading failures. However, our model allows one to identify "dynamic" strategies for step-by-step failure propagation, given initial seed node disruptions. Moreover, the proposed model accounts for backup arc capacity and node fortification to mitigate the impact of further failure cascades on network performance. The objective is to restore network performance during a finite recovery planning horizon at total minimal cost. We formulate this problem by mixedinteger optimization, and derive valid inequalities using the substructure of the problem. We report a summary of computational experiments to demonstrate the strength and effectiveness of the inequalities when compared to solving the problem with a commercial optimization solver.

关键词： Failure mitigation interdependent networks mixed-integer optimization restoration valid inequalities

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A Two-Timescale Duplex Neurodynamic Approach to mixed-integer optimization

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2021年第1期32卷 36-48页

作者： Che, Hangjun Wang, Jun Southwest Univ Coll Elect & Informat Engn Chongqing 400715 Peoples R China City Univ Hong Kong Dept Comp Sci Hong Kong Peoples R China City Univ Hong Kong Sch Data Sci Hong Kong Peoples R China City Univ Hong Kong Shenzhen Res Inst Shenzhen 518057 Peoples R China

This article presents a two-timescale duplex neurodynamic approach to mixed-integer optimization, based on a biconvex optimization problem reformulation with additional bilinear equality or inequality constraints. The proposed approach employs two recurrent neural networks operating concurrently at two timescales. In addition, particle swarm optimization is used to update the initial neuronal states iteratively to escape from local minima toward better initial states. In spite of its minimal system complexity, the approach is proven to be almost surely convergent to optimal solutions. Its superior performance is substantiated via solving five benchmark problems.

关键词： optimization Neurodynamics Biological neural networks Recurrent neural networks Collaboration Linear programming Eigenvalues and eigenfunctions Almost-sure convergence mixed-integer optimization neural networks

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A mixed-integer optimization for bifacial two-terminal perovskite-on-perovskite tandem solar cells

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SOLAR ENERGY 2023年第1期262卷

作者： Zhao, Xinhai Tan, Hu Quee Birgersson, Erik Xue, Hansong Natl Univ Singapore Solar Energy Res Inst Singapore SERIS 7 Engn Dr 1 Singapore 117574 Singapore Natl Univ Singapore Dept Mech Engn 9 Engn Dr 1 Singapore 117575 Singapore

We present a mixed-integer optimization for the optics of two-terminal perovskite-on-perovskite tandem solar cells, which accounts for thickness and material variations in all thin-film layers. The model, which is validated with experiments for both monofacial and bifacial devices, optimizes the short-circuit current density under six albedo conditions, identifying the best cell architectures and alternative material combinations that yield optimal and close-to-optimal results. In contrast to monofacial devices and low albedos, tandem devices with top subcells that are not semi-transparent are found to provide the highest short-circuit current densities under high albedos, such as albedos for white sand and snow. According to a sensitivity analysis, the short-circuit current densities in the optimized devices are not significantly affected by small perturbations in layer thicknesses. The device performances are also discussed in the context of the best optical structure. A power conversion efficiency of 34.6% and a bifacial equivalent efficiency of over 50% are predicted for the optimized monofacial device and bifacial device under high albedo values, respectively.

关键词： mixed-integer optimization Optics Two-terminal perovskite-on-perovskite tandem solar cell Bifacial solar cell Sensitivity analysis

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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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