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Interpretable clustering: an optimization approach

MACHINE LEARNING 2021年第1期110卷 89-138页

作者： Bertsimas, Dimitris Orfanoudaki, Agni Wiberg, Holly MIT Ctr Operat Res E40-111 Cambridge MA 02139 USA

State-of-the-art clustering algorithms provide little insight into the rationale for cluster membership, limiting their interpretability. In complex real-world applications, the latter poses a barrier to machine learning adoption when experts are asked to provide detailed explanations of their algorithms' recommendations. We present a new unsupervised learning method that leverages mixed integer optimization techniques to generate interpretable tree-based clustering models. Utilizing a flexible optimization-driven framework, our algorithm approximates the globally optimal solution leading to high quality partitions of the feature space. We propose a novel method which can optimize for various clustering internal validation metrics and naturally determines the optimal number of clusters. It successfully addresses the challenge of mixed numerical and categorical data and achieves comparable or superior performance to other clustering methods on both synthetic and real-world datasets while offering significantly higher interpretability.

关键词： Clustering Interpretability Unsupervised learning mixed integer optimization

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On modelling effects in the battery and thermal storage scheduling problem

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JOURNAL OF BUILDING PERFORMANCE SIMULATION 2021年第1期14卷 38-51页

作者： Murray, Alexander Schuetz, Thomas Pan, Guanru Mueller, Dirk Hagenmeyer, Veit Karlsruhe Inst Technol Inst Automat & Appl Informat Karlsruhe Germany Rhein Westfal TH Aachen Inst Energy Efficient Bldg & Indoor Climate Aachen Germany

The growing use of intermittent renewable energy sources requires an increased amount of storage capacity to match uncertain generation with uncertain demand. A possible solution is the use of thermal and electrical storages. This paper compares several model formulations: mixed integer linear programs (MILPs), nonlinear programs (NLPs), mixed integer nonlinear programs (MINLPs) for optimizing the operation of a multi-modal home energy system comprising heating and electricity subsystems. The respective optimization problems are then resolved within a model predictive control scheme and the final solutions are compared in terms of runtime and optimality. The results indicate that a thermocline-based thermal storage model leads to the overall lowest costs while not significantly impeding computing times. Additionally, the results show that a continuous heat pump model leads to reduced computing times without affecting the modelling accuracy.

关键词： Thermal storages battery scheduling model predictive control mixed integer optimization nonlinear optimization modelling

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Contention-resolving model predictive control for an intelligent intersection traffic model

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DISCRETE EVENT DYNAMIC SYSTEMS-THEORY AND APPLICATIONS 2021年第3期31卷 407-437页

作者： Yao, Ningshi Zhang, Fumin Georgia Inst Technol Sch Elect & Comp Engn Atlanta GA 30308 USA

We address the problem of optimally scheduling automated vehicles crossing an intelligent intersection by assigning vehicles with priorities and desired speed. An idealized intersection traffic model is established for the development and verification of the required algorithms. We formulate the intersection scheduling problem as a mixed integer programming (or MIP) problem which co-designs the priority and traveling speed for each vehicle. The co-design aims to minimize the vehicle waiting time at the intersection area, under a set of safety constraints. We derived a contention-resolving model predictive control (or MPC) algorithm to dynamically assign priorities and compute the vehicles' traveling speeds. A branch cost formulation is proposed for the decision tree constructed by contention-resolving MPC based on time instants when collisions might occur among vehicles. Based on the priority assignments, a decentralized control law is designed to control each vehicle to travel with an optimal speed given a specific priority assignment. The optimal priority assignment can be determined by searching the lowest cost path in the decision tree. The solution computed by contention-resolving MPC is proved to be optimal given the condition of immediate access (or CIA) required in real-time scheduling. The effectiveness of the proposed method is verified through simulation and compared with the first-come-first-serve (or FCFS) and highest-speed-first (or HSF) scheduling strategies.

关键词： Intersection scheduling Model predictive control mixed integer optimization Optimal control Event triggered system

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Interpretable algorithmic fairness in structured and unstructured data

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The Journal of Machine Learning Research 2024年第1期25卷 10406-10447页

作者： Hari Bandi Dimitris Bertsimas Thodoris Koukouvinos Sofie Kupiec Massachusetts Institute of Technology Cambridge MA Sloan School of Management Massachusetts Institute of Technology Cambridge MA Operations Research Center Massachusetts Institute of Technology Cambridge MA

Systemic bias with respect to gender and race is prevalent in datasets, making it challenging to train classification models that are accurate and alleviate bias. We propose a unified method for alleviating bias in structured and unstructured data, based on a novel optimization approach for optimally flipping outcome labels and training classification models simultaneously. In the case of structured data, we introduce constraints on selected objective measures of meritocracy, and present four case studies, demonstrating that our approach often outperforms state-of the art methods in terms of fairness and meritocracy. In the case of unstructured data, we present two case studies on image classification, demonstrating that our method outperforms state-of-the-art approaches in terms of fairness. Moreover, we note that the decrease in accuracy over the nominal model is 3:31% on structured data and 0:65% on unstructured data. Finally, we leverage Optimal Classification Trees (OCTs), to provide insights on which attributes of individuals lead to ipping of their labels and apply it to interpret the flipping decisions on structured data. Utilizing OCTs with auxiliary tabular data as well as Gradient-weighted Class Activation Mapping (Grad-CAM), we provide insights on the flipping decisions for unstructured data.

关键词： fair classification bias alleviation mixed integer optimization

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Adaptive stochastic and hybrid nonlinear optimization algorithms for improving the effectiveness of the biological processes at WWTP

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BIOCHEMICAL ENGINEERING JOURNAL 2023年第1期193卷

作者： Piotrowski, Robert Milewski, Krzysztof Maciag, Bartosz Gdansk Univ Technol Fac Elect & Control Engn EkoTech Ctr G Narutowicza 11-12 PL-80233 Gdansk Poland

Wastewater treatment plays an important factor in the modern world. Insufficient treatment may result in environmental pollution which can further lead to disasters and diseases. However, processes that take place inside wastewater treatment plants (WWTP) are highly complex in nature, therefore it is difficult to design an efficient, optimal control system. The problem regarding biochemical reactions inside Sequential Batch Reactor (SBR) is analyzed in this paper and the optimization problem is defined. Next, chosen optimization methods including both deterministic and metaheuristic algorithms are carefully described. Series of simulation experi-ments are conducted based on a verified model of SBR from Swarzewo Wastewater Treatment Plant (WWTP) -Northern Poland. The proposed optimal control system allowed to increase the efficiency, improve the quality of outflow (reduction of Chemical Oxygen Demand (COD), total nitrogen (Ntot) and total phosphorus (Ptot)) and reduce wastewater treatment costs.

关键词： Adaptive optimization Biological processes Deterministic algorithms Hybrid optimization Metaheuristic algorithms mixed integer optimization Nonlinear optimization Optimal control Wastewater treatment plant SBR

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MIP-BOOST: Efficient and Effective L₀ Feature Selection for Linear Regression

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JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 2021年第3期30卷 566-577页

作者： Kenney, Ana Chiaromonte, Francesca Felici, Giovanni Penn State Univ Dept Stat University Pk PA 16802 USA St Anna Sch Adv Studies Inst Econ Pisa Italy St Anna Sch Adv Studies EMbeDS Pisa Italy CNR Ist Anal Sistemi Informat Rome Italy

Recent advances in mathematical programming have made mixed integer optimization a competitive alternative to popular regularization methods for selecting features in regression problems. The approach exhibits unquestionable foundational appeal and versatility, but also poses important challenges. Here, we propose MIP-BOOST, a revision of standard mixed integer programming feature selection that reduces the computational burden of tuning the critical sparsity bound parameter and improves performance in the presence of feature collinearity and of signals that vary in nature and strength. The final outcome is a more efficient and effective L-0 feature selection method for applications of realistic size and complexity, grounded on rigorous cross-validation tuning and exact optimization of the associated mixed integer program. Computational viability and improved performance in realistic scenarios is achieved through three independent but synergistic proposals. including additional results, pseudocode, and computer code are available online.

关键词： Cross-validation Feature selection LASSO mixed integer optimization Regression Whitening

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FACTOR-DRIVEN TWO-REGIME REGRESSION

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ANNALS OF STATISTICS 2021年第3期49卷 1656-1678页

作者： Lee, Sokbae Liao, Yuan Seo, Myung Hwan Shin, Youngki Columbia Univ Dept Econ New York NY 10027 USA Rutgers State Univ Dept Econ New Brunswick NJ USA Seoul Natl Univ Dept Econ Seoul South Korea McMaster Univ Dept Econ Hamilton ON Canada

We propose a novel two-regime regression model where regime switching is driven by a vector of possibly unobservable factors. When the factors are latent, we estimate them by the principal component analysis of a panel data set. We show that the optimization problem can be reformulated as mixed integer optimization, and we present two alternative computational algorithms. We derive the asymptotic distribution of the resulting estimator under the scheme that the threshold effect shrinks to zero. In particular, we establish a phase transition that describes the effect of first-stage factor estimation as the cross-sectional dimension of panel data increases relative to the time-series dimension. Moreover, we develop bootstrap inference and illustrate our methods via numerical studies.

关键词： Threshold regression principal component analysis mixed integer optimization phase transition oracle properties

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A complete characterization of disjunctive conic cuts for mixed integer second order cone optimization

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DISCRETE optimization 2017年 24卷 3-31页

作者： Belotti, Pietro Goez, Julio C. Polik, Imre Ralphs, Ted K. Terlaky, Minas FICO Xpress Optimizer Team Birmingham W Midlands England NHH Norwegian Sch Econ Dept Business & Management Sci N-5045 Bergen Norway SAS Inst Cary NC 27513 USA Lehigh Univ Dept Ind & Syst Engn Bethlehem PA 18015 USA

We study the convex hull of the intersection of a disjunctive set defined by parallel hyperplanes and the feasible set of a mixed integer second order cone optimization (MISOCO) problem. We extend our prior work on disjunctive conic cuts (DCCs), which has thus far been restricted to the case in which the intersection of the hyperplanes and the feasible set is bounded. Using a similar technique, we show that one can extend our previous results to the case in which that intersection is unbounded. We provide a complete characterization in closed form of the conic inequalities required to describe the convex hull when the hyperplanes defining the disjunction are parallel. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Second order cone optimization mixed integer optimization Disjunctive programming Disjunctive conic cuts

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MDTri: robust and efficient global mixed integer search of spaces of multiple ternary alloys A DIRECT-inspired optimization algorithm for experimentally accessible computational material design

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COMPUTATIONAL optimization AND APPLICATIONS 2017年第3期68卷 671-687页

作者： Graf, Peter A. Billups, Stephen Natl Renewable Energy Lab Computat Sci Ctr Mail Stop ESIF 30115013 Denver West Pkwy Golden CO 80401 USA Univ Colorado Denver Dept Math & Stat Sci Denver CO USA

Computational materials design has suffered from a lack of algorithms formulated in terms of experimentally accessible variables. Here we formulate the problem of (ternary) alloy optimization at the level of choice of atoms and their composition that is normal for synthesists. Mathematically, this is a mixed integer problem where a candidate solution consists of a choice of three elements, and how much of each of them to use. This space has the natural structure of a set of equilateral triangles. We solve this problem by introducing a novel version of the DIRECT algorithm that (1) operates on equilateral triangles instead of rectangles and (2) works across multiple triangles. We demonstrate on a test case that the algorithm is both robust and efficient. Finally, we offer an explanation of the efficacy of DIRECT-specifically, its balance of global and local search-by showing that "potentially optimal rectangles" of the original algorithm are akin to the Pareto front of the "multi-component optimization" of global and local search.

关键词： mixed integer optimization DIRECT optimization Pareto front Sierpinski triangle Computational material design

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Integrated well pad development scheduling with steam injection control in steam-assisted gravity drainage

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JOURNAL OF PROCESS CONTROL 2020年 89卷 45-57页

作者： Mohankumar, Yashas Li, Zukui Huang, Biao Univ Alberta Dept Chem & Mat Engn Edmonton AB T6G 1H9 Canada

In this paper, we study the problem of integrated well pad development scheduling with nonlinear model predictive control based steam injection in steam-assisted gravity drainage (SAGD). The scheduling problem has been modeled as a mixed-integer program to find optimal development sequence and timing of multiple well-pads. Model predictive control problems are solved to find optimal steam injection profile such that the reservoir is under control. The integrated problem is solved using open-loop and closed-loop methods: (1) scheduling problem is only solved at the beginning of project operation, (2) Scheduling problem is solved every year with shrinking horizon implementation, and (3) shrinking horizon implementation of scheduling with reservoir model update based on feedback from control level. Simulation results demonstrate the benefits of closed-loop integrated scheduling and control: the NPV increase is 19%. (C) 2020 Elsevier Ltd. All rights reserved.

关键词： Integrated optimization Nonlinear model predictive control SAGD Nonlinear optimization mixed integer optimization

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