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

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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An approximation algorithm for multiobjective mixed-integer convex optimization

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MATHEMATICAL METHODS OF OPERATIONS RESEARCH 2024年第1期100卷 321-350页

作者： Lammel, Ina Kuefer, Karl-Heinz Suess, Philipp Fraunhofer Inst Ind Math ITWM Dept Optimizat Fraunhofer Pl 1 D-67663 Kaiserslautern Germany

In this article we introduce an algorithm that approximates the nondominated sets of multiobjective mixed-integer convex optimization problems. The algorithm constructs an inner and outer approximation of the front exploiting the convexity of the patches for problems with an arbitrary number of criteria. In the algorithm, the problem is decomposed into patches, which are multiobjective convex problems, by fixing the integer assignments. The patch problems are solved using (simplicial) Sandwiching. We identify parts of patches that are dominated by other patches and ensure that these patch parts are not refined further. We prove that the algorithm converges and show a bound on the reduction of the approximation error in the course of the algorithm. We illustrate the behaviour of our algorithm using some numerical examples and compare its performance to an algorithm from literature.

关键词： Multiobjective optimization mixed-integer optimization Approximation algorithm Convex optimization

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A hybrid patch decomposition approach to compute an enclosure for multi-objective mixed-integer convex optimization problems

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MATHEMATICAL METHODS OF OPERATIONS RESEARCH 2024年第1期100卷 291-320页

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

In multi-objective mixed-integer convex optimization, multiple convex objective functions need to be optimized simultaneously while some of the variables are restricted to take integer values. In this paper, we present a new algorithm to compute an enclosure of the nondominated set of such optimization problems. More precisely, we decompose the multi-objective mixed-integer convex optimization problem into several multi-objective continuous convex optimization problems, which we refer to as patches. We then dynamically compute and improve coverages of the nondominated sets of those patches to finally combine them to obtain an enclosure of the nondominated set of the multi-objective mixed-integer convex optimization problem. Additionally, we introduce a mechanism to reduce the number of patches that need to be considered in total. Our new algorithm is the first of its kind and guaranteed to return an enclosure of prescribed quality within a finite number of iterations. For selected numerical test instances we compare our new criterion space based approach to other algorithms from the literature and show that much larger instances can be solved with our new algorithm.

关键词： Multi-objective optimization mixed-integer optimization Enclosure Approximation algorithm

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Online mixed-integer optimization in Milliseconds

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INFORMS JOURNAL ON COMPUTING 2022年第4期34卷 2229-2248页

作者： Bertsimas, Dimitris Stellato, Bartolomeo MIT Operat Res Ctr Cambridge MA 02139 USA MIT Sloan Sch Management Cambridge MA 02139 USA Princeton Univ Dept Operat Res & Financial Engn Princeton NJ 08544 USA

We propose a method to approximate the solution of online mixed-integer optimization (MIO) problems at very high speed using machine learning. By exploiting the repetitive nature of online optimization, we can greatly speed up the solution time. Our approach encodes the optimal solution into a small amount of information denoted as strategy using the voice of optimization framework. In this way, the core part of the optimization routine becomes a multiclass classification problem that can be solved very quickly. In this work, we extend that framework to real-time and high-speed applications focusing on parametric mixed-integer quadratic optimization. We propose an extremely fast online optimization method consisting of a feedforward neural network evaluation and a linear system solution where the matrix has already been factorized. Therefore, this online approach does not require any solver or iterative algorithm. We show the speed of the proposed method both in terms of total computations required and measured execution time. We estimate the number of floating point operations required to completely recover the optimal solution as a function of the problem dimensions. Compared with state-of-the-art MIO routines, the online running time of our method is very predictable and can be lower than a single matrix factorization time. We benchmark our method against the state-of-the-art solver Gurobi obtaining up to two to three orders of magnitude speedups on examples from fuel cell energy management, sparse portfolio optimization, and motion planning with obstacle avoidance.

关键词： mixed-integer optimization computational methods artificial intelligence analysis of algorithms: computational complexity heuristic

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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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Tri-Level mixed-integer optimization for Two-Stage Microgrid Dispatch With Multi-Uncertainties

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IEEE TRANSACTIONS ON POWER SYSTEMS 2020年第5期35卷 3636-3647页

作者： Qiu, Haifeng Gu, Wei Xu, Yinliang Yu, Wenwu Pan, Guangsheng Liu, Pengxiang Southeast Univ Sch Elect Engn Nanjing 210096 Peoples R China Tsinghua Univ Tsinghua Berkeley Shenzhen Inst TBSI Tsinghua Shenzhen Int Grad Sch Tsinghua SIGS Shenzhen 518055 Peoples R China Southeast Univ Sch Math Nanjing 210096 Peoples R China

To meet the diverse operational requirements of microgrids with multi-uncertainties, a tri-level two-stage mixed-integer robust optimization model is proposed to solve the optimal scheduling problem with source-load power and unit N-1 uncertainties. In the first stage, the optimal goal is to obtain robust scheduling plans that can be applied effectively in satisfying the operation and adjustment constraints, as well as to find the minimal operating cost in the basic scenario. Considering the system may concern a specific operating factor (e.g., load satisfaction rate) under a N-1 contingency, the second stage of the model minimizes the total energy of load shedding and renewable energy abandonment in the worst scenario. As the proposed robust model is a multi-level coupled mixed-integer optimization problem, which is difficult to address by either the duality method or the Karush-Kuhn-Tucker (KKT) method, a novel nested reformulation-and-decomposition (R&D) algorithm is proposed to handle this complex problem. Numerical simulations and tests have verified the effectiveness and superiority of the proposed robust model and the solution algorithm. Furthermore, the robust optimal schemes satisfy the engineering application more reasonably.

关键词： Uncertainty optimization Load modeling Power systems Numerical models Indexes Scheduling Microgrid robust dispatch N-1 uncertainty mixed-integer optimization reformulation-and-decomposition

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Information Complexity of mixed-integer Convex optimization 1

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24th International Conference on integer Programming and Combinatorial optimization (IPCO)

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 14853 USA Microsoft Res Redmond WA 98052 USA Pontificia Univ Catolica Rio de Janeiro Comp Sci Dept Rio de Janeiro Brazil

ISBN: (数字)9783031327261

ISBN: (纸本)9783031327254;9783031327261

We investigate the information complexity of mixed-integer convex optimization under different kinds 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. Further, we prove the first set of results in the literature (to the best of our knowledge) on information complexity with respect to oracles based on first-order information but restricted to binary queries, and discuss various special cases of interest thereof.

关键词： mixed-integer optimization Information complexity

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A UNIFIED APPROACH TO mixed-integer optimization PROBLEMS WITH LOGICAL CONSTRAINTS

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SIAM JOURNAL ON optimization 2021年第3期31卷 2340-2367页

作者： Bertsimas, Dimitris Cory-Wright, Ryan Pauphilet, Jean MIT Sloan Sch Management 77 Massachusetts Ave Cambridge MA 02139 USA MIT 77 Massachusetts Ave Cambridge MA 02139 USA London Business Sch London NW1 4SA England

We propose a unified framework to address a family of classical mixed-integer optimization problems with logically constrained decision variables, including network design, facility pal component analysis, and sparse learning problems. These problems exhibit logical relationships between continuous and discrete variables, which are usually reformulated linearly using a big-M formulation. In this work, we challenge this long-standing modeling practice and express the logical constraints in a nonlinear way. By imposing a regularization condition, we reformulate these problems as convex binary optimization problems, which are solvable using an outer-approximation procedure. In numerical experiments, we establish that a general-purpose numerical strategy, which combines cutting-plane, first-order, and local search methods, solves these problems faster and at a larger scale than state-of-the-art mixed-integer linear or second-order cone methods. Our approach successfully solves network design problems with 100s of nodes and provides solutions up to 40\% better than the state of the art, sparse portfolio selection problems with up to 3,200 securities compared with 400 securities for previous attempts, and sparse regression problems with up to 100,000 covariates.

关键词： mixed-integer optimization branch and cut outer approximation

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PAPER mixed-integer Linear optimization Formulations for Feature Subset Selection in Kernel SVM Classification

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IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES 2024年第8期E107A卷 1151-1162页

作者： Tamura, Ryuta Takano, Yuichi Miyashiro, Ryuhei Tokyo Univ Agr & Technol Grad Sch Engn Koganei Tokyo 1848588 Japan October Sky Co Ltd Fuchu Tokyo 1830055 Japan Univ Tsukuba Inst Syst & Informat Engn Tsukuba Ibaraki 3058573 Japan Tokyo Univ Agr & Technol Inst Engn Koganei 1848588 Japan

We study the mixed-integer optimization (MIO) approach to feature subset selection in nonlinear kernel support vector machines (SVMs) for binary classification. To measure the performance of subset selection, we use the distance between two classes (DBTC) in a high- dimensional feature space based on the Gaussian kernel function. However, DBTC to be maximized as an objective function is nonlinear, nonconvex and nonconcave. Despite the difficulty of linearizing such a nonlinear function in general, our major contribution is to propose a mixed-integer linear optimization (MILO) formulation to maximize DBTC for feature subset selection, and this MILO problem can be solved to optimality using optimization software. We also derive a reduced version of the MILO problem to accelerate our MILO computations. Experimental results show good computational efficiency for our MILO formulation with the reduced problem. Moreover, our method can often outperform the linear-SVMbased MILO formulation and recursive feature elimination in prediction performance, especially when there are relatively few data instances.

关键词： key feature subset selection support vector machine mixed-integer optimization kernel-target alignment machine learning

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A neural network-based distributional constraint learning methodology for mixed-integer stochastic optimization

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EXPERT SYSTEMS WITH APPLICATIONS 2023年第1期232卷

作者： Alcantara, Antonio Ruiz, Carlos Univ Carlos III Madrid Dept Stat Avda Univ 30 Madrid 28911 Spain UC3M BS Inst Financial Big Data IFiBiD Madrid Spain

The use of machine-learning methods helps to improve decision-making in different fields. In particular, the idea of bridging predictions (predictive models) and prescriptions (optimization problems) is gaining attention within the scientific community. One of the main ideas to address this trade-off is the Constraint Learning (CL) methodology, where the structure of the machine learning model can be treated as a set of constraints to be embedded within the optimization problem, establishing the relationship between a direct decision variable x and a response variable y. However, most CL approaches have focused on making point predictions, not considering the statistical and external uncertainty faced in the modeling process. In this paper, we extend the CL methodology to deal with uncertainty in the response variable y. The novel Distributional Constraint Learning (DCL) methodology makes use of a piece-wise linearizable neural network-based model to estimate the parameters of the conditional distribution of y (dependent on decisions x and contextual information), which can be embedded within mixed-integer optimization problems. In particular, we formulate a stochastic optimization problem by sampling random values from the estimated distribution by using a linear set of constraints. In this sense, DCL combines both the predictive performance of the neural network method and the possibility of generating scenarios to account for uncertainty within a tractable optimization model. The behavior of the proposed methodology is tested in the context of electricity systems, where a Virtual Power Plant seeks to optimize its operation, subject to different forms of uncertainty, and with price-responsive consumers.

关键词： Stochastic optimization Constraint learning Distribution estimation Neural networks mixed-integer optimization

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