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piecewise linear function Fitting via Mixed-Integer linear Programming

INFORMS JOURNAL ON COMPUTING 2020年第2期32卷 507-530页

作者： Rebennack, Steffen Krasko, Vitally Karlsruhe Inst Technol Inst Operat Res D-76185 Karlsruhe Baden Wurttembe Germany Colorado Sch Mines Div Econ & Business Golden CO 80401 USA

piecewise linear (PWL) functions are used in a variety of applications. Computing such continuous PWL functions, however, is a challenging task. Software packages and the literature on PWL function fitting are dominated by heuristic methods. This is true for both fitting discrete data points and continuous univariate functions. The only exact methods rely on nonconvex model formulations. Exact methods compute continuous PWL function for a fixed number of breakpoints minimizing some distance function between the original function and the PWL function. An optimal PWL function can only be computed if the breakpoints are allowed to be placed freely and are not fixed to a set of candidate breakpoints. In this paper, we propose the first convex model for optimal continuous univariate PWL function fitting. Dependent on the metrics chosen, the resulting formulations are either mixed-integer linear programming or mixed-integer quadratic programming problems. These models yield optimal continuous PWL functions for a set of discrete data. On the basis of these convex formulations, we further develop an exact algorithm to fit continuous univariate functions. Computational results for benchmark instances from the literature demonstrate the superiority of the proposed convex models compared with state-of-the-art nonconvex models.

关键词： piecewise linear function linear spline splines of degree 1 polyhedral function mixed-integer linear programming (MILP) mixed-integer quadratic programming (MIQP) function fitting spline regression global optimization

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Modeling the memristor with piecewise linear function

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INTERNATIONAL JOURNAL OF NUMERICAL MODELLING-ELECTRONIC NETWORKS DEVICES AND FIELDS 2015年第1期28卷 96-106页

作者： Mu, Xiaomu Yu, Juntang Wang, Shuning Tsinghua Univ Tsinghua Natl Lab Informat Sci & Technol Beijing 100084 Peoples R China Tsinghua Univ Dept Automat Beijing 100084 Peoples R China

Memristor is introduced as the fourth basic circuit element. In this paper, we show that piecewise linear function can be used to model memristor. piecewise linear function is used as a window function to model the memristor device. The piecewise linear window function is linear at each subregion of the domain. Because of this good character, explicit relation between the thickness of the doped region and the charge can be delivered, together with the explicit memristance expression. Models using a general nonlinear window function may not have these explicit results. The piecewise linear window function is also more flexible than the existing window functions. piecewise linear function can also be used to model the more general memristive systems. Copyright (c) 2014 John Wiley & Sons, Ltd.

关键词： memristor piecewise linear function window function memristive system

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Reformulation of the modified goal programming for logarithmic piecewise linear function

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APPLIED MATHEMATICS AND COMPUTATION 2006年第1期174卷 13-23页

作者： Chang, CT Natl Changhua Univ Educ Informat Management Dept Changhua 50058 Taiwan

The purpose of this paper is to present a reformulation of the model presented by Chang [A modified goal programming model for piecewise linear function, European Journal of Operational Research 139 (2002) 62-67]. The modified goal programming (MGP) model of Chang has been accepted as an efficient method published for solving polynomial/posynomial (P/P) programming problems. However, it still cannot be used to fully practical applications because the problem with negative variable is undefined in logarithmic piecewise linear function (LPLF). This paper presents some appropriate strategies to overcome this undefined problem. The reformulation MGP model can be used to solve P/P programming problems with negative/positive variables. In addition, an illustrative example is included for demonstrating the correctness of the proposed method. (c) 2005 Elsevier Inc. All rights reserved.

关键词： goal programming piecewise linear function

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A Comparison of Two Mixed-Integer linear Programs for piecewise linear function Fitting

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INFORMS JOURNAL ON COMPUTING 2022年第2期34卷 1042-1047页

作者： Warwicker, John Alasdair Rebennack, Steffen Karlsruhe Inst Technol Inst Operat Res Stochast Optimizat D-76185 Karlsruhe Raden Wurttembe Germany

The problem of fitting continuous piecewise linear (PWL) functions to discrete data has applications in pattern recognition and engineering, amongst many other fields. To find an optimal PWL function, the positioning of the breakpoints connecting adjacent linear segments must not be constrained and should be allowed to be placed freely. Although the univariate PWL fitting problem has often been approached from a global optimisation perspective, recently, two mixed-integer linear programming approaches have been presented that solve for optimal PWL functions. In this paper, we compare the two approaches: the first was presented by Rebennack and Krasko [Rebennack S, Krasko V (2020) piecewise linear function fitting via mixed-integer linear programming. INFORMS J. Comput. 32(2):507-530] and the second by Kong and Maravelias [Kong L, Maravelias CT (2020) On the derivation of continuous piecewise linear approximating functions. INFORMS J. Comput. 32(3):531-5461 Both formulations are similar in that they use binary variables and logical implications modelled by big-M constructs to ensure the continuity of the PWL function, yet the former model uses fewer binary variables. We present experimental results comparing the time taken to find optimal PWL functions with differing numbers of breakpoints across 10 data sets for three different objective functions. Although neither of the two formulations is superior on all data sets, the presented computational results suggest that the formulation presented by Rebennack and Krasko is faster. This might be explained by the fact that it contains fewer complicating binary variables and sparser constraints. Summary of Contribution: This paper presents a comparison of the mixed-integer linear programming models presented in two recent studies published in the INFORMS Journal on Computing. Because of the similarity of the formulations of the two models, it is not clear which one is preferable. We present a detailed comparison of the two

关键词： piecewise linear function mixed-integer linear programming function fitting

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Matching localization algorithm of nonlinear T-S fuzzy system constructed by the piecewise linear function

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JOURNAL OF AMBIENT INTELLIGENCE AND HUMANIZED COMPUTING 2019年第2期10卷 417-427页

作者： Wang, Guijun Zhang, Guoying Tianjin Normal Univ Sch Math Sci Tianjin 300387 Peoples R China

The piecewise linear function is a generalization of a segmentation linear functions with one variable in the multivariate case, which is an important tool to describe the relationships between fuzzy system and approximation function. Firstly, aiming at the -integrable function, the concrete method of constructing piecewise linear functions and their analytical expressions were introduced, furthermore, based on the piecewise linear functions, we constructed the nonlinear T-S fuzzy systems with a singleton fuzzifier in this paper. And then, we put forward the matching-localization algorithm of the fuzzy system based on the spatial positioning mode for the input variables in the domain. Finally, the effectiveness of the matching localization algorithm of the nonlinear T-S fuzzy system was verified by the means of the simulation example. The results show that the approximating ability of the fuzzy system to a <-integrable function can be adjusted by changing the subdivision parameters.

关键词： piecewise linear function Singleton fuzzifier Nonlinear T-S fuzzy system (mu)over-cap-integrable function Matching-localization algorithm

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The resolution of excess investment in Fuzzy portfolio selection using piecewise linear function

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APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 2024年第1期32卷

作者： Huang, Yin-Yin Ma, Tongxia Pan, Jingchao Tsaur, Ruey-Chyn Nanchang Vocat Univ Sch Econ & Management Nanchang Peoples R China GongQing Inst Sci & Technol Jiujiang Peoples R China Tamkang Univ Dept Management Sci New Taipei City 25137 Taiwan

Fuzzy portfolio selection on the guaranteed return rate has been proposed to study excess investment in different dimensions of risk preferences. However, the theory of behavioural finance proposes that the investment proportions of securities are not only determined by the intrinsic value of the security but also largely affected by the behaviour of investors. Therefore, the investment behaviour of an investor in the fuzzy portfolio selection should be considered in the excess investment for different securities at different return rates. In this study, we propose a piecewise linear function for the fuzzy expected return rates and use the new decision variables for investment proportion in each security. In addition, considering the investment behaviours of decision-makers, we use the concept of fuzzy contingent strategy to take into account the relationships of investment proportion among the securities in the fuzzy portfolio selection. Analysis results indicated that we can find the investment proportions in the portfolio with the possible results of shortage investment, full investment, or excess investments by the fuzzy constraint of total investment proportions. Finally, in the illustration with two experiments, investors could individually select the portfolio by the piecewise linear function in shortage investment, full investment, or excess investment.

关键词： Fuzzy portfolio selection excess investment guaranteed return rate piecewise linear function contingent strategy 90-10

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A new chaotic Hopfield network with piecewise linear activation function

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Chinese Physics B 2010年第3期19卷 188-192页

作者：郑鹏升唐万生张建雄 Institute of Systems Engineering Tianjin University

This paper presents a new chaotic Hopfield network with a piecewise linear activation function. The dynamic of the network is studied by virtue of the bifurcation diagram, Lyapunov exponents spectrum and power spectru... 详细信息

关键词： Hopfield network chaos piecewise linear function

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Dynamical study of a second-order DPCM transmission system modeled by a piecewise-linear function

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-REGULAR PAPERS 2002年第11期49卷 1592-1609页

作者： Taralova, I Fournier-Prunaret, D Ecole Cent Nantes CNRS UMR 6597 IRCCyN F-44321 Nantes 3 France INSA DGEI LESIA SYD F-31077 Toulouse 4 France

This paper analyzes the behavior of a second-order differential pulse code modulation (DPCM) transmission system when the nonlinear characteristic of the quantizer is taken into consideration. In this way, qualitatively new properties of the DPCM system have been unraveled, which cannot be observed and explained if the nonlinearity of the quantizer is neglected. For the purpose of this study, a piecewise-linear nondifferentiable quantizer characteristic is considered. The resulting model of the DPCM is of the form of iteration equations (i.e., map), where the inverse iterate is not unique (i.e., noninvertible map). Therefore, the mathematical theory of noninvertible maps is particularly suitable for this analysis, together with the more classic tools of nonlinear dynamics. This study allowed us, in addition, to show, from a theoretical point of view, some new properties of nondifferentiable maps, in comparison with differentiable ones. After a short review of noninvertible maps, the presented methods and tools for noninvertible maps are applied to the DPCM system. An original algorithm for calculation of bifurcation curves for the DPCM map is proposed. Via the studies in the parameter and phase plane, different nonlinear phenomena such as the overlapping of bifurcation curves causing multistability, chaotic behavior, or multiple basins with fractal boundary are pointed out. All observed phenomena show a very complex dynamical behavior even in the constant input signal case, discussed here.

关键词： differential pulse code modulation (DPCM) piecewise linear function

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Smoothed piecewise linear Lyapunov function for the First Order Dynamical Systems

Smoothed Piecewise Linear Lyapunov Function for the First Or...

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International Scientific and Technical Conference on Integrated Computer Technologies in Mechanical Engineering - Synergetic Engineering (ICTM)

作者： Voliansky, Roman Shramko, Iurii Volianska, Nina Tolochko, Olga Sadovoi, Oleksandr Natl Tech Univ Ukraine Igor Sikorsky Kyiv Polytech Inst 37 Str Polytech UA-03056 Kiev Ukraine Dniprovsk State Tech Univ 2 Dniprobudivska Str UA-51918 Kamyanske Ukraine

ISBN: (纸本)9783031614149;9783031614156

The paper deals with the mathematical backgrounds of design piecewise single-variable Lyapunov functions. These backgrounds are based on the study of the main features of generalized Lyapunov function and using them to analyze the solution of dynamical programming problem. This analysis shows that the optimal control signal, which is supplied to the control plant, depends on the partial derivative of the Lyapunov function. This fact allows us to consider the Lyapunov function as a control signal integral over the perturbed motion variable. If the control signal is produced by the sliding mode controller, our approach defines the Lyapunov function as a non-quadratic absolute value function. Since this function is not differentiable in the origin, it is hard enough to use it while the optimal controller is being designed with optimal theory methods. We avoid this problem by applying a smoothing procedure, which is based on considering a small neighborhood near the fracture point. In this neighborhood, we replace the piecewise linear function with a polynomial one. Factors of the polynomial are defined to have the same function values and its derivatives in the boundary points. In our paper, we show that our approach can be extended to the case of a controller which produces a multilevel signal. Signal of such form causes occurring the piecewise linear Lyapunov function, which can be smoothed by using polynomials. Since such a piecewise function can have a lot of branches, we offer to consider it as two functions, which are defined in their intervals.

关键词： Lyapunov function piecewise linear function sliding mode control polynomial smoothing

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CALCULATION OF piecewise linear-APPROXIMATION TO A DISCRETE function

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IEEE TRANSACTIONS ON COMPUTERS 1975年第8期24卷 843-846页

作者： VANDEWALLE, J CATHOLIC UNIV LOUVAIN DEPT ELECT ENGNHEVERLEEBELGIUM

A repeated scanning algorithm for determining the minimax approximation to an arbitrary function by means of a piecewise linear function with a fixed number of variable knots is described, and it is compared to optima... 详细信息

关键词： Discrete Data Feature Generation Minimax Approximation piecewise linear function Waveform Seghnentation

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