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piecewise linear approximation using J1 compatible triangulations for efficient MILP representation

COMPUTERS & CHEMICAL ENGINEERING 2025年 196卷

作者： Birkelbach, Felix Inst Energy Syst & Thermodynam Getreidemarkt 9-E302 A-1060 Vienna Austria

For including piecewise linear (PWL) functions in MILP problems, the logarithmic convex combination (Log) formulation has been shown to yield very fast solving times. However, identifying approximations that can be used with Log is a big challenge since the approximation has to be compatible with a J1 triangulation. In this article, an algorithm is proposed that identifies approximations using J1 compatible triangulations. It seeks to satisfy the specified error tolerance with the minimum number of linear pieces, so that the MILP formulation is small. To evaluate the performance of the J1 approach it is applied to two sets of benchmark functions from literature and results are compared to state-of-the-art approaches. Overall the J1 approach is shown to efficiently approximate functions in up to 3 dimensions. Especially for tight error tolerances, these J1 approximations require fewer auxiliary variables in MILP compared to alternative approaches.

关键词： piecewise-linear approximation J1 triangulation Triangulation on grid Logarithmic formulation

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Global optimization of separable objective functions on convex polyhedra via piecewise-linear approximation

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2006年第1期197卷 212-217页

作者： Zhang, Hao Wang, Shuning Tsing Hua Univ Dept Automat Beijing 100084 Peoples R China

The problem of minimizing a separable nonlinear objective function under linear constraints is considered in this paper. A systematic approach is proposed to obtain an approximately globally optimal solution via piecewise-linear approximation. By means of the new approach a minimum point of the original problem confined in a region where more than one linear piece is needed for satisfactory approximation can be found by solving only one linear programming problem. Hence, the number of linear programming problems to be solved for finding the approximately globally optimal solution may be much less than that of the regions partitioned. In addition, zero-one variables are not introduced in this approach. These features are desirable for efficient computation. The practicability of the approach is demonstrated by an example. (c) 2005 Elsevier B.V. All rights reserved.

关键词： separable objective function nonlinear programming piecewise-linear approximation global optimization

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linearly constrained global optimization via piecewise-linear approximation

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2008年第1期214卷 111-120页

作者： Zhang, Hao Wang, Shuning Tsinghua Univ Dept Automat Beijing 100084 Peoples R China

This paper considers the problem of optimizing a continuous nonlinear objective function subject to linear constraints via a piecewise-linear approximation. A systematic approach is proposed, which uses a lattice piecewise-linear model to approximate the nonlinear objective function on a simplicial partition and determines an approximately globally optimal solution by solving a set of standard linear programs. The new approach is applicable to any continuous objective function rather than to sepal-able ones only and could be useful to treat more complex nonlinear problems. A numerical example is given to illustrate the practicability. (C) 2007 Elsevier B.V. All rights reserved.

关键词： global optimization piecewise-linear approximation lattice piecewise-linear model nonlinear programming

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Canonical piecewise-linear approximation of smooth functions

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-FUNDAMENTAL THEORY AND APPLICATIONS 1998年第5期45卷 567-571页

作者： Julian, P Jordan, M Desages, A Univ Nacl Sur Dept Ingn Elect RA-8000 Bahia Blanca Buenos Aires Argentina CONICET CRIBABB RA-8000 Bahia Blanca Buenos Aires Argentina CIC RA-1900 La Plata Argentina

This paper deals with the approximation of smooth functions using canonical piecewise-linear functions. The developing of tools in the field of analysis and control of nonlinear systems based on this kind of functions, as well as its efficiency in the representation of electronic devices, motivates the development of useful methods to obtain accurate approximations. A recursive method is proposed to obtain simultaneously all the parameters required and its convergence is studied. In addition, an iterative method to introduce new partitions on the domain, when the error obtained is not satisfactory, is described. This method takes advantage of the partitions already found to reduce the total number of parameters that the algorithm has to handle.

关键词： piecewise-linear approximation

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On the piecewise-linear approximation of the Polynomial Chaotic Dynamics

On the Piecewise-Linear Approximation of the Polynomial Chao...

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34th International Conference on Telecommunications and Signal Processing (TSP)

作者： Petrzela, Jiri Brno Univ Technol Dept Radio Elect Brno 60200 Czech Republic

ISBN: (纸本)9781457714115

This paper briefly shows the influence of handmade picewise-linear approximation on the global dynamics on the selected third-order systems with the quadratic vector fields. These effects are demonstrated using concept of the Ljapunov exponents calculation. The approximated systems are verified by means of the real electronic circuits derived using integrator synthesis and time-domain analysis in Orcad Pspice simulator. The universality of the proposed method is discussed and future perspectives are provided.

关键词： Dynamical system chaotic dynamics Ljapunov exponents piecewise-linear approximation

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Continuation Power Flow With Nonlinear Power Injection Variations: A piecewise linear approximation

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IEEE TRANSACTIONS ON POWER SYSTEMS 2008年第4期23卷 1637-1643页

作者： Li, Shao-Hua Chiang, Hsiao-Dong Cornell Univ Sch Elect & Comp Engn Ithaca NY 14853 USA Shanghai Jiao Tong Univ Sch Elect Informat & Elect Engn Shanghai 200240 Peoples R China

Power injection variations that occur in power systems are generally nonlinear. A continuation power flow method with multiple nonlinear power injection variations is presented in this paper. To model the nature of nonlinear power injection variations, a piecewise-linear model is proposed to approximate the nonlinear model of power injections. The piecewise-linear model is studied and incorporated into continuation power flow, making it capable of handling nonlinear power injection variations. The impacts of linear as well as nonlinear load variations on the P-V curve, Q-V curve, and P-Q-V curve are numerically analyzed on test power systems. Numerical studies indicate that the direction of power injection variation can significantly affect the load margin and that the differences between uniform linear power injection directions and multiple nonlinear power injection variations can be significant. The importance of accurately modeling power injection variations on simulated power system quasi-state behaviors is stressed.

关键词： Continuation power flow load margin nonlinear power injection variations piecewise-linear approximation

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α-Functions piecewise-linear approximation from Noisy and Hermite Data 22

α-Functions Piecewise-linear Approximation from Noisy and H...

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ACM SIGGRAPH 2022 Conference Proceedings

作者： Marc Alexa Computer Graphics Group TU Berlin Germany

ISBN: (纸本)9781450393379

We introduce α-functions, providing piecewise linear approximation to given data as the difference of two convex functions. The parameter α controls the shape of a paraboloid that is probing the data and may be used to filter out noise in the data. The use of convex functions enables tools for efficient approximation to the data, adding robustness to outliers, and dealing with gradient information. It also allows using the approach in higher dimension. We show that α-functions can be efficiently computed and demonstrate their versatility at the example of surface reconstruction from noisy surface samples.

关键词： convex-concave split piecewise-linear approximation weighted Delaunay triangulation Legendre transformation

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piecewise linear approximation for MILP leveraging piecewise convexity to improve performance

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COMPUTERS & CHEMICAL ENGINEERING 2024年 183卷

作者： Birkelbach, Felix Huber, David Hofmann, Rene Inst Energy Syst & Thermodynam Getreidemarkt 6-E302 A-1060 Vienna Austria

To realize adaptive operation planning with MILP unit commitment, piecewise-linear approximations of the functions that describe the operating behavior of devices in the energy system have to be computed. We present an algorithm to compute a piecewise-linear approximation of a multi-variate non-linear function. The algorithm splits the domain into two regions and approximates each region with a set of hyperplanes that can be translated to a convex set of constraints in MILP. The main advantage of this "piecewise-convex approximation"(PwCA) compared to more general piecewise-linear approximation with simplices is that the MILP representation of PwCA requires only one auxiliary binary variable. For this reason, PwCA yields significantly faster solving times in large MILP problems where the MILP representation of certain functions has to be replicated many times, such as in unit commitment. To quantify the impact on solving time, we compare the performance using PwCA with the performance of simplex approximation with logarithmic formulation and show that PwCA outperforms the latter by a big margin. For this reason, we conclude that PwCA will be a useful tool to set up and solve large MILP problems such as arise in unit commitment and similar engineering optimization problems.

关键词： MILP piecewise-linear approximation Unit commitment

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A max-piecewise-linear neural network for function approximation

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NEUROCOMPUTING 2008年第4-6期71卷 843-852页

作者： Wen, Chengtao Ma, Xiaoyan Tsinghua Univ Dept Automat Inst Syst Engn Beijing 100084 Peoples R China CEA Saclay DEN DMN SRMA LA2M F-91191 Gif Sur Yvette France

This paper proposes a Max-piecewise-linear (MPWL) Neural Network for function approximation. The MPWL network consists of a single hidden layer and employs the piecewise-linear (PWL) Basis Functions as the activation functions of hidden neurons. Since a PWL Basis Function possesses a simple functional form and universal representation capability, the MPWL network achieves a good balance between the computational simplicity and approximation accuracy. In addition, a PAIL version of Back-Propagation (PBP) algorithm is developed, whose computational complexity is lower than the training algorithms for the Canonical PWL network, and the Back-Propagation algorithm for the sigmoid network with same number of training cycles. Another advantage of the MPWL network is its amenability to hardware implementation. This facilitates many applications such as nonlinear circuit synthesis, dynamic identification and control. (c) 2007 Elsevier B.V. All rights reserved.

关键词： piecewise-linear approximation neural-network PWL basis function canonical PWL representation nonlinear circuit synthesis

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piecewise-linear approximations for a non-linear transmission expansion planning problem

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IET GENERATION TRANSMISSION & DISTRIBUTION 2015年第12期9卷 1235-1244页

作者： Camponogara, Eduardo de Almeida, Katia Campos Hardt Junior, Rubens Univ Fed Santa Catarina Dept Automat & Syst Engn BR-88040900 Florianopolis SC Brazil Univ Fed Santa Catarina Dept Elect Engn BR-88040900 Florianopolis SC Brazil

This study proposes a new approach to solve the transmission expansion planning (TEP) problem considering that the network is represented by a non-linear model. Non-linear constraints are approximated with piecewise-linear (PWL) functions, which renders TEP a mixed-integer linear programming (MILP) problem. This study presents MILP formulations obtained from the PWL approximation using the incremental, special ordered sets of type 2 (SOS2) and the logarithmic convex combination (Log) model. Three test systems are used to assess the effectiveness of the formulations. The computational analysis encompasses a baseline formulation, which accounts for the costs on line installation and transmission losses, and a variation of this formulation which minimises the operating costs rather than transmission losses. The results obtained by the MILP models are compared with those of a transmission planning model based on the DC representation of the network. The analysis of the results showed that expansion plans obtained from the MILP formulations are feasible, whereas the plans derived from the DC model are infeasible with respect to system constraints.

关键词： power transmission planning approximation theory piecewise linear techniques linear programming integer programming piecewise-linear approximation nonlinear transmission expansion planning problem nonlinear model nonlinear constraints mixed-integer linear programming problem MILP problem logarithmic convex combination model special ordered sets of type 2 line installation losses transmission losses

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