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Surrogate-model based MILP for the optimal design of ethylene production from shale gas

COMPUTERS & CHEMICAL ENGINEERING 2020年 141卷 107015-107015页

作者： Pedrozo, H. A. Rodriguez Reartes, S. B. Chen, Q. Diaz, M. S. Grossmann, I. E. Univ Nacl Sur UNS Dept Ingn Quim Bahia Blanca Buenos Aires Argentina Planta Piloto Ingn Quim Bahia Blanca Buenos Aires Argentina Carnegie Mellon Univ Dept Chem Engn Pittsburgh PA 15213 USA

We propose a novel algorithm for the optimal design of entire plants by refining piecewise linear surrogate models within an iterative framework. We apply this strategy to a superstructure for ethane-based ethylene production, including steam cracking and alternative technologies, and the separation, utility, carbon dioxide and hydrogen recovery systems. Multivariable piecewise linear surrogate models (SM) based on rigorous Aspen Plus models and capital cost correlations are obtained by solving Generalized Disjunctive Programming problems. Using these surrogates, a Master MILP problem is formulated to determine the optimal design. If convergence criteria are not met, SM are progressively refined in subsequent iterations. The optimal solution is the chemical looping oxidative dehydrogenation technology, whose net present value (NPV) is 12% higher than that of conventional steam cracking, while reducing the ethylene production cost by 15%. Finally, we validate the optimal design with Aspen Plus, obtaining an NPV error of less than 1%. (C) 2020 Elsevier Ltd. All rights reserved.

关键词： Surrogate models piecewise linear function Superstructure optimization Steam cracking Oxidative dehydrogenation Ethylene production

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RELU DEEP NEURAL NETWORKS AND linear FINITE ELEMENTS

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Journal of Computational Mathematics 2020年第3期38卷 502-527页

作者： Juncai He Lin Li Jinchao Xu Chunyue Zheng School of Mathematical Sciences Peking UniversityBeijing 100871China Beijing International Center for Mathematical Research Peking UniversityBeijing 100871China Department of Mathematics Pennsylvania State UniversityState CollegePA 16802USA

In this paper,we investigate the relationship between deep neural net works(DNN)with rectified linear unit(ReLU)function as the activation function and continuous piecewise linear(CPWL)functions,especially CPWL functions from the simplicial linear finite element method(FEM).We first consider the special case of *** exploring the DNN representation of its nodal basis functions,we present a ReLU DNN representation of CPWL in *** theoretically establish that at least 2 hidden layers are needed in a ReLU DNN to represent any linear finite element functions inΩ■R^2 when d≥***,for d=2,3 which are often encountered in scientific and engineering computing,the minimal number of two hidden layers are necessary and sufficient for any CPWL function to be represented by a ReLU *** we include a detailed account on how a general CPWL in R^d can be represented by a ReLU DNN with at most[log2(d+1)]|hidden layers and we also give an estimation of the number of neurons in DNN that are needed in such a represe ***,using the relationship bet ween DNN and FEM,we theoretically argue that a special class of DNN models with low bit-width are still expected to have an adequate representation power in ***,as a proof of concept,we present some numerical results for using ReLU DNNs to solve a two point boundary problem to demonstrate the potential of applying DNN for numerical solution of partial differential equations.

关键词： Finite element method Deep neural network piecewise linear function

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Smoothing Secant Line Slope Using Aggregation Fischer Burmeister function

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IEEE ACCESS 2020年 8卷 100521-100532页

作者： Ghazvini, Anahita Abdullah, Siti Norul Huda Sheikh Ayob, Masri Univ Kebangsaan Malaysia Fac Informat Sci & Technol Bangi 43600 Malaysia

As some of the objective functions are piecewise, so they are non-differentiable at specific points which have a significant impact on deep network rate and computational time. The non-differentiability issue increases the computational time dramatically. This issue is solved by the reformulation of the absolute value equation (AVE) through a parametrized single smooth equation. However, utilizing a single smoothing function is less effective to produce a better curve at the breaking points. Therefore, this work formulates a new smoothing function of Aggregation Fischer Burmister (AFB) via amalgamating of two popular smoothing functions: Aggregation (AGG) and Fischer-Burmeister (FB). These functions are having the ability of minimum estimation from both sides of the canonical piecewise function (CPF). If an amalgamation of smoothing functions can affect the differentiability of the piecewise objective function, then amalgamating the AGG and FB smoothing functions will produce a smooth secant line slope on both sides with less computational time. To evaluate the proposed technique, we implement a Newton algorithm using MATLAB, with random initial values. A new smoothing function is formulated by firstly converting the piecewise objective function to CPF. Then, we applied it to the Newton algorithm. Finally, to validate the AVE difficulty of the new piecewise function, we perform one run for each initial value, and 30 runs for time evaluation. The experimental analysis verified that the proposed technique outperformed other techniques of AGG and FB individually in terms of the natural logarithm, exponential, and square root. Hence, this novel technique yields promising smooth approximation for AVE with less computational time.

关键词： Smoothing methods linear programming Mathematical model Standards Optimization Estimation Computational modeling Absolute value equation artificial neural network objective function piecewise linear function smooth approximation

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P-SFA: Probability based Sigmoid function Approximation for Low-complexity Hardware Implementation

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MICROPROCESSORS AND MICROSYSTEMS 2020年 76卷 103105-103105页

作者： Wei, Linyu Cai, Jueping Vantruong Nguyen Chu, Jie Wen, Kailin Xidian Univ State Key Lab Wide Bandgap Semicond Technol Disci Xian 710071 Peoples R China

A probability-based sigmoid function approximation (P-SFA), which is based on piecewise linear function and neuron's values statistical probability distribution in each layer, is proposed to lower the complexity of neural network hardware implementation with only addition circuit. The sigmoid function is divided into three fixed regions, and the number of sub-regions with different sizes in each fixed region is adapted to neuron's values distribution in each layer to reduce the error between the sigmoid function and P-SFA function. The experimental results on FPGA show that the P-SFA function is efficient in terms of power and speed, and the recognition accuracies in DNN and CNN for MNIST with P-SFA are the highest among the state-of-the-art methods, up to 97.46% and 99.02% respectively. (C) 2020 Elsevier B.V. All rights reserved.

关键词： Sigmoid function Neural networks piecewise linear function

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High-precision linearized interpretation for fully connected neural network

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APPLIED SOFT COMPUTING 2021年 109卷 107572-107572页

作者： Lei, Xia Fan, Yongkai Li, Kuan-Ching Castiglione, Arcangelo Hu, Qian China Univ Petr Dept Automat Beijing Peoples R China Commun Univ China State Key Lab Media Convergence & Commun Beijing Peoples R China Providence Univ Dept Comp Sci & Informat Engn Taichung Taiwan Univ Salerno Dept Comp Sci I-84084 Fisciano Italy

Despite the widespread application of deep neural networks in finance, medical treatment, and autonomous driving, these networks face multiple security threats, such as maliciously constructed adversarial samples that can easily mislead deep neural network model classification, causing errors. Therefore, creating an interpretable model or designing an interpretation method is necessary to improve its security. This paper presents an interpretation scheme, named Convergent Interpretation for Deep Neural Networks (CIDNN), to obtain a provably convergent and consistent interpretation for deep neural networks. The main idea of CIDNN is to first convert the deep neural networks into a set of mathematically convergent piecewise linear Neural Networks (PLNN), then convert the PLNN into a set of equivalent linear classifiers. In this way, each linear classifier can be interpreted by its decision features. By analyzing the convergence of the local approximation interpretation scheme, we prove that this interpretable model can be sufficiently close to the deep neural network with certain conditions. Experiments show the convergence of CIDNN's interpretation, and the interpretation conforms with similar samples in the synthetic dataset. Besides, we demonstrate the semantical meaning of CIDNN in the Fashion-MNIST dataset. (C) 2021 Elsevier B.V. All rights reserved.

关键词： Deep neural network Convergent interpretation piecewise linear function Sigmoid function

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Topology design of remote patient monitoring system concerning qualitative and quantitative issues

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OMEGA-INTERNATIONAL JOURNAL OF MANAGEMENT SCIENCE 2021年 98卷 102137-102137页

作者： Zheng, Xiao-Xue Chang, Ching-Ter Minjiang Univ New Huadu Business Sch Fuzhou 350108 Fujian Peoples R China Fuzhou Univ Sch Econ & Management Fuzhou 350002 Peoples R China Chang Gung Univ Dept Informat Management Taoyuan Taiwan Chang Gung Mem Hosp Clin Trial Ctr Taoyuan Taiwan Ming Chi Univ Technol Dept Ind Engn & Management Taipei Taiwan

piecewise linear functions have been used to model and solve non-linear problems in science, social science, and technology fields since the 1950's. Multi-choice goal programming (MCGP) has been widely used to solve multiple objective decision-making (MODM) problems in which each goal mapping with multiple aspiration levels is allowed, expanding the original feasible region to obtain better solutions in the MODM problems. This paper integrates the efficient S-shaped penalty method, arbitrary piece-wise linear utility functions, trapezoidal utility functions, and MCGP to solve a topology design problem in a remote patient monitoring system (RPMS) by providing universal senior citizen coverage in which quantitative (e.g., patient satisfaction-related cost) and qualitative (e.g., satisfaction) issues are considered simultaneously. In addition, some novel utility functions such as the force utility function, indicator utility function, and arbitrary utility function are proposed to improve the usefulness of MCGP in the field of management science. These proposed utility functions can be easily used to model qualitative issues in real-world problems. A topology design problem for an RPMS is demonstrated to justify the feasibility, usefulness, and compatibility of the proposed methods. Further, sensitivity analysis and managerial implications are provided using an RPMS in Taiwan. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： Remote patient monitoring piecewise linear function Multi-choice Goal programming Utility function

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Deriving pack rules for hydro-photovoltaic hybrid power systems considering diminishing marginal benefit of energy

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APPLIED ENERGY 2021年 304卷 117858-117858页

作者： Gong, Yu Liu, Pan Ming, Bo Xu, Weifeng Huang, Kangdi Li, Xiao Wuhan Univ State Key Lab Water Resources & Hydropower Engn S Wuhan 430072 Peoples R China Wuhan Univ Hubei Prov Key Lab Water Syst Sci Sponge City Con Wuhan 430072 Peoples R China Wuhan Univ Res Inst Water Secur Wuhan 430072 Peoples R China Xian Univ Technol State Key Lab Ecohydraul Northwest Arid Reg China Xian 710048 Peoples R China

The hybrid operation of multiple renewables is a promising approach to enhance energy efficiency. The benefit of energy is influenced by a specified target. The energy exceeding the target is still beneficial, while the marginal benefit is diminishing. The diminishing marginal benefit of energy can significantly influence power system performance, however, was seldom considered in hybrid operation. To address this issue, we derive operating rules considering the diminishing marginal benefit of energy, namely the pack rules, for hydro-photovoltaic hybrid systems. First, a two-period hybrid operation model is built incorporating the diminishing marginal benefit of energy. After that, the total operation benefit is maximized and the optimal pack rules are derived to balance between the current and future energy. Finally, the sensitivity analysis of the optimal pack rules is conducted. Results using China's Ertan hydro-photovoltaic power plant indicate that: (1) the analytical pack rules illustrate that the power output is a piecewise linear function of the available energy, and the power output decreases with the increase in available energy when the power curtailment rate increases significantly;(2) the multi-point pack rules, which are based on two-point and three-point rules, outperform the standard operating policy by 6.92% and 18.76% in terms of operating benefit, respectively;and (3) the optimal water levels are more sensitive to forecast uncertainties for inflow and photovoltaic output in non-flood seasons compared with flood seasons. The derived pack rules could effectively improve the hybrid operation by considering the diminishing marginal benefit of energy.

关键词： Photovoltaic power Hydropower Operating rules Hybrid power system piecewise linear function

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Positive Definiteness of Complex piecewise linear functions and Some of Its Applications

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MATHEMATICAL NOTES 2018年第3-4期103卷 550-564页

作者： Zastavnyi, V. P. Manov, A. D. Donetsk Natl Univ Donetsk Ukraine

Given alpha a (0, 1) and c = h + i beta, h, beta a R, the function f (alpha,c) : R -> C defined as follows is considered: (1) f (alpha,c) is Hermitian, i.e., , x a R;(2) f (alpha,c) (x) = 0 for x > 1;moreover, on each of the closed intervals [0, alpha] and [alpha, 1], the function f (alpha,c) is linear and satisfies the conditions f (alpha,c) (0) = 1, f (alpha,c) (alpha) = c, and f (alpha,c) (1) = 0. It is proved that the complex piecewise linear function f (alpha,c) is positive definite on R if and only if m(alpha) h 1 - alpha and |beta| gamma(alpha, h), If m(alpha) h 1 - alpha and alpha a Q, then gamma(alpha, h) > 0;otherwise, gamma(alpha, h) = 0. This result is used to obtain a criterion for the complete monotonicity of functions of a special form and prove a sharp inequality for trigonometric polynomials.

关键词： positive definite function piecewise linear function completely monotone function Bochner-Khinchine theorem Bernstein's inequality

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Efficient continuous piecewise linear regression for linearising univariate non-linear functions

A I I E Transactions

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A I I E Transactions 2025年第3期57卷

作者： John Alasdair Warwicker Steffen Rebennack Stochastic Optimization Institute of Operations Research Karlsruhe Institute of Technology Karlsruhe Germany

Due to their flexibility and ability to incorporate non-linear relationships, Mixed-Integer Non-linear Programming (MINLP) approaches for optimization are commonly presented as a solution tool for real-world problems. Within this context, piecewise linear (PWL) approximations of non-linear continuous functions are useful, as opposed to non-linear machine learning-based approaches, since they enable the application of Mixed-Integer linear Programming techniques in the MINLP framework, as well as retaining important features of the approximated non-linear functions, such as convexity. In this work, we extend upon fast algorithmic approaches for modeling discrete data using PWL regression by tuning them to allow the modeling of continuous functions. We show that if the input function is convex, then the convexity of the resulting PWL function is guaranteed. An analysis of the runtime of the presented algorithm shows which function characteristics affect the efficiency of the model, and which classes of functions can be modeled very quickly. Experimental results show that the presented approach is significantly faster than five existing approaches for modeling non-linear functions from the literature, at least 11 times faster on the tested functions, and up to a maximum speedup of more than 328,000. The presented approach also solves six benchmark problems for the first time.

关键词： piecewise linear function continuous function function fitting piecewise linear regression convex function

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piecewise linear relationship between L1 norm objective functional values and L∞ norm constraint specifications

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DIGITAL SIGNAL PROCESSING 2019年 92卷 234-240页

作者： Miao, Qing Ling, Bingo Wing-Kuen Wang, Xiaoling Foshan Univ Dept Math Foshan 528000 Peoples R China Guangdong Univ Technol Sch Informat Engn Guangzhou 510006 Guangdong Peoples R China

For a sparse optimization problem with an L-1 norm objective function subject to an L-infinity norm inequality constraint, this paper finds that there is a piecewise linear relationship between the L-1 norm objective functional values and the L-infinity norm constraint specifications. This piecewise linear relationship is proved mathematically. Also, computer numerical simulations on a set of signal vectors verify the validity of this result. This result can provide a guidance for system analysts to define the specification on the L-infinity norm specification. (C) 2019 Elsevier Inc. All rights reserved.

关键词： L-1-minimization L-infinity-norm constraint piecewise linear function Sparse optimization

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