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John Lygeros  One of the best experts on this subject based on the ideXlab platform.

every continuous Piecewise affine function can be obtained by solving a parametric linear program
European Control Conference, 2013CoAuthors: Andreas B Hempel, Paul J Goulart, John LygerosAbstract:It is wellknown that solutions to parametric linear or quadratic programs are continuous Piecewise affine functions of the parameter. In this paper we prove the converse, i.e. that every continuous Piecewise affine function can be identified with the solution to a parametric linear program. In particular, we provide a constructive proof that every Piecewise affine function can be expressed as the linear mapping of the solution to a parametric linear program with at most twice as many variables as the dimension of the image of the Piecewise affine function. Our method is illustrated via two small numerical examples.

ECC  Every continuous Piecewise affine function can be obtained by solving a parametric linear program
2013 European Control Conference (ECC), 2013CoAuthors: Andreas B Hempel, Paul J Goulart, John LygerosAbstract:It is wellknown that solutions to parametric linear or quadratic programs are continuous Piecewise affine functions of the parameter. In this paper we prove the converse, i.e. that every continuous Piecewise affine function can be identified with the solution to a parametric linear program. In particular, we provide a constructive proof that every Piecewise affine function can be expressed as the linear mapping of the solution to a parametric linear program with at most twice as many variables as the dimension of the image of the Piecewise affine function. Our method is illustrated via two small numerical examples.
Wang Shuning  One of the best experts on this subject based on the ideXlab platform.

Dynamic behavior of Piecewiselinear approximations
Journal of Tsinghua University, 2011CoAuthors: Wang ShuningAbstract:The use of Piecewiselinear approximations to model nonlinear dynamic systems depends on the dynamic behavior of the Piecewiselinear approximations.When the Piecewiselinear approximation technique is used here to model the Logistic map,differences in the dynamic behavior are found between the approximation model and the original system.Even when the static error is very small,there are obvious differences between the bifurcation diagram of the original system and that of the Piecewiselinear approximation.The Piecewiselinear approximation is not as smooth as the original system,which causes the differences.A smoothed Piecewiselinear model gives the same dynamic behavior as the original system,so smoothed Piecewiselinear models are suitable for approximating nonlinear dynamic systems.

Piecewise Linear Approximation and its Application in Model Predictive Control
Control Engineering of China, 2010CoAuthors: Wang ShuningAbstract:The Piecewise linear approximation is introduced.The model of adaptive hinging hyperplanes(AHH) is a potential piecewie linear model and used in model predictive control(MPC).The nonlinear system to be controlled is modeled via AHH approximation,and the modeling is simple and efficient due to the adaptiveness of AHH identification algorithm.After the predictive model is built,an openloop optimization problem is formulated to get the optimal control sequence and the receding horizon control strategy is employed.The openloop optimization problem is actually a series of convex optimization subproblems and the existence of the global optimum is guaranteed for the original problem.In order to make an efficient application,a decent algorithm is proposed to search for a local optimum.Simulation results show the prospect of the AHHbased MPC.

Piecewise Linear Programming and its Application
Control Engineering of China, 2010CoAuthors: Wang ShuningAbstract:A new method for nonlinear optimization is proposed.Via Piecewise linear approximation and Piecewise linear programming,the nonlinear optimization can be transformed into a series of linear programming which can be solved by very efficient algorithms.It is proved that Piecewise linear programming can be solved by finite linear programming problems.Also,the necessary and sufficient condition for the local optimum is proposed.Using this condition,a descending algorithm is proposed for Piecewise linear programming.Combined with adaptive hinging hyperplanes,this algorithm is applied to the operation optimization for centrifugal chiller plants system.The energy performance of the system is improved significantly by Piecewise linear programming.This application shows the effectiveness of the proposed method.

Piecewise linear approximations with a hypercube partition
2008CoAuthors: Wang ShuningAbstract:A Piecewise linear approximation model with a hypercube partition is introduced for modeling complex nonlinear systems. The model partitions the domain into hypercubes, with one linear function used to approximate the origin nonlinear function in each hypercube. Lattice Piecewise linear expressions are used to connect all the local linear functions into a continuous Piecewise linear function. This model can approximate any twice differentiable nonlinear function to arbitrary precision. Since each hypercube was not partitioned into simplices, this model constructs simpler Piecewise linear functions for complex high dimensional functions.
Andreas B Hempel  One of the best experts on this subject based on the ideXlab platform.

every continuous Piecewise affine function can be obtained by solving a parametric linear program
European Control Conference, 2013CoAuthors: Andreas B Hempel, Paul J Goulart, John LygerosAbstract:It is wellknown that solutions to parametric linear or quadratic programs are continuous Piecewise affine functions of the parameter. In this paper we prove the converse, i.e. that every continuous Piecewise affine function can be identified with the solution to a parametric linear program. In particular, we provide a constructive proof that every Piecewise affine function can be expressed as the linear mapping of the solution to a parametric linear program with at most twice as many variables as the dimension of the image of the Piecewise affine function. Our method is illustrated via two small numerical examples.

ECC  Every continuous Piecewise affine function can be obtained by solving a parametric linear program
2013 European Control Conference (ECC), 2013CoAuthors: Andreas B Hempel, Paul J Goulart, John LygerosAbstract:It is wellknown that solutions to parametric linear or quadratic programs are continuous Piecewise affine functions of the parameter. In this paper we prove the converse, i.e. that every continuous Piecewise affine function can be identified with the solution to a parametric linear program. In particular, we provide a constructive proof that every Piecewise affine function can be expressed as the linear mapping of the solution to a parametric linear program with at most twice as many variables as the dimension of the image of the Piecewise affine function. Our method is illustrated via two small numerical examples.
Tomáš Masopust  One of the best experts on this subject based on the ideXlab platform.

Separability by Piecewise testable languages is PTimecomplete
Theoretical Computer Science, 2018CoAuthors: Tomáš MasopustAbstract:Abstract Piecewise testable languages form the first level of the Straubing–Therien hierarchy. The membership problem for this level is decidable and testing if the language of a DFA is Piecewise testable is NLcomplete. So far, this question has not been addressed for NFAs in the literature. We fill in this gap and show that it is PSpace complete. The main interest of this paper is, however, the lowerbound complexity of separability of regular languages by Piecewise testable languages. Two regular languages are separable by a Piecewise testable language if the Piecewise testable language includes one of them and is disjoint from the other. For languages represented by NFAs, separability by Piecewise testable languages is decidable in PTime . We show that it is PTime hard and that it remains PTime hard even if the input automata are minimal DFAs. As a result, it is unlikely that separability of regular languages by Piecewise testable languages can be solved in a restricted space or effectively parallelized.

Piecewise Testable Languages and Nondeterministic Automata
arXiv: Formal Languages and Automata Theory, 2016CoAuthors: Tomáš MasopustAbstract:A regular language is $k$Piecewise testable if it is a finite boolean combination of languages of the form $\Sigma^* a_1 \Sigma^* \cdots \Sigma^* a_n \Sigma^*$, where $a_i\in\Sigma$ and $0\le n \le k$. Given a DFA $A$ and $k\ge 0$, it is an NLcomplete problem to decide whether the language $L(A)$ is Piecewise testable and, for $k\ge 4$, it is coNPcomplete to decide whether the language $L(A)$ is $k$Piecewise testable. It is known that the depth of the minimal DFA serves as an upper bound on $k$. Namely, if $L(A)$ is Piecewise testable, then it is $k$Piecewise testable for $k$ equal to the depth of $A$. In this paper, we show that some form of nondeterminism does not violate this upper bound result. Specifically, we define a class of NFAs, called ptNFAs, that recognize Piecewise testable languages and show that the depth of a ptNFA provides an (up to exponentially better) upper bound on $k$ than the minimal DFA. We provide an application of our result, discuss the relationship between $k$Piecewise testability and the depth of NFAs, and study the complexity of $k$Piecewise testability for ptNFAs.

Piecewise Testable Languages and Nondeterministic Automata
2016CoAuthors: Tomáš MasopustAbstract:A regular language is kPiecewise testable if it is a finite boolean combination of languages of the form a1 ··· an , where ai2 and 0 n k. Given a DFAA and k 0, it is an NLcomplete problem to decide whether the language L(A) is Piecewise testable and, for k 4, it is coNPcomplete to decide whether the language L(A) is kPiecewise testable. It is known that the depth of the minimal DFA serves as an upper bound on k. Namely, if L(A) is Piecewise testable, then it is kPiecewise testable for k equal to the depth ofA. In this paper, we show that some form of nondeterminism does not violate this upper bound result. Specifically, we define a class of NFAs, called ptNFAs, that recognize Piecewise testable languages and show that the depth of a ptNFA provides an (up to exponentially better) upper bound on k than the minimal DFA. We provide an application of our result, discuss the relationship between kPiecewise testability and the depth of NFAs, and study the complexity of kPiecewise testability for ptNFAs. 1998 ACM Subject Classification F.1.1 Models of Computation, F.4.3 Formal Languages

MFCS  Piecewise Testable Languages and Nondeterministic Automata
2016CoAuthors: Tomáš MasopustAbstract:A regular language is kPiecewise testable if it is a finite boolean combination of languages of the form Sigma^* a_1 Sigma^* ... Sigma^* a_n Sigma^*, where a_i in Sigma and 0 = 0, it is an NLcomplete problem to decide whether the language L(A) is Piecewise testable and, for k >= 4, it is coNPcomplete to decide whether the language L(A) is kPiecewise testable. It is known that the depth of the minimal DFA serves as an upper bound on k. Namely, if L(A) is Piecewise testable, then it is kPiecewise testable for k equal to the depth of A. In this paper, we show that some form of nondeterminism does not violate this upper bound result. Specifically, we define a class of NFAs, called ptNFAs, that recognize Piecewise testable languages and show that the depth of a ptNFA provides an (up to exponentially better) upper bound on k than the minimal DFA. We provide an application of our result, discuss the relationship between kPiecewise testability and the depth of NFAs, and study the complexity of kPiecewise testability for ptNFAs.
Paul J Goulart  One of the best experts on this subject based on the ideXlab platform.

every continuous Piecewise affine function can be obtained by solving a parametric linear program
European Control Conference, 2013CoAuthors: Andreas B Hempel, Paul J Goulart, John LygerosAbstract:It is wellknown that solutions to parametric linear or quadratic programs are continuous Piecewise affine functions of the parameter. In this paper we prove the converse, i.e. that every continuous Piecewise affine function can be identified with the solution to a parametric linear program. In particular, we provide a constructive proof that every Piecewise affine function can be expressed as the linear mapping of the solution to a parametric linear program with at most twice as many variables as the dimension of the image of the Piecewise affine function. Our method is illustrated via two small numerical examples.

ECC  Every continuous Piecewise affine function can be obtained by solving a parametric linear program
2013 European Control Conference (ECC), 2013CoAuthors: Andreas B Hempel, Paul J Goulart, John LygerosAbstract:It is wellknown that solutions to parametric linear or quadratic programs are continuous Piecewise affine functions of the parameter. In this paper we prove the converse, i.e. that every continuous Piecewise affine function can be identified with the solution to a parametric linear program. In particular, we provide a constructive proof that every Piecewise affine function can be expressed as the linear mapping of the solution to a parametric linear program with at most twice as many variables as the dimension of the image of the Piecewise affine function. Our method is illustrated via two small numerical examples.