Data-driven individual and joint chance-constrained optimization via kernel smoothing

•Reformulation of individual and joint chance constraints using kernel smoothing.•Method to calculate the divergence tolerance based on kernel smoothing estimation.•Initialization scheme for joint chance-constrained problems.•Application in production planning with variability in production rates. W...

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Vydané v:	Computers & chemical engineering Ročník 78; s. 51 - 69
Hlavní autori:	Calfa, B.A., Grossmann, I.E., Agarwal, A., Bury, S.J., Wassick, J.M.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Elsevier Ltd 12.07.2015
Predmet:	Confidence Data-driven chance constraint Density Distribution functions Estimates Kernel smoothing Kernels Mathematical models Optimization Process systems engineering Smoothing ϕ-Divergence Data-driven chance constraint 90C90 90C15 ϕ-Divergence Process systems engineering Kernel smoothing
ISSN:	0098-1354, 1873-4375
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Abstract	•Reformulation of individual and joint chance constraints using kernel smoothing.•Method to calculate the divergence tolerance based on kernel smoothing estimation.•Initialization scheme for joint chance-constrained problems.•Application in production planning with variability in production rates. We propose a data-driven, nonparametric approach to reformulate (conditional) individual and joint chance constraints with right-hand side uncertainty into algebraic constraints. The approach consists of using kernel smoothing to approximate unknown true continuous probability density/distribution functions. Given historical data for continuous univariate or multivariate random variables (uncertain parameters in an optimization model), the inverse cumulative distribution function (quantile function) and the joint cumulative distribution function are estimated for the univariate and multivariate cases, respectively. The approach relies on the construction of a confidence set that contains the unknown true distribution. The distance between the true distribution and its estimate is modeled via ϕ-divergences. We propose a new way of specifying the size of the confidence set (i.e., the ϕ-divergence tolerance) based on point-wise standard errors of the smoothing estimates. The approach is illustrated with a motivating and an industrial production planning problem with uncertain plant production rates.
AbstractList	We propose a data-driven, nonparametric approach to reformulate (conditional) individual and joint chance constraints with right-hand side uncertainty into algebraic constraints. The approach consists of using kernel smoothing to approximate unknown true continuous probability density/distribution functions. Given historical data for continuous univariate or multivariate random variables (uncertain parameters in an optimization model), the inverse cumulative distribution function (quantile function) and the joint cumulative distribution function are estimated for the univariate and multivariate cases, respectively. The approach relies on the construction of a confidence set that contains the unknown true distribution. The distance between the true distribution and its estimate is modeled via -divergences. We propose a new way of specifying the size of the confidence set (i.e., the -divergence tolerance) based on point-wise standard errors of the smoothing estimates. The approach is illustrated with a motivating and an industrial production planning problem with uncertain plant production rates. •Reformulation of individual and joint chance constraints using kernel smoothing.•Method to calculate the divergence tolerance based on kernel smoothing estimation.•Initialization scheme for joint chance-constrained problems.•Application in production planning with variability in production rates. We propose a data-driven, nonparametric approach to reformulate (conditional) individual and joint chance constraints with right-hand side uncertainty into algebraic constraints. The approach consists of using kernel smoothing to approximate unknown true continuous probability density/distribution functions. Given historical data for continuous univariate or multivariate random variables (uncertain parameters in an optimization model), the inverse cumulative distribution function (quantile function) and the joint cumulative distribution function are estimated for the univariate and multivariate cases, respectively. The approach relies on the construction of a confidence set that contains the unknown true distribution. The distance between the true distribution and its estimate is modeled via ϕ-divergences. We propose a new way of specifying the size of the confidence set (i.e., the ϕ-divergence tolerance) based on point-wise standard errors of the smoothing estimates. The approach is illustrated with a motivating and an industrial production planning problem with uncertain plant production rates.
Author	Bury, S.J. Grossmann, I.E. Wassick, J.M. Calfa, B.A. Agarwal, A.
Author_xml	– sequence: 1 givenname: B.A. surname: Calfa fullname: Calfa, B.A. email: bacalfa@cmu.edu organization: Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA – sequence: 2 givenname: I.E. surname: Grossmann fullname: Grossmann, I.E. email: grossmann@cmu.edu organization: Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA – sequence: 3 givenname: A. surname: Agarwal fullname: Agarwal, A. email: AAgarwal2@dow.com organization: The Dow Chemical Company, Midland, MI 48674, USA – sequence: 4 givenname: S.J. surname: Bury fullname: Bury, S.J. email: SJBury@dow.com organization: The Dow Chemical Company, Midland, MI 48674, USA – sequence: 5 givenname: J.M. surname: Wassick fullname: Wassick, J.M. email: JMWassick@dow.com organization: The Dow Chemical Company, Midland, MI 48674, USA
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Snippet	•Reformulation of individual and joint chance constraints using kernel smoothing.•Method to calculate the divergence tolerance based on kernel smoothing... We propose a data-driven, nonparametric approach to reformulate (conditional) individual and joint chance constraints with right-hand side uncertainty into...
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SubjectTerms	Confidence Data-driven chance constraint Density Distribution functions Estimates Kernel smoothing Kernels Mathematical models Optimization Process systems engineering Smoothing ϕ-Divergence
Title	Data-driven individual and joint chance-constrained optimization via kernel smoothing
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