Gradient Estimation using Lagrange Interpolation Polynomials

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Abstract

In this paper we use Lagrange interpolation polynomials to obtain good gradient estimations.This is e.g. important for nonlinear programming solvers.As an error criterion we take the mean squared error.This error can be split up into a deterministic and a stochastic error.We analyze these errors using (N times replicated) Lagrange interpolation polynomials.We show that the mean squared error is of order N-1+ 1 2d if we replicate the Lagrange estimation procedure N times and use 2d evaluations in each replicate.As a result the order of the mean squared error converges to N-1 if the number of evaluation points increases to infinity.Moreover, we show that our approach is also useful for deterministic functions in which numerical errors are involved.Finally, we consider the case of a fixed budget of evaluations.For this situation we provide an optimal division between the number of replicates and the number of evaluations in a replicate.
Original languageEnglish
Place of PublicationTilburg
PublisherOperations research
Number of pages18
Volume2003-101
Publication statusPublished - 2003

Publication series

NameCentER Discussion Paper
Volume2003-101

Fingerprint

Gradient Estimation
Lagrange Interpolation
Polynomial
Mean Squared Error
Evaluation
Nonlinear Programming
Lagrange
Division
Infinity
Converge

Keywords

  • estimation
  • interpolation
  • polynomials
  • non linear programming

Cite this

Hamers, H. J. M., Brekelmans, R. C. M., Driessen, L., & den Hertog, D. (2003). Gradient Estimation using Lagrange Interpolation Polynomials. (CentER Discussion Paper; Vol. 2003-101). Tilburg: Operations research.
Hamers, H.J.M. ; Brekelmans, R.C.M. ; Driessen, L. ; den Hertog, D. / Gradient Estimation using Lagrange Interpolation Polynomials. Tilburg : Operations research, 2003. (CentER Discussion Paper).
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Hamers, HJM, Brekelmans, RCM, Driessen, L & den Hertog, D 2003 'Gradient Estimation using Lagrange Interpolation Polynomials' CentER Discussion Paper, vol. 2003-101, Operations research, Tilburg.

Gradient Estimation using Lagrange Interpolation Polynomials. / Hamers, H.J.M.; Brekelmans, R.C.M.; Driessen, L.; den Hertog, D.

Tilburg : Operations research, 2003. (CentER Discussion Paper; Vol. 2003-101).

Research output: Working paperDiscussion paperOther research output

TY - UNPB

T1 - Gradient Estimation using Lagrange Interpolation Polynomials

AU - Hamers, H.J.M.

AU - Brekelmans, R.C.M.

AU - Driessen, L.

AU - den Hertog, D.

N1 - Subsequently published in the Journal of Optimization Theory & Applications, 2008 Pagination: 18

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N2 - In this paper we use Lagrange interpolation polynomials to obtain good gradient estimations.This is e.g. important for nonlinear programming solvers.As an error criterion we take the mean squared error.This error can be split up into a deterministic and a stochastic error.We analyze these errors using (N times replicated) Lagrange interpolation polynomials.We show that the mean squared error is of order N-1+ 1 2d if we replicate the Lagrange estimation procedure N times and use 2d evaluations in each replicate.As a result the order of the mean squared error converges to N-1 if the number of evaluation points increases to infinity.Moreover, we show that our approach is also useful for deterministic functions in which numerical errors are involved.Finally, we consider the case of a fixed budget of evaluations.For this situation we provide an optimal division between the number of replicates and the number of evaluations in a replicate.

AB - In this paper we use Lagrange interpolation polynomials to obtain good gradient estimations.This is e.g. important for nonlinear programming solvers.As an error criterion we take the mean squared error.This error can be split up into a deterministic and a stochastic error.We analyze these errors using (N times replicated) Lagrange interpolation polynomials.We show that the mean squared error is of order N-1+ 1 2d if we replicate the Lagrange estimation procedure N times and use 2d evaluations in each replicate.As a result the order of the mean squared error converges to N-1 if the number of evaluation points increases to infinity.Moreover, we show that our approach is also useful for deterministic functions in which numerical errors are involved.Finally, we consider the case of a fixed budget of evaluations.For this situation we provide an optimal division between the number of replicates and the number of evaluations in a replicate.

KW - estimation

KW - interpolation

KW - polynomials

KW - non linear programming

M3 - Discussion paper

VL - 2003-101

T3 - CentER Discussion Paper

BT - Gradient Estimation using Lagrange Interpolation Polynomials

PB - Operations research

CY - Tilburg

ER -

Hamers HJM, Brekelmans RCM, Driessen L, den Hertog D. Gradient Estimation using Lagrange Interpolation Polynomials. Tilburg: Operations research. 2003. (CentER Discussion Paper).