Gradient Estimation Using Lagrange Interpolation Polynomials

Ruud Brekelmans, L. T. Driessen, H. J. M. Hamers () and D. Hertog
Additional contact information
L. T. Driessen: Philips Lightening
H. J. M. Hamers: Tilburg University
D. Hertog: Tilburg University

Journal of Optimization Theory and Applications, 2008, vol. 136, issue 3, No 3, 357 pages

Abstract: Abstract 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, which can be split up into a deterministic error 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 $N^{-1+\frac{1}{2d}}$ 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. We provide also an optimal division between the number of gridpoints and replicates in case the number of evaluations is fixed. Further, it is shown that the estimation of the derivatives is more robust when the number of evaluation points is increased. Finally, test results show the practical use of the proposed method.

Keywords: Gradient estimation; Lagrange interpolation; Mean squared error (search for similar items in EconPapers)
Date: 2008
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10957-007-9315-9 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:joptap:v:136:y:2008:i:3:d:10.1007_s10957-007-9315-9

Ordering information: This journal article can be ordered from
http://www.springer. ... cs/journal/10957/PS2

DOI: 10.1007/s10957-007-9315-9

Access Statistics for this article

Journal of Optimization Theory and Applications is currently edited by Franco Giannessi and David G. Hull

More articles in Journal of Optimization Theory and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().