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Computational Aspects of Primal Dual Proximal Algorithms for M-estimation with Constraints

  • M. L. Bougeard

Abstract

We summarize computational experience with algorithms based on the Spingarn Partial Inverse proximal method for Huber-M estimation. The result is a family of highly parallel primal-dual algorithms that are globally convergent and attractive for large scale problems. The approach is easily extended to handle constrained problems. To obtain an efficient implementation, remedies are introduced to ensure efficiency in case of highly degenerate situations. Here, several mechanisms are investigated for reducing the execution time. Problems with bundle of M-estimators are investigated. In computational practice, appropriate choice of start point and robust data pre-conditioning are shown to result in speed-up performance, even for box constrained problems.

Keywords

Huber-M regression proximal algorithm partial inverse method constraints pre-conditioning 

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References

  1. 1.
    Barrodale I. and Roberts F.D.K., (1974) An improved algorithm for L1 linear approximation, SIAM J. on Numerical Analysis, 10, 839–848CrossRefGoogle Scholar
  2. 2.
    Bougeard M.L., Bange J.-F., Caquineau C.-D. and Bec-Borsenberger A. (1997), ESA symposium Proceedings Hipparcos Venice’97, ESA SP-402, 165–169Google Scholar
  3. 3.
    Bougeard M.L. and Caquineau C.D. (1999), Parallel proximal decomposition algorithms for robust estimation, Annals of O. R. 90, 247–270 preprint Univ.Lyonl-UCBL, April 1996 CrossRefGoogle Scholar
  4. 4.
    Bougeard M.L., Gambis D., Ray R. (1999), Algorithms for box constrained M-estimation: fitting large data sets with application to EOP series, preprint Paris-observatory (submitted Physics and Chemistry of the Earth, May 1999)Google Scholar
  5. 5.
    Candahl E, (1995), Applications algorithmiques de l’analyse proximale, technical report D.E.A. University-Lyon 1, March 1995Google Scholar
  6. 6.
    Channes A., Cooper W.W. and Ferguson R.O. (1955), Optimal estimation of executive compensation by linear programming, Management Science, I:138–151CrossRefGoogle Scholar
  7. 7.
    Clark D.I. (1985), The mathematical structure of Huber’s M-estimator, SIAM J. on Scientific and Statistical Computing 6, 209–219CrossRefGoogle Scholar
  8. 8.
    Hampel P.W., Ronchetti, Rousseeuw P.J., and Stahel (1986), Robust Statistics, Wiley New YorkGoogle Scholar
  9. 9.
    Huber P.J. (1964), Robust estimation of a location parameter, Annals of Math. Statistics 35, 73–101CrossRefGoogle Scholar
  10. 10.
    Kennedy W.J. and Gentle J.E., (1980), Statistical Computing, ed M. DekkerGoogle Scholar
  11. 11.
    Michelot C. and Sougeard M.L. (1994), Duality results and proximal solutions of the Huber-M problem, J. Optimization Theory and Appl. 30, 203–221CrossRefGoogle Scholar
  12. 12.
    Ray R. (1999), Méthodes d’estimation robuste et Application au domaine de la rotation de la Terre, PhD technical report, Paris Observatory, June 1999Google Scholar
  13. 13.
    Rockafellar R.T. (1970), Convex analysis, Princeton University PressGoogle Scholar
  14. 14.
    Spingarn J.E.(1983), ‘Partial inverse of a monotone operator’, J.Appl. Math. Opt. 10, 247–265CrossRefGoogle Scholar
  15. 15.
    Thinking Machine (1994), CM-Scientific Subroutines Library for CM FortranGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • M. L. Bougeard
    • 1
    • 2
  1. 1.UMR8630 CNRSParis ObservatoryParisFrance
  2. 2.Université Lyon IVilleurbanneFrance

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