Skip to main content

Advertisement

SpringerLink
Log in

Best Approximation and Perturbation Property in Hilbert Spaces

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We study the perturbation property of best approximation to a set defined by an abstract nonlinear constraint system. We show that, at a normal point, the perturbation property of best approximation is equivalent to an equality expressed in terms of normal cones. This equality is related to the strong conical hull intersection property. Our results generalize many known results in the literature on perturbation property of best approximation established for a set defined by a finite system of linear/nonlinear inequalities. The connection to minimization problem is considered.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. Boor Particlede (1976) ArticleTitleOn Best Interpolation Journal of Approximation Theory 16 28–42 Occurrence Handle10.1016/0021-9045(76)90093-9

    Article  Google Scholar 

  2. C. K. Chui F. Deutsch J. D. Ward (1990) ArticleTitleConstrained Best Approximation in Hilbert Space Constructive Approximation 6 35–64 Occurrence Handle10.1007/BF01891408

    Article  Google Scholar 

  3. C. K. Chui F. Deutsch J. D. Ward (1992) ArticleTitleConstrained Best Approximation in Hilbert Space, II Journal of Approximation Theory 71 231–238

    Google Scholar 

  4. F. Deutsch W. Li J. D. Ward (1997) ArticleTitleA Dual Approach to Constrained Interpolation from a Convex Subset of Hilbert Space Journal of Approximation Theory 90 385–414 Occurrence Handle10.1006/jath.1996.3082

    Article  Google Scholar 

  5. F. Deutsch V. Ubhaya J. D. Ward Y. Xu (1996) ArticleTitleConstrained Best Approximation in Hilbert Space, III: Applications to n-Convex Functions Constructive Approximation 12 361–384 Occurrence Handle10.1007/s003659900019

    Article  Google Scholar 

  6. C. A. Micchelli P. W. Smith J. Swetits J. D. Ward (1985) ArticleTitleConstrained L p Approximation Constructive pproximation 1 93–102 Occurrence Handle10.1007/BF01890024

    Article  Google Scholar 

  7. C. A. Micchelli F. I. Utreras (1988) ArticleTitleSmoothing and Interpolation in a Convex Subset of a Hilbert Space SIAM Journal on Scientific and Statistical Computing 9 728–746 Occurrence Handle10.1137/0909048

    Article  Google Scholar 

  8. F. Deutsch W. Li J. D. Ward (1999) ArticleTitleBest Approximation from the Intersection of a Closed Convex Set and a Polyhedron in Hilbert Space, Weak Slater Conditions, and the Strong Conical Hull Intersection Property SIAM Journal on Optimization 10 252–268 Occurrence Handle10.1137/S1052623498337273

    Article  Google Scholar 

  9. J. F. Bonnans A. Shapiro (2000) Pertur ation Analysis of Optimization Problems, Springer Series in Operations Research Springer Verlag New York, NY

    Google Scholar 

  10. J. P. Aubin H. Frankowska (1990) Set-Valued Analysis Birkhäuser Boston, Massachusetts

    Google Scholar 

  11. F. H. Clarke (1983) Optimization and Nonsmooth Analysis John Wiley and Sons New York NY

    Google Scholar 

  12. F. H. Clarke Y. S. Ledyaev R. J. Stren P. R. Wolenski (1998) Nonsmooth Analysis and Control Theory Springer Verlag New York, NY

    Google Scholar 

  13. S. M. Robinson (1976) ArticleTitleStability Theory for Systems of Inequalities, II: Differentiable Nonlinear Systems SIAM Journal on Numerical Analysis 13 497–513 Occurrence Handle10.1137/0713043

    Article  Google Scholar 

  14. C. Li K. F. Ng (2002) ArticleTitleOn Best Approximation by Nonconvex Sets and Perturbation of Nonconvex Inequality Systems in Hilbert Spaces SIAM Journal on Optimization 13 726–744 Occurrence Handle10.1137/S1052623402401373

    Article  Google Scholar 

  15. C. Li XQ. Jin (2002) ArticleTitleNonlinearly Constrained Best Approximation in Hilbert Spaces: The Strong CHIP annd the Basic Constraint Qualification SIAM Journal on Optimization 13 228–239 Occurrence Handle10.1137/S1052623401385600

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Sichuan Normal University, Chengdu, Sichuan, PRC

    Y. R. He

  2. Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

    K. F. Ng

Authors
  1. Y. R. He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. F. Ng
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The authors thank the referees for valuable suggestions.

K.F. Ng - This author was partially supported by Grant A0324638 from the National Natural Science Foundation of China and Grants (2001) 01GY051-66 and SZD0406 from Sichuan Province.

Y.R. He -This author was supported by a Direct Grant (CUHK) and an Earmarked Grant from the Research Grant Council of Hong Kong.

Rights and permissions

Reprints and permissions

About this article

Cite this article

He, Y.R., Ng, K.F. Best Approximation and Perturbation Property in Hilbert Spaces. J Optim Theory Appl 126, 265–285 (2005). https://doi.org/10.1007/s10957-005-4714-2

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-005-4714-2

Keywords

Search

Navigation