Skip to main content
SpringerLink
Log in

Differentiability Properties of Metric Projections onto Convex Sets

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

Abstract

It is known that directional differentiability of metric projection onto a closed convex set in a finite-dimensional space is not guaranteed. In this paper, we discuss sufficient conditions ensuring directional differentiability of such metric projections. The approach is based on a general theory of sensitivity analysis of parameterized optimization problems.

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. Zarantonello, E.H.: Projections on convex sets in Hilbert space and spectral theory. In: contributions to Nonlinear Functional Analysis. pp. 237–424, Academic Press, New York (1971)

  2. Kruskal, J.: Two convex counterexamples: a discontinuous envelope function and a non-differentiable nearest point mapping. Proc. Amer. Math. Soc. 23, 697–703 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  3. Shapiro, A.: Directionally nondifferentiable metric projection. J. Optim. Theory Appl. 81, 203–204 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Holmes, R.B.: Smoothness of certain metric projections on Hilbert space. Trans. Amer. Math. Soc. 184, 87–100 (1973)

    Article  MathSciNet  Google Scholar 

  5. Akmal, S.S., Nam, N.M., Veerman, J.J.P.: On a convex set with nondifferentiable metric projection. Optim. Lett. 9, 1039–1052 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Shapiro, A.: On concepts of directional differentiability. J Optim. Theory Appl. 66, 447–487 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    Book  MATH  Google Scholar 

  8. Bonnans, J.F., Cominetti, R., Shapiro, A.: Sensitivity analysis of optimization problems under second order regular constraints. Math. Oper. Res. 23, 806–831 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bonnans, J.F., Shapiro, A.: Nondegeneracy and quantitative stability of parameterized optimization problems with multiple solutions. SIAM J. Optim. 8, 940–946 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Robinson, S.M.: Local structure of feasible sets in nonlinear programming. II. Nondegeneracy. Math. Program. Stud. 22, 217–230 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cominetti, R.: Metric regularity, tangent sets and second order optimality conditions. Appl. Math. Opt. 21, 265–287 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. Shapiro, A.: Sensitivity analysis of generalized equations. J. Math. Sci. 115, 2554–2565 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This research was partly supported by the NSF award CMMI 1232623.

Author information

Authors and Affiliations

  1. Georgia Institute of Technology, Atlanta, GA, USA

    Alexander Shapiro

Authors
  1. Alexander Shapiro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Shapiro.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shapiro, A. Differentiability Properties of Metric Projections onto Convex Sets. J Optim Theory Appl 169, 953–964 (2016). https://doi.org/10.1007/s10957-016-0871-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-016-0871-8

Keywords

Mathematics Subject Classification

Search

Navigation