Skip to main content
SpringerLink
Log in

Error Bounds for Eigenvalue and Semidefinite Matrix Inequality Systems

  • Published:
Mathematical Programming Submit manuscript

Abstract

In this paper we give sufficient conditions for existence of error bounds for systems expressed in terms of eigenvalue functions (such as in eigenvalue optimization) or positive semidefiniteness (such as in semidefinite programming).

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. Azé, D., Hiriart-Urruty, J.B.: Optimal Hoffman type estimates in eigenvalue and semidefinite inequality constraints. J. Global Optim. 24, 133–147 (2002)

    Article  Google Scholar 

  2. Bhatia, R.: Matrix analysis. Graduate Texts in Mathematics 169, Springer, 1996

  3. Bosch, P., Henion R., Jourani, A.: Sufficient conditions for error bounds and applications. Applied Math. Optim. 50, 161–181 (2004)

    Google Scholar 

  4. Burke, J.V., Ferris, M.C.: Weak sharp minima in mathematical programming. SIAM J. Contr. Optim. 31, 1340–1359 (1993)

    Article  Google Scholar 

  5. Clarke, F.H., Ledyaev, Yu.S., Stern, R.J., Wolenski, P.R.: Nonsmooth analysis and control theory. Springer, New York, 1998

  6. Jourani, A., Zălinescu, C.: Conditioning and upper-Lipschitz inverse suddifferentials in nonsmooth optimization problems. J. Optim. Th. Appl. 95, 127–148 (1997)

    Article  Google Scholar 

  7. Deng, S.: Computable error bounds for convex inequality systems in reflexive Banach spaces. SIAM J. Optim. 7, 274–279 (1997)

    Article  Google Scholar 

  8. Deng, S., Hu, H.: Computable error bounds for semidefinite programming. J. Global Optim. 14, 105–115 (1999)

    Article  Google Scholar 

  9. Fletcher, R.: Semi-definite matrix constraints in optimization. SIAM J. Control Optim. 23, 493–513 (1985)

    Article  Google Scholar 

  10. Hoffman, A.J.: On approximate solutions of systems of linear inequalities. J. Research Nat. Bur. Standards 49, 263–265 (1952)

    Google Scholar 

  11. Ioffe, A.D.: Regular points of Lipschitz functions. Trans. Amer. Math. Soc. 251, 61–69 (1979)

    Google Scholar 

  12. Jourani, A.: Hoffman's error bound, local controllability and sensitivity analysis. SIAM J. Control Optim. 38, 947–970 (2000)

    Article  Google Scholar 

  13. Jourani, A., Thibault, A.: Approximate subdifferential and metric regularity : Finite dimensional case. Math. Programming 47, 203–218 (1990)

    Article  Google Scholar 

  14. Lewis, A.S.: Nonsmooth Analysis of eigenvalues. Math. Programming 84, 1–24 (1999)

    Google Scholar 

  15. Lewis, A.S., Pang, J.S.: Error bounds for convex inequality systems. In: J.-P. Crouzeix, J.-E. Martinez-Legaz and M. Volle (eds), Generalized Convexity, Generalized Monotonicity, pp 75–110, 1998

  16. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. Vol.1: Basic Theory, Vol. 2: Applications, to appear in Springer

  17. Mordukhovich, B.S., Shao, Y.: Nonsmooth sequential analysis in Asplund space. Trans. Amer. Math. Soc. 348, 215–220 (1996)

    Article  Google Scholar 

  18. Ng, K.F., Zheng, X.Y.: Error bounds for lower semicontinuous functions in normed spaces. SIAM J. Optim. 12, 1–17 (2002)

    Article  Google Scholar 

  19. Overton, M.L., Womersley, R.S.: Optimality conditions and duality theory for minimizing sums of the largest eigenvalue of symmetric matrices. Math. Programming 62, 321–537 (1993)

    Article  Google Scholar 

  20. Pang, J.S.: Error bounds in mathematical programming,. Math. Programming 79, 299–332 (1997)

    Article  Google Scholar 

  21. Rockafellar, R.T., Wets, R.: Variational analysis. Springer, Berlin, 1998

  22. Shapiro, A.: First and second order analysis of nonlinear semidefinite programs. Math. Programming 77, 301–320 (1997)

    Article  Google Scholar 

  23. Wolkowicz, H., Saigal R., Vandenbergh, L. (eds.): Handbook of semidefinite programming. International Series in Operations Research & Management Science, 27, Kluwer Academic Publishers, Dordrecht, 2000

  24. Wu, Z., Ye, J.J.: Sufficient conditions for error bounds. SIAM J. Optim. 12, 421–435 (2001)

    Article  Google Scholar 

  25. Wu, Z., Ye, J.J.: On error bounds for lower semicontinuous functions,. Math. Programming, Ser. A 92, 301–314 (2002)

    Google Scholar 

  26. Wu, Z., Ye, J.J.: First-order and second-order conditions for error bounds. SIAM J. Optim. 14, 621–645 (2003)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, Analyse Appliquée et Optimisation, BP 47 870, 21078, Dijon Cedex, France

    A. Jourani

  2. Department of Mathematics and Statistics, University of Victoria, P.O. Box 3045, STN CSC, Victoria, BC, V8W 3P4, Canada

    J.J. Ye

Authors
  1. A. Jourani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J.J. Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J.J. Ye.

Additional information

The research of the author was partially supported by an NSERC grant.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jourani, A., Ye, J. Error Bounds for Eigenvalue and Semidefinite Matrix Inequality Systems. Math. Program. 104, 525–540 (2005). https://doi.org/10.1007/s10107-005-0627-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-005-0627-y

Mathematics Subject Classification (1991)

Search

Navigation