Skip to main content

Advertisement

SpringerLink
Log in

Metric regularity of semi-infinite constraint systems

  • Published:
Mathematical Programming Submit manuscript

Abstract

We obtain a formula for the modulus of metric regularity of a mapping defined by a semi-infinite system of equalities and inequalities. Based on this formula, we prove a theorem of Eckart-Young type for such set-valued infinite-dimensional mappings: given a metrically regular mapping F of this kind, the infimum of the norm of a linear function g such that F+g is not metrically regular is equal to the reciprocal to the modulus of regularity of F. The Lyusternik-Graves theorem gives a straightforward extension of these results to nonlinear systems. We also discuss the distance to infeasibility for homogeneous semi-infinite linear inequality systems.

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. Cánovas, M.J., López, M.A., Parra, J., Toledo, F.J.: Distance to ill-posedness and the consistency value of linear semi-infinite inequality systems. Math. Programming Ser. A, 103, 95–126 (2005)

    Google Scholar 

  2. Cheung, D., Cucker, F.: Probabilistic analysis of condition numbers for linear programming. J. Optim. Theory Appl. 114, 55–67 (2002)

    Article  Google Scholar 

  3. Dontchev, A.L.: The Graves theorem revisited. J. of Convex Analysis 3, 45–53 (1996)

    Google Scholar 

  4. Dontchev, A.L., Lewis, A.S., Rockafellar, R.T.: The radius of metric regularity. Trans. Amer. Math. Soc. 355, 493–517 (2003)

    Article  Google Scholar 

  5. Dontchev, A.L., Rockafellar, R.T.: Regularity properties and conditioning in variational analysis and optimization. Set-valued Analysis 12, 79–109 (2004)

    Article  Google Scholar 

  6. Dontchev, A.L., Lewis, A.S.: Perturbations and metric regularity. Preprint 2004

  7. Eckart, C., Young, G.: The approximation of one matrix by another of lower rank. Psychometrica 1, 211–218 (1936)

    Google Scholar 

  8. Goberna, M.A., López, M.A., Todorov, M.I.: Stability theory for linear inequality systems. SIAM J. Matrix Anal. Appl. 17, 730–743 (1996)

    Article  Google Scholar 

  9. Goberna, M.A., López, M.A.: Linear semi-infinite optimization. John Wiley and Sons, Chichester, 1998

  10. Hettich, R., Jongen, H.Th.: Semi-infinite programming: conditions of optimality and applications. In: J. Stoer, (ed) Optimization Techniques, Part 2, Lecture Notes in Control and Information Sciences, vol 7, Springer, Berlin, pp 1–11, 1978

  11. Hiriart-Urruty, J.-B., Lemarechal, C.: Convex Analysis and Minimization Algorithms I. Springer-Verlag, Berlin, 1993

  12. Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk 55, 333 (3), 103–162 (2000) English translation Math. Surveys. 55, 501–558 (2000)

    Google Scholar 

  13. Ioffe, A.D.: On stability estimates for the regularity of maps. In: Topological methods, variational methods and their applications (Taiyuan, 2002), World Sci. Publishing, River Edge, NJ, pp 133–142, 2003

  14. Ioffe, A.D.: On robustness of the regularity property of maps. Control and Cybern 32, 543–554 (2003)

    Google Scholar 

  15. Jongen, H.Th., Twilt, F., Weber, G.-W.: Semi-infinite optimization: structure and stability of the feasible set. J. Optim. Theory Appl. 72, 529–552 (1992)

    Article  Google Scholar 

  16. Jongen, H.Th., Rückmann, J.-J., Weber, G.-W.: One-parametric semi-infinite optimization: on the stability of the feasible set. SIAM J. Optim. 4 (3), 637–648 (1994)

    Article  Google Scholar 

  17. Klatte, D., Henrion, R.: Regularity and stability in nonlinear semi-infinite optimization. In: R. Reemtsen and J.-J. Rückmann (eds), Semi-Infinite Programming, Nonconvex Optimization and Its Applications, 25, Kluwer, Boston, pp 69–102, 1998

  18. Lewis, A.S.: Ill-conditioned convex processes and linear inequalities. Math. Oper. Res. 24, 829–834 (1999)

    Google Scholar 

  19. Luenberger, D.G.: Optimization by vector space methods. John Wiley and Sons, New York, 1969

  20. Mordukhovich, B.S.: Coderivative analysis of variational systems. J. Global Optim. 28, 347–362 (2004)

    Article  Google Scholar 

  21. Nayakkankuppam, M.H., Overton, M.L.: Conditioning of semidefinite programs. Math. Programming Ser. A 85, 525–540 (1999)

    Article  Google Scholar 

  22. Peña, J.: Conditioning of convex programs from a primal-dual perspective. Math. Oper. Res. 26, 206–220 (2001)

    Article  Google Scholar 

  23. Renegar, J.: Linear programming, complexity theory and elementary functional analysis. Math. Programming 70, 279–351 (1995)

    Google Scholar 

  24. Robinson, S.M.: Stability theorems for systems of inequalities. Part II: Differentiable nonlinear systems. SIAM J. Numer. Anal. 13, 497–513 (1976)

    Google Scholar 

  25. Rockafellar, R.T.: Convex analysis. Princeton University Press, Princeton, 1970

  26. Rockafellar, R.T., Wets, R. J.-B.: Variational analysis. Springer-Verlag, Berlin, 1997

  27. Shapiro, A., Homem-de-Mello, T., Kim, J.: Conditioning of convex piecewise linear stochastic programs. Math. Programming, Ser. A 94, 1–19 (2002)

    Google Scholar 

  28. Wright, M.H.: Ill-conditioning and computational error in interior methods for nonlinear programming. SIAM J. Optim. 9, 84–111 (1998)

    Article  Google Scholar 

  29. Zolezzi, T.: On the distance theorem in quadratic optimization. J. Convex Anal. 9, 693–700 (2002)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Operations Research Center, Miguel Hernández University of Elche, 03202, Elche (Alicante), Spain

    M.J. Cánovas & J. Parra

  2. Mathematical Reviews, Ann Arbor, MI, 48107-8604, USA

    A.L. Dontchev

  3. Department of Statistics and Operations Research, University of Alicante, 03071, Alicante, Spain

    M.A. López

Authors
  1. M.J. Cánovas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A.L. Dontchev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M.A. López
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. Parra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A.L. Dontchev.

Additional information

Dedicated to R. T. Rockafellar on his 70th Birthday

Research partially supported by grants BFM2002-04114-C02 (01-02) from MCYT (Spain) and FEDER (E.U.), GV04B-648 and GRUPOS04/79 from Generalitat Valenciana (Spain), and Bancaja-UMH (Spain).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cánovas, M., Dontchev, A., López, M. et al. Metric regularity of semi-infinite constraint systems. Math. Program. 104, 329–346 (2005). https://doi.org/10.1007/s10107-005-0618-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-005-0618-z

Keywords

Mathematics Subject Classification (1991)

Search

Navigation