Skip to main content
SpringerLink
Log in

Nonlinear regularity models

  • Full Length Paper
  • Series B
  • Published:
Mathematical Programming Submit manuscript

Abstract

The paper studies regularity properties of set-valued mappings between metric spaces. In the context of metric regularity, nonlinear models correspond to nonlinear dependencies of estimates of error bounds in terms of residuals. Among the questions addressed in the paper are equivalence of the corresponding concepts of openness and “pseudo-Hölder” behavior, general and local regularity criteria with special emphasis on “regularity of order \(k\)”, for local settings, and variational methods to extimate regularity moduli in case of length range spaces. The majority of the results presented in the paper are new.

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

Notes

  1. It is appropriate to mention here a very general result of Khan [22] which contains as particular case the results of Pták [24] and Dmitruk.

  2. That is for any \(y\in B(F(x),\mu (t))\cap V\) there is a \(v\in F(B(x,t))\) such that \(d(y,v)<\lambda \mu (t)\).

  3. A set is locally complete at a point if its intersection with a closed neighborhood of the point is a complete space in the induced metric.

References

  1. Ambrosio, A., Gigli, N., Savaré, G.: Gradient Flows, 2nd edn. Bürkhauser, Basel (2008)

    MATH  Google Scholar 

  2. Azé, D., Corvellec, J.-N.: On some variational properties of metric spaces. J. Fixed Point Theory. doi:10.1007/s11784-008-0054-9

  3. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, Berlin (2000)

    MATH  Google Scholar 

  4. Borwein, J.M.: Stability and regular points of inequality systems. J. Optim. Theory Appl. 48, 9–51 (1986)

    MathSciNet  MATH  Google Scholar 

  5. Borwein, J.M., Zhu, J.: Techniques of Variational Analysis, CMS Books in Mathematics, vol. 20. Springer, Berlin (2006)

    Google Scholar 

  6. Borwein, J.M., Zhuang, D.M.: Verifiable necessary and sufficient conditions for openness and regularity of set-valued and single-valued maps. J. Math. Anal. Appl 134, 441–459 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  7. Burago, D., Burago, Yu., Ivanov, S.: A Course in Metric Geometry, Graduate Studies in Math, vol. 33, AMS (2001)

  8. De Giorgi, E., Marino, A., Tosques, M.: Problemi di evoluzione in spazi metrici e curve di massima pendenza. Atti Acad. Nat. Lincei, Rend. Cl. Sci. Fiz. Mat. Natur. 68, 180–187 (1980)

    MathSciNet  MATH  Google Scholar 

  9. Dmitruk, A.V., Milyutin, A.A., Osmolovskii, N.P.: Lyusternik theorem and the theory of extrema. Russian Math. Surveys 35(6), 11–51 (1980)

    Article  MathSciNet  Google Scholar 

  10. Frankowska, H.: An open mapping principle for set-valued maps. J. Math. Anal. Appl. 127, 172–180 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  11. Frankowska, H.: High order inverse functions theorems. In: Attouch, H., Aubin, J.-P., Clarke, F.H., Ekeland, I. (eds.) Analyse Non Linéaire, pp. 283–304. Gauthier-Villars, Paris (1989)

    Google Scholar 

  12. Frankowska, H.: Some inverse mapping theorems. Ann. Inst. Henri Poincaré. Analyse Non Linéaire 7, 183–234 (1990)

    MathSciNet  MATH  Google Scholar 

  13. Frankowska, H.: Conical inverse mapping theorems. Bull. Australian Math. Soc. 45, 53–60 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. Frankowska, H., Quincampoix, M.: Hölder metric regularity of set-valued maps. Math. Programm. doi:10.1007/s10107-010-0401-7

  15. Hoffman, A.J.: On approximate solutions of systems of linear inequalities. J. Res. Nat. Bureau Stand. 49, 263–265 (1952)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Ioffe, A.D.: Metric regularity and subdifferential calculus, Uspehi Mat. Nauk 55(3), 103–162 (2000) (in Russian), English translation. Russian Math. Surveys 55(3), 501–558 (2000)

  18. Ioffe, A.D.: Towards metric theory of metric regularity, 501–558. In: Lassond, M. (ed.) Approximation, Optimization and Mathematical Economics, pp. 165–177. Physica Verlag, Heidelberg (2001)

    Chapter  Google Scholar 

  19. Ioffe, A.D.: On regularity estimates for mappings between embedded manifolds. Control Cybern. 36, 659–668 (2007)

    MathSciNet  MATH  Google Scholar 

  20. Ioffe, A.D.: Regularity on fixed sets, preprint (2011)

  21. Ioffe, A.D., Tikhomirov, V.M.: Theory of Extremal Problems, Nauka, Moscow (1974) (in Russian). North Holland, English translation (1979)

  22. Khanh, P.Q.: An induction theorem and general open mapping theorem. J. Math. Anal. Appl. 118, 519–534 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  23. Penot, J.-P.: Metric regularity, openness and Lipschitzean behavior of multifunctions. Nonlinear Anal. 13, 629–643 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pták, V.: A quantitative refinement of a closed graph theorem. Czechoslovak Math. J. 24, 503–506 (1974)

    MathSciNet  Google Scholar 

  25. Robinson, S.M.: Stability theory for system of inequalities. Part II: differentiable nonlinear systems. SIAM J. Numer. Anal. 13, 497–513 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  26. Robinson, S.M.: Regularity and stability for convex multivalued functions. Math. Oper. Res. 1, 130–143 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tziskaridze, KSh: Extremal problems in Banach spaces. In: Nekotorye Voprosy Matematicheskoy Theorii Optimalnogo Upravleniya (Some Problems of the Mathematical Theory of Optimal Control). Tbilisi State Univ, Inst. Appl. Math. (1975) (in Russian)

Download references

Acknowledgments

I wish to express my thanks to the reviewers for the detailed analysis of the text and many good suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Technion, 32000 , Haifa, Israel

    Alexander D. Ioffe

Authors
  1. Alexander D. Ioffe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander D. Ioffe.

Additional information

The paper is dedicated to Jon Borwein’s sixtieth anniversary.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ioffe, A.D. Nonlinear regularity models. Math. Program. 139, 223–242 (2013). https://doi.org/10.1007/s10107-013-0670-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-013-0670-z

Keywords

Mathematics Subject Classification (2000)

Search

Navigation