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Calmness and inverse image characterizations for Asplund spaces

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Abstract

We establish new characterizations of Asplund spaces in terms of conditions ensuring the calmness property for constraint set mappings and the validity of inverse image formula for a general constrained system.

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References

  1. Bonnans J.F., Shapiro A.: Perturbation analysis of optimization problems. Springer, Berlin (2000)

    MATH  Google Scholar 

  2. Chuong T.D., Kruger A.Y., Yao J.-C.: Calmness of efficient solution maps in parametric vector optimization. J. Global Optim. 51, 677–688 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Deng S.: Global error bounds for convex inequality systems in Banach spaces. SIAM J. Control. Optim. 36, 1240–1249 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Deville R., Godefroy G., Zizler V.: Smoothness and Renorming in Banach Spaces. Wiley, New York (1993)

    Google Scholar 

  5. Fabian M.: Gateaux differentiability of convex functions and topology. Weak asplund spaces. Wiley, New York (1997)

    MATH  Google Scholar 

  6. Giannessi F., Maugeri A., Pardalos M.P.: Equilibrium Problems: Nonsmooth optimization and variational inequality models. Kluwer Academic Publishers, Dordrecht (2002)

    Google Scholar 

  7. Henrion R., Outrata J.: A subdifferential condition for calmness of multifunctions. J. Math. Anal. Appl. 258, 110–130 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Henrion R., Jourani A., Outrata J.: On the calmness of a class of multifunctions. SIAM J. Optim. 13, 603–618 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Henrion R., Jourani A.: Subdfferential conditions for calmness of convex systems. SIAM J. Optim. 13, 520–534 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Henrion R., Outrata J.: Calmness of constraint systems with applications. Math. Program. 104, 437–464 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ioffe A.D.: Approximate subdifferentials and applications. I: The finite dimensional theory. Trans. Am. Math. Soc. 281, 389–415 (1984)

    MathSciNet  MATH  Google Scholar 

  12. Ioffe A.D.: Approximate subdifferentials and applications II. Mathematika 33, 11–128 (1986)

    Article  MathSciNet  Google Scholar 

  13. Ioffe A.D.: Approximate subdifferential and applications III. Mathematika 36, 1–38 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ioffe A.D.: Metric regularity and subdifferential calculus. Russ. Math. Surv. 55(3), 501–558 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ioffe A.D.: Codirectional compactness, metric regularity and subdifferential calculus. Can. Math. Soc. Conf. Proc. 27, 123–163 (2000)

    MathSciNet  Google Scholar 

  16. Ioffe A.D., Penot J.-P.: Subdifferentials of performance functions and calculus of coderivatives of set-valued mappings. Serdica Math. J. 22, 359–384 (1996)

    MathSciNet  MATH  Google Scholar 

  17. King A.J., Rockafellar R.T.: Sensitivity analysis for nonsmooth generalized equations. Math. Program. 55, 193–212 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mordukhovich, B.S.: Variational analysis and generalized differentiation. I. Basic theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 330. Springer, Berlin (2006)

  19. Mordukhovich, B.S.: Variational analysis and generalized differentiation. II. Applications. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 330, Springer, Berlin (2006)

  20. Ngai H.V., Théra M.: Metric inequality, subdifferential calculus and applications. Set-valued Anal. 9, 187–216 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pardalos P.M., Rassias T.M., Khan A.A.: Nonlinear analysis and variational problems. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  22. Penot J.-P.: Cooperative behavior for sets and relations. Math. Methods Oper. Res. 48, 229–246 (1988)

    MathSciNet  Google Scholar 

  23. Penot J.-P.: Subdifferential calculus without qualification conditions. J. Convex Anal. 3, 207–219 (1996)

    MathSciNet  MATH  Google Scholar 

  24. Penot J.-P: Compactness properties, openness criteria and coderivatives. Set-Valued Anal. 6, 363–380 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  25. Li S.J., Penot J.-P., Meng K.W.: Codifferential calculus. Set-Valued Var. Anal. 19(4), 505–536 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Phelps R.: Convex functions, monotone operators and differentiability, 2nd edn. Springer, Berlin (1993)

    MATH  Google Scholar 

  27. Song W.: Calmness and error bounds for convex constraint systems. SIAM J. Optim. 17, 353–371 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Yost D.: Asplund spaces for beginners. Acta Univ. Carolinae Ser. Math. Phys. 34, 159–177 (1993)

    MathSciNet  MATH  Google Scholar 

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Correspondence to Nooshin Movahedian.

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Movahedian, N. Calmness and inverse image characterizations for Asplund spaces. Optim Lett 7, 361–373 (2013). https://doi.org/10.1007/s11590-011-0424-x

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  • DOI: https://doi.org/10.1007/s11590-011-0424-x

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