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Implicit multifunction theorems in complete metric spaces

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Abstract

In this paper, we establish some new characterizations of metric regularity of implicit multifunctions in complete metric spaces by using lower semicontinuous envelopes of the distance functions to set-valued mappings. Through these new characterizations it is possible to investigate implicit multifunction theorems based on coderivatives and on contingent derivatives as well as the perturbation stability of implicit multifunctions.

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References

  1. Aubin, J.-P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)

    MATH  Google Scholar 

  2. Aubin, J.-P., Wets, R.J.-B.: Stable approximations of set-valued maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 5(6), 519–535 (1988)

    MathSciNet  MATH  Google Scholar 

  3. Azé, D.: A survey on error bounds for lower semicontinuous functions. In: Proceedings of 2003 MODE-SMAI Conference, vol. 13 of ESAIM Proc., EDP Sci., Les Ulis, pp. 1–17 (2003) (electronic)

  4. Azé, D.: A unified theory for metric regularity of multifunctions. J. Convex Anal. 13(2), 225–252 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Azé, D., Benahmed, S.: On implicit multifunction theorems. Set Valued Anal. 16(2–3), 129–155 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Azé, D., Corvellec, J.-N.: Characterizations of error bounds for lower semicontinuous functions on metric spaces. ESAIM Control Optim. Calc. Var. 10(3), 409–425 (2004) (electronic)

    Google Scholar 

  7. Azé, D., Corvellec, J.-N., Lucchetti, R.E.: Variational pairs and applications to stability in nonsmooth analysis. Nonlinear Anal. Ser. A Theory Methods 49(5), 643–670 (2002)

    Google Scholar 

  8. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer Series in Operations Research. Springer, New York (2000)

    Google Scholar 

  9. Borwein, J.M., Dontchev, A.L.: On the Bartle–Graves theorem. Proc. Am. Math. Soc. 131(8), 2553–2560 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Borwein, J.M., Zhu, Q.J.: Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity. SIAM J. Control. Optim. 34, 1568–1591 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. Springer, New York (2005)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Bosch, P., Jourani, A., Henrion, R.: Sufficient conditions for error bounds and applications. Appl. Math. Optim. 50(2), 161–181 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chuong, T.D., Kruger, A.Y., Yao, J.-C.: Calmness of efficient solution maps in parametric vector optimization. J. Glob. Optim. 51(4), 677–688 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Clarke, F.H.: Optimization and nonsmooth analysis, 2nd edn, vol. 5 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1990)

  16. Cominetti, R.: Metric regularity, tangent sets, and second-order optimality conditions. Appl. Math. Optim. 21(3), 265–287 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  17. De Giorgi, E., Marino, A., Tosques, M.: Problemi di evoluzione in spazi metrici e curve di massima pendenza. Atti. Accad. Naz. Lincei rend. Cl. Sci. fis. Mat. Natur. 68, 180–187 (1980)

    Google Scholar 

  18. Dmitruk, A.V., Kruger, A.Y.: Metric regularity and systems of generalized equations. J. Math. Anal. Appl. 342(2), 864–873 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dmitruk, A.V., Milyutin, A.A., Osmolovsky, N.P.: Lyusternik’s theorem and the theory of extrema. Uspekhy Mat. Nauk 35, 11–46 (1980) (in Russian)

    Google Scholar 

  20. Dontchev, A.L.: The Graves theorem revisited. J. Convex Anal. 3(1), 45–53 (1996)

    MathSciNet  MATH  Google Scholar 

  21. Dontchev, A.L., Lewis, A.S., Rockafellar, R.T.: The radius of metric regularity. Trans. Am. Math. Soc. 355(2), 49–517 (2003) (electronic)

    Google Scholar 

  22. Dontchev, A.L., Quincampoix, M., Zlateva, N.: Aubin criterion for metric regularity. J. Convex Anal. 13(2), 281–297 (2006)

    MathSciNet  MATH  Google Scholar 

  23. Dontchev, A.L., Rockafellar, R.T.: Robinson’s implicit function theorem and its extensions. Math. Program. Ser. B 117(1–2), 129–147 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  25. Graves, L.M.: Some mapping theorems. Duke Math. J. 17, 111–114 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ioffe, A.: Towards metric theory of metric regularity. In: Approximation, Optimization and Mathematical Economics (Pointe-à-Pitre, 1999). Physica, Heidelberg, pp. 165–176 (2001)

  27. Ioffe, A.: On regularity estimates for mappings between embedded manifolds. Control Cybern. 36(3), 659–668 (2007)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

  30. Ioffe, A.D.: On regularity concepts in variational analysis. J. Fixed Point Theory Appl. 8(2), 339–363 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Jourani, A.: Hoffman’s error bound, local controllability, and sensitivity analysis. SIAM J. Control Optim. 38(3), 947–970 (2000) (electronic)

    Google Scholar 

  32. Jourani, A., Thibault, L.: Metric regularity and subdifferential calculus in Banach spaces. Set Valued Anal. 3(1), 87–100 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  33. Jourani, A., Thibault, L.: Coderivatives of multivalued mappings, locally compact cones and metric regularity. Nonlinear Anal. Ser. A Theory Methods, 35(7), 925–945 (1999)

    Google Scholar 

  34. Kruger, A.Y.: A covering theorem for set-valued mappings. Optimization 19(6), 763–780 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ledyaev, Y.S., Zhu, Q.J.: Implicit multifunction theorems. Set Valued Anal. 7(3), 209–238 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lyusternik, L.A.: On conditional extrema of functionals. Math. Sbornik 41, 390–401 (1934) (in Russian)

    Google Scholar 

  37. Morduhovič, B.Š.: Metric approximations and necessary conditions for optimality for general classes of nonsmooth extremal problems. Dokl. Akad. Nauk SSSR 254(5), 1072–1076 (1980)

    MathSciNet  Google Scholar 

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

  39. Mordukhovich, B.S.: Variational analysis and generalized differentiation. II: Applications, vol. 331 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2006)

  40. Mordukhovich, B.S., Shao, Y.: Stability of set-valued mappings in infinite dimensions: point criteria and applications. SIAM J. Control Optim. 35(1), 285–314 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  41. Mordukhovich, B.S., Wang, B.: Restrictive metric regularity and generalized differential calculus in Banach spaces. Int. J. Math. Math. Sci. 49–52, 2653–2680 (2004)

    Article  MathSciNet  Google Scholar 

  42. Ng, K.F., Zheng, X.Y.: Error bounds for lower semicontinuous functions in normed spaces. SIAM J. Optim. 12(1), 1–17 (2001) (electronic)

    Google Scholar 

  43. Ngai, H.V., Théra, M.: Metric inequality, subdifferential calculus and applications. Set Valued Anal. 9(1–2), 187–216 (2001). Wellposedness in optimization and related topics (Gargnano, 1999)

    Article  MathSciNet  MATH  Google Scholar 

  44. Ngai, H.V., Théra, M.: Error bounds and implicit multifunction theorem in smooth Banach spaces and applications to optimization. Set Valued Anal. 12(1–2), 195–223 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  45. Ngai, H.V., Théra, M.: Error bounds in metric spaces and application to the perturbation stability of metric regularity. SIAM J. Optim. 19(1), 1–20 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  46. Ngai, H.V., Théra, M.: Error bounds for systems of lower semicontinuous functions in Asplund spaces. Math. Program. Ser. B 116(1–2), 397–427 (2009)

    Article  MATH  Google Scholar 

  47. Outrata, J., Kočvara, M., Zowe, J.: Nonsmooth approach to optimization problems with equilibrium constraints: theory, applications and numerical results, vol. 28 of Nonconvex Optimization and its Applications. Kluwer, Dordrecht (1998)

  48. Penot, J.-P.: Metric regularity, openness and Lipschitzian behavior of multifunctions. Nonlinear Anal. 13(6), 629–643 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  49. Robinson, S.M.: An inverse-function theorem for a class of multivalued functions. Proc. Am. Math. Soc. 41, 211–218 (1973)

    Article  MATH  Google Scholar 

  50. Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5(1), 43–62 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  51. Rockafellar, R.T.: Monotone processes of convex and concave type. Memoirs of the American Mathematical Society, No. 77. American Mathematical Society, Providence (1967)

    Google Scholar 

  52. Rockafellar, R.T., Wets, R.J.-B.: Variational analysis, vol. 317 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1998)

  53. Ursescu, C.: Multifunctions with convex closed graph. Czechoslov. Math. J. 25(100)(3), 438–441 (1975)

    MathSciNet  Google Scholar 

  54. Wu, Z., Ye, J.J.: On error bounds for lower semicontinuous functions. Math. Program. Ser. A 92(2), 301–314 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  55. Yen, N.D., Yao, J.-C.: Point-based sufficient conditions for metric regularity of implicit multifunctions. Nonlinear Anal. 70(7), 2806–2815 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  56. Zheng, X.Y., Ng, K.F.: Metric regularity and constraint qualifications for convex inequalities on Banach spaces. SIAM J. Optim. 14(3), 757–772 (2003) (electronic)

    Google Scholar 

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Acknowledgments

The authors are indebted to Professors Alexander Kruger and Jiří V. Outrata for their interest and for valuable discussions on this work. Research of Huynh Van Ngai was supported by NAFOSTED and partially by LIA Formath Vietnam and the project BQR Réseaux. B005/R1107005 from the university of Limoges. Research of Nguyen Huu Tron was supported by a doctoral grant from Région Limousin. Research of Michel Théra was partially supported by the project BQR Réseaux. B005/R1107005 from the university of Limoges, by the Australian Research Council under grant DP-110102011 and by the ECOS-SUD C10E08 project.

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Authors and Affiliations

  1. Department of Mathematics, University of Quy Nhon, Quy Nhon, Vietnam

    van Ngai Huynh & Huu Tron Nguyen

  2. Laboratoire XLIM, UMR-CNRS 7252, Université de Limoges, Limoges, France

    Michel Théra

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  1. van Ngai Huynh
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  2. Huu Tron Nguyen
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  3. Michel Théra
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Correspondence to Michel Théra.

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Dedicated to J. M. Borwein on the occasion of his sixtieth anniversary.

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Huynh, v.N., Nguyen, H.T. & Théra, M. Implicit multifunction theorems in complete metric spaces. Math. Program. 139, 301–326 (2013). https://doi.org/10.1007/s10107-013-0673-9

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