Advertisement

Journal of Optimization Theory and Applications

, Volume 171, Issue 3, pp 820–855 | Cite as

Nonlinear Metric Subregularity

  • Alexander Y. Kruger
Article

Abstract

In this article, we investigate nonlinear metric subregularity properties of set-valued mappings between general metric or Banach spaces. We demonstrate that these properties can be treated in the framework of the theory of (linear) error bounds for extended real-valued functions of two variables developed in Kruger (Error bounds and metric subregularity. Optimization 64(1):49–79, 2015). Several primal and dual space local quantitative and qualitative criteria of nonlinear metric subregularity are formulated. The relationships between the criteria are established and illustrated.

Keywords

Error bounds Slope Metric regularity Metric subregularity Hölder metric subregularity Calmness 

Mathematics Subject Classification

49J52 49J53 58C06 47H04 54C60 

Notes

Acknowledgments

The research was supported by the Australian Research Council, Project DP11010 2011. The author wishes to thank two of the three anonymous referees for the careful reading of the manuscript and many constructive comments and suggestions.

Conflict of interest

The author declares that he has no conflict of interest.

References

  1. 1.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  2. 2.
    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)Google Scholar
  3. 3.
    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. A View from Variational Analysis, 2nd edn. Springer Series in Operations Research and Financial Engineering. Springer, New York (2014)Google Scholar
  4. 4.
    Penot, J.P.: Calculus Without Derivatives, Graduate Texts in Mathematics, vol. 266. Springer, New York (2013). doi: 10.1007/978-1-4614-4538-8
  5. 5.
    Henrion, R., Outrata, J.V.: A subdifferential condition for calmness of multifunctions. J. Math. Anal. Appl. 258(1), 110–130 (2001)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Henrion, R., Jourani, A.: Subdifferential conditions for calmness of convex constraints. SIAM J. Optim. 13(2), 520–534 (2002)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Henrion, R., Jourani, A., Outrata, J.: On the calmness of a class of multifunctions. SIAM J. Optim. 13(2), 603–618 (2002)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dontchev, A.L., Rockafellar, R.T.: Regularity and conditioning of solution mappings in variational analysis. Set-Valued Anal. 12(1–2), 79–109 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Zheng, X.Y., Ng, K.F.: Metric subregularity and constraint qualifications for convex generalized equations in Banach spaces. SIAM J. Optim. 18, 437–460 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ioffe, A.D., Outrata, J.V.: On metric and calmness qualification conditions in subdifferential calculus. Set-Valued Anal. 16(2–3), 199–227 (2008)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Leventhal, D.: Metric subregularity and the proximal point method. J. Math. Anal. Appl. 360(2), 681–688 (2009). doi: 10.1016/j.jmaa.2009.07.012 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Penot, J.P.: Error bounds, calmness and their applications in nonsmooth analysis. In: Nonlinear analysis and optimization II. Optimization, Contemporary Mathematics, vol. 514, pp. 225–247. American Mathematical Society, Providence (2010). doi: 10.1090/conm/514/10110
  13. 13.
    Zheng, X.Y., Ng, K.F.: Metric subregularity and calmness for nonconvex generalized equations in Banach spaces. SIAM J. Optim. 20(5), 2119–2136 (2010). doi: 10.1137/090772174 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Zheng, X.Y., Ouyang, W.: Metric subregularity for composite-convex generalized equations in Banach spaces. Nonlinear Anal. 74(10), 3311–3323 (2011). doi: 10.1016/j.na.2011.02.008 MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Zheng, X.Y., Ng, K.F.: Metric subregularity for proximal generalized equations in Hilbert spaces. Nonlinear Anal. 75(3), 1686–1699 (2012). doi: 10.1016/j.na.2011.07.004 MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Apetrii, M., Durea, M., Strugariu, R.: On subregularity properties of set-valued mappings. Set-Valued Var. Anal. 21(1), 93–126 (2013). doi: 10.1007/s11228-012-0213-4 MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kruger, A.Y.: Error bounds and metric subregularity. Optimization 64(1), 49–79 (2015). doi: 10.1080/02331934.2014.938074 MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ioffe, A.D.: Nonlinear regularity models. Math. Program. 139(1–2), 223–242 (2013). doi: 10.1007/s10107-013-0670-z MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Frankowska, H.: An open mapping principle for set-valued maps. J. Math. Anal. Appl. 127(1), 172–180 (1987). doi: 10.1016/0022-247X(87)90149-1 MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    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(2), 441–459 (1988). doi: 10.1016/0022-247X(88)90034-0 MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Frankowska, H.: High order inverse function theorems. Ann. Inst. H. Poincaré Anal. Non Linéaire 6(suppl.), 283–303 (1989)Google Scholar
  22. 22.
    Penot, J.P.: Metric regularity, openness and Lipschitzian behavior of multifunctions. Nonlinear Anal. 13(6), 629–643 (1989). doi: 10.1016/0362-546X(89)90083-7 MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Jourani, A.: On metric regularity of multifunctions. Bull. Aust. Math. Soc. 44(1), 1–9 (1991). doi: 10.1017/S0004972700029403 MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Yen, N.D., Yao, J.C., Kien, B.T.: Covering properties at positive-order rates of multifunctions and some related topics. J. Math. Anal. Appl. 338(1), 467–478 (2008). doi: 10.1016/j.jmaa.2007.05.041 MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Ioffe, A.D.: Metric regularity and subdifferential calculus. Russ. Math. Surv. 55, 501–558 (2000)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Frankowska, H., Quincampoix, M.: Hölder metric regularity of set-valued maps. Math. Program. Ser. A 132(1–2), 333–354 (2012). doi: 10.1007/s10107-010-0401-7
  27. 27.
    Uderzo, A.: On mappings covering at a nonlinear rate and their perturbation stability. Nonlinear Anal. 75(3), 1602–1616 (2012). doi: 10.1016/j.na.2011.03.014 MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Kruger, A.Y.: Error bounds and Hölder metric subregularity. Set-Valued Var. Anal. 1–32 (2015). doi: 10.1007/s11228-015-0330-y
  29. 29.
    Kummer, B.: Inclusions in general spaces: Hoelder stability, solution schemes and Ekeland’s principle. J. Math. Anal. Appl. 358(2), 327–344 (2009). doi: 10.1016/j.jmaa.2009.04.060 MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Gaydu, M., Geoffroy, M.H., Jean-Alexis, C.: Metric subregularity of order \(q\) and the solving of inclusions. Cent. Eur. J. Math. 9(1), 147–161 (2011). doi: 10.2478/s11533-010-0087-3 MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Klatte, D., Kruger, A.Y., Kummer, B.: From convergence principles to stability and optimality conditions. J. Convex Anal. 19(4), 1043–1072 (2012)MathSciNetMATHGoogle Scholar
  32. 32.
    Li, G., Mordukhovich, B.S.: Hölder metric subregularity with applications to proximal point method. SIAM J. Optim. 22(4), 1655–1684 (2012). doi: 10.1137/120864660 MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Mordukhovich, B.S., Ouyang, W.: Higher-order metric subregularity and its applications. J. Glob. Optim. 1–19 (2015). doi: 10.1007/s10898-015-0271-x
  34. 34.
    Ngai, H.V., Tron, N.H., Théra, M.: Hölder metric subregularity via error bounds (2015, private communication)Google Scholar
  35. 35.
    Ngai, H.V., Tron, N.H., Théra, M.: Directional Hölder metric regularity. J. Optim. Theory Appl. 1–35 (2015). doi: 10.1007/s10957-015-0797-6
  36. 36.
    Ngai, H.V., Tinh, P.N.: Metric subregularity of multifunctions: first and second order infinitesimal characterizations. Math. Oper. Res. 40(3), 703–724 (2015). doi: 10.1287/moor.2014.0691 MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Klatte, D.: On quantitative stability for non-isolated minima. Control Cybernet. 23(1–2), 183–200 (1994)MathSciNetMATHGoogle Scholar
  38. 38.
    Cornejo, O., Jourani, A., Zălinescu, C.: Conditioning and upper-Lipschitz inverse subdifferentials in nonsmooth optimization problems. J. Optim. Theory Appl. 95(1), 127–148 (1997). doi: 10.1023/A:1022687412779 MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Pang, J.S.: Error bounds in mathematical programming. Math. Program. Ser. B 79(1–3), 299–332 (1997)MathSciNetMATHGoogle Scholar
  40. 40.
    Azé, D.: A survey on error bounds for lower semicontinuous functions. In: Proceedings of 2003 MODE-SMAI Conference, ESAIM Proceedings, vol. 13, pp. 1–17. EDP Sci., Les Ulis (2003)Google Scholar
  41. 41.
    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)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    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). doi: 10.1023/B:SVAN.0000023396.58424.98 MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Corvellec, J.N., Motreanu, V.V.: Nonlinear error bounds for lower semicontinuous functions on metric spaces. Math. Program. Ser. A 114(2), 291–319 (2008)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    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). doi: 10.1137/060675721 MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Fabian, M.J., Henrion, R., Kruger, A.Y., Outrata, J.V.: Error bounds: necessary and sufficient conditions. Set Valued Var. Anal. 18(2), 121–149 (2010)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Fabian, M.J., Henrion, R., Kruger, A.Y., Outrata, J.V.: About error bounds in metric spaces. In: Klatte, D., Lüthi, H.J., Schmedders, K. (eds.) Operations Research Proceedings 2011. Selected Papers of the International Conference Operations Research (OR 2011), August 30–September 2, 2011, Zurich, Switzerland, pp. 33–38. Springer, Berlin (2012)Google Scholar
  47. 47.
    De Giorgi, E., Marino, A., Tosques, M.: Problems of evolution in metric spaces and maximal decreasing curve. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 68(3), 180–187 (1980). (in Italian)MathSciNetMATHGoogle Scholar
  48. 48.
    Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. Springer, New York (2005)MATHGoogle Scholar
  49. 49.
    Fabian, M.: Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss. Acta Univ. Carolinae 30, 51–56 (1989)MathSciNetMATHGoogle Scholar
  50. 50.
    Kruger, A.Y.: On Fréchet subdifferentials. J. Math. Sci. 116(3), 3325–3358 (2003)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Ioffe, A.D., Tikhomirov, V.M.: Theory of Extremal Problems. Studies in Mathematics and its Applications. North-Holland Publishing, Amsterdam (1979)Google Scholar
  52. 52.
    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing, River Edge (2002). doi: 10.1142/9789812777096 CrossRefMATHGoogle Scholar
  53. 53.
    Lucchetti, R.: Convexity and Well-Posed Problems. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 22. Springer, New York (2006)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Centre for Informatics and Applied Optimization, Faculty of Science and TechnologyFederation University AustraliaBallaratAustralia

Personalised recommendations