Transversality Properties: Primal Sufficient Conditions

Abstract

The paper studies ‘good arrangements’ (transversality properties) of collections of sets in a normed vector space near a given point in their intersection. We target primal (metric and slope) characterizations of transversality properties in the nonlinear setting. The Hölder case is given a special attention. Our main objective is not formally extending our earlier results from the Hölder to a more general nonlinear setting, but rather to develop a general framework for quantitative analysis of transversality properties. The nonlinearity is just a simple setting, which allows us to unify the existing results on the topic. Unlike the well-studied subtransversality property, not many characterizations of the other two important properties: semitransversality and transversality have been known even in the linear case. Quantitative relations between nonlinear transversality properties and the corresponding regularity properties of set-valued mappings as well as nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe are also discussed.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Azé, D., Corvellec, J.N.: Nonlinear local error bounds via a change of metric. J. Fixed Point Theory Appl. 16(1-2), 351–372 (2014). https://doi.org/10.1007/s11784-015-0220-9

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Azé, D., Corvellec, J.N.: Nonlinear error bounds via a change of function. J. Optim. Theory Appl. 172(1), 9–32 (2017). https://doi.org/10.1007/s10957-016-1001-3

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Bakan, A., Deutsch, F., Li, W.: Strong CHIP, normality, and linear regularity of convex sets. Trans. Amer. Math. Soc. 357(10), 3831–3863 (2005). https://doi.org/10.1090/S0002-9947-05-03945-0

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38(3), 367–426 (1996). https://doi.org/10.1137/S0036144593251710

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Bauschke, H.H., Borwein, J.M., Li, W.: Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization. Math. Program., Ser. A 86(1), 135–160 (1999). https://doi.org/10.1007/s101070050083

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Bolte, J., Nguyen, T.P., Peypouquet, J., Suter, B.W.: From error bounds to the complexity of first-order descent methods for convex functions. Math. Program., Ser. A 165(2), 471–507 (2017). https://doi.org/10.1007/s10107-016-1091-6

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Borwein, J.M., Li, G., Tam, M.K.: Convergence rate analysis for averaged fixed point iterations in common fixed point problems. SIAM J. Optim. 27(1), 1–33 (2017). https://doi.org/10.1137/15M1045223

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Borwein, J.M., Li, G., Yao, L.: Analysis of the convergence rate for the cyclic projection algorithm applied to basic semialgebraic convex sets. SIAM J. Optim. 24(1), 498–527 (2014). https://doi.org/10.1137/130919052

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    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). https://doi.org/10.1016/0022-247X(88)90034-0

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Bui, H.T., Cuong, N.D., Kruger, A.Y.: Transversality of collections of sets: Geometric and metric characterizations. Vietnam J Math. https://doi.org/10.1007/s10013-020-00388-1 (2020)

  11. 11.

    Bui, H.T., Kruger, A.Y.: Extremality, stationarity and generalized separation of collections of sets. J. Optim. Theory Appl. 182(1), 211–264 (2019). https://doi.org/10.1007/s10957-018-01458-8

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Cibulka, R., Fabian, M., Kruger, A.Y.: On semiregularity of mappings. J. Math. Anal. Appl. 473(2), 811–836 (2019). https://doi.org/10.1016/j.jmaa.2018.12.071

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Cuong, N.D., Kruger, A.Y.: Dual sufficient characterizations of transversality properties. Positivity. https://doi.org/10.1007/s11117-019-00734-9 (2020)

  15. 15.

    Cuong, N.D., Kruger, A.Y.: Nonlinear transversality of collections of sets: Dual space necessary characterizations. J. Convex Anal. 27(1), 287–308 (2020)

    MATH  Google Scholar 

  16. 16.

    Cuong, N.D., Kruger, A.Y.: Primal space necessary characterizations of transversality properties. Preprint Optimization Online 2020-01-7579 (2020)

  17. 17.

    De Giorgi, E., Marino, A., Tosques, M.: Evolution problerns in in metric spaces and steepest descent curves. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 68(3), 180–187 (1980). In Italian. English translation: Ennio De Giorgi, Selected Papers, Springer, Berlin 2006, pp 527–533

    MATH  Google Scholar 

  18. 18.

    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. A View from Variational Analysis, 2 edn. Springer Series in Operations Research and Financial Engineering. Springer, New York (2014). https://doi.org/10.1007/978-1-4939-1037-3

    Google Scholar 

  19. 19.

    Drusvyatskiy, D., Ioffe, A.D., Lewis, A.S.: Transversality and alternating projections for nonconvex sets. Found. Comput. Math. 15(6), 1637–1651 (2015). https://doi.org/10.1007/s10208-015-9279-3

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Drusvyatskiy, D., Li, G., Wolkowicz, H.: A note on alternating projections for ill-posed semidefinite feasibility problems. Math. Program., Ser. A 162(1-2), 537–548 (2017). https://doi.org/10.1007/s10107-016-1048-9

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Frankowska, H.: An open mapping principle for set-valued maps. J. Math. Anal. Appl. 127(1), 172–180 (1987). https://doi.org/10.1016/0022-247X(87)90149-1

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Frankowska, H., Quincampoix, M.: Hölder metric regularity of set-valued maps. Math. Program., Ser. A 132(1-2), 333–354 (2012). https://doi.org/10.1007/s10107-010-0401-7

    MATH  Article  Google Scholar 

  23. 23.

    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). https://doi.org/10.2478/s11533-010-0087-3

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Hesse, R., Luke, D.R.: Nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problems. SIAM J. Optim. 23(4), 2397–2419 (2013). https://doi.org/10.1137/120902653

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Ioffe, A.D.: Metric regularity and subdifferential calculus. Russian Math. Surveys 55, 501–558 (2000). https://doi.org/10.1070/rm2000v055n03ABEH000292

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Ioffe, A.D.: Nonlinear regularity models. Math. Program. 139(1-2), 223–242 (2013). https://doi.org/10.1007/s10107-013-0670-z

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Ioffe, A.D.: Metric regularity – a survey. Part I. Theory. J. Aust. Math. Soc. 101(2), 188–243 (2016). https://doi.org/10.1017/S1446788715000701

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Ioffe, A.D.: Variational analysis of regular mappings. Theory and Applications. Springer Monographs in Mathematics Springer. https://doi.org/10.1007/978-3-319-64277-2 (2017)

  29. 29.

    Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications Nonconvex Optimization and its Applications, vol. 60. Kluwer Academic Publishers, Dordrecht (2002)

    Google Scholar 

  30. 30.

    Kruger, A.Y.: Stationarity and regularity of set systems. Pac. J. Optim. 1(1), 101–126 (2005)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Kruger, A.Y.: About regularity of collections of sets. Set-Valued Anal. 14(2), 187–206 (2006). https://doi.org/10.1007/s11228-006-0014-8

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Kruger, A.Y.: About stationarity and regularity in variational analysis. Taiwanese J. Math. 13(6A), 1737–1785 (2009). https://doi.org/10.11650/twjm/1500405612

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Kruger, A.Y.: Error bounds and metric subregularity. Optimization 64(1), 49–79 (2015). https://doi.org/10.1080/02331934.2014.938074

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Kruger, A.Y.: Error bounds and Hölder metric subregularity. Set-Valued Var. Anal. 23(4), 705–736 (2016). https://doi.org/10.1007/s11228-015-0330-y

    MATH  Article  Google Scholar 

  35. 35.

    Kruger, A.Y.: Nonlinear metric subregularity. J. Optim. Theory Appl. 171(3), 820–855 (2016). https://doi.org/10.1007/s10957-015-0807-8

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Kruger, A.Y.: About intrinsic transversality of pairs of sets. Set-Valued Var. Anal. 26(1), 111–142 (2018). https://doi.org/10.1007/s11228-017-0446-3

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Kruger, A.Y., López, M.A.: Stationarity and regularity of infinite collections of sets. J. Optim. Theory Appl. 154(2), 339–369 (2012). https://doi.org/10.1007/s10957-012-0043-4

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Kruger, A.Y., López, M.A.: Stationarity and regularity of infinite collections of sets. Applications to infinitely constrained optimization. J. Optim. Theory Appl. 155 (2), 390–416 (2012). https://doi.org/10.1007/s10957-012-0086-6

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Kruger, A.Y., Luke, D.R., Thao, N.H.: About subtransversality of collections of sets. Set-Valued Var. Anal. 25 (4), 701–729 (2017). https://doi.org/10.1007/s11228-017-0436-5

    MathSciNet  MATH  Article  Google Scholar 

  40. 40.

    Kruger, A.Y., Luke, D.R., Thao, N.H.: Set regularities and feasibility problems. Math. Program., Ser. B 168(1-2), 279–311 (2018). https://doi.org/10.1007/s10107-016-1039-x

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    Kruger, A.Y., Thao, N.H.: About uniform regularity of collections of sets. Serdica Math. J. 39(3-4), 287–312 (2013)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Kruger, A.Y., Thao, N.H.: About [q]-regularity properties of collections of sets. J. Math. Anal. Appl. 416(2), 471–496 (2014). https://doi.org/10.1016/j.jmaa.2014.02.028

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Kruger, A.Y., Thao, N.H.: Quantitative characterizations of regularity properties of collections of sets. J. Optim. Theory Appl. 164(1), 41–67 (2015). https://doi.org/10.1007/s10957-014-0556-0

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Kruger, A.Y., Thao, N.H.: Regularity of collections of sets and convergence of inexact alternating projections. J. Convex Anal. 23(3), 823–847 (2016)

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Kummer, B.: Inclusions in general spaces: Hoelder stability, solution schemes and Ekeland’s principle. J. Math. Anal. Appl. 358(2), 327–344 (2009). https://doi.org/10.1016/j.jmaa.2009.04.060

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Lewis, A.S., Luke, D.R., Malick, J.: Local linear convergence for alternating and averaged nonconvex projections. Found. Comput. Math. 9(4), 485–513 (2009). https://doi.org/10.1007/s10208-008-9036-y

    MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    Li, G.: Global error bounds for piecewise convex polynomials. Math. Program. 137(1-2, Ser. A), 37–64 (2013). https://doi.org/10.1007/s10107-011-0481-z

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

    Li, G., Mordukhovich, B.S.: Hölder metric subregularity with applications to proximal point method. SIAM J. Optim. 22(4), 1655–1684 (2012). https://doi.org/10.1137/120864660

    MathSciNet  MATH  Article  Google Scholar 

  49. 49.

    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 

  50. 50.

    Ng, K.F., Zang, R.: Linear regularity and ϕ-regularity of nonconvex sets. J. Math. Anal. Appl. 328 (1), 257–280 (2007). https://doi.org/10.1016/j.jmaa.2006.05.028

    MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Ngai, H.V., Théra, M.: Metric inequality, subdifferential calculus and applications. Set-Valued Anal. 9(1-2), 187–216 (2001). https://doi.org/10.1023/A:1011291608129

    MathSciNet  MATH  Article  Google Scholar 

  52. 52.

    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). https://doi.org/10.1137/060675721

    MathSciNet  MATH  Article  Google Scholar 

  53. 53.

    Noll, D., Rondepierre, A.: On local convergence of the method of alternating projections. Found. Comput. Math. 16(2), 425–455 (2016). https://doi.org/10.1007/s10208-015-9253-0

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Ouyang, W., Zhang, B., Zhu, J.: Hölder metric subregularity for constraint systems in Asplund spaces. Positivity 23(1), 161–175 (2019). https://doi.org/10.1007/s11117-018-0600-7

    MathSciNet  MATH  Article  Google Scholar 

  55. 55.

    Penot, J.P.: Calculus without derivatives graduate texts in mathematics, vol. 266. Springer, New York (2013). https://doi.org/10.1007/978-1-4614-4538-8

    Google Scholar 

  56. 56.

    Rockafellar, R.T., Wets, R.J.B.: Variational analysis. Springer, Berlin (1998)

    Google Scholar 

  57. 57.

    Thao, N.H., Bui, T.H., Cuong, N.D., Verhaegen, M.: Some new characterizations of intrinsic transversality in Hilbert spaces. Set-Valued Var. Anal. 28 (1), 5–39 (2020). https://doi.org/10.1007/s11228-020-00531-7

    MathSciNet  MATH  Article  Google Scholar 

  58. 58.

    Yao, J.C., Zheng, X.Y.: Error bound and well-posedness with respect to an admissible function. Appl. Anal. 95(5), 1070–1087 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  59. 59.

    Zheng, X.Y., Ng, K.F.: Linear regularity for a collection of subsmooth sets in Banach spaces. SIAM J. Optim. 19(1), 62–76 (2008). https://doi.org/10.1137/060659132

    MathSciNet  MATH  Article  Google Scholar 

  60. 60.

    Zheng, X.Y., Wei, Z., Yao, J.C.: Uniform subsmoothness and linear regularity for a collection of infinitely many closed sets. Nonlinear Anal. 73(2), 413–430 (2010). https://doi.org/10.1016/j.na.2010.03.032

    MathSciNet  MATH  Article  Google Scholar 

  61. 61.

    Zheng, X. Y., Zhu, J.: Generalized metric subregularity and regularity with respect to an admissible function. SIAM J. Optim. 26(1), 535–563 (2016). https://doi.org/10.1137/15M1016345

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgments

The authors wish to thank the referee and the handling editor for their careful reading of the manuscript and valuable comments and suggestions.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Alexander Y. Kruger.

Additional information

Dedicated to the memory of Prof. Rafail Gabasov, a great person and teacher

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The research was supported by the Australian Research Council, project DP160100854. The second author benefited from the support of the FMJH Program PGMO and from the support of EDF.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cuong, N.D., Kruger, A.Y. Transversality Properties: Primal Sufficient Conditions. Set-Valued Var. Anal (2020). https://doi.org/10.1007/s11228-020-00545-1

Download citation

Keywords

  • Transversality
  • Subtransversality
  • Semitransversality
  • Regularity
  • Subregularity
  • Semiregularity
  • Slope
  • Chain rule

Mathematics Subject Classification (2010)

  • Primary 49J52
  • 49J53
  • Secondary 49K40
  • 90C30
  • 90C46