Skip to main content
SpringerLink
Log in

Perturbation of error bounds

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

Abstract

Our aim in the current article is to extend the developments in Kruger et al. (SIAM J Optim 20(6):3280–3296, 2010. doi:10.1137/100782206) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations. We propose new concepts of (arbitrary, convex and linear) perturbations of the given function defining the system under consideration, which turn out to be a useful tool in our analysis. The characterizations of error bounds for families of perturbations can be interpreted as estimates of the ‘radius of error bounds’. The definitions and characterizations are illustrated by examples.

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. Private communication by J.-B. Hiriart-Urruty.

References

  1. Auslender, A., Crouzeix, J.P.: Global regularity theorems. Math. Oper. Res. 13(2), 243–253 (1988). doi:10.1287/moor.13.2.243

    Article  MathSciNet  MATH  Google Scholar 

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

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

    MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Beck, A., Teboulle, M.: Convergence rate analysis and error bounds for projection algorithms in convex feasibility problems. Optim. Methods Softw. 18(4), 377–394 (2003). doi:10.1080/10556780310001604977. The Second Japanese-Sino Optimization Meeting, Part II (Kyoto 2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bednarczuk, E.M., Kruger, A.Y.: Error bounds for vector-valued functions: necessary and sufficient conditions. Nonlinear Anal. 75(3), 1124–1140 (2012). doi:10.1016/j.na.2011.05.098

    Article  MathSciNet  MATH  Google Scholar 

  8. Bednarczuk, E.M., Kruger, A.Y.: Error bounds for vector-valued functions on metric spaces. Vietnam J. Math. 40(2–3), 165–180 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Beer, G.: Topologies on Closed and Closed Convex Sets, Mathematics and its Applications, vol. 268. Kluwer Academic Publishers Group, Dordrecht (1993). doi:10.1007/978-94-015-8149-3

    Book  MATH  Google Scholar 

  10. Bolte, J., Nguyen, T.P., Peypouquet, J., Suter, B.W.: From error bounds to the complexity of first order descent methods for convex functions. Preprint, arXiv:1510.08234 (2015)

  11. 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). doi:10.1137/130919052

    Article  MathSciNet  MATH  Google Scholar 

  12. Borwein, J.M., Vanderwerff, J.D.: Convex Functions: constructions, Characterizations and Counterexamples, Encyclopedia of Mathematics and its Applications, vol. 109. Cambridge University Press, Cambridge (2010). doi:10.1017/CBO9781139087322

    Book  MATH  Google Scholar 

  13. Burke, J.V., Deng, S.: Weak sharp minima revisited. II. Application to linear regularity and error bounds. Math. Program. Ser. B 104(2–3), 235–261 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cánovas, M.J., Hantoute, A., Parra, J., Toledo, F.J.: Boundary of subdifferentials and calmness moduli in linear semi-infinite optimization. Optim. Lett. 9(3), 513–521 (2015). doi:10.1007/s11590-014-0767-1

    Article  MathSciNet  MATH  Google Scholar 

  15. Cánovas, M.J., Henrion, R., López, M.A., Parra, J.: Outer limit of subdifferentials and calmness moduli in linear and nonlinear programming. J. Optim. Theory Appl. 169(3), 925–952 (2016). doi:10.1007/s10957-015-0793-x

    Article  MathSciNet  MATH  Google Scholar 

  16. Cánovas, M.J., Kruger, A.Y., López, M.A., Parra, J., Théra, M.A.: Calmness modulus of linear semi-infinite programs. SIAM J. Optim. 24(1), 29–48 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Censor, Y.: Iterative methods for the convex feasibility problem. In: Convexity and Graph Theory (Jerusalem 1981), North-Holland Math. Stud., vol. 87, pp. 83–91. North-Holland, Amsterdam (1984). doi:10.1016/S0304-0208(08)72812-3

  18. Combettes, P.L.: The convex feasibility problem in image recovery. In: Hawkes, P. (ed.) Advances in Imaging and Electron Physics, vol. 95, pp. 155–270. Academic Press, New York (1996)

    Google Scholar 

  19. 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

    Article  MathSciNet  MATH  Google Scholar 

  20. 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)

    Article  MathSciNet  MATH  Google Scholar 

  21. 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)

    MATH  Google Scholar 

  22. Drusvyatskiy, D., Ioffe, A.D., Lewis, A.S.: Curves of descent. SIAM J. Control Optim. 53(1), 114–138 (2015). doi:10.1137/130920216

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. 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)

    Article  MathSciNet  MATH  Google Scholar 

  25. 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-Verlag, Berlin (2012)

  26. Gfrerer, H.: First order and second order characterizations of metric subregularity and calmness of constraint set mappings. SIAM J. Optim. 21(4), 1439–1474 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gfrerer, H., Outrata, J.V.: On computation of generalized derivatives of the normal-cone mapping and their applications. Math. Oper. Res. 41(4), 1535–1556 (2016). doi:10.1287/moor.2016.0789

    Article  MathSciNet  MATH  Google Scholar 

  28. Henrion, R., Jourani, A.: Subdifferential conditions for calmness of convex constraints. SIAM J. Optim. 13(2), 520–534 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Henrion, R., Outrata, J.V.: A subdifferential condition for calmness of multifunctions. J. Math. Anal. Appl. 258(1), 110–130 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. 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). doi:10.1137/120902653

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  32. Huang, L.R., Ng, K.F.: On first- and second-order conditions for error bounds. SIAM J. Optim. 14(4), 1057–1073 (2004). doi:10.1137/S1052623401390549

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Ioffe, A.D.: Metric regularity—a survey. Part II. Applications. J. Aust. Math. Soc. 101(3), 376–417 (2016). doi:10.1017/S1446788715000695

    Article  MathSciNet  MATH  Google Scholar 

  36. Ioffe, A.D., Outrata, J.V.: On metric and calmness qualification conditions in subdifferential calculus. Set Valued Anal. 16(2–3), 199–227 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  38. Klatte, D., Li, W.: Asymptotic constraint qualifications and global error bounds for convex inequalities. Math. Program. Ser. A 84(1), 137–160 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  39. Kruger, A.Y.: Error bounds and Hölder metric subregularity. Set Valued Var. Anal. 23(4), 705–736 (2015). doi:10.1007/s11228-015-0330-y

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. Kruger, A.Y.: Nonlinear metric subregularity. J. Optim. Theory Appl. (2015). doi:10.1007/s10957-015-0807-8

  42. Kruger, A.Y., Luke, D.R., Thao, N.H.: Set regularities and feasibility problems. Math. Program, Ser. B. (2016). doi:10.1007/s10107-016-1039-x

  43. Kruger, A.Y., Ngai, H.V., Théra, M.: Stability of error bounds for convex constraint systems in Banach spaces. SIAM J. Optim. 20(6), 3280–3296 (2010). doi:10.1137/100782206

    Article  MathSciNet  MATH  Google Scholar 

  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. 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). doi:10.1007/s10208-008-9036-y

    Article  MathSciNet  MATH  Google Scholar 

  46. Lewis, A.S., Pang, J.S.: Error bounds for convex inequality systems. Generalized Convexity. Generalized Monotonicity: Recent Results (Luminy 1996), pp. 75–110. Kluwer Acad. Publ, Dordrecht (1998)

  47. 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

    Article  MathSciNet  MATH  Google Scholar 

  48. Li, M.H., Meng, K.W., Yang, X.Q.: On error bound moduli for locally Lipschitz and regular functions. Preprint 1608(03360), 1–26 (2016)

    Google Scholar 

  49. Mangasarian, O.L.: A condition number for differentiable convex inequalities. Math. Oper. Res. 10(2), 175–179 (1985). doi:10.1287/moor.10.2.175

    Article  MathSciNet  MATH  Google Scholar 

  50. Meng, K.W., Yang, X.Q.: Equivalent conditions for local error bounds. Set Valued Var. Anal. 20(4), 617–636 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  51. 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)

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

    Article  MathSciNet  MATH  Google Scholar 

  53. Ngai, H.V., Kruger, A.Y., Théra, M.: Stability of error bounds for semi-infinite convex constraint systems. SIAM J. Optim. 20(4), 2080–2096 (2010). doi:10.1137/090767819

    Article  MathSciNet  MATH  Google Scholar 

  54. 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

    Article  MathSciNet  MATH  Google Scholar 

  55. Ngai, H.V., Théra, M.: Error bounds for convex differentiable inequality systems in Banach spaces. Math. Program. Ser. B 104(2–3), 465–482 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  56. 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

    Article  MathSciNet  MATH  Google Scholar 

  57. 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  MathSciNet  MATH  Google Scholar 

  58. Ngai, H.V., Tron, N.H., Théra, M.: Implicit multifunction theorems in complete metric spaces. Math. Program. 139(1–2, Ser. B), 301–326 (2013). doi:10.1007/s10107-013-0673-9

    MathSciNet  MATH  Google Scholar 

  59. Ngai, H.V., Tron, N.H., Théra, M.: Metric regularity of the sum of multifunctions and applications. J. Optim. Theory Appl. 160(2), 355–390 (2014). doi:10.1007/s10957-013-0385-6

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  61. Pang, J.S.: Error bounds in mathematical programming. Math. Program.Ser. B 79(1–3), 299–332 (1997)

    MathSciNet  MATH  Google Scholar 

  62. Penot, J.P.: Error bounds, calmness and their applications in nonsmooth analysis. In: Nonlinear analysis and optimization II. Optimization, Contemp. Math., vol. 514, pp. 225–247. Amer. Math. Soc., Providence, RI (2010). doi:10.1090/conm/514/10110

  63. Robinson, S.M.: An application of error bounds for convex programming in a linear space. SIAM J. Control 13, 271–273 (1975)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  65. Rosenbloom, P.C.: Quelques classes de problémes extrémaux. Bulletin de la Société Mathématique de France 79, 1–58 (1951). http://eudml.org/doc/86848

  66. Wu, Z., Ye, J.J.: Sufficient conditions for error bounds. SIAM J. Optim. 12(2), 421–435 (2001/02)

  67. Zălinescu, C.: Sharp estimates for Hoffman’s constant for systems of linear inequalities and equalities. SIAM J. Optim. 14(2), 517–533 (2003). doi:10.1137/S1052623402403505

    Article  MathSciNet  MATH  Google Scholar 

  68. 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

    Article  MathSciNet  MATH  Google Scholar 

  69. 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

    Article  MathSciNet  MATH  Google Scholar 

  70. Zheng, X.Y., Wei, Z.: Perturbation analysis of error bounds for quasi-subsmooth inequalities and semi-infinite constraint systems. SIAM J. Optim. 22(1), 41–65 (2012). doi:10.1137/100806199

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the referees for the careful reading of the manuscript and many constructive comments and suggestions. We particularly thank one of the reviewers for attracting our attention to [26, Theorem 3.2] and [47, Theorem 4.4].

Author information

Authors and Affiliations

  1. Federation University Australia, Ballarat, Australia

    A. Y. Kruger, M. A. López & M. A. Théra

  2. Department of Statistics and Operations Research, University of Alicante, Alicante, Spain

    M. A. López

  3. Laboratoire XLIM, UMR-CNRS 6172, University of Limoges, Limoges, France

    M. A. Théra

Authors
  1. A. Y. Kruger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. A. López
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. A. Théra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. A. Théra.

Additional information

Dedicated to Professor Terry Rockafellar, one of the founders of the contemporary optimization theory, convex analysis and variational analysis, in honor of his 80th birthday.

The research is supported by the Australian Research Council: project DP160100854; EDF and the Jacques Hadamard Mathematical Foundation: Gaspard Monge Program for Optimization and Operations Research. The research of the second and third authors is also supported by MINECO of Spain and FEDER of EU: Grant MTM2014-59179-C2-1-P.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kruger, A.Y., López, M.A. & Théra, M.A. Perturbation of error bounds. Math. Program. 168, 533–554 (2018). https://doi.org/10.1007/s10107-017-1129-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-017-1129-4

Keywords

Mathematics Subject Classification

Search

Navigation