Mathematical Programming

, Volume 168, Issue 1–2, pp 533–554 | Cite as

Perturbation of error bounds

Full Length Paper Series B


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.


Error bound Feasibility problem Perturbation Subdifferential Metric regularity Metric subregularity 

Mathematics Subject Classification

49J52 49J53 90C30 



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].


  1. 1.
    Auslender, A., Crouzeix, J.P.: Global regularity theorems. Math. Oper. Res. 13(2), 243–253 (1988). doi: 10.1287/moor.13.2.243 MathSciNetCrossRefMATHGoogle Scholar
  2. 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)Google Scholar
  3. 3.
    Azé, D.: A unified theory for metric regularity of multifunctions. J. Convex Anal. 13(2), 225–252 (2006)MathSciNetMATHGoogle Scholar
  4. 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)MathSciNetCrossRefMATHGoogle Scholar
  5. 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 MathSciNetCrossRefMATHGoogle Scholar
  6. 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)MathSciNetCrossRefMATHGoogle Scholar
  7. 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/ MathSciNetCrossRefMATHGoogle Scholar
  8. 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)MathSciNetMATHGoogle Scholar
  9. 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 CrossRefMATHGoogle Scholar
  10. 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. 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 MathSciNetCrossRefMATHGoogle Scholar
  12. 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 CrossRefMATHGoogle Scholar
  13. 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)MathSciNetCrossRefMATHGoogle Scholar
  14. 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 MathSciNetCrossRefMATHGoogle Scholar
  15. 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 MathSciNetCrossRefMATHGoogle Scholar
  16. 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)MathSciNetCrossRefMATHGoogle Scholar
  17. 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. 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. 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 MathSciNetCrossRefMATHGoogle Scholar
  20. 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)MathSciNetCrossRefMATHGoogle Scholar
  21. 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)MATHGoogle Scholar
  22. 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 MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)MathSciNetCrossRefMATHGoogle Scholar
  24. 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)MathSciNetCrossRefMATHGoogle Scholar
  25. 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)Google Scholar
  26. 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)MathSciNetCrossRefMATHGoogle Scholar
  27. 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 MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Henrion, R., Jourani, A.: Subdifferential conditions for calmness of convex constraints. SIAM J. Optim. 13(2), 520–534 (2002)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Henrion, R., Outrata, J.V.: A subdifferential condition for calmness of multifunctions. J. Math. Anal. Appl. 258(1), 110–130 (2001)MathSciNetCrossRefMATHGoogle Scholar
  30. 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 MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Hoffman, A.J.: On approximate solutions of systems of linear inequalities. J. Res. Nat. Bur. Stand. 49, 263–265 (1952)MathSciNetCrossRefGoogle Scholar
  32. 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 MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Ioffe, A.D.: Metric regularity and subdifferential calculus. Russ. Math. Surv. 55, 501–558 (2000)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Ioffe, A.D.: Metric regularity—a survey. Part I. Theory. J. Aust. Math. Soc. 101(2), 188–243 (2016). doi: 10.1017/S1446788715000701 MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Ioffe, A.D.: Metric regularity—a survey. Part II. Applications. J. Aust. Math. Soc. 101(3), 376–417 (2016). doi: 10.1017/S1446788715000695 MathSciNetCrossRefMATHGoogle Scholar
  36. 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)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Jourani, A.: Hoffman’s error bound, local controllability, and sensitivity analysis. SIAM J. Control Optim. 38(3), 947–970 (2000)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Klatte, D., Li, W.: Asymptotic constraint qualifications and global error bounds for convex inequalities. Math. Program. Ser. A 84(1), 137–160 (1999)MathSciNetCrossRefMATHGoogle Scholar
  39. 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 MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Kruger, A.Y.: Error bounds and metric subregularity. Optimization 64(1), 49–79 (2015). doi: 10.1080/02331934.2014.938074 MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Kruger, A.Y.: Nonlinear metric subregularity. J. Optim. Theory Appl. (2015). doi: 10.1007/s10957-015-0807-8
  42. 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. 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 MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle Scholar
  45. 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 MathSciNetCrossRefMATHGoogle Scholar
  46. 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)Google Scholar
  47. 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 MathSciNetCrossRefMATHGoogle Scholar
  48. 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. 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 MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Meng, K.W., Yang, X.Q.: Equivalent conditions for local error bounds. Set Valued Var. Anal. 20(4), 617–636 (2012)MathSciNetCrossRefMATHGoogle Scholar
  51. 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)Google Scholar
  52. 52.
    Ng, K.F., Zheng, X.Y.: Error bounds for lower semicontinuous functions in normed spaces. SIAM J. Optim. 12(1), 1–17 (2001)MathSciNetCrossRefMATHGoogle Scholar
  53. 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 MathSciNetCrossRefMATHGoogle Scholar
  54. 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 MathSciNetCrossRefMATHGoogle Scholar
  55. 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)MathSciNetCrossRefMATHGoogle Scholar
  56. 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 MathSciNetCrossRefMATHGoogle Scholar
  57. 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)MathSciNetCrossRefMATHGoogle Scholar
  58. 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 MathSciNetMATHGoogle Scholar
  59. 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 MathSciNetCrossRefMATHGoogle Scholar
  60. 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 MathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    Pang, J.S.: Error bounds in mathematical programming. Math. Program.Ser. B 79(1–3), 299–332 (1997)MathSciNetMATHGoogle Scholar
  62. 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. 63.
    Robinson, S.M.: An application of error bounds for convex programming in a linear space. SIAM J. Control 13, 271–273 (1975)MathSciNetCrossRefMATHGoogle Scholar
  64. 64.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  65. 65.
    Rosenbloom, P.C.: Quelques classes de problémes extrémaux. Bulletin de la Société Mathématique de France 79, 1–58 (1951).
  66. 66.
    Wu, Z., Ye, J.J.: Sufficient conditions for error bounds. SIAM J. Optim. 12(2), 421–435 (2001/02)Google Scholar
  67. 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 MathSciNetCrossRefMATHGoogle Scholar
  68. 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 MathSciNetCrossRefMATHGoogle Scholar
  69. 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/ MathSciNetCrossRefMATHGoogle Scholar
  70. 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 MathSciNetCrossRefMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.Federation University AustraliaBallaratAustralia
  2. 2.Department of Statistics and Operations ResearchUniversity of AlicanteAlicanteSpain
  3. 3.Laboratoire XLIM, UMR-CNRS 6172University of LimogesLimogesFrance

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