A new notion of error bounds: necessary and sufficient conditions

Abstract

In this paper, we propose and study a new notion of local error bounds for a convex inequalities system defined in terms of a minimal time function. This notion is called generalized local error bounds with respect to F, where F is a closed convex subset of the Euclidean space \({\mathbb {R}}^n\) satisfying \(0\in F\). It is worth emphasizing that if F is a spherical sector with the apex at the origin then this notion becomes a new type of directional error bounds which is closely related to several directional regularity concepts in Durea et al. (SIAM J Optim 27:1204–1229, 2017), Gfrerer (Set Valued Var Anal 21:151–176, 2013), Ngai and Théra (Math Oper Res 40:969–991, 2015) and Ngai et al. (J Convex Anal 24:417–457, 2017). Furthermore, if F is the closed unit ball in \({\mathbb {R}}^n\) then the notion of generalized local error bounds with respect to F reduces to the concept of usual local error bounds. In more detail, firstly we establish several necessary conditions for the existence of these generalized local error bounds. Secondly, we show that these necessary conditions become sufficient conditions under various stronger conditions of F. Finally, we state and prove a generalized-invariant-point theorem and then use the obtained result to derive another sufficient condition for the existence of generalized local error bounds with respect to F.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Azé, D., Corvellec, J.-N.: On the sensitivity analysis of Hoffman constants for systems of linear inequalities. SIAM J. Optim. 12, 913–927 (2002)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Chuong, T.D., Jeyakumar, V.: Characterizing robust local error bounds for linear inequality systems under data uncertainty. Linear Algebra Appl. 489, 199–216 (2016)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Colombo, G., Goncharov, V.V., Mordukhovich Boris, S.: Well-posedness of minimal time problems with constant dynamics in Banach spaces. Set Valued Anal. 18, 349–372 (2010)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Colombo, G., Wolenski, P.R.: Variational analysis for a class of minimal time functions in Hilbert spaces. J. Convex Anal. 11, 335–361 (2004)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Colombo, G., Wolenski, P.R.: The subgradient formula for the minimal time function in the case of constant dynamics in Hilbert space. J. Glob. Optim. 28, 269–282 (2004)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Dancs, S., Hegedus, M., Medvegyev, P.: A general ordering and fixed-point principle in complete metric space. Acta Sci. Math. Szeged. 46, 381–388 (1983)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Durea, M., Panţiruc, M., Strugariu, R.: Minimal time function with respect to a set of directions: basic properties and applications. Optim. Methods Softw. 31, 535–561 (2016)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Durea, M., Panţiruc, M., Strugariu, R.: A new of directional regularity for mappings and applications to optimization. SIAM J. Optim. 27, 1204–1229 (2017)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Fabian, M.J., Henrion, R., Kruger, A.Y., Outrata, J.V.: Error bounds: necessary and sufficient conditions. Set Valued Var. Anal. 18, 121–149 (2010)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Gfrerer, H.: On directional metric regularity, subregularity and optimality conditions for nonsmooth mathematical programs. Set Valued Var. Anal. 21, 151–176 (2013)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Ha, T.X.D.: Slopes, error bounds and weak sharp pareto minima of a vector-valued map. J. Optim. Theory Appl. 176, 634–649 (2018)

    MathSciNet  Article  Google Scholar 

  12. 12.

    He, Y., Ng, K.F.: Subdifferentials of a minimum time function in Banach spaces. J. Math. Anal. Appl. 321, 896–910 (2006)

    MathSciNet  Article  Google Scholar 

  13. 13.

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

    MathSciNet  Article  Google Scholar 

  14. 14.

    Hu, H.: Characterizations of the strong basic constraint qualifications. Math. Oper. Res. 30, 956–965 (2005)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Hu, H.: Characterizations of local and global error bounds for convex inequalities in Banach spaces. SIAM J. Optim. 18, 309–321 (2007)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Ioffe, A.D.: Variational Analysis of Regular Mappings. Springer Monographs in Mathematics. Theory and applications. Springer, Cham (2017)

    Google Scholar 

  17. 17.

    Ivanov, G.E., Thibault, L.: Infimal convolution and optimal time control problem III: minimal time projection set. SIAM J. Optim. 28, 30–44 (2018)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Jiang, Y., He, Y.: Subdifferentials of a minimum time function in normed spaces. J. Math. Anal. Appl. 358, 410–418 (2009)

    MathSciNet  Article  Google Scholar 

  19. 19.

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

    MathSciNet  Article  Google Scholar 

  20. 20.

    Kruger, A.Y.: Error bounds and metric subregularity. Optimization 64, 49–79 (2015)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Kruger, A.Y., López, M.A., Théra, M.: Perturbation of error bounds. Math. Program. Ser. A 168, 533–554 (2018)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Lewis, A.S., Pang, J.S.: Error bounds for convex inequality systems. In: Crouzeix, J.P., Martinez Legaz, J.E., Volle, M. (eds.) Generalized Convexity, Generalized Monotonicity, pp. 75–110. Springer, New York (1998)

    Google Scholar 

  23. 23.

    Li, G.: On the asymptotic well behaved functions and global error bound for convex polynomials. SIAM J. Optim. 20, 1923–1943 (2010)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Li, M.H., Meng, K.W., Yang, X.Q.: On error bound moduli for locally Lipschitz and regular functions. Math. Program. Ser. A 171, 463–487 (2018)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Mordukhovich, B.S., Nam, N.M.: Applications of variational analysis to a generalized Fermat–Torricelli problem. J. Optim. Theory Appl. 148, 431–454 (2011)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Mordukhovich, B.S., Nam, N.M.: An Easy Path to Convex Analysis and Applications. Synthesis Lectures on Mathematics and Statistics, vol. 14. Morgan & Claypool Publishers, Williston (2014)

    Google Scholar 

  27. 27.

    Nam, N.M., Villalobos, M.C., An, N.T.: Minimal time functions and the smallest intersecting ball problem with unbounded dynamics. J. Optim. Theory Appl. 154, 768–791 (2012)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Nam, N.M., Zalinescu, C.: Variational analysis of directional minimal time functions and applications to location problems. Set Valued Var. Anal. 21, 405–430 (2013)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Ngai, H.V.: Global error bounds for systems of convex polynomials over polyhedral constraints. SIAM J. Optim. 25, 521–539 (2015)

    MathSciNet  Article  Google Scholar 

  30. 30.

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

    MathSciNet  Article  Google Scholar 

  31. 31.

    Ngai, H.V., Théra, M.: Directional metric regularity of multifunctions. Math. Oper. Res. 40, 969–991 (2015)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Ngai, H.V., Tron, N.H., Tinh, P.N.: Directional Holder metric subregularity and application to tangent cones. J. Convex Anal. 24, 417–457 (2017)

    MathSciNet  MATH  Google Scholar 

  33. 33.

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

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Rockafellar, R.R., Wets, R.J.-B.: Variational Analysis. Grundlehren Series (Fundamental Principles of Mathematical Sciences), vol. 317. Springer, Berlin (1998)

    Google Scholar 

  35. 35.

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

    MathSciNet  Article  Google Scholar 

  36. 36.

    Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

    Google Scholar 

  37. 37.

    Zheng, X.Y., Ng, K.F.: Metric regularity and constraint qualifications for convex inequalities on Banach spaces. SIAM J. Optim. 14, 757–772 (2004)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The author would like to express his sincere thanks to the reviewers and the associate editor for their constructive comments and valuable suggestions which have contributed to the preparation of this paper. This research is funded by Vietnam National University-Hochiminh City under the Grant Number C2019-18-01.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Vo Si Trong Long.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Long, V.S.T. A new notion of error bounds: necessary and sufficient conditions. Optim Lett 15, 171–188 (2021). https://doi.org/10.1007/s11590-020-01578-z

Download citation

Keywords

  • Generalized local error bounds
  • The end sets
  • Invariant-point theorem
  • Necessary and sufficient conditions for error bounds