Skip to main content

Advertisement

Log in

Nonlinear error bounds for quasiconvex inequality systems

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

The error bound is an inequality that restricts the distance from a vector to a given set by a residual function. The error bound has so many useful applications, for example in variational analysis, in convergence analysis of algorithms, in sensitivity analysis, and so on. For convex inequality systems, Lipschitzian error bounds are studied mainly. If an inequality system is not convex, it is difficult to show the existence of a Lipschitzian global error bound in general. Hence for nonconvex inequality systems, Hölderian error bounds and nonlinear error bounds have been investigated. For quasiconvex inequality systems, there are so many examples such that systems do not have Lipschitzian and Hölderian error bounds. However, the research of nonlinear error bounds for quasiconvex inequality systems have not been investigated yet as far as we know. In this paper, we study nonlinear error bounds for quasiconvex inequality systems. We show the existence of a global nonlinear error bound by a generator of a quasiconvex function and a constraint qualification. We show well-posedness of a quasiconvex function by the error bound.

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

Access this article

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

References

  1. Azé, D.: A survey on error bounds for lower semicontinuous functions. ESAIM Proc. 13, 1–17 (2003)

    MathSciNet  MATH  Google Scholar 

  2. Azé, D., Corvellec, J.N.: Characterizations of error bounds for lower semicontinuous functions on metric spaces. ESAIM Control Optim. Calc. Var. 10, 409–425 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bosch, P., Jourani, A., Henrion, R.: Sufficient conditions for error bounds and applications. Appl. Math. Optim. 50, 161–181 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chao, M., Cheng, C.: Linear and nonlinear error bounds for lower semicontinuous functions. Optim. Lett. 8, 1301–1312 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Corvellec, J.N., Motreanu, V.V.: Nonlinear error bounds for lower semicontinuous functions on metric spaces. Math. Program. 114, 291–319 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Le Thi, H.A., Pham Dinh, T., Ngai, H.V.: Exact penalty and error bounds in DC programming. J. Glob. Optim. 52, 509–535 (2012)

  11. Lewis, A.S., Pang, J.S.: Error bounds for convex inequality systems. Nonconvex Optim. Appl. 27, 75–110 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, G.: Global error bounds for piecewise convex polynomials. Math. Program. Ser. A 137, 37–64 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  14. Penot, J.P.: Well-behavior, well-posedness and nonsmooth analysis. Pliska Stud. Math. Bulgar. 12, 141–190 (1998)

    MathSciNet  MATH  Google Scholar 

  15. Penot, J.P.: Error bounds, calmness and their applications in nonsmooth analysis. Contemp. Math. 514, 225–247 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Penot, J.P.: What is quasiconvex analysis? Optimization 47, 35–110 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Penot, J.P.: Genericity of well-posedness, perturbations and smooth variational principles. Set Valued Anal. 9, 131–157 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Penot, J.P.: Calmness and stability properties of marginal and performance functions. Numer. Funct. Anal. Optim. 25, 287–308 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Suzuki, S., Kuroiwa, D.: On set containment characterization and constraint qualification for quasiconvex programming. J. Optim. Theory Appl. 149, 554–563 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Suzuki, S., Kuroiwa, D.: Optimality conditions and the basic constraint qualification for quasiconvex programming. Nonlinear Anal. 74, 1279–1285 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Suzuki, S., Kuroiwa, D.: Necessary and sufficient conditions for some constraint qualifications in quasiconvex programming. Nonlinear Anal. 75, 2851–2858 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Suzuki, S., Kuroiwa, D.: Some constraint qualifications for quasiconvex vector-valued systems. J. Glob. Optim. 55, 539–548 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Suzuki, S., Kuroiwa, D.: Generators and constraint qualifications for quasiconvex inequality systems. J. Nonlinear Convex Anal. (2015, accepted)

  24. Martínez-Legaz, J.E.: Quasiconvex duality theory by generalized conjugation methods. Optimization 19, 603–652 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  25. Penot, J.P., Volle, M.: On quasi-convex duality. Math. Oper. Res. 15, 597–625 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  26. Suzuki, S., Kuroiwa, D.: Necessary and sufficient constraint qualification for surrogate duality. J. Optim. Theory Appl. 152, 366–367 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Suzuki, S., Kuroiwa, D., Lee, G.M.: Surrogate duality for robust optimization. Eur. J. Oper. Res. 231, 257–262 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors are grateful to anonymous referees for many comments and suggestions improved the quality of the paper. This work was supported by JSPS KAKENHI Grant Numbers 15K17588, 25400205.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Satoshi Suzuki.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Suzuki, S., Kuroiwa, D. Nonlinear error bounds for quasiconvex inequality systems. Optim Lett 11, 107–120 (2017). https://doi.org/10.1007/s11590-015-0992-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-015-0992-2

Keywords

Navigation