Skip to main content

Sufficient conditions for error bounds of difference functions and applications

Journal of Global Optimization volume 66, pages 439–456 (2016)Cite this article

Abstract

This paper establishes verifiable sufficient conditions for the existence of error bounds for the sub-level set of a difference function over an abstract constraint by applying a technique used by A. D. Ioffe. As a consequence, error bounds for constraint systems defined by d.c. inequalities and their applications in studying of exactness of the associated \(\ell _1\) penalty function and existence of Lagrange multipliers as necessary optimality conditions are also investigated.

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

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Auslender, A.A., Crouzeix, J.-P.: Global regularity theorems. Math. Oper. Res. 13(2), 243–253 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bector, C.R., Chandra, S., Dutta, J.: Principles of Optimization Theory. Alpha Science International, Oxford (2005)

    Google Scholar 

  3. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems, Springer Series in Operations Research. Springer, New York (2000)

    Book  MATH  Google Scholar 

  4. Clarke, F.H.: A new approach to Lagrange multipliers. Math. Oper. Res. 1, 165–174 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  5. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Gao, Y.: Optimality conditions with Lagrange multipliers for inequality constrained quasidifferentiable optimization. In: Demy’anov, V.F., Rubinov, A.M. (eds.) Quasidifferentiability and Related Topics. Nonconvex Optimization and Its Applications, vol. 43, pp. 151–162. Kluwer, Dordrecht (2000)

    Google Scholar 

  8. Hang, N.T.V., Yen, N.D.: On the problem of minimizing a difference of polyhedral convex functions under linear constraints. J. Optim. Theory Appl. (2015). doi:10.1007/s10957-015-0769-x

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Ioffe, A.D.: Regular points of Lipschitz functions. Trans. Am. Math. Soc. 251, 61–69 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ioffe, A.D.: Necessary and sufficient conditions for a local minimum. 1: A reduction theorem and first order conditions. SIAM J. Control Optim. 17(2), 245–250 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland, Amsterdam (1979)

    MATH  Google Scholar 

  13. 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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Luo, Z.-Q., Tseng, P.: On the linear convergence of descent methods for convex essentially smooth minimization. SIAM J. Control Optim. 30, 408–425 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. Luo, Z.-Q., Tseng, P.: Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem. SIAM J. Optim. 2, 43–54 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Luo, Z.-Q., Tseng, P.: Error bound and reduced-gradient projection algorithms for convex minimization over a polyhedral set. SIAM J. Optim. 3, 43–59 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mangasarian, O.L.: A condition number for differentiable convex inequalities. Math. Oper. Res. 10, 175–179 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mangasarian, O.L.: Error bounds for nondifferentiable convex inequalities under a strong Slater constraint qualification. Math. Program. 83, 187–194 (1998)

    MathSciNet  MATH  Google Scholar 

  19. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)

    Google Scholar 

  20. Mordukhovich, B.S., Nam, N.M., Yen, N.D.: Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming. Optimization 55(5–6), 685–708 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Ngai, H.V., Théra, M.A.: 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 

  23. Phelps, R.R.: Convex functions, monotone operators and differentiability. Lecture Notes in Mathematics, vol. 1364. Springer, Berlin (1993)

  24. Robinson, S.M.: An application of error bounds for convex programing in a linear space. SIAM J. Control Optim. 13(2), 271–273 (1975)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the reviewers for making relationships between their results and known ones more clear. They would particularly like to thank Prof. Nguyen Dong Yen for valuable discussions. The work of Nguyen Thi Van Hang was partially supported by National Foundation for Science and Technology Development (NAFOSTED, Vietnam) under Grant 101.01-2014.37 and, the work of Jen-Chih Yao was partially supported by the Grant MOST 102-2221-E-039 -017 -MY3. The first author gratefully acknowledges hospitality and support from Kaohsiung Medical University and Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan during her internship in 2014.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, 10307, Vietnam

    Nguyen Thi Van Hang

  2. Center for General Education, China Medical University, Taichung, 40402, Taiwan

    Jen-Chih Yao

Authors
  1. Nguyen Thi Van Hang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jen-Chih Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jen-Chih Yao.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Van Hang, N.T., Yao, JC. Sufficient conditions for error bounds of difference functions and applications. J Glob Optim 66, 439–456 (2016). https://doi.org/10.1007/s10898-016-0410-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-016-0410-z

Keywords

  • d.c. inequalities system
  • Error bounds
  • Subdifferential
  • Exact penalty function
  • Lagrange multipliers

Mathematics Subject Classification

  • 49J52
  • 90C26
  • 90C46