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Journal of Global Optimization

, Volume 66, Issue 3, pp 439–456 | Cite as

Sufficient conditions for error bounds of difference functions and applications

  • Nguyen Thi Van Hang
  • Jen-Chih YaoEmail author
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.

Keywords

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

Mathematics Subject Classification

49J52 90C26 90C46 

Notes

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.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam
  2. 2.Center for General EducationChina Medical UniversityTaichungTaiwan

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