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.
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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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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
- d.c. inequalities system
- Error bounds
- Exact penalty function
- Lagrange multipliers
Mathematics Subject Classification