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Relaxed constant positive linear dependence constraint qualification and its application to bilevel programs

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Abstract

Relaxed constant positive linear dependence constraint qualification (RCPLD) for a system of smooth equalities and inequalities is a constraint qualification that is weaker than the usual constraint qualifications such as Mangasarian Fromovitz constraint qualification and the linear constraint qualification. Moreover RCPLD is known to induce an error bound property. In this paper we extend RCPLD to a very general feasibility system which may include Lipschitz continuous inequality constraints, complementarity constraints and abstract constraints. We show that this RCPLD for the general system is a constraint qualification for the optimality condition in terms of limiting subdifferential and limiting normal cone and it is a sufficient condition for the error bound property under the strict complementarity condition for the complementarity system and Clarke regularity conditions for the inequality constraints and the abstract constraint set. Moreover we introduce and study some sufficient conditions for RCPLD including the relaxed constant rank constraint qualification. Finally we apply our results to the bilevel program.

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Acknowledgements

The authors would like to thank the anonymous referees for their helpful suggestions and comments which help us to improve the presentation of the paper.

Funding

Funding was provided by National Natural Science Foundation of China (Grant No. 11601376 and No. 11901556).

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Authors and Affiliations

  1. Institute of Mathematics, Hebei University of Technology, Tianjin, 300401, China

    Mengwei Xu

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 2Y2, Canada

    Jane J. Ye

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  1. Mengwei Xu
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  2. Jane J. Ye
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Correspondence to Jane J. Ye.

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M. Xu: The research of this author was supported by the National Natural Science Foundation of China under Projects No. 11601376 and No. 11901556. J. J. Ye: The research of this author was partially supported by NSERC.

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Xu, M., Ye, J.J. Relaxed constant positive linear dependence constraint qualification and its application to bilevel programs. J Glob Optim 78, 181–205 (2020). https://doi.org/10.1007/s10898-020-00907-x

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