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On Bilevel Programs with a Convex Lower-Level Problem Violating Slater’s Constraint Qualification

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Abstract

This paper focuses on bilevel programs with a convex lower-level problem violating Slater’s constraint qualification. We relax the constrained domain of the lower-level problem. Then, an approximate solution of the original bilevel program can be obtained by solving this perturbed bilevel program. As the lower-level problem of the perturbed bilevel program satisfies Slater’s constraint qualification, it can be reformulated as a mathematical program with complementarity constraints which can be solved by standard algorithms. The lower convergence properties of the constraint set mapping and the solution set mapping of the lower-level problem of the perturbed bilevel program are expanded. We show that the solutions of a sequence of the perturbed bilevel programs are convergent to that of the original bilevel program under some appropriate conditions. And this convergence result is applied to simple trilevel programs.

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Acknowledgements

This work was supported by the Natural Science Foundation of China (71471140,11871383), High Level Introduction of Talent Research Start-up Fund No. 1856009, and Scientific and Technological Research Program of Chongqing Municipal Education Commission No. KJQN201800810.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, China

    Gaoxi Li

  2. School of Mathematics and Statistics, Wuhan University, Wuhan, China

    Gaoxi Li & Zhongping Wan

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  1. Gaoxi Li
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  2. Zhongping Wan
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Correspondence to Gaoxi Li.

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Li, G., Wan, Z. On Bilevel Programs with a Convex Lower-Level Problem Violating Slater’s Constraint Qualification. J Optim Theory Appl 179, 820–837 (2018). https://doi.org/10.1007/s10957-018-1392-4

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  • DOI: https://doi.org/10.1007/s10957-018-1392-4

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