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Simple bilevel programming and extensions

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Abstract

In this paper we discuss the simple bilevel programming problem (SBP) and its extension, the simple mathematical programming problem under equilibrium constraints (SMPEC). Here we first define both these problems and study their interrelations. Next we study the various types of necessary and sufficient optimality conditions for the (SMPEC) problems, which occur under various reformulations. The optimality conditions for (SBP) are special cases of the results obtained for (SMPEC) when the lower level objective is the gradient of a convex function. Among the various optimality conditions presented in this article are the sequential optimality conditions, which do not need any constraint qualification. We also present a schematic algorithm for (SMPEC), where the sequential optimality conditions play a key role in the convergence analysis.

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Acknowledgements

Work of Stephan Dempe has been supported by Deutsche Forschungsgemeinschaft, work of Nguyen Dinh was supported by the Project B2019-28-02: Generalized scalar and vector Farkas-type results with applications to optimization theory from Vietnam National University—Ho Chi Minh city, Vietnam. The authors are indebted to anonymous referees and the editor for very careful reading of the presentation and making useful suggestions and remarks, which resulted in essential improvements to the paper.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, TU Bergakademie Freiberg, Freiberg, Germany

    Stephan Dempe

  2. Department of Mathematics, International University, Vietnam National University-Ho Chi Minh City, Ho Chi Minh City, Vietnam

    Nguyen Dinh

  3. Department of Economic Sciences, Indian Institute of Technology, Kanpur, India

    Joydeep Dutta

  4. Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, India

    Tanushree Pandit

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  1. Stephan Dempe
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  2. Nguyen Dinh
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  3. Joydeep Dutta
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  4. Tanushree Pandit
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Correspondence to Stephan Dempe.

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Dempe, S., Dinh, N., Dutta, J. et al. Simple bilevel programming and extensions. Math. Program. 188, 227–253 (2021). https://doi.org/10.1007/s10107-020-01509-x

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