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Global efficiency for multiobjective bilevel programming problems under generalized invexity

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Abstract

In this paper, we consider a nonlinear optimistic bilevel programming problem \(({\textit{NMBP}})\) where the upper-level is a vector optimization problem and the lower-level is a scalar optimization problem. By using the Karush–Kuhn–Tucker conditions associated to the lower-level problem, we reformulate the bilevel programming problem into a nonlinear multiobjective single-level programming problem with equality and inequality constraints \(({\textit{MP}})\). Similarly to Dempe and Dutta (2012), we establish relationships between the problems \(({\textit{NMBP}})\) and \(({\textit{MP}})\). We prove that under appropriate constraint qualification and convexity assumptions, global (weakly or properly) efficient solutions of \(({\textit{MP}})\) correspond to global (weakly or properly) efficient solutions of \(({\textit{NMBP}})\). We establish Fritz John type necessary efficiency conditions for \(({\textit{NMBP}})\) without using any constraint qualification. Furthermore, we obtain (Fritz John type) sufficient efficiency conditions for a feasible point of \(({\textit{MP}})\) corresponds to a (weakly or properly) efficient solution for the bilevel problem \(({\textit{NMBP}})\) under various forms of generalized invexity and infineness. Moreover, a linear multiobjective bilevel programming problem is studied and sufficient efficiency conditions are derived. To illustrate the obtained results some examples are given.

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Acknowledgments

The authors are grateful to all the anonymous referees of the previous versions of the paper for their valuable suggestions and comments which have served to improve its quality.

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Authors and Affiliations

  1. Faculty of Sciences, Mathematics Department, University of Tizi-ouzou, 15000, Tizi-ouzou, Algeria

    Karima Bouibed

  2. LaMOS Research Unit, Operational Research Department, University of Bejaia, 06000, Bejaia, Algeria

    Karima Bouibed & Mohammed Said Radjef

  3. LaMOS Research Unit, Computer Science Department, University of Bejaia, 06000, Bejaia, Algeria

    Hachem Slimani

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  1. Karima Bouibed
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Correspondence to Karima Bouibed.

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Bouibed, K., Slimani, H. & Radjef, M.S. Global efficiency for multiobjective bilevel programming problems under generalized invexity. J. Appl. Math. Comput. 53, 507–530 (2017). https://doi.org/10.1007/s12190-015-0979-2

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  • DOI: https://doi.org/10.1007/s12190-015-0979-2

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