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An interior point technique for solving bilevel programming problems

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Abstract

This paper deals with bilevel programs with strictly convex lower level problems. We present the theoretical basis of a kind of necessary and sufficient optimality conditions that involve a single-level mathematical program satisfying the linear independence constraint qualification. These conditions are obtained by replacing the inner problem by their optimality conditions and relaxing their inequality constraints. An algorithm for the bilevel program, based on a well known technique for classical smooth constrained optimization, is also studied. The algorithm obtains a solution of this problem with an effort similar to that required by a classical well-behaved nonlinear constrained optimization problem. Several illustrative problems which include linear, quadratic and general nonlinear functions and constraints are solved, and very good results are obtained for all cases.

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Acknowledgements

The authors thank to CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil) and FAPERJ (Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro) for the support provided to this research.

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Authors and Affiliations

  1. Mechanical Engineering Program, COPPE, Federal University of Rio de Janeiro, P.O. Box 68503, CEP 21945-970, CT, Cidade Universitária, Ilha do Fundão, Rio de Janeiro, Brazil

    José Herskovits & Mario Tanaka Filho

  2. Mathematical Program—UFOPA, Federal University of West of Pará, Santarém, PA, Brazil

    Mario Tanaka Filho

  3. Instituto de Matemática, Federal University of Rio de Janeiro, Cidade Universitária, Ilha do Fundão, CEP 21945-970, P.O. Box 68503, Rio de Janeiro, Brazil

    Anatoli Leontiev

Authors
  1. José Herskovits
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  2. Mario Tanaka Filho
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  3. Anatoli Leontiev
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Correspondence to Mario Tanaka Filho.

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Herskovits, J., Tanaka Filho, M. & Leontiev, A. An interior point technique for solving bilevel programming problems. Optim Eng 14, 381–394 (2013). https://doi.org/10.1007/s11081-012-9192-4

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  • DOI: https://doi.org/10.1007/s11081-012-9192-4

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