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Optimality conditions and duality for semi-infinite mathematical programming problems with equilibrium constraints, using convexificators

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Abstract

In this paper, we consider semi-infinite mathematical programming problems with equilibrium constraints (SIMPEC). We establish necessary and sufficient optimality conditions for the SIMPEC, using convexificators. We study the Wolfe type dual problem for the SIMPEC under the \(\partial ^{*}\)-convexity assumption. A Mond–Weir type dual problem is also formulated and studied for the SIMPEC under the \(\partial ^{*}\)-convexity, \(\partial ^{*}\)-pseudoconvexity and \(\partial ^{*}\)-quasiconvexity assumptions. Weak duality theorems are established to relate the SIMPEC and two dual programs in the framework of convexificators. Further, strong duality theorems are obtained under generalized standard Abadie constraint qualification.

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Acknowledgements

The authors are grateful to anonymous referees for careful reading of the manuscript, which improved the paper in its present form. The first author is supported by the Science and Engineering Research Board, a statutory body of the Department of Science and Technology (DST), Government of India, through file no. PDF/2016/001113.

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Authors and Affiliations

  1. Department of HSS, Indian Institute of Technology, Kanpur, India

    Yogendra Pandey

  2. Department of Mathematics, Banaras Hindu University, Varanasi, 221005, India

    S. K. Mishra

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  1. Yogendra Pandey
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  2. S. K. Mishra
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Correspondence to S. K. Mishra.

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Pandey, Y., Mishra, S.K. Optimality conditions and duality for semi-infinite mathematical programming problems with equilibrium constraints, using convexificators. Ann Oper Res 269, 549–564 (2018). https://doi.org/10.1007/s10479-017-2422-6

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  • DOI: https://doi.org/10.1007/s10479-017-2422-6

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