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Some Results on Mathematical Programs with Equilibrium Constraints

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Abstract

Mathematical programs with equilibrium constraints (MPEC) are special class of constrained optimization problems. The feasible set of MPEC violates most of the standard constraint qualifications. Thus, the Karush-Kuhn-Tucker conditions are not necessarily satisfied at minimizers, and the convergence assumptions of many methods for solving constrained optimization problems are not fulfilled. Thus, it is necessary, from a theoretical and numerical point of view, to consider suitable optimality conditions for solving such optimization problems. In this paper, we show that M-stationary condition is sufficient for global or local optimality under some mathematical programming problem with equilibrium constraints and generalized invexity assumptions. Further, we formulate and study, Wolfe-type and Mond-Weir-type dual models for the MPEC and we establish weak and strong duality theorems relating to the MPEC and the two dual models under invexity and generalized invexity assumptions. The main purpose of this manuscript is to study the Mathematical programs with equilibrium constraints under the framework of differentiable generalized invex functions and to obtain optimality conditions and duality results.

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Acknowledgements

The author is thankful to the anonymous reviewers for their insightful comments and suggestions.

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  1. Department of Mathematics, Graphic Era University, Dehradun, 248002, India

    Bhuwan Chandra Joshi

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  1. Bhuwan Chandra Joshi
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Correspondence to Bhuwan Chandra Joshi.

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Joshi, B.C. Some Results on Mathematical Programs with Equilibrium Constraints. Oper. Res. Forum 2, 53 (2021). https://doi.org/10.1007/s43069-021-00061-4

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