Abstract
In this chapter, we consider a class of multiobjective optimization problems with inequality, equality and vanishing constraints. For the scalar case, this class of problems reduces to the class of mathematical programs with vanishing constraints recently appeared in literature. We show that under fairly mild assumptions some constraint qualifications like Cottle constraint qualification, Slater constraint qualification, Mangasarian-Fromovitz constraint qualification, linear independence constraint qualification, linear objective constraint qualification and linear constraint qualification do not hold at an efficient solution, whereas the standard generalized Guignard constraint qualification is sometimes satisfied. We introduce suitable modifications of above mentioned constraint qualifications, establish relationships among them and derive the Karush-Kuhn-Tucker type necessary optimality conditions for efficiency.
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Acknowledgements
The authors are thankful to the anonymous referees for their valuable comments and suggestions which helped to improve this chapter in its present form. This work was done when Vinay Singh was a Post Doctoral Fellow of National Board of Higher Mathematics (NBHM), Department of Atomic Energy (DAE), Government of India and Vivek Laha was a Senior Research Fellow of the Council of Scientific and Industrial Research (CSIR), New Delhi, Ministry of Human Resources Development, Government of India at Department of Mathematics, Banaras Hindu University.
Currently, Vivek Laha is supported by the Postdoctoral Fellowship of National Board of Higher Mathematics, Department of Atomic Energy, Government of India (Ref. No. 2/40(47)/2014/R & D-II/1170).
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Mishra, S.K., Singh, V., Laha, V., Mohapatra, R.N. (2015). On Constraint Qualifications for Multiobjective Optimization Problems with Vanishing Constraints. In: Xu, H., Wang, S., Wu, SY. (eds) Optimization Methods, Theory and Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47044-2_6
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DOI: https://doi.org/10.1007/978-3-662-47044-2_6
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