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Stability and Scalarization of Weak Efficient, Efficient and Henig Proper Efficient Sets Using Generalized Quasiconvexities

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Abstract

The aim of this paper is to establish the stability of weak efficient, efficient and Henig proper efficient sets of a vector optimization problem, using quasiconvex and related functions. We establish the Kuratowski–Painlevé set-convergence of the minimal solution sets of a family of perturbed problems to the corresponding minimal solution set of the vector problem, where the perturbations are performed on both the objective function and the feasible set. This convergence is established by using gamma convergence of the sequence of the perturbed objective functions and Kuratowski–Painlevé set-convergence of the sequence of the perturbed feasible sets. The solution sets of the vector problem are characterized in terms of the solution sets of a scalar problem, where the scalarization function satisfies order preserving and order representing properties. This characterization is further used to establish the Kuratowski–Painlevé set-convergence of the solution sets of a family of scalarized problems to the solution sets of the vector problem.

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Acknowledgements

Research of C.S. Lalitha was supported by \(\operatorname{R}\&\operatorname{D}\) Doctoral Research Programme funds for university faculty. The authors are grateful to the reviewer for the valuable comments and suggestions which helped in improving the paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Delhi South Campus, Benito Jaurez Road, New Delhi, 110021, India

    C. S. Lalitha

  2. Department of Mathematics, St. Stephen’s College, University of Delhi, Delhi, 110007, India

    Prashanto Chatterjee

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  1. C. S. Lalitha
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  2. Prashanto Chatterjee
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Correspondence to Prashanto Chatterjee.

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Lalitha, C.S., Chatterjee, P. Stability and Scalarization of Weak Efficient, Efficient and Henig Proper Efficient Sets Using Generalized Quasiconvexities. J Optim Theory Appl 155, 941–961 (2012). https://doi.org/10.1007/s10957-012-0106-6

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  • DOI: https://doi.org/10.1007/s10957-012-0106-6

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