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

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  • C. S. Lalitha

    (University of Delhi South Campus)

  • Prashanto Chatterjee

    (University of Delhi)

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.

Suggested Citation

  • C. S. Lalitha & Prashanto Chatterjee, 2012. "Stability and Scalarization of Weak Efficient, Efficient and Henig Proper Efficient Sets Using Generalized Quasiconvexities," Journal of Optimization Theory and Applications, Springer, vol. 155(3), pages 941-961, December.
    • Handle: RePEc:spr:joptap:v:155:y:2012:i:3:d:10.1007_s10957-012-0106-6
    DOI: 10.1007/s10957-012-0106-6
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    References listed on IDEAS

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    1. X. X. Huang, 2000. "Stability in vector-valued and set-valued optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 52(2), pages 185-193, November.
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    4. C. S. Lalitha & Prashanto Chatterjee, 2012. "Stability for Properly Quasiconvex Vector Optimization Problem," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 492-506, November.
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    7. Gutiérrez, C. & Jiménez, B. & Novo, V., 2010. "Optimality conditions via scalarization for a new [epsilon]-efficiency concept in vector optimization problems," European Journal of Operational Research, Elsevier, vol. 201(1), pages 11-22, February.
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    Cited by:

    1. Xiao-Bing Li & Qi-Lin Wang & Zhi Lin, 2016. "Stability of Set-Valued Optimization Problems with Naturally Quasi-Functions," Journal of Optimization Theory and Applications, Springer, vol. 168(3), pages 850-863, March.
    2. Shiva Kapoor & C. S. Lalitha, 2019. "Stability in unified semi-infinite vector optimization," Journal of Global Optimization, Springer, vol. 74(2), pages 383-399, June.
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    4. C. S. Lalitha & Prashanto Chatterjee, 2015. "Stability and Scalarization in Vector Optimization Using Improvement Sets," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 825-843, September.

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