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Proper Efficiency in Vector Optimization on Real Linear Spaces

Author

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  • M. Adán

    (Universidad de Castilla-La Mancha)

  • V. Novo

    (Universidad Nacional de Educación a Distancia)

Abstract

In this paper, we use an algebraic type of closure, which is called vector closure, and through it we introduce some adaptations to the proper efficiency in the sense of Hurwicz, Benson, and Borwein in real linear spaces without any particular topology. Scalarization, multiplier rules, and saddle-point theorems are obtained in order to characterize the proper efficiency in vector optimization with and without constraints. The usual convexlikeness concepts used in such theorems are weakened through the vector closure.

Suggested Citation

  • M. Adán & V. Novo, 2004. "Proper Efficiency in Vector Optimization on Real Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 121(3), pages 515-540, June.
    • Handle: RePEc:spr:joptap:v:121:y:2004:i:3:d:10.1023_b:jota.0000037602.13941.ed
    DOI: 10.1023/B:JOTA.0000037602.13941.ed
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    References listed on IDEAS

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    1. J. B. G. Frenk & G. Kassay, 1999. "On Classes of Generalized Convex Functions, Gordan–Farkas Type Theorems, and Lagrangian Duality," Journal of Optimization Theory and Applications, Springer, vol. 102(2), pages 315-343, August.
    2. G. Y. Chen & W. D. Rong, 1998. "Characterizations of the Benson Proper Efficiency for Nonconvex Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 98(2), pages 365-384, August.
    3. Adan, M. & Novo, V., 2003. "Weak efficiency in vector optimization using a closure of algebraic type under cone-convexlikeness," European Journal of Operational Research, Elsevier, vol. 149(3), pages 641-653, September.
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    Citations

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    Cited by:

    1. Elisabeth Köbis & Markus A. Köbis & Xiaolong Qin, 2020. "An Inequality Approach to Approximate Solutions of Set Optimization Problems in Real Linear Spaces," Mathematics, MDPI, vol. 8(1), pages 1-17, January.
    2. Z. A. Zhou & J. W. Peng, 2012. "Scalarization of Set-Valued Optimization Problems with Generalized Cone Subconvexlikeness in Real Ordered Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 830-841, September.
    3. M. Chinaie & F. Fakhar & M. Fakhar & H. R. Hajisharifi, 2019. "Weak minimal elements and weak minimal solutions of a nonconvex set-valued optimization problem," Journal of Global Optimization, Springer, vol. 75(1), pages 131-141, September.
    4. Zhi-Ang Zhou & Xin-Min Yang, 2014. "Scalarization of $$\epsilon $$ ϵ -Super Efficient Solutions of Set-Valued Optimization Problems in Real Ordered Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 680-693, August.
    5. Elham Kiyani & Majid Soleimani-damaneh, 2014. "Algebraic Interior and Separation on Linear Vector Spaces: Some Comments," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 994-998, June.
    6. Christian Günther & Bahareh Khazayel & Christiane Tammer, 2022. "Vector Optimization w.r.t. Relatively Solid Convex Cones in Real Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 408-442, June.
    7. Vicente Novo & Constantin Zălinescu, 2021. "On Relatively Solid Convex Cones in Real Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 188(1), pages 277-290, January.
    8. C. Gutiérrez & L. Huerga & B. Jiménez & V. Novo, 2018. "Approximate solutions of vector optimization problems via improvement sets in real linear spaces," Journal of Global Optimization, Springer, vol. 70(4), pages 875-901, April.

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