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On Approximate Efficiency in Multiobjective Programming

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  • C. Gutiérrez
  • B. Jiménez
  • V. Novo

Abstract

This paper is focused on approximate ( $$\varepsilon$$ -efficient) solutions of multiobjective mathematical programs. We introduce a new $$\varepsilon$$ -efficiency concept which extends and unifies different notions of approximate solution defined in the literature. We characterize these $$\varepsilon$$ -efficient solutions in convex multiobjective programs through approximate solutions of linear scalarizations, which allow us to obtain parametric representations of different $$\varepsilon$$ -efficiency sets. Several classical $$\varepsilon$$ -efficiency notions are considered in order to show the concepts introduced and the results obtained. Copyright Springer-Verlag 2006

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  • C. Gutiérrez & B. Jiménez & V. Novo, 2006. "On Approximate Efficiency in Multiobjective Programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(1), pages 165-185, August.
    • Handle: RePEc:spr:mathme:v:64:y:2006:i:1:p:165-185
    DOI: 10.1007/s00186-006-0078-0
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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. Bienvenido Jiménez & Vicente Novo, 2003. "Second order necessary conditions in set constrained differentiable vector optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 58(2), pages 299-317, November.
    3. White, D.J., 1998. "Epsilon-dominating solutions in mean-variance portfolio analysis," European Journal of Operational Research, Elsevier, vol. 105(3), pages 457-466, March.
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    Cited by:

    1. Jian-Wen Peng & Wen-Bin Wei & Refail Kasimbeyli, 2025. "Linear and Nonlinear Scalarization Methods for Vector Optimization Problems with Variable Ordering Structures," Journal of Optimization Theory and Applications, Springer, vol. 206(1), pages 1-21, July.
    2. C. Gutiérrez & L. Huerga & B. Jiménez & V. Novo, 2020. "Optimality conditions for approximate proper solutions in multiobjective optimization with polyhedral cones," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 526-544, July.
    3. L. P. Hai & L. Huerga & P. Q. Khanh & V. Novo, 2019. "Variants of the Ekeland variational principle for approximate proper solutions of vector equilibrium problems," Journal of Global Optimization, Springer, vol. 74(2), pages 361-382, June.
    4. César Gutiérrez & Lidia Huerga & Vicente Novo & Lionel Thibault, 2015. "Chain Rules for a Proper $$\varepsilon $$ ε -Subdifferential of Vector Mappings," Journal of Optimization Theory and Applications, Springer, vol. 167(2), pages 502-526, November.
    5. P. Kesarwani & P. K. Shukla & J. Dutta & K. Deb, 2022. "Approximations for Pareto and Proper Pareto solutions and their KKT conditions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(1), pages 123-148, August.
    6. Le Phuoc Hai, 2021. "Ekeland variational principles involving set perturbations in vector equilibrium problems," Journal of Global Optimization, Springer, vol. 79(3), pages 733-756, March.
    7. Tran Su, 2024. "Optimality analysis for $$\epsilon $$ ϵ -quasi solutions of optimization problems via $$\epsilon $$ ϵ -upper convexificators: a dual approach," Journal of Global Optimization, Springer, vol. 90(3), pages 651-669, November.
    8. C. Gutiérrez & L. Huerga & V. Novo & C. Tammer, 2016. "Duality related to approximate proper solutions of vector optimization problems," Journal of Global Optimization, Springer, vol. 64(1), pages 117-139, January.
    9. Abbas Sayadi-bander & Latif Pourkarimi & Refail Kasimbeyli & Hadi Basirzadeh, 2017. "Coradiant sets and $$\varepsilon $$ ε -efficiency in multiobjective optimization," Journal of Global Optimization, Springer, vol. 68(3), pages 587-600, July.
    10. C. Gutiérrez & B. Jiménez & V. Novo, 2012. "Equivalent ε-efficiency notions in vector optimization," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(2), pages 437-455, July.
    11. Ying Gao & Xin-Min Yang, 2019. "Properties of the nonlinear scalar functional and its applications to vector optimization problems," Journal of Global Optimization, Springer, vol. 73(4), pages 869-889, April.
    12. Y. Gao & S. H. Hou & X. M. Yang, 2012. "Existence and Optimality Conditions for Approximate Solutions to Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 152(1), pages 97-120, January.
    13. A. Engau & M. M. Wiecek, 2007. "Cone Characterizations of Approximate Solutions in Real Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 134(3), pages 499-513, September.
    14. Gutiérrez, C. & Jiménez, B. & Novo, V., 2012. "Improvement sets and vector optimization," European Journal of Operational Research, Elsevier, vol. 223(2), pages 304-311.
    15. 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.
    16. C. Gutiérrez & B. Jiménez & V. Novo, 2011. "A generic approach to approximate efficiency and applications to vector optimization with set-valued maps," Journal of Global Optimization, Springer, vol. 49(2), pages 313-342, February.

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