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Existence and Optimality Conditions for Approximate Solutions to Vector Optimization Problems

Author

Listed:
  • Y. Gao

    (Chongqing Normal University)

  • S. H. Hou

    (The Hong Kong Polytechnic University)

  • X. M. Yang

    (Chongqing Normal University)

Abstract

In this paper, we introduce a new concept of ϵ-efficiency for vector optimization problems. This extends and unifies various notions of approximate solutions in the literature. Some properties for this new class of approximate solutions are established, and several existence results, as well as nonlinear scalarizations, are obtained by means of the Ekeland’s variational principle. Moreover, under the assumption of generalized subconvex functions, we derive the linear scalarization and the Lagrange multiplier rule for approximate solutions based on the scalarization in Asplund spaces.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:152:y:2012:i:1:d:10.1007_s10957-011-9891-6
    DOI: 10.1007/s10957-011-9891-6
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    References listed on IDEAS

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    1. X. M. Yang & D. Li & S. Y. Wang, 2001. "Near-Subconvexlikeness in Vector Optimization with Set-Valued Functions," Journal of Optimization Theory and Applications, Springer, vol. 110(2), pages 413-427, August.
    2. 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.
    3. J. B. Hiriart-Urruty, 1979. "Tangent Cones, Generalized Gradients and Mathematical Programming in Banach Spaces," Mathematics of Operations Research, INFORMS, vol. 4(1), pages 79-97, February.
    4. M. Durea & J. Dutta & C. Tammer, 2010. "Lagrange Multipliers for ε-Pareto Solutions in Vector Optimization with Nonsolid Cones in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 145(1), pages 196-211, April.
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    Cited by:

    1. Jiawei Chen & Qamrul Hasan Ansari & Yeong-Cheng Liou & Jen-Chih Yao, 2016. "A proximal point algorithm based on decomposition method for cone constrained multiobjective optimization problems," Computational Optimization and Applications, Springer, vol. 65(1), pages 289-308, September.
    2. C. Gutiérrez & B. Jiménez & V. Novo, 2015. "Optimality Conditions for Quasi-Solutions of Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 167(3), pages 796-820, December.
    3. C. Gutiérrez & R. López & J. Martínez, 2022. "Generalized $${\varepsilon }$$ ε -quasi solutions of set optimization problems," Journal of Global Optimization, Springer, vol. 82(3), pages 559-576, March.
    4. 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.
    5. Qamrul Hasan Ansari & Pradeep Kumar Sharma, 2022. "Some Properties of Generalized Oriented Distance Function and their Applications to Set Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 247-279, June.
    6. Wang Chen & Xinmin Yang & Yong Zhao, 2023. "Conditional gradient method for vector optimization," Computational Optimization and Applications, Springer, vol. 85(3), pages 857-896, July.

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