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Optimality Conditions for Quasi-Solutions of Vector Optimization Problems

Author

Listed:
  • C. Gutiérrez

    (Universidad de Valladolid)

  • B. Jiménez

    (Universidad Nacional de Educación a Distancia)

  • V. Novo

    (Universidad Nacional de Educación a Distancia)

Abstract

In this paper, we deal with quasi-solutions of constrained vector optimization problems. These solutions are a kind of approximate minimal solutions and they are motivated by the Ekeland variational principle. We introduce several notions of quasi-minimality based on free disposal sets and we characterize these solutions through scalarization and Lagrange multiplier rules. When the problem fulfills certain convexity assumptions, these results are obtained by using linear separation and the Fenchel subdifferential. In the nonconvex case, they are stated by using the so-called Gerstewitz (Tammer) nonlinear separation functional and the Mordukhovich subdifferential.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:167:y:2015:i:3:d:10.1007_s10957-013-0393-6
    DOI: 10.1007/s10957-013-0393-6
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    References listed on IDEAS

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    1. 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.
    2. M. Chicco & F. Mignanego & L. Pusillo & S. Tijs, 2011. "Vector Optimization Problems via Improvement Sets," Journal of Optimization Theory and Applications, Springer, vol. 150(3), pages 516-529, September.
    3. Gutiérrez, C. & Jiménez, B. & Novo, V., 2012. "Improvement sets and vector optimization," European Journal of Operational Research, Elsevier, vol. 223(2), pages 304-311.
    4. 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.
    5. 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.
    6. 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.
    7. 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. Thai Doan Chuong, 2022. "Approximate solutions in nonsmooth and nonconvex cone constrained vector optimization," Annals of Operations Research, Springer, vol. 311(2), pages 997-1015, April.
    2. 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.
    3. 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.

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