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Improvement sets and vector optimization

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

Abstract

In this paper we focus on minimal points in linear spaces and minimal solutions of vector optimization problems, where the preference relation is defined via an improvement set E. To be precise, we extend the notion of E-optimal point due to Chicco et al. in [4] to a general (non-necessarily Pareto) quasi ordered linear space and we study its properties. In particular, we relate the notion of improvement set with other similar concepts of the literature and we characterize it by means of sublevel sets of scalar functions. Moreover, we obtain necessary and sufficient conditions for E-optimal solutions of vector optimization problems through scalarization processes by assuming convexity assumptions and also in the general (nonconvex) case. By applying the obtained results to certain improvement sets we generalize well-known results of the literature referred to efficient, weak efficient and approximate efficient solutions of vector optimization problems.

Suggested Citation

  • Gutiérrez, C. & Jiménez, B. & Novo, V., 2012. "Improvement sets and vector optimization," European Journal of Operational Research, Elsevier, vol. 223(2), pages 304-311.
    • Handle: RePEc:eee:ejores:v:223:y:2012:i:2:p:304-311
    DOI: 10.1016/j.ejor.2012.05.050
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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. 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., 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.
    4. 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.
    5. 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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    Citations

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    Cited by:

    1. Pirro Oppezzi & Anna Rossi, 2015. "Improvement Sets and Convergence of Optimal Points," Journal of Optimization Theory and Applications, Springer, vol. 165(2), pages 405-419, May.
    2. Ovidiu Bagdasar & Nicolae Popovici, 2018. "Unifying local–global type properties in vector optimization," Journal of Global Optimization, Springer, vol. 72(2), pages 155-179, October.
    3. 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.
    4. Jiawei Chen & La Huang & Shengjie Li, 2018. "Separations and Optimality of Constrained Multiobjective Optimization via Improvement Sets," Journal of Optimization Theory and Applications, Springer, vol. 178(3), pages 794-823, September.
    5. Zhiang Zhou & Wang Chen & Xinmin Yang, 2019. "Scalarizations and Optimality of Constrained Set-Valued Optimization Using Improvement Sets and Image Space Analysis," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 944-962, December.
    6. Fabián Flores-Bazán & Fernando Flores-Bazán & Sigifredo Laengle, 2015. "Characterizing Efficiency on Infinite-dimensional Commodity Spaces with Ordering Cones Having Possibly Empty Interior," Journal of Optimization Theory and Applications, Springer, vol. 164(2), pages 455-478, February.
    7. Nguyen Xuan Hai & Nguyen Hong Quan & Vo Viet Tri, 2023. "Some saddle-point theorems for vector-valued functions," Journal of Global Optimization, Springer, vol. 86(1), pages 141-161, May.
    8. Maurizio Chicco & Anna Rossi, 2015. "Existence of Optimal Points Via Improvement Sets," Journal of Optimization Theory and Applications, Springer, vol. 167(2), pages 487-501, November.
    9. C. Gutiérrez & L. Huerga & E. Köbis & C. Tammer, 2021. "A scalarization scheme for binary relations with applications to set-valued and robust optimization," Journal of Global Optimization, Springer, vol. 79(1), pages 233-256, January.
    10. 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.
    11. 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.
    12. 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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