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Improvement sets and vector optimization

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

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    Bibliographic Info

    Article provided by Elsevier in its journal European Journal of Operational Research.

    Volume (Year): 223 (2012)
    Issue (Month): 2 ()
    Pages: 304-311

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    Handle: RePEc:eee:ejores:v:223:y:2012:i:2:p:304-311

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    Web page: http://www.elsevier.com/locate/eor

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    Keywords: Improvement set; Minimal point; Vector optimization; ε-Efficiency; Scalarization;

    References

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    1. 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.
    2. C. Gutiérrez & B. Jiménez & V. Novo, 2006. "On Approximate Efficiency in Multiobjective Programming," Computational Statistics, Springer, vol. 64(1), pages 165-185, August.
    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.
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