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New Farkas-Type Results for Vector-Valued Functions: A Non-abstract Approach

Author

Listed:
  • Nguyen Dinh

    (International University, Vietnam National University-HCM)

  • Miguel A. Goberna

    (University of Alicante)

  • Dang H. Long

    (VNUHCM-University of Science
    Tien Giang University)

  • Marco A. López-Cerdá

    (University of Alicante
    Federation University)

Abstract

This paper provides new Farkas-type results characterizing the inclusion of a given set, called contained set, into a second given set, called container set, both of them are subsets of some locally convex space, called decision space. The contained and the container sets are described here by means of vector functions from the decision space to other two locally convex spaces which are equipped with the partial ordering associated with given convex cones. These new Farkas lemmas are obtained via the complete characterization of the conic epigraphs of certain conjugate mappings which constitute the core of our approach. In contrast with a previous paper of three of the authors (Dinh et al. in J Optim Theory Appl 173:357–390, 2017), the aimed characterizations of the containment are expressed here in terms of the data.

Suggested Citation

  • Nguyen Dinh & Miguel A. Goberna & Dang H. Long & Marco A. López-Cerdá, 2019. "New Farkas-Type Results for Vector-Valued Functions: A Non-abstract Approach," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 4-29, July.
    • Handle: RePEc:spr:joptap:v:182:y:2019:i:1:d:10.1007_s10957-018-1352-z
    DOI: 10.1007/s10957-018-1352-z
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    References listed on IDEAS

    as
    1. N. Dinh & G. Vallet & M. Volle, 2014. "Functional inequalities and theorems of the alternative involving composite functions," Journal of Global Optimization, Springer, vol. 59(4), pages 837-863, August.
    2. 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.
    3. N. Dinh & V. Jeyakumar, 2014. "Rejoinder on: Farkas’ lemma: three decades of generalizations for mathematical optimization," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 41-44, April.
    4. N. Dinh & V. Jeyakumar, 2014. "Farkas’ lemma: three decades of generalizations for mathematical optimization," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 1-22, April.
    5. David Bartl, 2012. "A very short algebraic proof of the Farkas Lemma," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 75(1), pages 101-104, February.
    6. N. Dinh & M. A. Goberna & M. A. López & T. H. Mo, 2017. "Farkas-Type Results for Vector-Valued Functions with Applications," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 357-390, May.
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    Cited by:

    1. Nithirat Sisarat & Rabian Wangkeeree & Gue Myung Lee, 2020. "On Set Containment Characterizations for Sets Described by Set-Valued Maps with Applications," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 824-841, March.
    2. Nguyen Dinh & Dang Hai Long, 2022. "A Perturbation Approach to Vector Optimization Problems: Lagrange and Fenchel–Lagrange Duality," Journal of Optimization Theory and Applications, Springer, vol. 194(2), pages 713-748, August.
    3. Xiaopeng Zhao & Markus A. Köbis & Yonghong Yao & Jen-Chih Yao, 2021. "A Projected Subgradient Method for Nondifferentiable Quasiconvex Multiobjective Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 82-107, July.

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