IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v173y2017i2d10.1007_s10957-016-1055-2.html
   My bibliography  Save this article

Farkas-Type Results for Vector-Valued Functions with Applications

Author

Listed:
  • N. Dinh

    (International University, Vietnam National University - HCMC)

  • M. A. Goberna

    (University of Alicante)

  • M. A. López

    (University of Alicante
    Federation University)

  • T. H. Mo

    (Tien Giang University)

Abstract

The main purpose of this paper consists of providing characterizations of the inclusion of the solution set of a given conic system posed in a real locally convex topological space into a variety of subsets of the same space defined by means of vector-valued functions. These Farkas-type results are used to derive characterizations of the weak solutions of vector optimization problems (including multiobjective and scalar ones), vector variational inequalities, and vector equilibrium problems.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:173:y:2017:i:2:d:10.1007_s10957-016-1055-2
    DOI: 10.1007/s10957-016-1055-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-016-1055-2
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-016-1055-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    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. N. Dinh & V. Jeyakumar & G. M. Lee, 2005. "Sequential Lagrangian Conditions for Convex Programs with Applications to Semidefinite Programming," Journal of Optimization Theory and Applications, Springer, vol. 125(1), pages 85-112, April.
    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. Radu Ioan Bot, 2010. "Conjugate Duality in Convex Optimization," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-642-04900-2, July.
    6. N. Dinh & J. Strodiot & V. Nguyen, 2010. "Duality and optimality conditions for generalized equilibrium problems involving DC functions," Journal of Global Optimization, Springer, vol. 48(2), pages 183-208, October.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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. D. H. Fang & Y. Zhang, 2018. "Extended Farkas’s Lemmas and Strong Dualities for Conic Programming Involving Composite Functions," Journal of Optimization Theory and Applications, Springer, vol. 176(2), pages 351-376, February.
    3. 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.
    4. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. 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.
    3. P. Montiel López & M. Ruiz Galán, 2016. "Nonlinear Programming via König’s Maximum Theorem," Journal of Optimization Theory and Applications, Springer, vol. 170(3), pages 838-852, September.
    4. 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.
    5. Thai Doan Chuong & Vaithilingam Jeyakumar, 2020. "Generalized Farkas Lemma with Adjustable Variables and Two-Stage Robust Linear Programs," Journal of Optimization Theory and Applications, Springer, vol. 187(2), pages 488-519, November.
    6. Meijia Yang & Shu Wang & Yong Xia, 2022. "Toward Nonquadratic S-Lemma: New Theory and Application in Nonconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 194(1), pages 353-363, July.
    7. Yingrang Xu & Shengjie Li, 2022. "Optimality and Duality for DC Programming with DC Inequality and DC Equality Constraints," Mathematics, MDPI, vol. 10(4), pages 1-14, February.
    8. Fabián Flores-Bazán & William Echegaray & Fernando Flores-Bazán & Eladio Ocaña, 2017. "Primal or dual strong-duality in nonconvex optimization and a class of quasiconvex problems having zero duality gap," Journal of Global Optimization, Springer, vol. 69(4), pages 823-845, December.
    9. 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.
    10. D. H. Fang & Y. Zhang, 2018. "Extended Farkas’s Lemmas and Strong Dualities for Conic Programming Involving Composite Functions," Journal of Optimization Theory and Applications, Springer, vol. 176(2), pages 351-376, February.
    11. Glaydston Carvalho Bento & Sandro Dimy Barbosa Bitar & João Xavier Cruz Neto & Antoine Soubeyran & João Carlos Oliveira Souza, 2020. "A proximal point method for difference of convex functions in multi-objective optimization with application to group dynamic problems," Computational Optimization and Applications, Springer, vol. 75(1), pages 263-290, January.
    12. Xiangkai Sun & Hongyong Fu & Jing Zeng, 2018. "Robust Approximate Optimality Conditions for Uncertain Nonsmooth Optimization with Infinite Number of Constraints," Mathematics, MDPI, vol. 7(1), pages 1-14, December.
    13. Hocine Mokhtar-Kharroubi, 2017. "Convex and convex-like optimization over a range inclusion problem and first applications," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 40(1), pages 277-299, November.
    14. Wenyu Sun & Chengjin Li & Raimundo Sampaio, 2011. "On duality theory for non-convex semidefinite programming," Annals of Operations Research, Springer, vol. 186(1), pages 331-343, June.
    15. Fabián Flores-Bazán & Filip Thiele, 2022. "On the Lower Semicontinuity of the Value Function and Existence of Solutions in Quasiconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 390-417, November.
    16. Xiangkai Sun & Wen Tan & Kok Lay Teo, 2023. "Characterizing a Class of Robust Vector Polynomial Optimization via Sum of Squares Conditions," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 737-764, May.
    17. T. Q. Son & J. J. Strodiot & V. H. Nguyen, 2009. "ε-Optimality and ε-Lagrangian Duality for a Nonconvex Programming Problem with an Infinite Number of Constraints," Journal of Optimization Theory and Applications, Springer, vol. 141(2), pages 389-409, May.
    18. N. Dinh & J. Strodiot & V. Nguyen, 2010. "Duality and optimality conditions for generalized equilibrium problems involving DC functions," Journal of Global Optimization, Springer, vol. 48(2), pages 183-208, October.
    19. William B. Haskell & Alejandro Toriello, 2018. "Modeling Stochastic Dominance as Infinite-Dimensional Constraint Systems via the Strassen Theorem," Journal of Optimization Theory and Applications, Springer, vol. 178(3), pages 726-742, September.
    20. Yldenilson Torres Almeida & João Xavier Cruz Neto & Paulo Roberto Oliveira & João Carlos de Oliveira Souza, 2020. "A modified proximal point method for DC functions on Hadamard manifolds," Computational Optimization and Applications, Springer, vol. 76(3), pages 649-673, July.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:173:y:2017:i:2:d:10.1007_s10957-016-1055-2. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.