IDEAS home Printed from https://ideas.repec.org/a/spr/mathme/v78y2013i1p1-34.html
   My bibliography  Save this article

Variational convergence for vector-valued functions and its applications to convex multiobjective optimization

Author

Listed:
  • Rubén López

Abstract

The aim of this work is to study a notion of variational convergence for vector-valued functions. We show that it is suitable for obtaining existence and stability results in convex multiobjective optimization. We obtain various of properties of the variational convergence. We characterize it via the set convergence of epigraphs, coepigraphs, level sets, and some infima. We also characterize it by means of two metrics. We compare it with other notions of convergence for vector-valued functions from the literature and we show that it is more general than most of them. For obtaining the existence and stability results we employ an asymptotic method that has shown to be very useful in optimization theory. In this method we couple the variational convergence with notions of asymptotic analysis (asymptotic cones and functions). Copyright Springer-Verlag Berlin Heidelberg 2013

Suggested Citation

  • Rubén López, 2013. "Variational convergence for vector-valued functions and its applications to convex multiobjective optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 78(1), pages 1-34, August.
    • Handle: RePEc:spr:mathme:v:78:y:2013:i:1:p:1-34
    DOI: 10.1007/s00186-013-0430-0
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s00186-013-0430-0
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s00186-013-0430-0?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. S. Deng, 1998. "On Efficient Solutions in Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 96(1), pages 201-209, January.
    2. Regina S. Burachik & Alfredo N. Iusem, 2008. "Set-Valued Mappings and Enlargements of Monotone Operators," Springer Optimization and Its Applications, Springer, number 978-0-387-69757-4, December.
    3. X. X. Huang, 2000. "Stability in vector-valued and set-valued optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 52(2), pages 185-193, November.
    4. Regina S. Burachik & Alfredo N. Iusem, 2008. "Enlargements of Monotone Operators," Springer Optimization and Its Applications, in: Set-Valued Mappings and Enlargements of Monotone Operators, chapter 0, pages 161-220, Springer.
    5. S. W. Xiang & W. S. Yin, 2007. "Stability Results for Efficient Solutions of Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 134(3), pages 385-398, September.
    Full references (including those not matched with items on IDEAS)

    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. Huynh Van Ngai & Nguyen Huu Tron & Michel Théra, 2014. "Metric Regularity of the Sum of Multifunctions and Applications," Journal of Optimization Theory and Applications, Springer, vol. 160(2), pages 355-390, February.
    2. Dawan Chumpungam & Panitarn Sarnmeta & Suthep Suantai, 2021. "A New Forward–Backward Algorithm with Line Searchand Inertial Techniques for Convex Minimization Problems with Applications," Mathematics, MDPI, vol. 9(13), pages 1-20, July.
    3. Walaa M. Moursi & Lieven Vandenberghe, 2019. "Douglas–Rachford Splitting for the Sum of a Lipschitz Continuous and a Strongly Monotone Operator," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 179-198, October.
    4. Sedi Bartz & Minh N. Dao & Hung M. Phan, 2022. "Conical averagedness and convergence analysis of fixed point algorithms," Journal of Global Optimization, Springer, vol. 82(2), pages 351-373, February.
    5. Bello Cruz, J.Y. & Iusem, A.N., 2015. "Full convergence of an approximate projection method for nonsmooth variational inequalities," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 114(C), pages 2-13.
    6. Regina S. Burachik & Minh N. Dao & Scott B. Lindstrom, 2021. "Generalized Bregman Envelopes and Proximity Operators," Journal of Optimization Theory and Applications, Springer, vol. 190(3), pages 744-778, September.
    7. Warunun Inthakon & Suthep Suantai & Panitarn Sarnmeta & Dawan Chumpungam, 2020. "A New Machine Learning Algorithm Based on Optimization Method for Regression and Classification Problems," Mathematics, MDPI, vol. 8(6), pages 1-17, June.
    8. Jonathan M. Borwein & Liangjin Yao, 2013. "Structure Theory for Maximally Monotone Operators with Points of Continuity," Journal of Optimization Theory and Applications, Springer, vol. 157(1), pages 1-24, April.
    9. Ludovic Nagesseur, 2016. "A bundle method using two polyhedral approximations of the $$\varepsilon $$ ε -enlargement of a maximal monotone operator," Computational Optimization and Applications, Springer, vol. 64(1), pages 75-100, May.
    10. Juan Pablo Luna & Claudia Sagastizábal & Mikhail Solodov, 2020. "A class of Benders decomposition methods for variational inequalities," Computational Optimization and Applications, Springer, vol. 76(3), pages 935-959, July.
    11. Edvaldo E. A. Batista & Glaydston de Carvalho Bento & Orizon P. Ferreira, 2016. "Enlargement of Monotone Vector Fields and an Inexact Proximal Point Method for Variational Inequalities in Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 170(3), pages 916-931, September.
    12. J. Y. Bello Cruz & R. Díaz Millán, 2016. "A relaxed-projection splitting algorithm for variational inequalities in Hilbert spaces," Journal of Global Optimization, Springer, vol. 65(3), pages 597-614, July.
    13. Heinz H. Bauschke & Warren L. Hare & Walaa M. Moursi, 2016. "On the Range of the Douglas–Rachford Operator," Mathematics of Operations Research, INFORMS, vol. 41(3), pages 884-897, August.
    14. Regina S. Burachik & C. Yalçın Kaya & Shoham Sabach, 2012. "A Generalized Univariate Newton Method Motivated by Proximal Regularization," Journal of Optimization Theory and Applications, Springer, vol. 155(3), pages 923-940, December.
    15. Hsien-Chung Wu, 2018. "Near Fixed Point Theorems in Hyperspaces," Mathematics, MDPI, vol. 6(6), pages 1-15, May.
    16. Walaa M. Moursi, 2018. "The Forward–Backward Algorithm and the Normal Problem," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 605-624, March.
    17. Henri Bonnel & Julien Collonge, 2014. "Stochastic Optimization over a Pareto Set Associated with a Stochastic Multi-Objective Optimization Problem," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 405-427, August.
    18. Hsien-Chung Wu, 2019. "Informal Norm in Hyperspace and Its Topological Structure," Mathematics, MDPI, vol. 7(10), pages 1-20, October.
    19. Phan Vuong & Jean Strodiot & Van Nguyen, 2014. "Projected viscosity subgradient methods for variational inequalities with equilibrium problem constraints in Hilbert spaces," Journal of Global Optimization, Springer, vol. 59(1), pages 173-190, May.
    20. Dawan Chumpungam & Panitarn Sarnmeta & Suthep Suantai, 2022. "An Accelerated Convex Optimization Algorithm with Line Search and Applications in Machine Learning," Mathematics, MDPI, vol. 10(9), pages 1-20, April.

    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:mathme:v:78:y:2013:i:1:p:1-34. 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.