IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v251y2017i1d10.1007_s10479-015-1813-9.html
   My bibliography  Save this article

Quasiconvexity of set-valued maps assures well-posedness of robust vector optimization

Author

Listed:
  • Giovanni P. Crespi

    (Università degli Studi dell’Insubria)

  • Daishi Kuroiwa

    (Shimane University)

  • Matteo Rocca

    (Università degli Studi dell’Insubria)

Abstract

Robust optimization is a fast growing methodology to study optimization problems with uncertain data. An uncertain vector optimization problem can be studied through its robust or optimistic counterpart, as in Ben-Tal and Nemirovski (Math Oper Res 23:769–805, 1998) and Beck and Ben-Tal (Oper Res Lett 37: 1–6, 2009). In this paper we formulate the counterparts as set optimization problems. This setting appears to be more natural, especially when the uncertain problem is a non-linear vector optimization problem. Under this setting we study the well-posedness of both the robust and the optimistic counterparts, using the embedding technique for set optimization developed in Kuroiwa and Nuriya (Proceedings of the fourth international conference on nonlinear and convex analysis, pp 297–304, 2006). To prove our main results we also need to study the notion of quasiconvexity for set-valued maps, that is the property of convexity of level set. We provide a general scheme to define the notion of level set and we study the relations among different subsequent definitions of quasi-convexity. We prove some existing notions arise as a special case in the proposed scheme.

Suggested Citation

  • Giovanni P. Crespi & Daishi Kuroiwa & Matteo Rocca, 2017. "Quasiconvexity of set-valued maps assures well-posedness of robust vector optimization," Annals of Operations Research, Springer, vol. 251(1), pages 89-104, April.
    • Handle: RePEc:spr:annopr:v:251:y:2017:i:1:d:10.1007_s10479-015-1813-9
    DOI: 10.1007/s10479-015-1813-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10479-015-1813-9
    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/s10479-015-1813-9?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. G. P. Crespi & A. Guerraggio & M. Rocca, 2007. "Well Posedness in Vector Optimization Problems and Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 132(1), pages 213-226, January.
    2. Ryoichi Nishimura & Shunsuke Hayashi & Masao Fukushima, 2013. "SDP reformulation for robust optimization problems based on nonconvex QP duality," Computational Optimization and Applications, Springer, vol. 55(1), pages 21-47, May.
    3. Joël Benoist & Nicolae Popovici, 2003. "Characterizations of convex and quasiconvex set-valued maps," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 57(3), pages 427-435, August.
    4. G. P. Crespi & M. Papalia & M. Rocca, 2009. "Extended Well-Posedness of Quasiconvex Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 141(2), pages 285-297, May.
    5. A. Ben-Tal & A. Nemirovski, 1998. "Robust Convex Optimization," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 769-805, November.
    6. E. Miglierina & E. Molho & M. Rocca, 2005. "Well-Posedness and Scalarization in Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 126(2), pages 391-409, August.
    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. Kuntal Som & V. Vetrivel, 2021. "On robustness for set-valued optimization problems," Journal of Global Optimization, Springer, vol. 79(4), pages 905-925, April.
    2. Kuntal Som & V. Vetrivel, 2023. "Global well-posedness of set-valued optimization with application to uncertain problems," Journal of Global Optimization, Springer, vol. 85(2), pages 511-539, February.
    3. Giovanni Paolo Crespi & Andreas H. Hamel & Matteo Rocca & Carola Schrage, 2021. "Set Relations via Families of Scalar Functions and Approximate Solutions in Set Optimization," Mathematics of Operations Research, INFORMS, vol. 46(1), pages 361-381, February.
    4. L. Q. Anh & T. Q. Duy & D. V. Hien, 2020. "Well-posedness for the optimistic counterpart of uncertain vector optimization problems," Annals of Operations Research, Springer, vol. 295(2), pages 517-533, December.
    5. Yu Han & Kai Zhang & Nan-jing Huang, 2020. "The stability and extended well-posedness of the solution sets for set optimization problems via the Painlevé–Kuratowski convergence," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(1), pages 175-196, February.
    6. S. Khoshkhabar-amiranloo, 2021. "Scalarization of Multiobjective Robust Optimization Problems," SN Operations Research Forum, Springer, vol. 2(3), pages 1-16, September.
    7. Giovanni P. Crespi & Daishi Kuroiwa & Matteo Rocca, 2020. "Robust Nash equilibria in vector-valued games with uncertainty," Annals of Operations Research, Springer, vol. 289(2), pages 185-193, June.

    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. Meenakshi Gupta & Manjari Srivastava, 2019. "Well-posedness and scalarization in set optimization involving ordering cones with possibly empty interior," Journal of Global Optimization, Springer, vol. 73(2), pages 447-463, February.
    2. Giovanni P. Crespi & Mansi Dhingra & C. S. Lalitha, 2018. "Pointwise and global well-posedness in set optimization: a direct approach," Annals of Operations Research, Springer, vol. 269(1), pages 149-166, October.
    3. C. S. Lalitha & Prashanto Chatterjee, 2012. "Stability for Properly Quasiconvex Vector Optimization Problem," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 492-506, November.
    4. C. Lalitha & Prashanto Chatterjee, 2014. "Levitin–Polyak well-posedness for constrained quasiconvex vector optimization problems," Journal of Global Optimization, Springer, vol. 59(1), pages 191-205, May.
    5. Rocca Matteo & Papalia Melania, 2008. "Well-posedness in vector optimization and scalarization results," Economics and Quantitative Methods qf0807, Department of Economics, University of Insubria.
    6. Giovanni Paolo Crespi & Andreas H. Hamel & Matteo Rocca & Carola Schrage, 2021. "Set Relations via Families of Scalar Functions and Approximate Solutions in Set Optimization," Mathematics of Operations Research, INFORMS, vol. 46(1), pages 361-381, February.
    7. M. Darabi & J. Zafarani, 2015. "Tykhonov Well-Posedness for Quasi-Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 165(2), pages 458-479, May.
    8. Gang Xiao & Hong Xiao & Sanyang Liu, 2011. "Scalarization and pointwise well-posedness in vector optimization problems," Journal of Global Optimization, Springer, vol. 49(4), pages 561-574, April.
    9. L. Q. Anh & T. Q. Duy & D. V. Hien, 2020. "Well-posedness for the optimistic counterpart of uncertain vector optimization problems," Annals of Operations Research, Springer, vol. 295(2), pages 517-533, December.
    10. Onetti Alberto & Verma Sameer, 2008. "Licensing and Business Models," Economics and Quantitative Methods qf0806, Department of Economics, University of Insubria.
    11. Crespi, Giovanni P. & Kuroiwa, Daishi & Rocca, Matteo, 2018. "Robust optimization: Sensitivity to uncertainty in scalar and vector cases, with applications," Operations Research Perspectives, Elsevier, vol. 5(C), pages 113-119.
    12. A. Takeda & S. Taguchi & R. H. Tütüncü, 2008. "Adjustable Robust Optimization Models for a Nonlinear Two-Period System," Journal of Optimization Theory and Applications, Springer, vol. 136(2), pages 275-295, February.
    13. Laura J. Kettner & Sien Deng, 2012. "On Well-Posedness and Hausdorff Convergence of Solution Sets of Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 153(3), pages 619-632, June.
    14. G. P. Crespi & M. Papalia & M. Rocca, 2009. "Extended Well-Posedness of Quasiconvex Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 141(2), pages 285-297, May.
    15. San-hua Wang & Nan-jing Huang & Donal O’Regan, 2013. "Well-posedness for generalized quasi-variational inclusion problems and for optimization problems with constraints," Journal of Global Optimization, Springer, vol. 55(1), pages 189-208, January.
    16. César Gutiérrez & Enrico Miglierina & Elena Molho & Vicente Novo, 2016. "Convergence of Solutions of a Set Optimization Problem in the Image Space," Journal of Optimization Theory and Applications, Springer, vol. 170(2), pages 358-371, August.
    17. Wenqing Chen & Melvyn Sim & Jie Sun & Chung-Piaw Teo, 2010. "From CVaR to Uncertainty Set: Implications in Joint Chance-Constrained Optimization," Operations Research, INFORMS, vol. 58(2), pages 470-485, April.
    18. Stefan Mišković, 2017. "A VNS-LP algorithm for the robust dynamic maximal covering location problem," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 39(4), pages 1011-1033, October.
    19. Chuong, T.D. & Jeyakumar, V., 2017. "Convergent hierarchy of SDP relaxations for a class of semi-infinite convex polynomial programs and applications," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 381-399.
    20. Chassein, André & Dokka, Trivikram & Goerigk, Marc, 2019. "Algorithms and uncertainty sets for data-driven robust shortest path problems," European Journal of Operational Research, Elsevier, vol. 274(2), pages 671-686.

    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:annopr:v:251:y:2017:i:1:d:10.1007_s10479-015-1813-9. 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.