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

Author

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  • 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
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    References listed on IDEAS

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    1. A. Ben-Tal & A. Nemirovski, 1998. "Robust Convex Optimization," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 769-805, November.
    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. 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.
    4. 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.
    5. 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.
    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.
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    Citations

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    Cited by:

    1. 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.
    2. 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.
    3. S. Khoshkhabar-amiranloo, 2021. "Scalarization of Multiobjective Robust Optimization Problems," SN Operations Research Forum, Springer, vol. 2(3), pages 1-16, September.
    4. Kuntal Som & V. Vetrivel, 2021. "On robustness for set-valued optimization problems," Journal of Global Optimization, Springer, vol. 79(4), pages 905-925, April.
    5. 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.
    6. 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.
    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.

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