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Extended and strongly extended well-posedness of set-valued optimization problems

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  • X. X. Huang

Abstract

In this paper, we deal with the extended well-posedness and strongly extended well-posedness of set-valued optimization problems. These two concepts are generalizations of the extended well-posedness of real-valued optimization probems defined by Zolezzi. We obtain some criteria and characterizations of these two types of extended well-posedness, further generalizing most results obtained by Zolezzi for the extended well-posedness of scalar optimization problems. In the mean time, many results obtained by us for the extended well-posedness of vector optimization problems have been generalized to set-valued optimization. Finally, we present an approximate variational principle for set-valued maps, derive a necessary approximate optimality condition for set-valued optimization, based on which we introduce a condition, which is somewhat analogous to the Palais-Smale condition (C), and provide sufficient conditions for the extended and strongly extended well-posedness of set-valued optimization problems. Copyright Springer-Verlag Berlin Heidelberg 2001

Suggested Citation

  • X. X. Huang, 2001. "Extended and strongly extended well-posedness of set-valued optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 53(1), pages 101-116, April.
    • Handle: RePEc:spr:mathme:v:53:y:2001:i:1:p:101-116
    DOI: 10.1007/s001860000100
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    Citations

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    Cited by:

    1. Fang, Ya-Ping & Huang, Nan-Jing & Yao, Jen-Chih, 2010. "Well-posedness by perturbations of mixed variational inequalities in Banach spaces," European Journal of Operational Research, Elsevier, vol. 201(3), pages 682-692, March.
    2. Savin Treanţă, 2021. "On Well-Posedness of Some Constrained Variational Problems," Mathematics, MDPI, vol. 9(19), pages 1-12, October.
    3. Onetti Alberto & Verma Sameer, 2008. "Licensing and Business Models," Economics and Quantitative Methods qf0806, Department of Economics, University of Insubria.
    4. Elisa Mastrogiacomo & Matteo Rocca, 2021. "Set optimization of set-valued risk measures," Annals of Operations Research, Springer, vol. 296(1), pages 291-314, January.
    5. S. Khoshkhabar-amiranloo & E. Khorram, 2015. "Pointwise well-posedness and scalarization in set optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 82(2), pages 195-210, October.
    6. X. J. Long & J. W. Peng, 2013. "Generalized B-Well-Posedness for Set Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 157(3), pages 612-623, June.
    7. Ya-Ping Fang & Rong Hu, 2007. "Estimates of approximate solutions and well-posedness for variational inequalities," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(2), pages 281-291, April.
    8. Mircea Sofonea & Yi-bin Xiao, 2021. "Well-Posedness of Minimization Problems in Contact Mechanics," Journal of Optimization Theory and Applications, Springer, vol. 188(3), pages 650-672, March.
    9. Mircea Sofonea & Yi-bin Xiao, 2019. "On the Well-Posedness Concept in the Sense of Tykhonov," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 139-157, October.
    10. 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.
    11. Miglierina Enrico & Molho Elena & Rocca Matteo, 2004. "Well-posedness and scalarization in vector optimization," Economics and Quantitative Methods qf0403, Department of Economics, University of Insubria.
    12. Yi-bin Xiao & Nan-jing Huang, 2011. "Well-posedness for a Class of Variational–Hemivariational Inequalities with Perturbations," Journal of Optimization Theory and Applications, Springer, vol. 151(1), pages 33-51, October.
    13. 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.
    14. Y. P. Fang & R. Hu & N. J. Huang, 2007. "Extended B-Well-Posedness and Property (H) for Set-Valued Vector Optimization with Convexity," Journal of Optimization Theory and Applications, Springer, vol. 135(3), pages 445-458, December.
    15. 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.
    16. Rocca Matteo & Papalia Melania, 2008. "Well-posedness in vector optimization and scalarization results," Economics and Quantitative Methods qf0807, Department of Economics, University of Insubria.
    17. 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.

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