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Well-posedness and convexity in vector optimization

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  • E. Miglierina
  • E. Molho

Abstract

We study a notion of well-posedness in vector optimization through the behaviour of minimizing sequences of sets, defined in terms of Hausdorff set-convergence. We show that the notion of strict efficiency is related to the notion of well-posedness. Using the obtained results we identify a class of well-posed vector optimization problems: the convex problems with compact efficient frontiers. Copyright Springer-Verlag 2003

Suggested Citation

  • E. Miglierina & E. Molho, 2003. "Well-posedness and convexity in vector optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 58(3), pages 375-385, December.
    • Handle: RePEc:spr:mathme:v:58:y:2003:i:3:p:375-385
    DOI: 10.1007/s001860300310
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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. Miglierina Enrico & Molho Elena & Patrone Fioravante & Steff H. Tijs, 2005. "An axiomatic approach to approximate solutions in vector optimization," Economics and Quantitative Methods qf0507, Department of Economics, University of Insubria.
    3. 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.
    4. E. Miglierina & E. Molho & F. Patrone & S. Tijs, 2008. "Axiomatic approach to approximate solutions in multiobjective optimization," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 31(2), pages 95-115, November.
    5. 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.
    6. 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.
    7. Xian-Jun Long & Jian-Wen Peng & Zai-Yun Peng, 2015. "Scalarization and pointwise well-posedness for set optimization problems," Journal of Global Optimization, Springer, vol. 62(4), pages 763-773, August.
    8. Ya-ping Fang & Nan-jing Huang, 2007. "Increasing-along-rays property, vector optimization and well-posedness," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(1), pages 99-114, February.
    9. Onetti Alberto & Verma Sameer, 2008. "Licensing and Business Models," Economics and Quantitative Methods qf0806, Department of Economics, University of Insubria.
    10. 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.
    11. 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.
    12. 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.
    13. 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.
    14. Rocca Matteo & Papalia Melania, 2008. "Well-posedness in vector optimization and scalarization results," Economics and Quantitative Methods qf0807, Department of Economics, University of Insubria.
    15. 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.
    16. M. Bianchi & G. Kassay & R. Pini, 2009. "Well-posedness for vector equilibrium problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 70(1), pages 171-182, August.

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