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Convergence of Solutions of a Set Optimization Problem in the Image Space

Journal of Optimization Theory and Applications volume 170, pages 358–371 (2016)Cite this article

Abstract

The present work is devoted to the study of stability in set optimization. In particular, a sequence of perturbed set optimization problems, with a fixed objective map, is studied under suitable continuity assumptions. A formulation of external and internal stability of the solutions is considered in the image space, in such a way that the convergence of a sequence of solutions of perturbed problems to a solution of the original problem is studied under appropriate compactness assumptions. Our results can also be seen as an extension to the set-valued framework of known stability results in vector optimization.

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References

  1. Chen, G.Y., Huang, X., Yang, X.: Vector optimization. Set-valued and variational analysis. Springer, Berlin (2005)

  2. Jahn, J.: Vector Optimization. Theory, Applications, and Extensions. Springer, Berlin (2011)

    MATH  Google Scholar 

  3. Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989)

    Book  Google Scholar 

  4. Kuroiwa, D.: Some duality theorems of set-valued optimization with natural criteria. In: Takahashi, W., Tanaka, T. (eds.) Nonlinear Analysis and Convex Analysis, pp. 221–228. World Scientific Publishing, River Edge (1999)

  5. Kuroiwa, D.: On set-valued optimization. Nonlinear Anal. 47, 1395–1400 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ide, J., Köbis, E., Kuroiwa, D., Schöbel, A., Tammer, C.: The relationship between multi-objective robustness concepts and set-valued optimization. Fixed Point Theory Appl. 83, 1–20 (2014)

    MathSciNet  MATH  Google Scholar 

  7. Ehrgott, M., Ide, J., Schöbel, A.: Minmax robustness for multi-objective optimization problems. Eur. J. Oper. Res. 239, 17–31 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ide, J., Köbis, E.: Concepts of efficiency for uncertain multi-objective optimization problems based on set order relations. Math. Methods Oper. Res. 80, 99–127 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kuroiwa, D., Tanaka, T., Ha, T.X.D.: On cone convexity of set-valued maps. Nonlinear Anal. 30, 1487–1496 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gutiérrez, C., Jiménez, B., Miglierina, E., Molho, E.: Scalarization in set optimization with solid and nonsolid ordering cones. J. Global Optim. 61, 525–552 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gutiérrez, C., Jiménez, B., Novo, V., Thibault, L.: Strict approximate solutions in set-valued optimization with applications to the approximate Ekeland variational principle. Nonlinear Anal. 73, 3842–3855 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gutiérrez, C., Miglierina, E., Molho, E., Novo, V.: Pointwise well-posedness in set optimization with cone proper sets. Nonlinear Anal. 75, 1822–1833 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hamel, A., Löhne, A.: Minimal element theorems and Ekeland’s principle with set relations. J. Nonlinear Convex Anal. 7, 19–37 (2006)

    MathSciNet  MATH  Google Scholar 

  14. Hernández, E., Rodríguez-Marín, L.: Existence theorems for set optimization problems. Nonlinear Anal. 67, 1726–1736 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hernández, E., Rodríguez-Marín, L.: Nonconvex scalarization in set optimization with set-valued maps. J. Math. Anal. Appl. 325, 1–18 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jahn, J., Ha, T.X.D.: New order relations in set optimization. J. Optim. Theory Appl. 148, 209–236 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bednarczuk, E.: Stability Analysis for Parametric Vector Optimization Problems. Dissertationes Math. 442, Institute of Mathematics, Polish Academy of Sciences, Warsaw (2007)

  18. Luc, D.T., Lucchetti, R., Malivert, C.: Convergence of the efficient sets. Set-Valued Anal. 2, 207–218 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lucchetti, R., Miglierina, E.: Stability for convex vector optimization problems. Optimization 53, 517–528 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Miglierina, E., Molho, E.: Convergence of minimal sets in convex vector optimization. SIAM J. Optim. 15, 513–526 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Penot, J.-P., Sterna-Karwat, A.: Parametrized multicriteria optimization: continuity and closedness of optimal multifunctions. J. Math. Anal. Appl. 120, 150–168 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  22. Xu, Y.D., Li, Y.D.: Continuity of the solution set mappings to a parametric set optimization problem. Optim. Lett. 8, 2315–2327 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Giannessi, F.: Constrained optimization and image space analysis. Vol. 1: separation of sets and optimality conditions. Springer, New York (2005)

  24. Loridan, P., Morgan, J., Raucci, R.: Convergence of minimal and approximate minimal elements of sets in partially ordered vector spaces. J. Math. Anal. Appl. 239, 427–439 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Göpfert, A., Rihai, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, Berlin (2003)

    MATH  Google Scholar 

  26. Crespi, G.P., Papalia, M., Rocca, M.: Extended well-posedness of quasiconvex vector optimization problems. J. Optim. Theory Appl. 141, 285–297 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kutateladze, S.S.: Convex \(\varepsilon \)-programming. Soviet Math. Dokl. 20, 301–303 (1979)

    MathSciNet  Google Scholar 

  28. Crespi, G.P., Guerraggio, A., Rocca, M.: Well posedness in vector optimization problems and vector variational inequalities. J. Optim. Theory Appl. 132, 213–226 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Li, S.J., Zhang, W.Y.: Hadamard well-posed vector optimization problems. J. Global Optim. 46, 383–393 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This research was partially supported by the Ministerio de Economía y Competitividad (Spain) under project MTM2012-30942. The third author was also partially supported by MIUR PRIN MISURA Project, 2013-2015, Italy. The authors would like to thank the Editor and the anonymous referee for their useful comments.

Author information

Authors and Affiliations

  1. Universidad de Valladolid, Paseo de Belén 15, Campus Miguel Delibes, 47011, Valladolid, Spain

    César Gutiérrez

  2. Università Cattolica del Sacro Cuore, Via Necchi 9, 20123, Milan, Italy

    Enrico Miglierina

  3. Università degli Studi di Pavia, Via S. Felice 5, 27100, Pavia, Italy

    Elena Molho

  4. Universidad Nacional de Educación a Distancia (UNED), c/ Juan del Rosal 12, Ciudad Universitaria, 28040, Madrid, Spain

    Vicente Novo

Authors
  1. César Gutiérrez
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  2. Enrico Miglierina
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  3. Elena Molho
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  4. Vicente Novo
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Corresponding author

Correspondence to Vicente Novo.

Additional information

Communicated by Lionel Thibault.

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Gutiérrez, C., Miglierina, E., Molho, E. et al. Convergence of Solutions of a Set Optimization Problem in the Image Space. J Optim Theory Appl 170, 358–371 (2016). https://doi.org/10.1007/s10957-016-0942-x

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  • DOI: https://doi.org/10.1007/s10957-016-0942-x

Keywords

  • Set optimization
  • Set relations
  • Minimal solutions
  • Stability
  • Set convergence

Mathematics Subject Classification

  • 49J53
  • 90C31
  • 90C29