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Polar Conic Set-Valued Map in Vector Optimization. Continuity and Derivability

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Abstract

In the context of vector optimization, several results are stated mainly about the continuity and the derivability of a conic set-valued map (the polar conic function) having a close relation with the positive efficient points, the ideal points and other distinguished elements of the efficient line. The contingent cone to the set of the general positive quasiefficient points at a point x 0 is also related with the frontier of the dual cone of the image at x 0 of the polar conic function.

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References

  1. Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Academic Press, New York (1985)

    MATH  Google Scholar 

  2. Wierzbicki, A.P.: Reference point approaches. Multicriteria decision making. In: Gal, T., Steward, T.J., Hanne, T. (eds.) Internat. Ser. Oper. Res. Management Sci., vol. 21. Kluwer Academic, Boston (1999)

    Google Scholar 

  3. Gearhart, W.B.: Compromise solutions and estimation of the noninferior set. J. Optim. Theory Appl. 28, 29–47 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  4. Yu, P.L., Leitmann, G.: Compromise solutions, domination structures, and Salukvadze’s solution. Differential games: a collection of articles. J. Optim. Theory Appl. 13, 362–378 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  5. Balbás, A., Ballvé, M.E., Jiménez Guerra, P.: Density theorems for ideal points in vector optimization. Eur. J. Oper. Res. 133, 260–266 (2001)

    Article  MATH  Google Scholar 

  6. Ballvé, M.E., Jiménez Guerra, P.: Some geometrical aspects of the efficient line in vector optimization. Eur. J. Oper. Res. 162, 497–502 (2005)

    Article  MATH  Google Scholar 

  7. Jiménez Guerra, P., Melguizo, M.A., Muñoz-Bouzo, M.J.: Conic set-valued maps in vector optimization. Set-Valued Anal. 15, 47–59 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gutev, V.G.: Weak factorizations of continuous set-valued mappings. Topol. Appl. 102, 33–51 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Michael, E.: Continuous selections, III. Ann. Math. 65, 375–390 (1957)

    Article  MathSciNet  Google Scholar 

  10. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  11. Shaefer, H.H.: Topological Vector Spaces. Springer, Berlin (1971)

    Google Scholar 

  12. Rockafellar, R.T.: Proto-differentiability of set-valued mappings and its applications in optimization. Ann. Inst. Henri Poincaré. Anal. Non Linéaire 6, 449–482 (1989)

    MathSciNet  Google Scholar 

  13. Levy, A.B., Rockafellar, R.T.: Sensitivity analysis of solutions to generalized equations. Trans. Am. Math. Soc. 345(2), 661–671 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  14. Poliquin, R.A., Rockafellar, R.T.: Proto-derivative formulas for basic subgradient mappings in mathematical programming. Set convergence in nonlinear analysis and optimization. Set-Valued Anal. 2(1–2), 275–290 (1994)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  16. Jahn, J.: Vector Optimization. Springer, Berlin (2004)

    MATH  Google Scholar 

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Correspondence to P. Jiménez Guerra.

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This work was partially supported by Grant SEJ2006–15401–C04–02 of Spanish Ministerio de Ciencia y Tecnología and Grant S-0505/tic/0230-D3 of Comunidad Autónoma de Madrid. The authors are grateful to the referees for suggestions which led to improving the paper.

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Jiménez Guerra, P., Melguizo, M.A. & Muñoz-Bouzo, M.J. Polar Conic Set-Valued Map in Vector Optimization. Continuity and Derivability. J Optim Theory Appl 142, 343–354 (2009). https://doi.org/10.1007/s10957-009-9542-3

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