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On Second-Order Proto-Differentiability of Perturbation Maps

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Abstract

In this paper, second-order sensitivity analysis in vector optimization problems is considered. We prove that the efficient solution map and the efficient frontier map of a parameterized vector optimization problem are second-order proto-differentiable under some appropriate qualification conditions. Some sufficient conditions for inner and outer approximation of the second-order proto-derivative are also provided.

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References

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

    MATH  Google Scholar 

  2. Bonnans, J.F., Shapiro, A.: Pertubation Analysis of Optimization Problem. Springer, New York (2000)

    Book  Google Scholar 

  3. Chuong, T.D.: Derivatives of the efficient point multifunction in parametric vector optimization problems. J. Optim. Theory Appl. 156, 247–265 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chuong, T.D., Yao, J.-C.: Generalized Clarke epiderivatives of parametric vector optimization problems. J. Optim. Theory Appl. 146, 77–94 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Durea, M., Strugariu, R.: Calculus of tangent sets and derivatives of set-valued maps under metric subregularity conditions. J. Glob. Optim. 56, 587–603 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Huy, N.Q., Lee, G.M.: Sensitivity of solutions to a parametric generalized equation. Set-Valued Anal. 16, 805–820 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kalashnikov, V., Jadamba, B., Khan, A.A.: First and second-order optimality conditions in set optimization. In: Dempe, S., Kalashnikov, V. (eds.) Optimization with Multivalued Mappings, pp 265–276 (2006)

  8. Kuk, H., Tanino, T., Tanaka, M.: Sensitivity analysis in vector optimization. J. Optim. Theory Appl. 89, 713–730 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kuk, H., Tanino, T., Tanaka, M.: Sensitivity analysis in parameterized convex vector optimization. J. Math. Anal. Appl. 202, 511–522 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lee, G.M., Huy, N.Q.: On proto-differentiablility of generalized perturbation maps. J. Math. Anal. Appl. 324, 1297–1309 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lee, G.M., Huy, N.Q.: On sensitivity analysis in vector optimization. Taiwan. J. Math. 11, 945–958 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Levy, A.B., Poliquin, R.A., Rockafellar, R.T.: Stability of locally optimal solutions. SIAM J. Optim. 10, 580–604 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Levy, A.B., Rockafellar, R.T.: Variational conditions and the proto-differentiation of partial subgradient mappings. Nonlinear Anal. 26, 1951–1964 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, S.J., Teo, K.L., Yang, X.Q.: Higher-order optimality conditions for set-valued optimization. J. Optim. Theory Appl. 137, 533–553 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, S.J., Liao, C.M.: Second-order differentiability of generalized perturbation maps. J. Global Optim. 52, 243–252 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, S.J., Sun, X.K., Zhai, J.: Second-order contingent derivatives of set-valued mappings with application to set-valued optimization. Appl. Math. Comput. 218, 6874–6886 (2012)

    MathSciNet  MATH  Google Scholar 

  18. Li, S.J., Zhu, S.K., Li, X.B.: Second-order optimality conditions for strict efficiency of constrained set-valued optimization. J. Optim. Theory Appl. 155, 534–557 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, vol. I: Basic Theory. Springer, Berlin (2006)

    Google Scholar 

  20. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, vol. II: Applications. Springer, Berlin (2006)

    Google Scholar 

  21. Mordukhovich, B.S., Outrata, J.V.: On second-order subdifferentials and their applications. SIAM J. Optim. 12, 139–169 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mordukhovich, B.S., Rockafellar, R.T.: Second-order subdifferential calculus with applications to tilt stability in optimization. SIAM J. Optim. 22, 953–986 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mordukhovich, B.S., Rockafellar, R.T., Sarabi, M.E.: Characterizations of full stability in constrained optimization. SIAM J. Optim. 23, 1810–1849 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mordukhovich, B.S., Nghia, T.T.A., Rockafellar, R.T.: Full stability in finite-dimensional optimization. Math. Oper. Res. 40, 226–252 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Penot, J.P.: Second-order conditions for optimization problems with constraints. SIAM J. Control Optim. 37, 303–318 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. Poliquin, R.A., Rockafellar, R.T.: Tilt stability of a local minimum. SIAM J. Optim. 8, 287–299 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ren, J.J., Sen, P.K.: Second-order Hadamard differentiability in statistical applications. J. Multivariate Anal. 77, 187–228 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, 3rd edn. Springer, Berlin (2009)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  31. Shi, D.S: Contingent derivative of the perturbation map in multiobjective optimization. J. Optim. Theory Appl. 70, 385–396 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  32. Shi, D.S: Sensitivity analysis in convex vector optimization. J. Optim. Theory Appl. 77, 145–159 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tanino, T.: Sensitivity analysis in multiobjective optimization. J. Optim. Theory Appl. 56, 479–499 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tanino, T.: Stability and sensitivity analysis in convex vector optimization. SIAM J. Control Optim. 26, 521–536 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  35. Tung, L.T.: Second-order radial-asymptotic derivatives and applications in set-valued vector optimization. Pacific J. Optim., accepted for publication

  36. Wang, Q.L., Li, S.J.: Sensitivity and stability for the second-order contingent derivative of the proper perturbation map in vector optimization. Optim. Lett. 6, 731–748 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, Q.L., Li, S.J.: Second-order contingent derivative of the pertubation map in multiobjective optimization. Fixed Point Theory Appl. 2011. Article ID 857520 (2011)

  38. Ward, D.: A chain rule for first and second order epiderivatives and hypoderivatives. J. Math. Anal. Appl. 348, 324–336 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, College of Natural Sciences, Cantho University, Cantho, Vietnam

    L. T. Tung

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  1. L. T. Tung
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Tung, L.T. On Second-Order Proto-Differentiability of Perturbation Maps. Set-Valued Var. Anal 26, 561–579 (2018). https://doi.org/10.1007/s11228-016-0397-0

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  • DOI: https://doi.org/10.1007/s11228-016-0397-0

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