Second-Order Conditions for Open-Cone Minimizers and Firm Minimizers in Set-Valued Optimization Subject to Mixed Constraints

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Abstract

We consider second-order optimality conditions for set-valued optimization problems subject to mixed constraints. Such optimization models are useful in a wide range of practical applications. By using several kinds of derivatives, we obtain second-order necessary conditions for local Q-minimizers and local firm minimizers with attention to the envelope-like effect. Under the second-order Abadie constraint qualification, we get stronger necessary conditions. When the second-order Kurcyusz–Robinson–Zowe constraint qualification is imposed, our multiplier rules are of the Karush–Kuhn–Tucker type. Sufficient conditions for firm minimizers are established without any convexity assumptions. As an application, we extend and improve some recent existing results for nonsmooth mathematical programming.

Keywords

Set-valued optimization Abadie constraint qualification Kurcyusz–Robinson–Zowe constraint qualification Open-cone minimizer Firm minimizer 

Mathematics Subject Classification

90C29 49J52 90C46 90C48 

References

  1. 1.
    Abadie, J.: On the Kuhn–Tucker theorem in Nonlinear Programming (NATO Summer School, Menton, 1964). North-Holland, Amsterdam (1967)Google Scholar
  2. 2.
    Bannans, J.F., Cominetti, R., Shapiro, A.: Second order optimality conditions based on parabolic second order tangent sets. SIAM J. Optim. 9, 466–492 (1999)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cominetti, R.: Metric regularity, tangent sets, and second-order optimality conditions. Appl. Math. Optim. 21, 265–287 (1990)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Gfrerer, H.: On directional metric regularity, subregularity and optimality conditions for nonsmooth mathematical programs. Set-Valued Var. Anal. 21, 151–176 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Gfrerer, H.: On directional metric subregularity and second-order optimality conditions for a class of nonsmooth mathematical programs. SIAM J. Optim. 23, 632–665 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Kawasaki, H.: An envelope-like effect of infinitely many inequality constraints on second order necessary conditions for minimization problems. Math. Program. 41, 73–96 (1988)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Penot, J.P.: Second order conditions for optimization problems with constraints. SIAM J. Control Optim. 37, 303–318 (1999)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gutiérrez, C., Jiménez, B., Novo, N.: On second order Fritz John type optimality conditions in nonsmooth multiobjective programming. Math. Program. Ser. B 123, 199–223 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Jiménez, B., Novo, V.: Optimality conditions in differentiable vector optimization via second-order tangent sets. Appl. Math. Optim. 49, 123–144 (2004)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Khanh, P.Q., Tuan, N.D.: Optimality conditions for nonsmooth multiobjective optimization using Hadamard directional derivatives. J. Optim. Theory Appl. 133, 341–357 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Khanh, P.Q., Tuan, N.D.: Second order optimality conditions with the envelope-like effect in nonsmooth multiobjective programming II: optimality conditions. J. Math. Anal. Appl. 403, 703–714 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Khanh, P.Q., Tuan, N.D.: Second-order optimality conditions with the envelope-like effect for nonsmooth vector optimization in infinite dimensions. Nonlinear Anal. 77, 130–148 (2013)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Taa, A.: Second order conditions for nonsmooth multiobjective optimization problems with inclusion constraints. J. Glob. Optim. 50, 271–291 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Zhu, S., Li, S.: Optimality conditions of strict minimality in optimization problems under inclusion constraints. Appl. Math. Comput. 219, 4816–4825 (2013)MathSciNetMATHGoogle Scholar
  15. 15.
    Durea, M.: First and second order Lagrange claims for set-valued maps. J. Optim. Theory Appl. 133, 111–116 (2007)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Jahn, J., Khan, A.A., Zeilinger, P.: Second-order optimality conditions in set optimization. J. Optim. Theory Appl. 125, 331–347 (2005)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    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)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Khan, A.A., Tammer, C.: Second-order optimality conditions in set-valued optimization via asymptotic derivatives. Optimization 62, 743–758 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Zhu, S.K., Li, S.J., Teo, K.L.: Second-order Karush–Kuhn–Tucker optimality conditions for set-valued optimization. J. Glob. Optim. 58, 673–679 (2014)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Khanh, P.Q., Tung, N.M.: First and second-order optimality conditions without differentiability in multivalued vector optimization. Positivity 19, 817–841 (2015)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Khanh, P.Q., Tung, N.M.: Second-order optimality conditions with the envelope-like effect for set-valued optimization. J. Optim. Theory Appl. 167, 68–90 (2015)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Studniarski, M.: Necessary and sufficient conditions for isolated local minima of nonsmooth functions. SIAM J. Control Optim. 24, 1044–1049 (1986)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Jiménez, B.: Strict efficiency in vector optimization. J. Math. Anal. Appl. 265, 264–284 (2002)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Flores-Bazán, F., Jiménez, B.: Strict efficiency in set-valued optimization. SIAM J. Control Optim. 48, 881–908 (2009)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Penot, J.P.: Differentiability of relations and differential stability of perturbed optimization problems. SIAM J. Control Optim. 22, 529–551 (1984)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Ward, D.E.: A chain rule for first and second order epiderivatives and hypoderivatives. J. Math. Anal. Appl. 348, 324–336 (2008)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Khan, A.A., Tammer, C., Zalinescu, C.: Set-Valued Optimization: An Introduction with Application. Springer, Berlin (2014)MATHGoogle Scholar
  28. 28.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhauser, Boston (1990)MATHGoogle Scholar
  29. 29.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Applications, vol. II. Springer, Berlin (2006)Google Scholar
  30. 30.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, 3rd edn. Springer, Berlin (2009)MATHGoogle Scholar
  31. 31.
    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, Berlin (2009)CrossRefMATHGoogle Scholar
  32. 32.
    Khanh, P.Q., Kruger, A.K., Thao, N.H.: An induction theorem and nonlinear regularity models. SIAM J. Optim. 25, 2561–2588 (2015)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Ha, T.X.D.: Optimality conditions for several types of efficient solutions of set-valued optimization problems. In: Pardalos, P., Rassis, ThM, Khan, A.A. (eds.) Chapter 21, Nonlinear Analysis and Variational Problems, pp. 305–324. Springer, Heidelberg (2009)Google Scholar
  34. 34.
    Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, New Jersey/London (2002)CrossRefMATHGoogle Scholar
  35. 35.
    Robinson, S.M.: Regularity and stability for convex multivalued functions. Math. Oper. Res. 1, 130–143 (1976)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Ursescu, C.: Multifunctions with closed convex graph. Czechoslov. Math. J. 25, 438–441 (1975)MathSciNetMATHGoogle Scholar
  37. 37.
    Jahn, J.: Introduction to the Theory of Nonlinear Optimization, 2nd edn. Springer, Berlin (1996)CrossRefMATHGoogle Scholar
  38. 38.
    Robinson, S.M.: Stability theorems for systems of inequalities. Part II: Differentiable nonlinear systems. SIAM J. Numer. Anal. 13, 497–513 (1976)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsInternational University, Vietnam National University Hochiminh CityHochiminh CityVietnam
  2. 2.Department of Mathematics and ComputingUniversity of Science, Vietnam National University Hochiminh CityHochiminh CityVietnam

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