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

  • Phan Quoc Khanh
  • Nguyen Minh Tung


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.


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

Mathematics Subject Classification

90C29 49J52 90C46 90C48 



This work was supported by the Vietnam National University Hochiminh City under Grant Number B2015-28-03. A part of the work was completed during a scientific stay of the authors at Vietnam Institute for Advance Study in Mathematics (VIASM), whose hospitality is appreciated. The authors are very grateful to the editors and referees for their valuable comments and suggestions.


  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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cominetti, R.: Metric regularity, tangent sets, and second-order optimality conditions. Appl. Math. Optim. 21, 265–287 (1990)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Penot, J.P.: Second order conditions for optimization problems with constraints. SIAM J. Control Optim. 37, 303–318 (1999)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Taa, A.: Second order conditions for nonsmooth multiobjective optimization problems with inclusion constraints. J. Glob. Optim. 50, 271–291 (2011)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Durea, M.: First and second order Lagrange claims for set-valued maps. J. Optim. Theory Appl. 133, 111–116 (2007)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Khan, A.A., Tammer, C.: Second-order optimality conditions in set-valued optimization via asymptotic derivatives. Optimization 62, 743–758 (2013)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Studniarski, M.: Necessary and sufficient conditions for isolated local minima of nonsmooth functions. SIAM J. Control Optim. 24, 1044–1049 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jiménez, B.: Strict efficiency in vector optimization. J. Math. Anal. Appl. 265, 264–284 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Flores-Bazán, F., Jiménez, B.: Strict efficiency in set-valued optimization. SIAM J. Control Optim. 48, 881–908 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Penot, J.P.: Differentiability of relations and differential stability of perturbed optimization problems. SIAM J. Control Optim. 22, 529–551 (1984)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Khan, A.A., Tammer, C., Zalinescu, C.: Set-Valued Optimization: An Introduction with Application. Springer, Berlin (2014)zbMATHGoogle Scholar
  28. 28.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhauser, Boston (1990)zbMATHGoogle 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)zbMATHGoogle Scholar
  31. 31.
    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, Berlin (2009)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)CrossRefzbMATHGoogle Scholar
  35. 35.
    Robinson, S.M.: Regularity and stability for convex multivalued functions. Math. Oper. Res. 1, 130–143 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Ursescu, C.: Multifunctions with closed convex graph. Czechoslov. Math. J. 25, 438–441 (1975)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Jahn, J.: Introduction to the Theory of Nonlinear Optimization, 2nd edn. Springer, Berlin (1996)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar

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© 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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