On Necessary Optimality Conditions for Nonsmooth Vector Optimization Problems with Mixed Constraints in Infinite Dimensions

Article

Abstract

In this note, we develop first- and second-order necessary optimality conditions for local weak solutions in nonsmooth vector optimization problems subject to mixed constraints in infinite-dimensional settings. To this aim, we use some set-valued directional derivatives of the Hadamard type and tangent sets, and impose (first-order) Hadamard differentiability assumptions of the data at the point of consideration.

Keywords

Nonsmooth vector optimization necessary optimality conditions weak solutions second-order tangent sets second-order directional derivatives 

Mathematics Subject Classification

90C29 90C46 49K27 26B05 

References

  1. 1.
    Allali, K., Amahroq, T.: Second-order approximations and primal and dual necessary optimality conditions. Optimization 40, 229–246 (1997)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bednařík, D., Pastor, K.: On second-order conditions in unconstrained optimization. Math. Program. Ser. A 113, 283–298 (2008)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bednařík, D., Pastor, K.: Decrease of \(C^{1,1}\) property in vector optimization. RAIRO Oper. Res. 43, 359–372 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bednařík, D., Pastor, K.: On second-order optimality conditions in constrained multiobjective optimization. Nonlinear Anal. TMA 74, 1372–1382 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bigi, G., Castellani, M.: Second-order optimality conditions for differentiable multiobjective problems. RAIRO Oper. Res. 34, 411–426 (2000)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bonnans, 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
  7. 7.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer Series in Operations Research. Springer, New York (2000)CrossRefMATHGoogle Scholar
  8. 8.
    Borwein, J., Goebel, R.: Notions of relative interior in Banach spaces. J. Math. Sci. 115, 2542–2553 (2003)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cambini, A., Martein, L., Vlach, M.: Second-order tangent sets and optimality conditions. Math. Jpn. 49, 451–461 (1999)MathSciNetMATHGoogle Scholar
  10. 10.
    Cominetti, R.: Metric regularity, tangent sets and second-order optimality conditions. Appl. Math. Optim. 21, 265–287 (1990)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Dhara, A., Luc, D.T., Tinh, P.N.: On second-order conditions for nonsmooth problems with constraints. Vietnam J. Math. 40, 201–229 (2012)MathSciNetMATHGoogle Scholar
  12. 12.
    Dhara, A., Mehra, A.: Second-order optimality conditions in minimax optimization problems. J. Optim. Theory Appl. 156, 567–590 (2013)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Donchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. A View from Variational Analysis. Springer Monographs in Mathematics. Springer, Dordrecht (2009)CrossRefGoogle Scholar
  14. 14.
    Georgiev, P.G., Zlateva, N.P.: Second-order subdifferentials of \(C^{1, 1}\) constrained vector optimization. Set Valued Anal. 4, 101–117 (1996)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gfrerer, H.: Second-order optimality conditions for scalar and vector optimization problems in Banach spaces. SIAM J. Control. Optim. 45, 972–997 (2006)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Gfrerer, H.: On directional metric regularity, subregularity and optimality conditions for nonsmooth mathematical programs. Set Valued Var. Anal. 21, 151–176 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    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
  18. 18.
    Ginchev, I., Guerraggio, A., Rocca, M.: Second-order conditions for \(C^{1, 1}\) constrained vector optimization. Math. Program. Ser. B 104, 389–405 (2005)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ginchev, I., Guerraggio, A., Rocca, M.: From scalar to vector optimization. Appl. Math. 51, 5–36 (2006)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Ginchev, I., Ivanov, V.I.: Second-order optimality conditions for problems with \(C^{1}\) data. J. Math. Anal. Appl. 340, 646–657 (2008)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Giorgi, G., Jiménez, B., Novo, V.: An overview of second-order tangent sets and their application to vector optimization. Bol. Soc. Esp Mat. Apl. 52, 73–96 (2010)MathSciNetMATHGoogle Scholar
  22. 22.
    Gutiérrez, C., Jiménez, B., Novo, V.: New second-order directional derivative and optimality conditions in scalar and vector optimization. J. Optim. Theory Appl. 142, 85–106 (2009)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Gutiérrez, C., Jiménez, B., Novo, V.: On second-order Fritz John type optimality conditions in nonsmooth multiobjective programming. Math. Program. Ser. B 123, 199–223 (2010)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Hiriart-Urruty, J.B., Strodiot, J.J., Nguyen, V.H.: Generalized Hessian matrix and second-order optimality conditions for problem with \(C^{1, 1}\) data. Appl. Math. Optim. 11, 43–56 (1984)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Ivanov, V.I.: Second- and first-order optimality conditions in vector optimization. Int. J. Inf. Technol. Decis. Mak. 14, 1–21 (2015)CrossRefGoogle Scholar
  26. 26.
    Ivanov, V.I.: Second-order optimality conditions with arbitrary nondifferentiable function in scalar and vector optimization. Nonlinear Anal. TMA 125, 270–289 (2015)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Ivanov, V.I.: Second-order optimality conditions for vector problems with continuously Fréchet differentiable data and second order constraint qualifications. J. Optim. Theory Appl. 166, 777–790 (2015)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Jeyakumar, V., Luc, D.T.: Nonsmooth vector functions and continuous optimization. Springer Optimization and Its Applications, vol. 10. Springer, New York (2008)MATHGoogle Scholar
  29. 29.
    Jiménez, B., Novo, V.: Second-order necessary conditions in set constrained differentiable vector optimization. Math. Methods Oper. Res. 58, 299–317 (2003)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Jiménez, B., Novo, V.: Optimality conditions in differentiable vector optimization via second-order tangent sets. Appl. Math. Optim. 49, 123–144 (2004)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Jiménez, B., Novo, V.: First-order optimality conditions in vector optimization involving stable functions. Optimization 57, 449–471 (2008)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    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
  33. 33.
    Kawasaki, H.: The upper and lower second-order directional derivatives of a sup-type function. Math. Program. 41, 327–339 (1988)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Kawasaki, H.: Second-order necessary optimality conditions for minimizing a sup-type function. Math. Program. 49, 213–229 (1991)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Khan, A.A., Tammer, C., Zălinescu, C.: Set-valued Optimization. An Introduction with Applications. Vector Optimization. Springer, Heidelberg (2015)MATHGoogle Scholar
  36. 36.
    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
  37. 37.
    Khanh, P.Q., Tuan, N.D.: Second-order optimality conditions with envelope-like effect for nonsmooth vector optimization in infinite dimensions. Nonlinear Anal. TMA 77, 130–148 (2013)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Khanh, P.Q., Tuan, N.D.: Second-order optimality conditions with the envelope-like effect in nonsmooth multiobjective mathematical programming, I: \(l\)-stability and set-valued directional derivatives. J. Math. Anal. Appl. 403, 695–702 (2013)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Khanh, P.Q., Tuan, N.D.: Second-order optimality conditions with the envelope-like effect in nonsmooth multiobjective mathematical programming, II: Optimality conditions. J. Math. Anal. Appl. 403, 703–714 (2013)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Liu, L., Neittaanmäki, P., Křížek, M.: Second-order optimality conditions for nondominated solutions of multiobjective programming with \(C^{1,1}\) data. Appl. Math. 45, 381–397 (2000)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Maruyama, Y.: Second-order necessary conditions for nonlinear optimization problems in Banach spaces and their applications to an optimal control problem. Math. Oper. Res. 15, 467–482 (1990)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Maruyama, Y.: Second-order necessary conditions for nonlinear optimization problems in Banach spaces by the use of Neustadt derivative. Math. Jpn. 40, 509–522 (1994)MathSciNetMATHGoogle Scholar
  43. 43.
    Michel, P., Penot, J.P.: A generalized derivative for calm and stable functions. Differ. Integral Eq. 5, 433–454 (1992)MathSciNetMATHGoogle Scholar
  44. 44.
    De Oliveira, V.A., Rojas-Medar, M.A.: Multiobjective infinite programming. Comput. Math. Appl. 55, 1907–1922 (2008)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Páles, Z., Zeidan, V.M.: Nonsmooth optimum problems with constraints. SIAM J. Control Optim. 32, 1476–1502 (1994)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Pastor, K.: Differentiability properties of \(l\)-stable vector functions in infinite-dimensional normed spaces. Taiwan. J. Math. 18, 187–197 (2014)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Penot, J.P.: Optimality conditions in mathematical programming and composite optimization. Math. Program. 67, 225–245 (1994)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Penot, J.P.: Optimality conditions for mildly nonsmooth constrained optimization. Optimization 43, 323–337 (1998)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Penot, J.P.: Second-order conditions for optimization problems with constraints. SIAM J. Control Optim. 37, 303–318 (1998)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Penot, J.P.: Recent advances on second-order optimality conditions. In: Nguyen, V.H., Strodiot, J.J., Tossings, P. (eds.) Optimization, pp. 357–380. Springer, Berlin (2000)CrossRefGoogle Scholar
  51. 51.
    Santos, L.B., Osuna-Gómez, R., Hernánder-Jiménez, B., Rojas-Medar, M.A.: Necessary and sufficient second-order optimality conditions for multiobjective problems with \(C^{1}\) data. Nonlinear Anal. TMA 85, 192–203 (2013)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Shapiro, A.: Semi-infinite programming, duality, discretization and optimality conditions. Optimization 58, 133–161 (2009)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Taa, A.: Second-order conditions for nonsmooth multiobjective optimization problems with inclusion constraints. J. Glob. Optim. 50, 271–291 (2011)MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Tuan, N.D.: First and second-order optimality conditions for nonsmooth vector optimization using set-valued directional derivatives. Appl. Math. Comput. 251, 300–317 (2015)MathSciNetMATHGoogle Scholar
  55. 55.
    Ward, D.E.: Calculus for parabolic second-order derivatives. Set Valued Anal. 1, 213–246 (1993)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Economics-Ho Chi Minh CityHo Chi Minh CityVietnam

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