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

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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.

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References

  1. Allali, K., Amahroq, T.: Second-order approximations and primal and dual necessary optimality conditions. Optimization 40, 229–246 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bednařík, D., Pastor, K.: On second-order conditions in unconstrained optimization. Math. Program. Ser. A 113, 283–298 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bednařík, D., Pastor, K.: Decrease of \(C^{1,1}\) property in vector optimization. RAIRO Oper. Res. 43, 359–372 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bednařík, D., Pastor, K.: On second-order optimality conditions in constrained multiobjective optimization. Nonlinear Anal. TMA 74, 1372–1382 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bigi, G., Castellani, M.: Second-order optimality conditions for differentiable multiobjective problems. RAIRO Oper. Res. 34, 411–426 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer Series in Operations Research. Springer, New York (2000)

    Book  MATH  Google Scholar 

  8. Borwein, J., Goebel, R.: Notions of relative interior in Banach spaces. J. Math. Sci. 115, 2542–2553 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cambini, A., Martein, L., Vlach, M.: Second-order tangent sets and optimality conditions. Math. Jpn. 49, 451–461 (1999)

    MathSciNet  MATH  Google Scholar 

  10. Cominetti, R.: Metric regularity, tangent sets and second-order optimality conditions. Appl. Math. Optim. 21, 265–287 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  12. Dhara, A., Mehra, A.: Second-order optimality conditions in minimax optimization problems. J. Optim. Theory Appl. 156, 567–590 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Donchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. A View from Variational Analysis. Springer Monographs in Mathematics. Springer, Dordrecht (2009)

    Book  Google Scholar 

  14. Georgiev, P.G., Zlateva, N.P.: Second-order subdifferentials of \(C^{1, 1}\) constrained vector optimization. Set Valued Anal. 4, 101–117 (1996)

    Article  MathSciNet  Google Scholar 

  15. Gfrerer, H.: Second-order optimality conditions for scalar and vector optimization problems in Banach spaces. SIAM J. Control. Optim. 45, 972–997 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gfrerer, H.: On directional metric regularity, subregularity and optimality conditions for nonsmooth mathematical programs. Set Valued Var. Anal. 21, 151–176 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ginchev, I., Guerraggio, A., Rocca, M.: From scalar to vector optimization. Appl. Math. 51, 5–36 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ginchev, I., Ivanov, V.I.: Second-order optimality conditions for problems with \(C^{1}\) data. J. Math. Anal. Appl. 340, 646–657 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ivanov, V.I.: Second- and first-order optimality conditions in vector optimization. Int. J. Inf. Technol. Decis. Mak. 14, 1–21 (2015)

    Article  Google Scholar 

  26. Ivanov, V.I.: Second-order optimality conditions with arbitrary nondifferentiable function in scalar and vector optimization. Nonlinear Anal. TMA 125, 270–289 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  28. Jeyakumar, V., Luc, D.T.: Nonsmooth vector functions and continuous optimization. Springer Optimization and Its Applications, vol. 10. Springer, New York (2008)

    MATH  Google Scholar 

  29. Jiménez, B., Novo, V.: Second-order necessary conditions in set constrained differentiable vector optimization. Math. Methods Oper. Res. 58, 299–317 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Jiménez, B., Novo, V.: Optimality conditions in differentiable vector optimization via second-order tangent sets. Appl. Math. Optim. 49, 123–144 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Jiménez, B., Novo, V.: First-order optimality conditions in vector optimization involving stable functions. Optimization 57, 449–471 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kawasaki, H.: The upper and lower second-order directional derivatives of a sup-type function. Math. Program. 41, 327–339 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kawasaki, H.: Second-order necessary optimality conditions for minimizing a sup-type function. Math. Program. 49, 213–229 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  35. Khan, A.A., Tammer, C., Zălinescu, C.: Set-valued Optimization. An Introduction with Applications. Vector Optimization. Springer, Heidelberg (2015)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  43. Michel, P., Penot, J.P.: A generalized derivative for calm and stable functions. Differ. Integral Eq. 5, 433–454 (1992)

    MathSciNet  MATH  Google Scholar 

  44. De Oliveira, V.A., Rojas-Medar, M.A.: Multiobjective infinite programming. Comput. Math. Appl. 55, 1907–1922 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  45. Páles, Z., Zeidan, V.M.: Nonsmooth optimum problems with constraints. SIAM J. Control Optim. 32, 1476–1502 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  46. Pastor, K.: Differentiability properties of \(l\)-stable vector functions in infinite-dimensional normed spaces. Taiwan. J. Math. 18, 187–197 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  47. Penot, J.P.: Optimality conditions in mathematical programming and composite optimization. Math. Program. 67, 225–245 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  48. Penot, J.P.: Optimality conditions for mildly nonsmooth constrained optimization. Optimization 43, 323–337 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  52. Shapiro, A.: Semi-infinite programming, duality, discretization and optimality conditions. Optimization 58, 133–161 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  53. Taa, A.: Second-order conditions for nonsmooth multiobjective optimization problems with inclusion constraints. J. Glob. Optim. 50, 271–291 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  55. Ward, D.E.: Calculus for parabolic second-order derivatives. Set Valued Anal. 1, 213–246 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This work was supported by a grant of the UEH Foundation for Academic Research. The final part of working on the paper was completed during a stay of the author as research visitor at the Vietnam Institute for Advanced Study in Mathematics (VIASM), whose hospitality is gratefully acknowledged. The author would like to thank the Editors and Anonymous Referee for their valuable remarks and suggestions, which have helped him to improve the paper.

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Tuan, N.D. On Necessary Optimality Conditions for Nonsmooth Vector Optimization Problems with Mixed Constraints in Infinite Dimensions. Appl Math Optim 77, 515–539 (2018). https://doi.org/10.1007/s00245-016-9383-z

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  • DOI: https://doi.org/10.1007/s00245-016-9383-z

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