Opuscula Math. 43, no. 3 (2023), 335-391
https://doi.org/10.7494/OpMath.2023.43.3.335

 
Opuscula Mathematica

On efficiency and duality for a class of nonconvex nondifferentiable multiobjective fractional variational control problems

Tadeusz Antczak
Manuel Arana-Jimenéz
Savin Treanţă

Abstract. In this paper, we consider the class of nondifferentiable multiobjective fractional variational control problems involving the nondifferentiable terms in the numerators and in the denominators. Under univexity and generalized univexity hypotheses, we prove optimality conditions and various duality results for such nondifferentiable multiobjective fractional variational control problems. The results established in the paper generalize many similar results established earlier in the literature for such nondifferentiable multiobjective fractional variational control problems.

Keywords: nondifferentiable multiobjective fractional variational control problem, efficient solution, optimality conditions, (generalized) univexity, Mond-Weir duality, Wolfe duality.

Mathematics Subject Classification: 65K10, 90C32, 90C46, 90C30, 90C26.

Full text (pdf)

  1. B. Aghezzaf, K. Khazafi, Sufficient optimality conditions and duality in multiobjective variational programming problems with generalized, Control Cyber. 33 (2004), 1-14.
  2. I. Ahmad, S. Sharma, Sufficiency and duality for multiobjective variational control problems with generalized \((F,\alpha, \rho, \theta)\)-V-convexity, Nonlinear Anal. 72 (2010), 2564-2579. https://doi.org/10.1016/j.na.2009.11.005
  3. T. Antczak, Duality for multiobjective variational control problems with \((\Phi,\rho)\)-invexity, Calcolo 51 (2014), 393-421.
  4. T. Antczak, Sufficient optimality criteria and duality for multiobjective variational control problems with \(G\)-type I objective and constraint functions, J. Global Optim. 61 (2015), 695-720. https://doi.org/10.1007/s10898-014-0203-1
  5. T. Antczak, A. Jayswal, S. Jha, The modified objective function method for univex multiobjective variational problems, Bull. Iranian Math. Soc. 45 (2019), 267-282. https://doi.org/10.1007/s41980-018-0131-9
  6. M. Arana-Jiménez, G. Ruiz-Garzón, A. Rufián-Lizana, R. Osuna-Gómez, A necessary and sufficient condition for duality in multiobjective variational problems, Eur. J. Oper. Res. 201 (2010), 672-681. https://doi.org/10.1016/j.ejor.2009.03.047
  7. M. Arana-Jiménez, G. Ruiz-Garzón, A. Rufián-Lizana, R. Osuna-Gómez, Weak efficiency in multiobjective variational problems under generalized convexity, J. Global Optim. 52 (2012), 109-121. https://doi.org/10.1007/s10898-011-9689-y
  8. C.R. Bector, S. Chandra, S. Gupta, S.K. Suneja, Univex sets, functions and univex nonlinear programming, [in:] S. Komolosi, T. Rapcsák, S. Schaible, Generalized Convexity, Lecture Notes in Econom. and Math. Systems, vol. 405, Springer Verlag, Berlin, 1994.
  9. D. Bhatia, P. Kumar, Multiobjective control problem with generalized invexity, J. Math. Anal. Appl. 189 (1995), 676-692. https://doi.org/10.1006/jmaa.1995.1045
  10. D. Bhatia, A. Mehra, Optimality conditions and duality for multiobjective variational problems with generalized \(B\)-invexity, J. Math. Anal. Appl. 234 (1999), 341-360. https://doi.org/10.1006/jmaa.1998.6256
  11. T.-M. Ding, K.-W. Ding, T. Liu, Duality in multiobjective fractional variational control problems with generalized \(F\)-invexity, Nonlinear Anal. Forum 14 (2009), 191-199.
  12. S. Gramatovici, Optimality conditions in multiobjective control problems with generalized invexity, Ann. Univ. Craiova Math. Comp. Sci. Ser. 32 (2005), 150-157.
  13. M.A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl. 80 (1981), 545-550. https://doi.org/10.1016/0022-247X(81)90123-2
  14. D.G. Hull, Optimal Control Theory for Applications, Mech. Engrg. Ser., Springer 2003.
  15. I. Husain, V.K. Jain, A class of nondifferentiable multiobjective control problems, Stud. Math. Sci. 6 (2013), 1-17.
  16. I. Husain, R.G. Mattoo, On mixed type duality for nondifferentiable multiobjective variational problems, Eur. J. Pure Appl. Math. 3 (2010), 81-97.
  17. A. Jayswal, I. Ahmad, A.K. Prasad, Duality in multiobjective fractional programming problems involving \((H_p,r)\)-invex functions, J. Appl. Math. & Informatics 32 (2014), 99-111. https://doi.org/10.14317/jami.2014.099
  18. K. Khazafi, N. Rueda, Multiobjective variational programming under generalized type I univexity, J. Optim. Theory Appl. 142 (2009), 363-376. https://doi.org/10.1007/s10957-009-9526-3
  19. D.S. Kim, A.L. Kim, Optimality and duality for nondifferentiable multiobjective variational problems, J. Math. Anal. Appl. 274 (2002), 255-278. https://doi.org/10.1016/S0022-247X(02)00298-6
  20. G.M. Lee, D.S. Kim, Ch.L. Jo, Duality for multiobjective variational problems with generalized invexity, J. Inf. Optim. Sci. 19 (1998), 13-23.
  21. J.C. Liu, Duality for nondifferentaible static multiobjective variational problems involving generalized \((F,\rho)\)-convex functions, Comput. Math. Appl. 31 (1996), 77-89.
  22. K. Miettinen, Nonlinear Multiobjective Optimization, International Series in Operations Research & Management Science, Kluwer Academic Publishers, Massachusetts, 2004.
  23. S.K. Mishra, R.N. Mukherjee, Duality for multiobjective fractional variational problems, J. Math. Anal. Appl. 186 (1994), 711-725. https://doi.org/10.1006/jmaa.1994.1328
  24. S.K. Mishra, R.N. Mukherjee, On efficiency and duality for multiobjective variational problems, J. Math. Anal. Appl. 187 (1994), 40-54. https://doi.org/10.1006/jmaa.1994.1343
  25. S.K. Mishra, R.N. Mukherjee, Generalized continuous nondifferentiable fractional programming problems with invexity, J. Math. Anal. Appl. 195 (1995), 191-213. https://doi.org/10.1006/jmaa.1995.1350
  26. S.K. Mishra, R.N. Mukherjee, Multiobjective control problem with \(V\)-invexity, J. Math. Anal. Appl. 235 (1999), 1-12. https://doi.org/10.1006/jmaa.1998.6110
  27. Ş. Mititelu, Effciency conditions for multiobjective fractional variational problems, Appl. Sci. 10 (2008), 162-175.
  28. Ş. Mititelu, M. Postolache, Mond-Weir dualities with Lagrangians for multiobjective fractional and non-fractional variational problems, J. Adv. Math. Stud. 3 (2010), 41-58.
  29. Ş. Mititelu, I.M. Stancu-Minasian, Efficiency and duality for multiobjective fractional variational problems with \((\rho,b)\)-quasiinvexity, Yugosl. J. Oper. Res. 19 (2009), 85-99.
  30. Ş. Mititelu, S. Treanţă, Efficiency conditions in vector control problems governed by multiple integrals, J. Appl. Math. Comput. 57 (2018), 647-665.
  31. B. Mond, S. Chandra, I. Husain, Duality for variational problems with invexity, J. Math. Anal. Appl. 134 (1988), 322-328. https://doi.org/10.1016/0022-247X(88)90026-1
  32. B. Mond, M.A. Hanson, Duality for variational problems, J. Math. Anal. Appl. 18 (1967), 355-364. https://doi.org/10.1016/0022-247X(67)90063-7
  33. C. Nahak, Duality for multiobjective variational control and multiobjective fractional variational control problems with pseudoinvexity, J. Appl. Math. Stoch. Anal. 2006 (2006), Article ID 62631. https://doi.org/10.1155/JAMSA/2006/62631
  34. C. Nahak, S. Nanda, Duality for multiobjective fractional control problems with generalized invexity, Korean J. Comput. & Appl. Math. 5 (1998), 433-446.
  35. J.Y. Park, J.U. Jeong, Duality for multiobjective fractional generalized control problems with \((F,\rho)\)-convexity, Nonlinear Anal. 59 (2004), 1311-1332. https://doi.org/10.1016/j.na.2004.08.016
  36. L.V. Reddy, R.N. Mukherjee, Efficiency and duality of multiobjective fractional control problems under \((F,\rho)\)-convexity, Indian J. Pure Appl. Math. 30 (1999), 51-69.
  37. S. Sharma, A. Jayswal, S. Choudhury, Sufficiency and mixed type duality for multiobjective variational control problems involving \(\alpha\)-\(V\)-univexity, Evol. Equ. Control Theory 6 (2017), 93-109. https://doi.org/10.3934/eect.2017006
  38. C. Singh, Optimality conditions in multiobjective differentiable programming, J. Optim. Theory Appl. 53 (1987), 115-123. https://doi.org/10.1007/BF00938820
  39. S. Treanţă, Efficiency in generalised \(V\)-\(KT\)-pseudoinvex control problems, Internat. J. Control 93 (2020), 611-618. https://doi.org/10.1080/00207179.2018.1483082
  40. S. Treanţă, M. Arana-Jimenèz, \(KT\)-pseudoinvex multidimensional control problem, Optimal Control Appl. Methods 39 (2018), 1291-1300. https://doi.org/10.1002/oca.2410
  41. G.J. Zalmai, Proper efficiency and duality for a class of constrained multiobjective fractional optimal control problems containing arbitrary norms, J. Optim. Theory Appl. 90 (1996), 435-556. https://doi.org/10.1007/BF02190007
  42. G.J. Zalmai, Optimality conditions and duality models for a class of nonsmooth constrained fractional variational problems, Optimization 30 (1994), 15-51. https://doi.org/10.1080/02331939408843969
  • Communicated by Marek Galewski.
  • Received: 2021-12-03.
  • Revised: 2023-01-16.
  • Accepted: 2023-02-16.
  • Published online: 2023-05-17.
Opuscula Mathematica - cover

Cite this article as:
Tadeusz Antczak, Manuel Arana-Jimenéz, Savin Treanţă, On efficiency and duality for a class of nonconvex nondifferentiable multiobjective fractional variational control problems, Opuscula Math. 43, no. 3 (2023), 335-391, https://doi.org/10.7494/OpMath.2023.43.3.335

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

We advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.