Skip to main content
SpringerLink
Log in

Efficiency conditions in vector control problems governed by multiple integrals

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this paper, we formulate and prove necessary and sufficient optimality conditions in multiobjective control problems which involve multiple integrals. Under \( \displaystyle (\rho , b) \)-quasiinvexity assumptions, sufficient efficiency conditions for a feasible solution are derived.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agarwal, R.P., Ahmad, I., Iqbal, A., Ali, S.: Generalized invex sets and preinvex functions on Riemannian manifolds. Taiwan. J. Math. 16(5), 1719–1732 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Antczak, T.: \(G\)-pre-invex functions in mathematical programming. J. Comput. Appl. Math. 217(1), 212–226 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arana-Jiménez, M., Osuna-Gómez, R., Rufián-Lizana, A., Ruiz-Garzón, G.: KT-invex control problem. Appl. Math. Comput. 197(2), 489–496 (2008)

    MathSciNet  MATH  Google Scholar 

  4. Barani, A., Pouryayevali, M.R.: Invex sets and preinvex functions on Riemannian manifolds. J. Math. Anal. Appl. 328(2), 767–779 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bector, C.R., Cambini, R.: Generalized B-invex vector valued functions. J. Stat. Manag. Syst. 5(1–3), 141–173 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Berkovitz, L.D.: Variational methods in problems of control and programming. J. Math. Anal. Appl. 3(1), 145–169 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bhatia, D., Kumar, P.: Multiobjective control problem with generalized invexity. J. Math. Anal. Appl. 189(3), 676–692 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chandra, S., Craven, B.D., Husain, I.: A class of nondifferentiable control problems. J. Optim. Theory Appl. 56(2), 227–243 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, X.: Duality for a class of multiobjective control problems. J. Math. Anal. Appl. 267(1), 377–394 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ferrara, M., Mititelu, Şt.: Mond–Weir duality in vector programming with generalized invex functions on differentiable manifolds. Balkan J. Geom. Appl. 11(1), 80–87 (2006)

  11. Gramatovici, S.: Optimality conditions in multiobjective control problems with generalized invexity. Ann. Univ. Craiova Math. Comp. Sci. Ser. 32, 150–157 (2005)

    MathSciNet  MATH  Google Scholar 

  12. Hanson, M.A.: Bounds for functionally convex optimal control problems. J. Math. Anal. Appl. 8(1), 84–89 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hanson, M.A.: On sufficiency of Kuhn–Tucker conditions. J. Math. Anal. Appl. 80(2), 545–550 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jagannathan, R.: Duality for nonlinear fractional programming. Z. Oper. Res. 17, 1–3 (1973)

    MathSciNet  MATH  Google Scholar 

  15. Jeyakumar, V.: Strong and weak invexity in mathematical programming. Math. Oper. Res. 55, 109–125 (1985)

    MathSciNet  MATH  Google Scholar 

  16. Martin, D.H.: The essence of invexity. J. Optim. Theory Appl. 47(1), 65–76 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mititelu, Şt.: Efficiency and duality for multiobjective fractional problems in optimal control with \(\rho \)-quasiinvexity. In: International Conference on Trends and Challenges in Applied Mathematics—ICTCAM, Bucharest, Romania, 20–23 June 2007

  18. Mititelu, Şt: Optimality and duality for invex multi-time control problems with mixed constraints. J. Adv. Math. Stud. 2(1), 25–34 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Mititelu, Şt., Stancu-Minasian, I.M.: Efficiency and duality for multiobjective fractional variational problems with \((\rho , b)\)-quasiinvexity. Yugosl. J. Oper. Res 19(1), 85–99 (2009)

  20. Mond, B., Hanson, M.A.: Duality for variational problems. J. Math. Anal. Appl. 18(2), 355–364 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mond, B., Smart, I.: Duality and sufficiency in control problems with invexity. J. Math. Anal. Appl. 136(1), 325–333 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  22. Nahak, C., Nanda, S.: On efficiency and duality for multiobjective variational control problems with \((F, \rho )\)-convexity. J. Math. Anal. Appl. 209(2), 415–434 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. Preda, V.: On duality and sufficiency in control problems with general invexity. Bull. Math. de la Soc. Sci. Math de Roum. 35(83)(3/4), 271–280 (1991)

    MathSciNet  MATH  Google Scholar 

  24. Rapcsák, T.: Smooth Nonlinear Optimization in \( R^{n} \). Nonconvex Optimization and Its Applications. Kluwer Academic, Dordrecht (1997)

    Book  MATH  Google Scholar 

  25. Tang, W., Yang, X.: The sufficiency and necessity conditions of strongly preinvex functions. OR Trans. 10(3), 50–58 (2006)

    Google Scholar 

  26. Treanţă, S., Udrişte, C.: Optimal control problems with higher order ODEs constraints. Balkan J. Geom. Appl. 18(1), 71–86 (2013)

    MathSciNet  MATH  Google Scholar 

  27. Treanţă, S.: PDEs of Hamilton–Pfaff type via multi-time optimization problems. U.P.B. Sci. Bull. Ser. A Appl. Math. Phys. 76(1), 163–168 (2014)

    MathSciNet  MATH  Google Scholar 

  28. Treanţă, S.: Optimal control problems on higher order jet bundles. In: International Conference on Differential Geometry—Dynamical Systems, 10–13 October 2013, Bucharest-Romania, pp. 181-192. Balkan Society of Geometers, Geometry Balkan Press (2014)

  29. Treanţă, S.: Multiobjective fractional variational problem on higher-order jet bundles. Commun. Math. Stat. 4(3), 323–340 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Udrişte, C.: Convex Functions and Optimization Methods on Riemannian Manifolds, Mathematics and Its Applications. Kluwer Academic, Dordrecht (1994)

    Book  MATH  Google Scholar 

  31. Udrişte, C.: Simplified multitime maximum principle. Balkan J. Geom. Appl. 14(1), 102–119 (2009)

    MathSciNet  MATH  Google Scholar 

  32. Weir, T., Mond, B.: Pre-invex functions in multiple objective optimization. J. Math. Anal. Appl. 136(1), 29–38 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhian, L., Qingkai, Y.: Duality for a class of multiobjective control problems with generalized invexity. J. Math. Anal. Appl. 256(2), 446–461 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Technical University of Civil Engineering, 124 Lacul Tei, 020396, Bucharest, Romania

    Ştefan Mititelu

  2. Faculty of Applied Sciences, Department of Applied Mathematics, University Politehnica of Bucharest, 313 Splaiul Independentei, 060042, Bucharest, Romania

    Savin Treanţă

Authors
  1. Ştefan Mititelu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Savin Treanţă
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Savin Treanţă.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mititelu, Ş., Treanţă, S. Efficiency conditions in vector control problems governed by multiple integrals. J. Appl. Math. Comput. 57, 647–665 (2018). https://doi.org/10.1007/s12190-017-1126-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-017-1126-z

Keywords

Mathematics Subject Classification

Search

Navigation