Skip to main content
Log in

Efficiency for variational control problems on Riemann manifolds with geodesic quasiinvex curvilinear integral functionals

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, we formulate and prove necessary and sufficient conditions of geodesic efficiency associated with a new class of multiobjective fractional variational control problems governed by geodesic quasiinvex path-independent curvilinear integral functionals and mixed constraints involving first order PDE of m-flow type. Under \( \displaystyle (\rho , b) \)-geodesic quasiinvexity assumptions, by using the new notion of (normal) geodesic efficient solution, we set sufficient conditions of geodesic efficiency for a feasible solution in the considered variational control problems.

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

Access this article

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. Arana-Jiménez, M., Antczak, T.: The minimal criterion for the equivalence between local and global optimal solutions in nondifferentiable optimization problem. Math. Methods Appl. Sci. 40, 6556–6564 (2017)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  3. Clarke, F.H.: Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, vol. 264. Springer, London (2013)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. Gupta, P., Mishra, S.K., Mohapatra, R.N.: Duality models for multiobjective semiinfinite fractional programming problems involving type-I and related functions. Quaest. Math. https://doi.org/10.2989/16073606.2018.1509906

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

    MathSciNet  MATH  Google Scholar 

  7. Ivanov, V.I.: Second-order Kuhn–Tucker invex constrained problems. J. Global Optim. 50, 519–529 (2011)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  11. Mititelu, Ş., Stancu-Minasian, I.M.: Invexity at a point: generalisations and classifications. Bull. Aust. Math. Soc. 48, 117–126 (1993)

    MathSciNet  MATH  Google Scholar 

  12. Mititelu, Ş.: Invex sets. Math. Rep. 46(5), 529–532 (1994)

    MathSciNet  MATH  Google Scholar 

  13. Mititelu, Ş.: Invex functions. Rev. Roum. Math. Pures Appl. 49(5–6), 529–544 (2005)

    MathSciNet  MATH  Google Scholar 

  14. Mititelu, Ş.: Invex sets and nonsmooth invex functions. Rev. Roum. Math. Pures Appl. 52(6), 665–672 (2007)

    MathSciNet  MATH  Google Scholar 

  15. Mititelu, Ş.: Generalized Convexities. Fair Partners, Bucharest (2011)

    MATH  Google Scholar 

  16. Noor, M.A., Noor, K.I.: Some characterizations of strongly preinvex functions. J. Math. Anal. Appl. 316(2), 697–706 (2006)

    MathSciNet  MATH  Google Scholar 

  17. Awan, M.U., Noor, M.A., Mihai, M.V., Noor, K.I., AlMohsen, B.A.: Two dimensional extensions of Hermite–Hadamard’s inequalities via preinvex functions. Rev. de la Real Acad. de Ciencias Exactas Fisicas y Nat. Ser. A Mat. (2018). https://doi.org/10.1007/s13398-018-0492-1

  18. de Oliveira, V.A., Silva, G.N.: On sufficient optimality conditions for multiobjective control problems. J. Glob. Optim. 64(4), 721–744 (2016)

    MathSciNet  MATH  Google Scholar 

  19. Pini, R.: Convexity along curves and indunvexity. Optimization 29(4), 301–309 (1994)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

  22. Treanţă, S.: Optimal control problems on higher order jet bundles, BSG Proceedings 21, The International Conference “Differential Geometry—Dynamical Systems”, DGDS-2013, October 10–13, 2013, vol. 21, pp. 181–192 . Bucharest (2014)

  23. Treanţă, S.: PDEs of Hamilton-Pfaff type via multi-time optimization problems. UPB Sci. Bull. Ser. A 76(1), 163–168 (2014)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  25. Treanţă, S.: Sufficient efficiency conditions associated with a multidimensional multiobjective fractional variational problem. J. Multidiscip. Model. Optim. 1(1), 1–13 (2018)

    MathSciNet  Google Scholar 

  26. Olteanu, O., Treanţă, S.: Convexity, Optimization and Approximation, with some Applications. LAP Lambert Academic Publishing, Riga (2018)

    Google Scholar 

  27. Treanţă, S.: Higher-order Hamilton dynamics and Hamilton–Jacobi divergence PDE. Comput. Math. Appl. 75(2), 547–560 (2018)

    MathSciNet  MATH  Google Scholar 

  28. Treanţă, S.: PDE-constrained vector variational problems governed by curvilinear integral functionals. Appl. Anal. Optim. 3(1), 83–101 (2019)

    MathSciNet  Google Scholar 

  29. Treanţă, S.: Constrained variational problems governed by second-order Lagrangians. Appl. Anal. (2020). https://doi.org/10.1080/00036811.2018.1538501

  30. Treanţă, S., Arana-Jiménez, M.: On generalized KT-pseudoinvex control problems involving multiple integral functionals. Eur. J. Control 43, 39–45 (2018)

    MathSciNet  MATH  Google Scholar 

  31. Udrişte, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Mathematics and its Applications, vol. 297. Kluwer Academic, New York (1994)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Savin Treanţă.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Treanţă, S., Mititelu, Ş. Efficiency for variational control problems on Riemann manifolds with geodesic quasiinvex curvilinear integral functionals. RACSAM 114, 113 (2020). https://doi.org/10.1007/s13398-020-00842-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13398-020-00842-2

Keywords

Mathematics Subject Classification

Navigation