Skip to main content
SpringerLink
Log in

On Unconstrained Fuzzy-Valued Optimization Problems

  • Published:
International Journal of Fuzzy Systems Aims and scope Submit manuscript

Abstract

In this paper, based on the parametric representation of fuzzy-valued function, an unconstrained fuzzy-valued optimization problem is converted to a general unconstrained optimization problem. Two solutions for the unconstrained fuzzy-valued optimization problem are proposed which are parallel to the concept of efficient solution in the case of multi-objective programming problem. Also it is proven that the optimal solution of its corresponding general unconstrained optimization problem is the optimal solution of original problem. Finally, some numerical examples are given to illustrate the discussed suitability scheme. In the first example, the various solutions are discussed in details and a classic motivating example in the mathematical study of variational inequalities, namely the Elliptic Obstacle Problem, is expressed in the second one. Non-convex Fuzzy Rosenbrock Function have been solved in third example.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Einstein, A.: Ideas and Opinions. Crown Publishers, New York (1954)

    Google Scholar 

  2. Courant, R., Hilbert, D.: Method of Mathematical Physics, vol. 1. Interscience Publishers, New York (1953) (Translated and revised from the 1937 German original)

  3. Biegler, L.T.: Nonlinear Programming, Concepts, Algorithms and Applications to Chemical Processes. SIAM, Philadelphia (2010)

    Book  MATH  Google Scholar 

  4. Torres, N.V., Voit, E.O.: Pathway Analysis and Optimization in Metabolic Engineering. Cambridge University Press, New York (2002)

    Book  Google Scholar 

  5. Floudas, C.A., Pardalos, P.M.: Optimization in Computational Chemistry and Molecular Biology: Local and Global Approaches. Kluwer Academic Publishers, Dordrecht (2000)

    Book  MATH  Google Scholar 

  6. Buckley, J.J., Jowers, L.J.: Simulating Continuous Fuzzy Systems. Springer, Berlin (2006)

    MATH  Google Scholar 

  7. Hanss, M.: Applied Fuzzy Arithmetic: An Introduction with Engineering Applications. Springer, Berlin (2005)

    MATH  Google Scholar 

  8. Reddy, B.D.: Introductory Functional Analysis with Applications to Boundary Value Problems and Finite Elements. Springer, New York (1998)

    MATH  Google Scholar 

  9. Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM Studies in Applied Mathematics, vol. 8. SIAM, Philadelphia (1988)

    Book  MATH  Google Scholar 

  10. Rodrigues, J.F.: Obstacle Problems in Mathematical Physics. North-Holland Mathematics Studies, vol. 134. North-Holland, Amsterdam (1987)

    Book  MATH  Google Scholar 

  11. Bhurjee, A.K., Panda, G.: Efficient solution of interval optimization problem. Math. Methods Oper. Res. 76, 273–288 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bellman, R.E., Zadeh, L.A.: Decision making in a fuzzy environment. Manag. Sci. 17, 141–164 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cadenas, J.M., Verdegay, J.L.: Towards a new strategy for solving fuzzy optimization problems. Fuzzy Optim. Decis. Making 8, 231–244 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Klerk, E., Roos, C., Terlaky, T.: Lecture Notes in Nonlinear Optimization (CO 367). University of Waterloo (2004)

  15. Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice-Hall of India, Upper Saddle River (1995)

    MATH  Google Scholar 

  16. Nehi, H.M., Daryab, A.: Saddle point optimality conditions in fuzzy optimization problems. Int. J. Fuzzy Syst. 14(1), 11–21 (2012)

    MathSciNet  Google Scholar 

  17. Negoita, C.V., Ralescu, D.A.: Applications of Fuzzy Sets to Systems Analysis. Wiley, New York (1975)

    Book  MATH  Google Scholar 

  18. Pathak, V.D., Pirzada, U.M.: Necessary and sufficient optimality conditions for nonlinear fuzzy optimization problem. Int. J. Math. Sci. Educ. 4(1), 1–16 (2011)

    Google Scholar 

  19. Pirzada, U.M., Pathak, V.D.: Newton method for solving the multi-variable fuzzy optimization problem. J. Optim. Theory Appl. 156, 867–881 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pirzada, U.M., Pathak, V.D.: First and second-order optimaility conditions for unconstrained L-fuzzy optimization problems. Adv. Model. Optim. 13(1) (2011)

  21. Wu, H.C.: Duality theory in fuzzy optimization problems. Fuzzy Optim. Decis. Making 3, 345–365 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wu, H.C.: The Karush-Kuhn-Tucker optimality conditions for multi-objective programming problems with fuzzy-valued objective functions. Fuzzy Optim. Decis. Making 8, 1–28 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wu, H.C.: The optimality conditions for optimization problems with fuzzy-valued objective functions. Optimization 57(3), 473–489 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  25. Burger, M., Matevosyan, N., Wolfram, M.T.: A level set based shape optimization method for an elliptic obstacle problem. Math. Models Methods Appl. Sci. 21(4), 619–649 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Gudi, T., Porwal, K.: A posteriori error control of discontinuous Galerkin methods for elliptic obstacle problems. Math. Comput. 83, 579–602 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Computer Science, Damghan University, Damghan, Iran

    M. Heidari, M. R. Zadeh, O. S. Fard & A. H. Borzabadi

Authors
  1. M. Heidari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. R. Zadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. O. S. Fard
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. H. Borzabadi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Heidari.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Heidari, M., Zadeh, M.R., Fard, O.S. et al. On Unconstrained Fuzzy-Valued Optimization Problems. Int. J. Fuzzy Syst. 18, 270–283 (2016). https://doi.org/10.1007/s40815-016-0165-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-016-0165-1

Keywords

Search

Navigation