Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

A Davidon-Fletcher-Powell Type Quasi-Newton Method to Solve Fuzzy Optimization Problems

  • Conference paper
  • First Online:
Mathematics and Computing (ICMC 2017)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 655))

Included in the following conference series:

  • 983 Accesses

Abstract

In this article, a Davidon-Fletcher-Powell type quasi-Newton method is proposed to capture nondominated solutions of fuzzy optimization problems. The functions that we attempt to optimize here are multivariable fuzzy-number-valued functions. The decision variables are considered to be crisp. Towards developing the quasi-Newton method, the notion of generalized Hukuhara difference between fuzzy numbers, and hence generalized Hukuhara differentiability for multi-variable fuzzy-number-valued functions are used. In order to generate the iterative points, the proposed technique produces a sequence of positive definite inverse Hessian approximations. The convergence result and an algorithm of the developed method are also included. It is found that the sequence in the proposed method has superlinear convergence rate. To illustrate the developed technique, a numerical example is exhibited.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Andrei, N.: Hybrid conjugate gradient algorithm for unconstrained optimization. J. Optim. Theor. Appl. 141, 249–264 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anastassiou, G.A.: Fuzzy Mathematics: Approximation Theory, vol. 251. Springer, Heidelberg (2010)

    Book  MATH  Google Scholar 

  3. Banks, H.T., Jacobs, M.Q.: A differential calculus for multifunctions. J. Math. Anal. Appl. 29, 246–272 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley-interscience, 3rd edn. Wiley, New York (2006)

    Book  MATH  Google Scholar 

  5. Bede, B., Stefanini, L.: Generalized differentiability of fuzzy-valued functions. Fuzzy Sets Syst. 230, 119–141 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bellman, R.E., Zadeh, L.A.: Decision making in a fuzzy environment. Manag. Sci. 17, B141–B164 (1970)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Chalco-Cano, Y., Silva, G.N., Rufián-Lizana, A.: On the Newton method for solving fuzzy optimization problems. Fuzzy Sets Syst. 272, 60–69 (2015)

    Article  MathSciNet  Google Scholar 

  9. Chakraborty, D., Ghosh, D.: Analytical fuzzy plane geometry II. Fuzzy Sets Syst. 243, 84–109 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dubois, D., Prade, H.: Towards fuzzy differential calculus part 3: differentiation. Fuzzy Sets Syst. 8(3), 225–233 (1982)

    Article  MATH  Google Scholar 

  11. Ghosh, D.: Newton method to obtain efficient solutions of the optimization problems with interval-valued objective functions. J. Appl. Math. Comput. 53, 709–731 (2016). (Accepted Manusctipt)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ghosh, D.: A Newton method for capturing efficient solutions of interval optimization problems. Opsearch 53, 648–665 (2016)

    Article  MathSciNet  Google Scholar 

  13. Ghosh, D.: A quasi-newton method with rank-two update to solve interval optimization problems. Int. J. Appl. Comput. Math. (2016). doi:10.1007/s40819-016-0202-7 (Accepted Manusctipt)

  14. Ghosh, D., Chakraborty, D.: Analytical fuzzy plane geometry I. Fuzzy Sets Syst. 209, 66–83 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ghosh, D., Chakraborty, D.: Analytical fuzzy plane geometry III. Fuzzy Sets Syst. 283, 83–107 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ghosh, D., Chakraborty, D.: A method to capture the entire fuzzy non-dominated set of a fuzzy multi-criteria optimization problem. Fuzzy Sets Syst. 272, 1–29 (2015)

    Article  Google Scholar 

  17. Ghosh, D., Chakraborty, D.: A new method to obtain fuzzy Pareto set of fuzzy multi-criteria optimization problems. Int. J. Intel. Fuzzy Syst. 26, 1223–1234 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Ghosh, D., Chakraborty, D.: Quadratic interpolation technique to minimize univariable fuzzy functions. Int. J. Appl. Comput. Math. (2016). doi:10.1007/s40819-015-0123-x (Accepted Manuscript)

  19. Ghosh, D., Chakraborty, D.: A method to obtain complete fuzzy non-dominated set of fuzzy multi-criteria optimization problems with fuzzy parameters. In: Proceedings of IEEE International Conference on Fuzzy Systems, FUZZ IEEE, IEEE Xplore, pp. 1–8 (2013)

    Google Scholar 

  20. Lai, Y.-J., Hwang, C.-L.: Fuzzy Mathematical Programming: Methods and Applications. Lecture Notes in Economics and Mathematical Systems, vol. 394. Springer, New York (1992)

    MATH  Google Scholar 

  21. Lai, Y.-J., Hwang, C.-L.: Fuzzy Multiple Objective Decision Making: Methods and Applications. Lecture Notes in Economics and Mathematical Systems, vol. 404. Springer, New York (1994)

    MATH  Google Scholar 

  22. Lodwick, W.A., Kacprzyk, J.: Fuzzy Optimization: Recent Advances and Applications, vol. 254. Physica-Verlag, New York (2010)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Puri, M.L., Ralescu, D.A.: Differentials of fuzzy functions. J. Math. Anal. Appl. 91(2), 552–558 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lee-Kwang, H., Lee, J.-H.: A method for ranking fuzzy numbers and its application to decision-making. IEEE Trans. Fuzzy Syst. 7(6), 677–685 (1999)

    Article  Google Scholar 

  26. Luhandjula, M.K.: Fuzzy optimization: milestones and perspectives. Fuzzy Sets Syst. 274, 4–11 (2015)

    Article  MathSciNet  Google Scholar 

  27. Ramík, J., Rimanek, J.: Inequality relation between fuzzy numbers and its use in fuzzy optimization. Fuzzy Sets Syst. 16(2), 123–138 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  28. Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, New York (2006)

    MATH  Google Scholar 

  29. Słowínski, R.: Fuzzy Sets in Decision Analysis, Operations Research and Statistics. Kluwer, Boston (1998)

    Book  MATH  Google Scholar 

  30. Stefanini, L.: A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets Syst. 161, 1564–1584 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang, X., Kerre, E.E.: Reasonable properties for the ordering of fuzzy quantities. Fuzzy Sets Syst. 118, 375–385 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, Z.-X., Liu, Y.-L., Fan, Z.-P., Feng, B.: Ranking L-R fuzzy number based on deviation degree. Inform. Sci. 179, 2070–2077 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgement

The author is truly thankful to the anonymous reviewers and editors for their valuable comments and suggestions. The financial support through Early Career Research Award (ECR/2015/000467), Science and Engineering Research Board, Government of India is also gratefully acknowledged.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Varanasi, 221005, Uttar Pradesh, India

    Debdas Ghosh

Authors
  1. Debdas Ghosh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Debdas Ghosh .

Editor information

Editors and Affiliations

  1. Haldia Institute of Technology, Haldia, India

    Debasis Giri

  2. University of Central Florida, Orlando, Florida, USA

    Ram N. Mohapatra

  3. Freie Universität Berlin, Berlin, Germany

    Heinrich Begehr

  4. Fordham University, Bronx, New York, USA

    Mohammad S. Obaidat

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer Nature Singapore Pte Ltd.

About this paper

Cite this paper

Ghosh, D. (2017). A Davidon-Fletcher-Powell Type Quasi-Newton Method to Solve Fuzzy Optimization Problems. In: Giri, D., Mohapatra, R., Begehr, H., Obaidat, M. (eds) Mathematics and Computing. ICMC 2017. Communications in Computer and Information Science, vol 655. Springer, Singapore. https://doi.org/10.1007/978-981-10-4642-1_20

Download citation

  • DOI: https://doi.org/10.1007/978-981-10-4642-1_20

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-4641-4

  • Online ISBN: 978-981-10-4642-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation