Skip to main content
SpringerLink
Log in

A Quasi-Newton Method with Rank-Two Update to Solve Interval Optimization Problems

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

In this study, a quasi-Newton method is developed to obtain efficient solutions of interval optimization problems. The idea of generalized Hukuhara differentiability for multi-variable interval-valued functions is employed to derive the quasi-Newton method. Through an inverse-Hessian approximation with rank-two modification, the proposed technique sidesteps the high computational cost for the computation of inverse-Hessian in Newton method for interval optimization problems. The rank-two modification of inverse-Hessian approximation is applied to generate the iterative points in the quasi-Newton technique. A sequential algorithm and the convergence result of the derived method are also presented. It is obtained that the sequence in the proposed method has superlinear convergence rate. The method is also found to have quadratic termination property. Two numerical examples are provided to illustrate the developed technique.

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

    Book  MATH  Google Scholar 

  2. Bhurjee, A.K., Panda, G.: Efficient solution of interval optimization problem. Math. Methods Oper. Res. 76(3), 273–288 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boggs, P.T., Byrd, R.H., Schnabel, R.B.: Numerical optimization 1984. In: Proceedings of the SIAM Conference on Numerical Optimization. Boulder, Colorado, June 12–14, 1984, Vol. 20, Siam (1985)

  4. Chakraborty, D., Ghosh, D.: Analytical fuzzy plane geometry II. Fuzzy Sets Syst. 243, 84–100 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chalco-Cano, Y., Rufian-Lizana, A., Roman-Flores, H., Jimenez-Gamero, M.D.: Calculus for interval-valued functions using generalized Hukuhara derivative and applications. Fuzzy Sets Syst. 219, 49–67 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Ghosh, D.: A Newton method for capturing efficient solutions of interval optimization problems. Opsearch (2016). doi:10.1007/s12597-016-0249-6

  8. Ghosh, D.: Newton method to obtain efficient solutions of the optimization problems with interval-valued objective functions. J. Appl. Math. Comput. (2016). doi:10.1007/s12190-016-0990-2

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

    Article  MathSciNet  Google Scholar 

  10. Ghosh, D., Chakraborty, D.: A method for capturing the entire fuzzy non-dominated set of a fuzzy multi-criteria optimization problem. J. Intel. Fuzzy Syst. 26, 1223–1234 (2014)

    MATH  Google Scholar 

  11. Ghosh, D., Chakraborty, D.: Quadratic interpolation technique to minimize univariable fuzzy functions. Int. J. Appl. Comput. Math. (2015). doi:10.1007/s40819-015-0123-x

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Ghosh, D., Chakraborty, D.: On general form of fuzzy lines and its application in fuzzy line fitting. J. Intel. Fuzzy Syst. 29, 659–671 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ghosh, D., Chakraborty, D.: A direction based classical method to obtain complete Pareto set of multi-criteria optimization problems. Opsearch 52(2), 340–366 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ghosh, D., Chakraborty, D.: A new Pareto set generating method for multi-criteria optimization problems. Oper. Res. Lett. 42, 514–521 (2014)

    Article  MathSciNet  Google Scholar 

  17. Ghosh, D., Chakraborty, D.: Ideal Cone: a new method to generate complete pareto set of multi-criteria optimization problems. In: Mathematics and Computing 2013, Vol. 91, Springer, Proceedings in Mathematics and Statistics, pp. 171–190 (2013)

  18. Hansen, W.G.E.: Global Optimization Using Interval Analysis. Marcel Dekker Inc., New York (2004)

    MATH  Google Scholar 

  19. Higham, N.J.: Accuracy and Stability of Numerical Algorithms. SIAM (2002)

  20. Hladik, M.: Interval Linear Programming: A Survey. Nova Science Publishers, New York (2012)

    MATH  Google Scholar 

  21. Hu, B., Wang, S.: A novel approach in uncertain programming, part I: new arithmetic and order relation of interval numbers. J. Ind. Manag. Optim. 2(4), 351–371 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hu, B., Wang, S.: A novel approach in uncertain programming, part II: a class of constrained nonlinear programming with interval objective function. J. Ind. Manag. Optim. 2(4), 373–385 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hukuhara, M.: Integration des applications mesurables dont la valeur est un compact convexe. Funkc Ekvacioj 10, 205–223 (1967)

    MathSciNet  MATH  Google Scholar 

  24. Ishibuchi, H., Tanaka, H.: Multiobjective programming in optimization of the interval objective function. Eur. J. Oper. Res. 48(2), 219–225 (1990)

    Article  MATH  Google Scholar 

  25. Jayswal, A., Stancu-Minasian, I., Ahmed, I.: On sufficiency and duality for a class of interval-valued programming problems. Appl. Math. Comput. 218(8), 4119–4127 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Jeyakumar, V., Li, G.Y.: Robust duality for fractional programming problems with constraint-wise data uncertainty. Eur. J. Oper. Res. 151(2), 292–303 (2011)

    MathSciNet  MATH  Google Scholar 

  27. Jiang, C., Han, X., Liu, G.R.: A nonlinear interval number programming method for uncertain optimization problems. Eur. J. Oper. Res. 188(1), 1–13 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Li, W., Tian, X.: Numerical solution method for general interval quadratic programming. Appl. Math. Comput. 202(2), 589–595 (2008)

    MathSciNet  MATH  Google Scholar 

  29. Liu, S.T., Wang, R.T.: A numerical solution method to interval quadratic programming. Appl. Math. Comput. 189(2), 1274–1281 (2007)

    MathSciNet  MATH  Google Scholar 

  30. Luciana, T.G., Barrosb, L.C.: A note on the generalized difference and the generalized differentiability. Fuzzy Sets Syst. 280, 142–145 (2015)

    Article  MathSciNet  Google Scholar 

  31. Marin, M.: On existence and uniqueness in thermoelasticity of micropolar bodies. C. R. Acad. Sci. Paris Ser. II 321(12), 475–480 (1995)

    MATH  Google Scholar 

  32. Marin, M.: An evolutionary equation in thermoelasticity of dipolar bodies. J. Math. Phys. 40(3), 1391–1399 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  33. Marin, M., Agarwal, R.P., Mahmoud, S.R.: Nonsimple material problems addressed by the Lagrange’s identity. Bound. Value Prob. 1–14, Article No. 135 (2013)

  34. Markov, S.: Calculus for interval functions of a real variable. Computing 22, 325–337 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  35. Moore, R.: Interval Anal. Prentice-Hall, Englewood Cliffs (1966)

    Google Scholar 

  36. Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM (2009)

  37. Neumaier, A.: Interval Methods for Systems of Equation, Encyclopedia of Mathematics and Its Applications, vol. 37. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  38. Nocedal, J.: Updating quasi-Newton matrices with limited storage. Math. Comput. 35, 773–782 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  39. Nocedal, J., Stephen, W.: Numerical Optimization, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  40. 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 

  41. Rohn, J.: Positive definiteness and stability of interval matrices. SIAM J. Matrix Anal. Appl. 15, 175–184 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  42. Sengupta, A., Pal, T.K.: Fuzzy Preference Ordering of Interval Numbers in Decision Problems. Series on Studies in Fuzziness and Soft Computing, vol. 238. Springer, Berlin (2009)

  43. Stahl, T.: Interval methods for bounding the range of Polynomials and solving sytems of nonlinear equation. PhD thesis. Johannes Kepler University Linz, Austria (1994)

  44. Stefanini, L.: A Generalization of Hukuhara Difference, Soft Methods for Handling Variability and Imprecision. Series on Advances in Soft Computing, vol. 48, pp. 203–210. Springer, Berlin (2008)

    Book  Google Scholar 

  45. 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 

  46. Wang, H., Zhang, R.: Optimality conditions and duality for arcwise connected interval optimization problems. Opsearch 52(4), 870–883 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  47. Wu, H.C.: On interval-valued nonlinear programing problem. J. Math. Anal. Appl. 338(1), 299–316 (2008)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

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

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Information Technology Kalyani, Kalyani, 741235, West Bengal, 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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ghosh, D. A Quasi-Newton Method with Rank-Two Update to Solve Interval Optimization Problems. Int. J. Appl. Comput. Math 3, 1719–1738 (2017). https://doi.org/10.1007/s40819-016-0202-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40819-016-0202-7

Keywords

Mathematics Subject Classification

Search

Navigation