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

  • Debdas GhoshEmail author
Original Paper


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.


Interval optimization Quasi-Newton method Generalized-Hukuhara differentiability Efficient solution 

Mathematics Subject Classification

90C30 65K05 



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.


  1. 1.
    Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, 3rd edn. Wiley, New York (2006)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bhurjee, A.K., Panda, G.: Efficient solution of interval optimization problem. Math. Methods Oper. Res. 76(3), 273–288 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 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)Google Scholar
  4. 4.
    Chakraborty, D., Ghosh, D.: Analytical fuzzy plane geometry II. Fuzzy Sets Syst. 243, 84–100 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ghosh, D.: A Newton method for capturing efficient solutions of interval optimization problems. Opsearch (2016). doi: 10.1007/s12597-016-0249-6
  8. 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. 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)MathSciNetCrossRefGoogle Scholar
  10. 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)zbMATHGoogle Scholar
  11. 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. 12.
    Ghosh, D., Chakraborty, D.: Analytical fuzzy plane geometry I. Fuzzy Sets Syst. 209, 66–83 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ghosh, D., Chakraborty, D.: Analytical fuzzy plane geometry III. Fuzzy Sets Syst. 283, 83–107 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ghosh, D., Chakraborty, D.: A new Pareto set generating method for multi-criteria optimization problems. Oper. Res. Lett. 42, 514–521 (2014)MathSciNetCrossRefGoogle Scholar
  17. 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)Google Scholar
  18. 18.
    Hansen, W.G.E.: Global Optimization Using Interval Analysis. Marcel Dekker Inc., New York (2004)zbMATHGoogle Scholar
  19. 19.
    Higham, N.J.: Accuracy and Stability of Numerical Algorithms. SIAM (2002)Google Scholar
  20. 20.
    Hladik, M.: Interval Linear Programming: A Survey. Nova Science Publishers, New York (2012)zbMATHGoogle Scholar
  21. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hukuhara, M.: Integration des applications mesurables dont la valeur est un compact convexe. Funkc Ekvacioj 10, 205–223 (1967)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Ishibuchi, H., Tanaka, H.: Multiobjective programming in optimization of the interval objective function. Eur. J. Oper. Res. 48(2), 219–225 (1990)CrossRefzbMATHGoogle Scholar
  25. 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)MathSciNetzbMATHGoogle Scholar
  26. 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)MathSciNetzbMATHGoogle Scholar
  27. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Li, W., Tian, X.: Numerical solution method for general interval quadratic programming. Appl. Math. Comput. 202(2), 589–595 (2008)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Liu, S.T., Wang, R.T.: A numerical solution method to interval quadratic programming. Appl. Math. Comput. 189(2), 1274–1281 (2007)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Luciana, T.G., Barrosb, L.C.: A note on the generalized difference and the generalized differentiability. Fuzzy Sets Syst. 280, 142–145 (2015)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Marin, M.: On existence and uniqueness in thermoelasticity of micropolar bodies. C. R. Acad. Sci. Paris Ser. II 321(12), 475–480 (1995)zbMATHGoogle Scholar
  32. 32.
    Marin, M.: An evolutionary equation in thermoelasticity of dipolar bodies. J. Math. Phys. 40(3), 1391–1399 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 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)Google Scholar
  34. 34.
    Markov, S.: Calculus for interval functions of a real variable. Computing 22, 325–337 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Moore, R.: Interval Anal. Prentice-Hall, Englewood Cliffs (1966)Google Scholar
  36. 36.
    Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM (2009)Google Scholar
  37. 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. 38.
    Nocedal, J.: Updating quasi-Newton matrices with limited storage. Math. Comput. 35, 773–782 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Nocedal, J., Stephen, W.: Numerical Optimization, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  40. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Rohn, J.: Positive definiteness and stability of interval matrices. SIAM J. Matrix Anal. Appl. 15, 175–184 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 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)Google Scholar
  43. 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)Google Scholar
  44. 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)CrossRefGoogle Scholar
  45. 45.
    Stefanini, L.: A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets Syst. 161, 1564–1584 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Wang, H., Zhang, R.: Optimality conditions and duality for arcwise connected interval optimization problems. Opsearch 52(4), 870–883 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Wu, H.C.: On interval-valued nonlinear programing problem. J. Math. Anal. Appl. 338(1), 299–316 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer India Pvt. Ltd. 2016

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Information Technology KalyaniKalyaniIndia

Personalised recommendations