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Journal of Applied Mathematics and Computing

, Volume 58, Issue 1–2, pp 193–217 | Cite as

A saddle point characterization of efficient solutions for interval optimization problems

  • Debdulal Ghosh
  • Debdas GhoshEmail author
  • Sushil Kumar Bhuiya
  • Lakshmi Kanta Patra
Original Research
  • 190 Downloads

Abstract

In this article, we attempt to characterize efficient solutions of constrained interval optimization problems. Towards this aim, at first, we study a scalarization characterization to capture efficient solutions. Then, with the help of saddle point of a newly introduced Lagrangian function, we investigate efficient solutions of an interval optimization problem. Several parts of the results are supported with numerical and pictorial illustration.

Keywords

Interval optimization Efficient solution Lagrangian function Saddle point 

Mathematics Subject Classification

90C30 65K05 

Notes

Acknowledgements

Debdas Ghosh gratefully acknowledges the financial support of Early Career Research Award (ECR/2015/000467), Science and Engineering Research Board, Government of India. The authors are thankful to the anonymous reviewers’ suggestion to improve the perfection of the article.

References

  1. 1.
    Bhurjee, A.K., Panda, G.: Efficient solution of interval optimization problems. Math. Methods Oper. Res. 76(3), 273–288 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chakraborty, D., Ghosh, D.: Analytical fuzzy plane geometry II. Fuzzy Sets Syst. 243, 84–10 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chalco-Cano, Y., Lodwick, W.A., Rufian-Lizana, A.: Optimality conditions of type KKT for optimization problem with interval-valued objective function via generalized derivative. Fuzzy Optim. Decis. Mak. 12(3), 305–322 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chen, S.H., Wu, J., Chen, Y.D.: Interval optimization for uncertain structures. Finite Elem. Anal. Des. 40, 1379–1398 (2004)CrossRefGoogle Scholar
  5. 5.
    Cheng, J., Liu, Z., Wu, Z., Tang, M., Tan, J.: Direct optimization of uncertain structures based on degree of interval constraint violation. Comput. Struct. 164, 83–94 (2016)CrossRefGoogle Scholar
  6. 6.
    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
  7. 7.
    Frits, E.R., Rev, E., Lelkes, Z., Markot, M., Csendes, T.: Application of an interval optimization method for studying feasibility of batch extractive distillation. In: Proceedings of the International Workshop on Global Optimization, Almeria, pp. 103–108 (2005)Google Scholar
  8. 8.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ghosh, D.: A Newton method for capturing efficient solutions of interval optimization problems. Opesearch 53, 648–665 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ghosh, D., Chakraborty, D.: Analytical fuzzy plane geometry I. Fuzzy Sets Syst. 209, 66–83 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ghosh, D., Chakraborty, D.: Analytical fuzzy plane geometry III. Fuzzy Sets Syst. 283, 83–107 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ghosh, D., Chakraborty, D.: A method for capturing the entire fuzzy non-dominated set of a fuzzy multi-criteria optimization problem. J. Intell. Fuzzy Syst. 26, 1223–1234 (2014)zbMATHGoogle Scholar
  13. 13.
    Ghosh, D., Chakraborty, D.: A new Pareto set generating method for multi-criteria optimization problems. Oper. Res. Lett. 42, 514–521 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ghosh, D., Chakraborty, D.: Quadratic interpolation technique to minimize univariable fuzzy functions. Int. J. Appl. Comput. Math. 3(2), 527–547 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hong, F.-X., Li, D.-F.: Nonlinear programming method for interval-valued \(n\)-person cooperative games. Oper. Res. Int. J. 17(2), 479–497 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kumar, P.: Inventory model with price-dependent demand rate and no shortages: an interval-valued linear fractional programming approach. Oper. Res. Appl. Int. J. 2(4), 1–14 (2015)Google Scholar
  18. 18.
    Oliveria, C., Antunes, C.H.: Multiple objective linear programming models with interval coefficients-an illustrated review. Eur. J. Oper. Res. 181, 1434–1463 (2007)CrossRefzbMATHGoogle Scholar
  19. 19.
    Singh, D., Dar, B.A., Goyal, A.: KKT optimality conditions for interval-valued optimization problems. J. Nonlinear Anal. Optim. 5(2), 91–103 (2014)MathSciNetGoogle Scholar
  20. 20.
    Wang, H., Zhang, R.: Optimality conditions and duality for arcwise connected interval optimization problems. Opsearch 52(4), 870–883 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Wu, H.C.: The Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function. Eur. J. Oper. Res. 176(1), 46–59 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wu, H.C.: Dulity theory for optimization problems with interval-valued objective fuction. J. Optim. Theory Appl. 144(3), 615–628 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Wu, H.C.: Duality theory in interval-valued linear programming problems. J. Optim. Theory Appl. 150, 298–316 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ying, Z.-Y., Xi, Y.-G., Zhang, Z.-J.: Test on reachability of a Robot to an object. In: The Proceedings of IEEE International Conference on Robotics and Automation, IEEE Xplore (2002)Google Scholar
  25. 25.
    Zhang, J., Liu, S., Li, L., Feng, Q.: The KKT optimality conditions in a class of generalized convex optimization problems with an interval-valued objective function. Optim. Lett. 8(2), 607–631 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2017

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology KharagpurKharagpurIndia
  2. 2.Department of Mathematical SciencesIndian Institute of Technology (BHU)VaranasiIndia

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