Skip to main content
SpringerLink
Log in

Testing weak optimality of a given solution in interval linear programming revisited: NP-hardness proof, algorithm and some polynomially-solvable cases

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

We address the problem of testing weak optimality of a given solution of a given interval linear program. The problem was recently wrongly stated to be polynomially solvable. We disprove it. We show that the problem is NP-hard in general. We propose a new algorithm for the problem, based on orthant decomposition and solving linear systems. Running time of the algorithm is exponential in the number of equality constraints. In particular, the proposed algorithm runs in polynomial time for interval linear programs with no equality constraints.

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

Similar content being viewed by others

References

  1. Allahdadi, M., Mishmast Nehi, H.: The optimal solution set of the interval linear programming problems. Optim. Lett. 7(8), 1893–1911 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ashayerinasab, H.A., Nehi, H.M., Allahdadi, M.: Solving the interval linear programming problem: a new algorithm for a general case. Expert Syst. Appl. 93(Suppl. C), 39–49 (2018)

    Article  Google Scholar 

  3. Balas, E.: Disjunctive programming: properties of the convex hull of feasible points. Discrete Appl. Math. 89(1–3), 3–44 (1998). https://doi.org/10.1016/S0166-218X(98)00136-X

    Article  MathSciNet  MATH  Google Scholar 

  4. Beeck, H.: Linear programming with inexact data. technical report TUM-ISU-7830, Technical University of Munich, Munich (1978)

  5. Chinneck, J.W., Ramadan, K.: Linear programming with interval coefficients. J. Oper. Res. Soc. 51(2), 209–220 (2000)

    Article  MATH  Google Scholar 

  6. Gabrel, V., Murat, C.: Robustness and duality in linear programming. J. Oper. Res. Soc. 61(8), 1288–1296 (2010)

    Article  MATH  Google Scholar 

  7. Garajová, E., Hladík, M., Rada, M.: Interval Linear Programming Under Transformations: Optimal Solutions and Optimal Value Range (2018). arXiv:1802.09872

  8. Hladík, M.: Interval linear programming: a survey. In: Mann, Z.Á. (ed.) Linear Programming-New Frontiers in Theory and Applications, Mathematics Research Developments, pp. 85–120. Nova Science Publishers, Hauppauge (2012)

    Google Scholar 

  9. Hladík, M.: Weak and strong solvability of interval linear systems of equations and inequalities. Linear Algebra Appl. 438(11), 4156–4165 (2013). https://doi.org/10.1016/j.laa.2013.02.012

    Article  MathSciNet  MATH  Google Scholar 

  10. Hladík, M.: How to determine basis stability in interval linear programming. Optim. Lett. 8(1), 375–389 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hladík, M.: Robust optimal solutions in interval linear programming with forall-exists quantifiers. Eur. J. Oper. Res. 254(3), 705–714 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hladík, M.: On strong optimality of interval linear programming. Optim. Lett. 11(7), 1459–1468 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hladík, M.: Transformations of interval linear systems of equations and inequalities. Linear Multilinear Algebra 65(2), 211–223 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Koníčková, J.: Sufficient condition of basis stability of an interval linear programming problem. Z. Angew. Math. Mech. 81(Suppl. 3), 677–678 (2001)

    MATH  Google Scholar 

  15. Krawczyk, R.: Fehlerabschätzung bei linearer Optimierung. In: Nickel, K. (ed.) Interval Mathemantics: Proceedings of the International Symposium, Karlsruhe, West Germany, May 20–24, 1975, LNCS, vol. 29, pp. 215–222. Springer (1975)

  16. Li, H.: Necessary and sufficient conditions for unified optimality of interval linear program in the general form. Linear Algebra Appl. 484, 154–174 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, W.: A note on dependency between interval linear systems. Optim. Lett. 9(4), 795–797 (2015). https://doi.org/10.1007/s11590-014-0791-1

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, W., Liu, P., Li, H.: Checking weak optimality of the solution to interval linear program in the general form. Optim. Lett. 10(1), 77–88 (2016). https://doi.org/10.1007/s11590-015-0856-9

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, W., Liu, X., Li, H.: Generalized solutions to interval linear programmes and related necessary and sufficient optimality conditions. Optim. Methods Softw. 30(3), 516–530 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Luo, J., Li, W., Wang, Q.: Checking strong optimality of interval linear programming with inequality constraints and nonnegative constraints. J. Comput. Appl. Math. 260, 180–190 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Machost, B.: Numerische Behandlung des Simplexverfahrens mit intervallanalytischen Methoden. Technical Report 30, Berichte der Gesellschaft für Mathematik und Datenverarbeitung, 54 pages, Bonn (1970)

  22. Mráz, F.: Calculating the exact bounds of optimal values in LP with interval coefficients. Ann. Oper. Res. 81, 51–62 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Novotná, J., Hladík, M., Masařík, T.: Duality gap in interval linear programming. In: Zadnik Stirn, L. et al. (ed.) Proceedings of the 14th International Symposium on Operational Research SOR’17, pp. 501–506. Slovenian Society Informatika, Ljubljana, Slovenia (2017)

  24. Rohn, J.: Stability of the optimal basis of a linear program under uncertainty. Oper. Res. Lett. 13(1), 9–12 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rohn, J.: Complexity of some linear problems with interval data. Reliab. Comput. 3(3), 315–323 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rohn, J.: Interval linear programming. In: Fiedler, M., et al. (eds.) Linear Optimization Problems with Inexact Data, Chap. 3, pp. 79–100. Springer, New York (2006)

    Chapter  Google Scholar 

Download references

Acknowledgements

The work of M. Rada was supported by the Czech Science Foundation Grant P403/17-13086S. The work of M. Hladík and E. Garajová was supported by the Czech Science Foundation Grant P403/18-04735S. The work of E. Garajová was also supported by Grant No. 156317 of Grant Agency of Charles University.

Author information

Authors and Affiliations

  1. Department of Financial Accounting and Auditing, Faculty of Finance and Accounting, University of Economics, W. Churchill’s Sq. 4, 130 67, Prague, Czech Republic

    Miroslav Rada

  2. Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, 118 00, Prague, Czech Republic

    Milan Hladík & Elif Garajová

Authors
  1. Miroslav Rada
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Milan Hladík
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Elif Garajová
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miroslav Rada.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rada, M., Hladík, M. & Garajová, E. Testing weak optimality of a given solution in interval linear programming revisited: NP-hardness proof, algorithm and some polynomially-solvable cases. Optim Lett 13, 875–890 (2019). https://doi.org/10.1007/s11590-018-1289-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-018-1289-z

Keywords

Search

Navigation