Skip to main content
Springer Nature Link
Log in

An objective space cut and bound algorithm for convex multiplicative programmes

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

Multiplicative programming problems are global optimisation problems known to be NP-hard. In this paper we propose an objective space cut and bound algorithm for approximately solving convex multiplicative programming problems. This method is based on an objective space approximation algorithm for convex multi-objective programming problems. We show that this multi-objective optimisation algorithm can be changed into a cut and bound algorithm to solve convex multiplicative programming problems. We use an illustrative example to demonstrate the working of the algorithm. Computational experiments illustrate the superior performance of our algorithm compared to other methods from the literature.

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

Access this article

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

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Benson, H.P.: An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem. J. Glob. Optim. 13, 1–24 (1998)

    Article  Google Scholar 

  2. Benson, H.P.: An outcome space branch and bound-outer approximation algorithm for convex multiplicative programming. J. Glob. Optim. 15, 315–342 (1999)

    Article  Google Scholar 

  3. Benson, H.P., Boger, G.M.: Multiplicative programming problems: analysis and efficient point search heuristic. J. Optim. Theory Appl. 94, 487–510 (1997)

    Article  Google Scholar 

  4. Benson, H.P., Boger, G.M.: Outcome-space cutting-plane algorithm for linear multiplicative programming. J. Optim. Theory Appl. 104, 301–322 (2000)

    Article  Google Scholar 

  5. Chen, P.C., Hansen, P., Jaumard, B.: On-line and off-line vertex enumeration by adjacency lists. Oper. Res. Lett. 10, 403–409 (1991)

    Article  Google Scholar 

  6. Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005)

    Google Scholar 

  7. Ehrgott, M., Shao, L., Schöbel, A.: An approximation algorithm for convex multi-objective programming problems. J. Glob. Optim. 50, 397–416 (2011)

    Article  Google Scholar 

  8. Falk, J.E., Palocsay, S.W.: Image space analysis of generalized fractional programs. J. Glob. Optim. 4, 63–88 (1994)

    Article  Google Scholar 

  9. Gao, Y., Xu, C., Yang, Y.: An outcome-space finite algorithm for solving linear multiplicative programming. Appl. Math. Comput. 179, 494–505 (2006)

    Article  Google Scholar 

  10. Gao, Y., Wu, G., Ma, W.: A new global optimization approach for convex multiplicative programming. Appl. Math. Comput. 216, 1206–1218 (2010)

    Article  Google Scholar 

  11. Geoffrion, A.: Solving bicriterion mathematical programs. Oper. Res. 15, 39–54 (1967)

    Article  Google Scholar 

  12. Henderson, J.M., Quandt, R.E.: Microeconomic Theory: A Mathematical Approach. McGraw-Hill, New York (1971)

    Google Scholar 

  13. Jaumard, B., Meyer, C., Tuy, H.: Generalized convex multiplicative programming via quasiconcave minimization. J. Glob. Optim. 10, 229–256 (1997)

    Article  Google Scholar 

  14. Konno, H., Kuno, T.: Generalized linear multiplicative and fractional programming. Ann. Oper. Res. 25, 147–162 (1990)

    Article  Google Scholar 

  15. Konno, H., Kuno, T.: Linear multiplicative programming. Math. Program. 56, 51–64 (1992)

    Article  Google Scholar 

  16. Konno, H., Kuno, T., Yajima, Y.: Global minimization of a generalized convex multiplicative function. J. Glob. Optim. 4, 47–62 (1994)

    Article  Google Scholar 

  17. Kuno, T.: A finite branch-and-bound algorithm for linear multiplicative programming. Comput. Optim. Appl. 20, 119–135 (2001)

    Article  Google Scholar 

  18. Kuno, T., Yajima, Y., Konno, H.: An outer approximation method for minimizing the product of several convex functions on a convex set. J. Glob. Optim. 3, 325–335 (1993)

    Article  Google Scholar 

  19. Maling, K., Mueller, S.H., Heller, W.R.: On finding most optimal optional rectangular package plans. In: Proceedings of the ACM IEEE 19th Design Automation Conference, pp. 663–670. IEEE Service Center, Piscataway (1982)

  20. Matsui, T.: NP-hardness of linear multiplicative programming and related problems. J. Glob. Optim. 9, 113–119 (1996)

    Article  Google Scholar 

  21. Miettinen, K., Mäkelä, M.M., Kaario, K.: Experiments with classification-based scalarizing functions in interactive multiobjective optimization. Eur. J. Oper. Res. 175, 931–947 (2006)

    Article  Google Scholar 

  22. Oliveira, R.M., Ferreira, P.A.V.: A convex analysis approach for convex multiplicative programming. J. Glob. Optim. 41, 579–592 (2008)

    Article  Google Scholar 

  23. Thoai, N.V.: A global optimization approach for solving the convex multiplicativeprogramming problem. J. Optim. Theory Appl. 1, 341–357 (1991)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Key Laboratory of Advanced Control of Iron and Steel Process (Ministry of Education), School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing, 100083, China

    Lizhen Shao

  2. Department of Management Science, Lanacaster University Management School, Bailrigg, Lancaster, LA1 4YX, UK

    Matthias Ehrgott

  3. Department of Engineering Science, The University of Auckland, Private Bag 92019, Auckland, 1142, New Zealand

    Matthias Ehrgott

Authors
  1. Lizhen Shao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matthias Ehrgott
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lizhen Shao.

Additional information

This work has been partially supported by the National Natural Science Foundation of China (No. 81000650), the Ph.D. Programs Foundation of Ministry of Education of China (No. 20100006120016), Beijing Key Discipline Development Program (No. XK100080537) and National High-tech Research Development Program of China (863 Program) (No. 2013AA040705).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shao, L., Ehrgott, M. An objective space cut and bound algorithm for convex multiplicative programmes. J Glob Optim 58, 711–728 (2014). https://doi.org/10.1007/s10898-013-0102-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-013-0102-x

Keywords