[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Global algorithm for solving linear multiplicative programming problems

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

Abstract

This article presents a rectangular branch-and-bound algorithm with standard bisection rule for solving linear multiplicative problem (LMP). In this algorithm, a novel linear relaxation technique is presented for deriving the linear relaxation programming of problem LMP, which has separable characteristics and can be used to acquire the upper bound of the optimal value to problem LMP. Thus, to obtain a global optimal solution for problem LMP, the main computational work of the algorithm involves the solutions of a sequence of linear programming problems. Moreover, the proof of its convergence property and the numerical result show the feasibility and efficiency of the presented algorithm.

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
£29.99 /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 includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Konno, H., Wantanabe, H.: Bond portfolio optimization problems and their applications to index tracking. J. Oper. Res. Soc. Jpn. 39(3), 295–306 (1994)

    Article  Google Scholar 

  2. Shen, P.P., Yang, L.P., Liang, Y.C.: Range division and contraction algorithm for a class of global optimization problems. Appl. Math. Comput. 242, 116–126 (2014)

    MathSciNet  MATH  Google Scholar 

  3. Maranas, C.D., Androulakis, I.P., Flounda, C.A., Berger, A.J., Mulvey, J.M.: Solving long-term financial planning problems via global optimization. J. Econ. Dyn. Control 21, 1405–1425 (1997)

    Article  MathSciNet  Google Scholar 

  4. Kahl, F., Agarwal, S., Chandraker, M.K., Kriegman, D., Belongies, S.: Practical global optimization for multiview geometry. Int. J. Comput. Vis. 79(3), 271–284 (2008)

    Article  Google Scholar 

  5. Mulvey, J.M., Vanderbei, R.J., Zenios, S.A.: Robust optimization of large-scale systems. Oper. Res. 43, 264–281 (1995)

    Article  MathSciNet  Google Scholar 

  6. Cambini, A., Martein, L.: Generalized Convexity and Optimization: Theory and Applications. Lecture Notes in Economics and Mathematical Systems. Springer, Berlin (2009)

    MATH  Google Scholar 

  7. Cambini, R., Sodini, C.: On the minimization of a class of generalized linear functions on a flow polytope. Optimization 63(10), 1449–1464 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Tuy, H.: Convex Analysis and Global Optimization, 2nd edn. Springer Nature, Berlin (2016)

    Book  Google Scholar 

  10. Zhao, Y.F., Liu, S.Y.: Global optimization algorithm for mixed integer quadratically constrained quadratic program. J. Comput. Appl. Math. 319, 159–169 (2017)

    Article  MathSciNet  Google Scholar 

  11. Wang, C.F., Liu, S.Y., Shen, P.P.: Global minimization of a generalized linear multiplicative programming. Appl. Math. Model. 36, 2446–2451 (2012)

    Article  MathSciNet  Google Scholar 

  12. Jiao, H.W., Liu, S.Y., Chen, Y.Q.: Global optimization algorithm of a generalized linear multiplicative programming. J. Appl. Math. Comput. 40, 551–568 (2012)

    Article  MathSciNet  Google Scholar 

  13. Oliveira, R.M., Ferreira, A.V.P.: An outcome space approach for generalized convex multiplicative programs. J. Glob. Optim. 47, 107–118 (2010)

    Article  MathSciNet  Google Scholar 

  14. Gao, Y.L., Xu, C.X., Yang, Y.T.: Outcome-space branch and bound algorithm for solving linear multiplicative programming. Comput. Intell. Secur. 3801, 675–681 (2005)

    Article  Google Scholar 

  15. Zhou, X.G., Wu, K.: A method of acceleration for a class of multiplicative programming with exponent. J. Comput. Appl. Math. 223, 975–982 (2009)

    Article  MathSciNet  Google Scholar 

  16. Shen, P.P., Zhang, T.L., Wang, C.F.: Solving a class of generalized fractional programming problems using the feasibility of linear programs. J. Inequal. Appl. 2017, 147 (2017). https://doi.org/10.1186/s13660-017-1420-1

    Article  MathSciNet  MATH  Google Scholar 

  17. Cambini, R., Sodini, C.: A unifying approach to solve some classes of rank-three multiplicative and fractional programs involving linear functions. Eur. J. Oper. Res. 207, 25–29 (2010)

    Article  MathSciNet  Google Scholar 

  18. Cambini, R., Sodini, C.: Global optimization of a rank-two nonconvex program. Math. Methods Oper. Res. 71(1), 165–180 (2010)

    Article  MathSciNet  Google Scholar 

  19. Shen, P.P., Wang, C.F.: Linear decomposition approach for a class of nonconvex programming problems. J. Inequal. Appl. 2017, 74 (2017). https://doi.org/10.1186/s13660-017-1342-y

    Article  MathSciNet  MATH  Google Scholar 

  20. Yang, L.P., Shen, P.P., Pei, Y.G.: A global optimization approach for solving generalized nonlinear multiplicative programming problem. Abstr. Appl. Anal. (2014). https://doi.org/10.1155/2014/641909

  21. Wang, C.F., Bai, Y.Q., Shen, P.P.: A practicable branch-and-bound algorithm for globally solving multiplicative programming. Optimization 66(3), 397–405 (2017)

    Article  MathSciNet  Google Scholar 

  22. Liu, S.Y., Zhao, Y.F.: An efficient algorithm for globally solving generalized linear multiplicative programming. J. Comput. Appl. Math. 296, 840–847 (2016)

    Article  MathSciNet  Google Scholar 

  23. Benson, H.P., Boger, G.M.: Outcome-space cutting-plane algorithm for linear multiplicative programming. J. Optim. Theory Appl. 104, 301–332 (2000)

    Article  MathSciNet  Google Scholar 

  24. Cambini, R., Salvi, F.: A branch and reduce approach for solving a class of low rank DC programs. J. Comput. Appl. Math. 233, 492–501 (2009)

    Article  MathSciNet  Google Scholar 

  25. Cambini, R., Salvi, F.: Solving a class of low rank DC programs via a branch and bound approach: a computational experience. Oper. Res. Lett. 38(5), 354–357 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the responsible editor and the anonymous referees for their valuable comments and suggestions, which have greatly improved the earlier version of this paper. This paper is supported by National Natural Science Foundation of China (11671122, 11871196, 11171094), and by the Key Scientific Research Project for Universities in Henan Province (17A110006).

Author information

Authors and Affiliations

  1. College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, People’s Republic of China

    Peiping Shen & Bingdi Huang

Authors
  1. Peiping Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bingdi Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peiping Shen.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shen, P., Huang, B. Global algorithm for solving linear multiplicative programming problems. Optim Lett 14, 693–710 (2020). https://doi.org/10.1007/s11590-018-1378-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-018-1378-z

Keywords

Search

Navigation