Skip to main content
SpringerLink
Log in

Branch-reduction-bound algorithm for generalized geometric programming

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

Abstract

This article presents a branch-reduction-bound algorithm for globally solving the generalized geometric programming problem. To solve the problem, an equivalent monotonic optimization problem whose objective function is just a simple univariate is proposed by exploiting the particularity of this problem. In contrast to usual branch-and-bound methods, in the algorithm the upper bound of the subproblem in each node is calculated easily by arithmetic expressions. Also, a reduction operation is introduced to reduce the growth of the branching tree during the algorithm search. The proposed algorithm is proven to be convergent and guarantees to find an approximative solution that is close to the actual optimal solution. Finally, numerical examples are given to illustrate the feasibility and efficiency of the present algorithm.

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

Similar content being viewed by others

References

  1. Hansen P., Jaumard B.: Reduction of indefinite quadratic programs to bilinear programs. J. Global Optim. 2(1), 41–60 (1992)

    Article  Google Scholar 

  2. Beightler C.S., Phillips D.T.: Applied Geometric Programming. Wiley, New York, NY (1976)

    Google Scholar 

  3. Avriel M., Williams A.C.: An extension of geometric programming with applications in engineering optimization. J. Eng. Math. 5(3), 187–199 (1971)

    Article  Google Scholar 

  4. Jefferson T.R., Scott C.H.: Generalized geometric programming applied to problems of optimal control: I.theory. J. Optim. Theory Appl. 26, 117–129 (1978)

    Article  Google Scholar 

  5. Nand K.J.: Geometric programming based robot control design. Comput. Ind. Eng. 29(1–4), 631–635 (1995)

    Google Scholar 

  6. Das K., Roy T.K., Maiti M.: Multi-item inventory model with under imprecise objective and restrictions: a geometric programming approach. Prod. Plan. Control 11(8), 781–788 (2000)

    Article  Google Scholar 

  7. Jae Chul C., Bricker Dennis L.: Effectiveness of a geometric programming algorithm for optimization of machining economics models. Comput. Oper. Res. 23(10), 957–961 (1996)

    Article  Google Scholar 

  8. EI Barmi H., Dykstra R.L.: Restricted multinomial maximum likelihood estimation based upon Fenchel duality. Stat. Probab. Lett. 21, 121–130 (1994)

    Article  Google Scholar 

  9. Bricker, D.L., Kortanek, K.O., Xu, L.: Maximum linklihood estimates with order restrictions on probabilities and odds ratios: a geometric programming approach. Applied Mathematical and Computational Sciences, University of IA, Iowa City, IA (1995)

  10. Jagannathan R.: A stochastic geometric programming problem with multiplicative recourse. Oper. Res. Lett. 9, 99–104 (1990)

    Article  Google Scholar 

  11. Maranas C.D., Floudas C.A.: Global optimization in generalized geometric programming. Comput. Chem. Eng. 21(4), 351–369 (1997)

    Article  Google Scholar 

  12. Rijckaert M.J., Matens X.M.: Analysis and optimization of the Williams-Otto process by geometric programming. AICHE J. 20(4), 742–750 (1974)

    Article  Google Scholar 

  13. Ecker J.G.: Geometric programming: methods, computations and applications. SIAM Rev. 22(3), 338–362 (1980)

    Article  Google Scholar 

  14. Kortanek K.O., Xiaojie X., Yinyu Y.: An infeasible interior-point algorithm for solving primal and dual geometric programs. Math. Program. 76, 155–181 (1996)

    Google Scholar 

  15. Passy U.: Generalized weighted mean programming. SIAM J. Appl. Math. 20, 763–778 (1971)

    Article  Google Scholar 

  16. Passy U., Wilde D.J.: Generalized polynomial optimization. J. Appl. Math. 15(5), 1344–1356 (1967)

    Google Scholar 

  17. Wang Y., Zhang K., Gao Y.: Global optimization of generalized geometric programming. Appl. Math. Comput. 48, 1505–1516 (2004)

    Article  Google Scholar 

  18. Qu S., Zhang K., Wang F.: A global optimization using linear relaxation for generalized geometric programming. Eur. J. Oper. Res. 190, 345–356 (2008)

    Article  Google Scholar 

  19. Shen P., Zhang K.: Global optimization of signomial geometric programming using linear relaxation. Appl. Math. Comput. 150, 99–114 (2004)

    Article  Google Scholar 

  20. Qu S., Zhang K., Ji Y.: A new global optimization algorithm for signomial geometric programming via Lagrangian relaxation. Appl. Math. Comput. 182(2), 886–894 (2007)

    Article  Google Scholar 

  21. Wang Y., Liang Z.: A deterministic global optimization algorithm for generalized geometric programming. Appl. Math. Comput. 168, 722–737 (2005)

    Article  Google Scholar 

  22. Shen P., Jiao H.: A new rectangle branch-and-pruning approach for generalized geometric programming. Appl. Math. Comput. 183, 1027–1038 (2006)

    Article  Google Scholar 

  23. Sherali H.D., Tuncbilek C.H.: A global optimization algorithm for polynomial programming problems using a formulation-linearzation technique. J. Glob. Optim. 2, 101–112 (1992)

    Article  Google Scholar 

  24. Sherali H.D.: Global optimization of nonconvex polynomial programming problems having rational exponents. J. Glob. Optim. 12, 267–283 (1998)

    Article  Google Scholar 

  25. Gounaris C.E., Floudas C.A.: Convexity of products of univariate functions and convexification transformations for geometric programming. J. Optim. Theory Appl. 138, 407–427 (2008)

    Article  Google Scholar 

  26. Lu H.C., Floudas C.A.: Convex relaxation for solving posynomial programs. J. Glob. Optim. 46, 147–154 (2010)

    Article  Google Scholar 

  27. Tsai J.F., Lin M.H.: An efficient global approach for posynomial geometric programming problems. INFORMS J. Comput. 23(3), 483–492 (2011)

    Article  Google Scholar 

  28. Wang Y., Li T., Liang Z.: A general algorithm for solving generalized geometric programming with nonpositive degree of difficulty. Comput. Optim. Appl. 44, 139–158 (2009)

    Article  Google Scholar 

  29. Shen P., Ma Y., Chen Y.Y.: A robust algorithm for generalized geometric programming. J. Glob. Optim. 41, 593–612 (2008)

    Article  Google Scholar 

  30. Tuy H.: Polynomial optimization: a robust approach. Pac. J. Optim. 1, 357–374 (2005)

    Google Scholar 

  31. Porn R., Bjork K.M., Westerlund T.: Global solution of optimization of problems with signomial parts. Discrete Optim. 5, 108–120 (2008)

    Article  Google Scholar 

  32. Lundell A., Westerlund T.: Convex underestimation strategies for signomial functions. Optim. Methods Softw. 24, 505–522 (2009)

    Article  Google Scholar 

  33. Lundell A., Westerlund J., Westerlund T.: Some transformation techniques with applications in global optimization. J. Glob. Optim. 43, 391–405 (2009)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Peiping Shen & Xiaoai Li

Authors
  1. Peiping Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoai Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peiping Shen.

Additional information

Research supported by NSFC (11171094; 11171368) and Innovation Scientists and Technicians Troop Construction Projects of Henan Province (114200510011).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shen, P., Li, X. Branch-reduction-bound algorithm for generalized geometric programming. J Glob Optim 56, 1123–1142 (2013). https://doi.org/10.1007/s10898-012-9933-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-012-9933-0

Keywords

Search

Navigation