Journal of Global Optimization

, Volume 56, Issue 3, pp 1123–1142

Branch-reduction-bound algorithm for generalized geometric programming

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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.

Keywords

Generalized geometric programming Global optimization Monotonic function Reduction operation 

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References

  1. 1.
    Hansen P., Jaumard B.: Reduction of indefinite quadratic programs to bilinear programs. J. Global Optim. 2(1), 41–60 (1992)CrossRefGoogle Scholar
  2. 2.
    Beightler C.S., Phillips D.T.: Applied Geometric Programming. Wiley, New York, NY (1976)Google Scholar
  3. 3.
    Avriel M., Williams A.C.: An extension of geometric programming with applications in engineering optimization. J. Eng. Math. 5(3), 187–199 (1971)CrossRefGoogle Scholar
  4. 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)CrossRefGoogle Scholar
  5. 5.
    Nand K.J.: Geometric programming based robot control design. Comput. Ind. Eng. 29(1–4), 631–635 (1995)Google Scholar
  6. 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)CrossRefGoogle Scholar
  7. 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)CrossRefGoogle Scholar
  8. 8.
    EI Barmi H., Dykstra R.L.: Restricted multinomial maximum likelihood estimation based upon Fenchel duality. Stat. Probab. Lett. 21, 121–130 (1994)CrossRefGoogle Scholar
  9. 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)Google Scholar
  10. 10.
    Jagannathan R.: A stochastic geometric programming problem with multiplicative recourse. Oper. Res. Lett. 9, 99–104 (1990)CrossRefGoogle Scholar
  11. 11.
    Maranas C.D., Floudas C.A.: Global optimization in generalized geometric programming. Comput. Chem. Eng. 21(4), 351–369 (1997)CrossRefGoogle Scholar
  12. 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)CrossRefGoogle Scholar
  13. 13.
    Ecker J.G.: Geometric programming: methods, computations and applications. SIAM Rev. 22(3), 338–362 (1980)CrossRefGoogle Scholar
  14. 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. 15.
    Passy U.: Generalized weighted mean programming. SIAM J. Appl. Math. 20, 763–778 (1971)CrossRefGoogle Scholar
  16. 16.
    Passy U., Wilde D.J.: Generalized polynomial optimization. J. Appl. Math. 15(5), 1344–1356 (1967)Google Scholar
  17. 17.
    Wang Y., Zhang K., Gao Y.: Global optimization of generalized geometric programming. Appl. Math. Comput. 48, 1505–1516 (2004)CrossRefGoogle Scholar
  18. 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)CrossRefGoogle Scholar
  19. 19.
    Shen P., Zhang K.: Global optimization of signomial geometric programming using linear relaxation. Appl. Math. Comput. 150, 99–114 (2004)CrossRefGoogle Scholar
  20. 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)CrossRefGoogle Scholar
  21. 21.
    Wang Y., Liang Z.: A deterministic global optimization algorithm for generalized geometric programming. Appl. Math. Comput. 168, 722–737 (2005)CrossRefGoogle Scholar
  22. 22.
    Shen P., Jiao H.: A new rectangle branch-and-pruning approach for generalized geometric programming. Appl. Math. Comput. 183, 1027–1038 (2006)CrossRefGoogle Scholar
  23. 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)CrossRefGoogle Scholar
  24. 24.
    Sherali H.D.: Global optimization of nonconvex polynomial programming problems having rational exponents. J. Glob. Optim. 12, 267–283 (1998)CrossRefGoogle Scholar
  25. 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)CrossRefGoogle Scholar
  26. 26.
    Lu H.C., Floudas C.A.: Convex relaxation for solving posynomial programs. J. Glob. Optim. 46, 147–154 (2010)CrossRefGoogle Scholar
  27. 27.
    Tsai J.F., Lin M.H.: An efficient global approach for posynomial geometric programming problems. INFORMS J. Comput. 23(3), 483–492 (2011)CrossRefGoogle Scholar
  28. 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)CrossRefGoogle Scholar
  29. 29.
    Shen P., Ma Y., Chen Y.Y.: A robust algorithm for generalized geometric programming. J. Glob. Optim. 41, 593–612 (2008)CrossRefGoogle Scholar
  30. 30.
    Tuy H.: Polynomial optimization: a robust approach. Pac. J. Optim. 1, 357–374 (2005)Google Scholar
  31. 31.
    Porn R., Bjork K.M., Westerlund T.: Global solution of optimization of problems with signomial parts. Discrete Optim. 5, 108–120 (2008)CrossRefGoogle Scholar
  32. 32.
    Lundell A., Westerlund T.: Convex underestimation strategies for signomial functions. Optim. Methods Softw. 24, 505–522 (2009)CrossRefGoogle Scholar
  33. 33.
    Lundell A., Westerlund J., Westerlund T.: Some transformation techniques with applications in global optimization. J. Glob. Optim. 43, 391–405 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2012

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceHenan Normal UniversityXinxiangPeople’s Republic of China

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