Journal of Global Optimization

, Volume 55, Issue 2, pp 227–251 | Cite as

Multi-parametric disaggregation technique for global optimization of polynomial programming problems

  • João P. Teles
  • Pedro M. CastroEmail author
  • Henrique A. Matos


This paper discusses a power-based transformation technique that is especially useful when solving polynomial optimization problems, frequently occurring in science and engineering. The polynomial nonlinear problem is primarily transformed into a suitable reformulated problem containing new sets of discrete and continuous variables. By applying a term-wise disaggregation scheme combined with multi-parametric elements, an upper/lower bounding mixed-integer linear program can be derived for minimization/maximization problems. It can then be solved to global optimality through standard methods, with the original problem being approximated to a certain precision level, which can be as tight as desired. Furthermore, this technique can also be applied to signomial problems with rational exponents, after a few effortless algebraic transformations. Numerical examples taken from the literature are used to illustrate the effectiveness of the proposed approach.


Polynomial Signomial Optimization Mixed-integer linear programming Parameterization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Biegler L.T., Grossmann I.E.: Retrospective on optimization. Comput. Chem. Eng. 28, 1169–1192 (2004)CrossRefGoogle Scholar
  2. 2.
    Grossmann I.E., Biegler L.T.: Part II. Future perspective on optimisation. Comput. Chem. Eng. 28, 1193–1218 (2004)CrossRefGoogle Scholar
  3. 3.
    Floudas C.A., Akrotirianakis I.G., Caratzoulas S., Meyer C.A., Kallrath J.: Global optimization in the 21st century: advances and challenges. Comput. Chem. Eng. 29, 1185–1202 (2005)CrossRefGoogle Scholar
  4. 4.
    Floudas C.A., Pardalos P.M., Adjiman C.S., Esposito W.R., Gumus Z.H., Harding S.T., Klepeis J.L., Meyer C.A., Schweiger C.A.: Handbook of Test Problems in Local and Global Optimization. Kluwer, Boston (1999)Google Scholar
  5. 5.
    Dembo R.S.: A set of geometric programming test problems and their solutions. Math. Prog. 10, 192–213 (1976)CrossRefGoogle Scholar
  6. 6.
    Avriel M., Williams A.C.: An extension of geometric programming with applications in engineering optimization. J. Eng. Math. 5, 187–194 (1971)CrossRefGoogle Scholar
  7. 7.
    Karuppiah R., Grossmann I.E.: Global optimization for the synthesis of integrated water systems in chemical processes. Comput. Chem. Eng. 30, 650–673 (2006)CrossRefGoogle Scholar
  8. 8.
    Glinos K., Marquez F., Douglas J.: Simple, analytical criteria for the sequencing of distillation columns. AIChE J. 31, 683 (1985)CrossRefGoogle Scholar
  9. 9.
    Westerberg A.W.: The synthesis of distillation based separation systems. Comput. Chem. Eng. 9, 421 (1985)CrossRefGoogle Scholar
  10. 10.
    Haverly C.A.: Studies of the behaviour of recursion for the pooling problem. ACM SIGMAP Bull. 25, 19–28 (1978)CrossRefGoogle Scholar
  11. 11.
    Misener R., Gounaris C.E., Floudas C.A.: Mathematical modeling and global optimization of large-scale extended pooling problems with the (EPA) complex emission constraints. Comput. Chem. Eng. 34, 1432–1456 (2010)CrossRefGoogle Scholar
  12. 12.
    Zener C.: A mathematical aid in optimizing engineering designs. Proc. Natl. Acad. Sci. USA 47, 537–539 (1961)CrossRefGoogle Scholar
  13. 13.
    Zener C.: A further mathematical aid in optimizing engineering designs. Proc. Natl. Acad. Sci. USA 48, 512–522 (1962)CrossRefGoogle Scholar
  14. 14.
    Duffin R.J., Peterson E.L.: A mathematical aid in optimizing engineering designs. SIAM J. Appl. Math. 14, 1307–1349 (1966)CrossRefGoogle Scholar
  15. 15.
    Duffin R.J.: Linearizing geometric programs. SIAM Rev. 12, 211–227 (1970)CrossRefGoogle Scholar
  16. 16.
    Passy U., Wilde D.J.: Generalized polynomial optimization. SIAM J. Appl. Math. 15, 1344 (1967)CrossRefGoogle Scholar
  17. 17.
    Blau G.E., Wilde D.J.: Generalized polynomial programming. Can. J. Chem. Eng. 47, 317 (1969)CrossRefGoogle Scholar
  18. 18.
    Avriel M., Williams A.C.: Complementary geometric programs. SIAM J. Appl. Math. 19, 125–141 (1970)CrossRefGoogle Scholar
  19. 19.
    Duffin R.J., Peterson E.L.: Reversed geometric programs treated by harmonic means. Indiana Univ. Math. J. 22, 531–549 (1972)CrossRefGoogle Scholar
  20. 20.
    Duffin R.J., Peterson E.L.: Geometric programming with signomials. J. Optim. Theory Appl. 11, 3–35 (1973)CrossRefGoogle Scholar
  21. 21.
    Maranas C.D., Floudas C.A.: Finding all solutions of nonlinearly constrained systems of equations. J. Glob. Optim. 7, 143–182 (1995)CrossRefGoogle Scholar
  22. 22.
    Maranas C.D., Floudas C.A.: Global optimization in generalized geometric programming. Comput. Chem. Eng. 21, 351–370 (1997)CrossRefGoogle Scholar
  23. 23.
    Floudas C.A.: Deterministic Global Optimization: Theory, Algorithms and Applications. Kluwer, Dordrecht (2000)CrossRefGoogle Scholar
  24. 24.
    Floudas C.A., Pardalos P.M.: State of the Art in Global Optimization—Computational Methods and Applications. Kluwer, Dordrecht (1996)CrossRefGoogle Scholar
  25. 25.
    Shen P.: Linearization method of global optimization for generalized geometric programming. Appl. Math. Comput. 162, 353–370 (2005)CrossRefGoogle Scholar
  26. 26.
    Wang Y., Liang Z.: A deterministic global optimization algorithm for generalized geometric programming. Appl. Math. Comput. 168, 722–737 (2005)CrossRefGoogle Scholar
  27. 27.
    Meyer C.A., Floudas C.A.: Trilinear monomials with mixed sign domains: facets of the convex and concave envelopes. J. Glob. Optim. 29, 125–155 (2004)CrossRefGoogle Scholar
  28. 28.
    Li H.L., Tsai J.F.: Treating free variables in generalized geometric global optimization programs. J. Glob. Optim. 33, 1–13 (2005)CrossRefGoogle Scholar
  29. 29.
    Tsai J.F., Lin M.H.: An optimization approach for solving signomial discrete programming problems with free variables. Comput. Chem. Eng. 30, 1256–1263 (2006)CrossRefGoogle Scholar
  30. 30.
    Tsai J.F., Lin M.H., Hu Y.C.: On generalized geometric programming problems with nonpositive variables. Eur. J. Oper. Res. 178, 10–17 (2007)CrossRefGoogle Scholar
  31. 31.
    Li H.L., Tsai J.F., Floudas C.A.: Convex underestimation for posynomial functions of positive variables. Optim. Lett. 2, 333–340 (2008)CrossRefGoogle Scholar
  32. 32.
    Björk K.M., Lindberg P.O., Westerlund T.: Some convexifications in global optimization of problems containing signomial terms. Comput. Chem. Eng. 27, 669–679 (2003)CrossRefGoogle Scholar
  33. 33.
    Westerlund T.: . In: Liberti, L., Maculan, N. (eds) Global Optimization: From Theory to Implementation, pp. 45–74. Springer, Berlin (2006)Google Scholar
  34. 34.
    Lundell A., Westerlund T.: Convex underestimation strategies for signomial functions. Optim. Methods Softw. 24(4–5), 505–522 (2009)CrossRefGoogle Scholar
  35. 35.
    Pörn R., Björkb K.-M., Westerlund T.: Global solution of optimization problems with signomial parts. Discret. Optim. 5, 108–120 (2008)CrossRefGoogle Scholar
  36. 36.
    Shen P.-P., Li X., Jiao H.-W.: Accelerating method of global optimization for signomial geometric programming. J. Comput. Appl. Math. 214, 66–77 (2008)CrossRefGoogle Scholar
  37. 37.
    Lundell A., Westerlund J., Westerlund T.: Some transformation techniques with applications in global optimization. J. Glob. Optim. 43, 391–405 (2008)CrossRefGoogle Scholar
  38. 38.
    Lu H.-C., Li H.-L., Gounaris C.E., Floudas C.A.: Convex relaxation for solving posynomial programs. J. Glob. Optim. 46, 147–154 (2010)CrossRefGoogle Scholar
  39. 39.
    Westerlund T.: Some transformation techniques in global optimization. In: Liberti, L., Maculan, N. (eds) Global Optimization: From Theory to Implementation, pp. 47–74. Springer, Berlin (2005)Google Scholar
  40. 40.
    Li H.-L., Chang C.-T.: An approximate approach of global optimization for polynomial programming problems. Eur. J. Oper. Res. 107, 625–632 (1998)CrossRefGoogle Scholar
  41. 41.
    McCormick G.P.: Computability of global solutions to factorable nonconvex programs. Part I. Convex underestimating problems. Math. Program. 10, 146–175 (1976)CrossRefGoogle Scholar
  42. 42.
    Lasserre J.B.: Global optimization and the problem of moments. SIAM J. Optim. 11, 796–817 (2001)CrossRefGoogle Scholar
  43. 43.
    Shen P., Zhang K.: Global optimization of signomial geometric programming using linear relaxation. Appl. Math. Comput. 150(1), 99–114 (2004)CrossRefGoogle Scholar
  44. 44.
    Kesavan P. et al.: Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs. Math. Program. 100, 517–535 (2004)CrossRefGoogle Scholar
  45. 45.
    You F., Castro P.M., Grossmann I.E.: Dinkelbach’s algorithm as an efficient method to solve a class of MINLP models for large-scale cyclic scheduling problems. Comput. Chem. Eng. 33, 1879–1889 (2009)CrossRefGoogle Scholar
  46. 46.
    Tsai J.F., Lin M.H.: Global Optimization of signomial mixed-integer nonlinear programming problems with free variables. J. Glob. Optim. 42, 39–49 (2008)CrossRefGoogle Scholar
  47. 47.
    Floudas C.A., Pardalos P.M.: A Collection of Test Problems for Constrained Global Optimization Algorithms, Volume 455 of Lecture Notes in Computer Science. Springer, Berlin (1990)CrossRefGoogle Scholar
  48. 48.
    Adjiman C.S., Androulakis I.P., Floudas C.A.: A global optimization method, αBB, for general twice-differentiable constrained NLPs-II. Implementation and computational results. Comput. Chem. Eng. 22, 1159–1179 (1998)CrossRefGoogle Scholar
  49. 49.
    Teles, J.P., Castro, P.M., Matos, H.A.: Global optimization of water networks design using multiparametric disaggregation. Comput. Chem. Eng. (Submitted) (2011)Google Scholar
  50. 50.
    Teles J., Castro P., Matos H.: Parametric programming technique for global optimization of wastewater treatment systems. In: Pierucci, S., Buzzi Ferraris, G. (eds) Computer-Aided Chemical Engineering, vol. 28, pp. 1093–1098. Elsevier, Amsterdam (2010)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  • João P. Teles
    • 1
    • 2
  • Pedro M. Castro
    • 1
    Email author
  • Henrique A. Matos
    • 2
  1. 1.Laboratório Nacional de Energia e GeologiaUnidade de Modelação e Optimização de Sistemas EnergéticosLisboaPortugal
  2. 2.Departamento de Engenharia Química e BiológicaInstituto Superior TécnicoLisboaPortugal

Personalised recommendations