Convex relaxation for solving posynomial programs

  • Hao-Chun Lu
  • Han-Lin Li
  • Chrysanthos E. Gounaris
  • Christodoulos A. Floudas
Article

Abstract

Convex underestimation techniques for nonlinear functions are an essential part of global optimization. These techniques usually involve the addition of new variables and constraints. In the case of posynomial functions \({x_1^{\alpha _1 } x_2^{\alpha _2 }\ldots x_n^{\alpha _n } ,}\) logarithmic transformations (Maranas and Floudas, Comput. Chem. Eng. 21:351–370, 1997) are typically used. This study develops an effective method for finding a tight relaxation of a posynomial function by introducing variables yj and positive parameters βj, for all αj > 0, such that \({y_j =x_j^{-\beta _j }}\) . By specifying βj carefully, we can find a tighter underestimation than the current methods.

Keywords

Convex underestimation Posynomial functions 

References

  1. 1.
    Akrotirianakis I.G., Floudas C.A.: Computational experience with a new class of convex underestimators: box constrained NLP problems. J. Glob. Optim. 29, 249–264 (2004). doi:10.1023/B:JOGO.0000044768.75992.10 CrossRefGoogle Scholar
  2. 2.
    Akrotirianakis I.G., Floudas C.A.: A new class of improved convex underestimators for twice continuously differentiable constrained NLPs. J. Glob. Optim. 30, 367–390 (2004). doi:10.1007/s10898-004-6455-4 CrossRefGoogle Scholar
  3. 3.
    Brooke A., Kendrick D., Meeraus A., Raman R.: GAMS: a users guide. GAMS Development Corporation, Washington, DC (2005)Google Scholar
  4. 4.
    Björk K.J., Lindberg P.O., Westerlund T.: Some convexifications in global optimization of problems containing signomial terms. Comput. Chem. Eng. 27, 669–679 (2003). doi:10.1016/S0098-1354(02)00254-5 CrossRefGoogle Scholar
  5. 5.
    Caratzoulas S., Floudas C.A.: A trigonometric convex underestimator for the base functions in Fourier space. J. Optim. Theory Appl. 124, 339–362 (2005). doi:10.1007/s10957-004-0940-2 CrossRefGoogle Scholar
  6. 6.
    Czyzyk J., Mesnier M., More J.: The NEOS server. IEEE J. Comput. Sci. Eng. 5, 68–75 (1998). doi:10.1109/99.714603 CrossRefGoogle Scholar
  7. 7.
    Drud A.S.: CONOPT: a system for large-scale nonlinear optimization. Reference manual for CONOPT subroutine library. ARKI Consulting and Development A/S, Bagsvaerd, Denmark (1996)Google Scholar
  8. 8.
    Floudas C.A.: Global optimization in design and control of chemical process systems. J. Process Control 20, 125–134 (2000). doi:10.1016/S0959-1524(99)00019-0 CrossRefGoogle Scholar
  9. 9.
    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). doi:10.1016/j.compchemeng.2005.02.006 CrossRefGoogle Scholar
  10. 10.
    Gounaris C.E., Floudas C.A.: Tight convex underestimators for C2-continuous problems: I. Univariate functions. J. Glob. Optim. 42, 51–67 (2008)CrossRefGoogle Scholar
  11. 11.
    Gounaris C.E., Floudas C.A.: Tight convex underestimators for C2-continuous problems: II. Multivariate functions. J. Glob. Optim. 42, 69–89 (2008)Google Scholar
  12. 12.
    Hamed, A.S.E.: Calculation of bounds on variables and underestimating convex functions for nonconvex functions. Ph.D thesis. The George Washington University (1991)Google Scholar
  13. 13.
    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
  14. 14.
    Liberti L., Pantelides C.C.: Convex envelops of monomials of odd degree. J. Glob. Optim. 25, 157–168 (2003). doi:10.1023/A:1021924706467 CrossRefGoogle Scholar
  15. 15.
    Maranas C.D., Floudas C.A.: Finding all solutions of nonlinearly constrained systems of equations. J. Glob. Optim. 7, 143–182 (1995). doi:10.1007/BF01097059 CrossRefGoogle Scholar
  16. 16.
    Maranas C.D., Floudas C.A.: Global optimization in generalized geometric programming. Comput. Chem. Eng. 21, 351–370 (1997). doi:10.1016/S0098-1354(96)00282-7 CrossRefGoogle Scholar
  17. 17.
    Meyer, C.A., Floudas, C.A.: Trilinear monomials with positive or negative domains: facets of the convex and concave envelopes. In : Floudas, C.A., Pardalos, P.M. (eds.) Frontiers in Global Optimization. Kluwer Academic Publishers, Santorini, Greece (2003)Google Scholar
  18. 18.
    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). doi:10.1023/B:JOGO.0000042112.72379.e6 CrossRefGoogle Scholar
  19. 19.
    Meyer C.A., Floudas C.A.: Convex envelopes for edge concave functions. Math. Program. Ser. B. 103, 207–224 (2005). doi:10.1007/s10107-005-0580-9 CrossRefGoogle Scholar
  20. 20.
    Pörn R., Harjunkoski I., Westerlund T.: Convexification of different classes of non-convex MINLP problems. Comput. Chem. Eng. 23, 439–448 (1999). doi:10.1016/S0098-1354(98)00305-6 CrossRefGoogle Scholar
  21. 21.
    Ryoo H.S., Sahinidis N.V.: Analysis of bounds for multilinear functions. J. Glob. Optim. 19, 403–424 (2001). doi:10.1023/A:1011295715398 CrossRefGoogle Scholar
  22. 22.
    Tardella F.: On the existence of polyhedral convex envelopes. In: Floudas, C.A., Pardalos, P.M. (eds) Frontiers in Global Optimization, pp. 563–573. Kluwer Academic Publishers, Santorini, Greece (2003)Google Scholar
  23. 23.
    Tawarmalani, M., Sahinidis, N.V.: BARON 7.2.5: Global optimization of mixed-integer nonlinear programs, User’s manual (2005)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  • Hao-Chun Lu
    • 1
  • Han-Lin Li
    • 2
  • Chrysanthos E. Gounaris
    • 3
  • Christodoulos A. Floudas
    • 3
  1. 1.Department of Information ManagementCollege of Management Fu Jen Catholic UniversitySinjhuang, Taipei Taiwan
  2. 2.Institute of Information ManagementNational Chiao Tung UniversityHsinchuTaiwan
  3. 3.Department of Chemical EngineeringPrinceton UniversityPrincetonUSA

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