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Convexity of Products of Univariate Functions and Convexification Transformations for Geometric Programming

Author

Listed:
  • C. E. Gounaris

    (Princeton University)

  • C. A. Floudas

    (Princeton University)

Abstract

We investigate the characteristics that have to be possessed by a functional mapping f:ℝ↦ℝ so that it is suitable to be employed in a variable transformation of the type x→f(y) in the convexification of posynomials. We study first the bilinear product of univariate functions f 1(y 1), f 2(y 2) and, based on convexity analysis, we derive sufficient conditions for these two functions so that ℱ2(y 1,y 2)=f 1(y 1)f 2(y 2) is convex for all (y 1,y 2) in some box domain. We then prove that these conditions suffice for the general case of products of univariate functions; that is, they are sufficient conditions for every f i (y i ), i=1,2,…,n, so as ℱ n (y 1,y 2,…,y n )=∏ i=1 n f i (y i ) to be convex. In order to address the transformation of variables that are exponentiated to some power κ≠1, we investigate under which further conditions would the function (f) κ be also suitable. The results provide rigorous reasoning on why transformations that have already appeared in the literature, like the exponential or reciprocal, work properly in convexifying posynomial programs. Furthermore, a useful contribution is in devising other transformation schemes that have the potential to work better with a particular formulation. Finally, the results can be used to infer the convexity of multivariate functions that can be expressed as products of univariate factors, through conditions on these factors on an individual basis.

Suggested Citation

  • C. E. Gounaris & C. A. Floudas, 2008. "Convexity of Products of Univariate Functions and Convexification Transformations for Geometric Programming," Journal of Optimization Theory and Applications, Springer, vol. 138(3), pages 407-427, September.
    • Handle: RePEc:spr:joptap:v:138:y:2008:i:3:d:10.1007_s10957-008-9402-6
    DOI: 10.1007/s10957-008-9402-6
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    References listed on IDEAS

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    1. Elmor L. Peterson, 1976. "Fenchel's Duality Thereom in Generalized Geometric Programming," Discussion Papers 252, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    2. Elmor L. Peterson, 1976. "Optimality Conditions in Generalized Geometric Programming," Discussion Papers 221, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
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    Cited by:

    1. Jung-Fa Tsai & Ming-Hua Lin, 2011. "An Efficient Global Approach for Posynomial Geometric Programming Problems," INFORMS Journal on Computing, INFORMS, vol. 23(3), pages 483-492, August.
    2. S. Fanelli, 2011. "A New Algorithm for Box-Constrained Global Optimization," Journal of Optimization Theory and Applications, Springer, vol. 149(1), pages 175-196, April.
    3. Lin, Ming-Hua & Tsai, Jung-Fa, 2012. "Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming problems," European Journal of Operational Research, Elsevier, vol. 216(1), pages 17-25.
    4. Ruth Misener & Christodoulos A. Floudas, 2014. "A Framework for Globally Optimizing Mixed-Integer Signomial Programs," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 905-932, June.
    5. Boukouvala, Fani & Misener, Ruth & Floudas, Christodoulos A., 2016. "Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO," European Journal of Operational Research, Elsevier, vol. 252(3), pages 701-727.

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