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

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 ℱ 2 ( 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.