On the Relationship between Power and Exponential Transformations for Positive Signomial Functions

Global optimization of mixed integer nonlinear programming (MINLP) problems containing signomial terms is in many cases a difficult task, and many different approaches to solve these problems have been devised. In Westerlund (2005) a method where a relaxed convex relaxation of the original problem is obtained by approximating single-variable transformations with piecewise linear functions (PLFs) is described. What transformations are applicable depend on the sign of the signomial terms. For positive terms power transformations (PTs) (Lundell et al., 2007) and the exponential transformation (ET) (Lundell and Westerlund, 2008) can be used. In Lundell and Westerlund (2009) it was shown that the ET always results in a tighter underestimator than the negative power transformation (NPT). These results are in this paper extended to also include the positive power transformation (PPT); we will show that the PPT is generally better than the NPT, and also describe the relationship between the PPT and the ET.