Optimization of Transformations for Convex Relaxations of MINLP Problems Containing Signomial Functions

In this paper, a method for determining an optimized set of transformations for signomial functions in a nonconvex mixed integer nonlinear programming (MINLP) problem is described. Through the proposed mixed integer linear programming (MILP) problem formulation, a set of single-variable transformations is obtained. By varying the parameters in the MILP problem, different sets of transformations are obtained. Using these transformations and some approximation techniques, a nonconvex MINLP problem can be transformed into a convex overestimated form. What transformations are used have a direct effect on the combinatorial complexity and approximation quality of these problems, so it is of great importance to find the best possible transformations. Variants of the method have previously been presented in Lundell et al. (2007) and Lundell and Westerlund (2008). Here, the scope of the procedure is extended to also allow for minimization of the number of required transformation variables, as well as, favor transformations with better numerical properties. These improvements can have a significant impact on the computational effort needed when solving the transformed MINLP problems.