Computational implementation of non-linear convex hull reformulation

Lee and Grossmann [Lee, S., & Grossmann, I. E. (2000). New algorithms for nonlinear generalized disjunctive programming. Computers and Chemical Engineering, 24, 2125–2141] have developed a reformulation for nonlinear Generalized Disjunctive Programming (GDP) problems that obtains from the intersection of the convex hulls of every disjunction. In order to computationally implement this method, it is necessary to reformulate the problem in such a way so as to avoid division by zero in the nonlinear inequalities present amongst the convex hull constraints, while preserving the convex nature of the problem. To accomplish this, we propose to replace the original set of nonlinear constraints by two different sets of convex constraints that circumvent the aforementioned problem and that approximate the original set of constraints exactly at their limit. Furthermore, we compare the two sets of approximating constraints against each other and give rigorous theoretical conditions under which one is superior to the other. Finally, we illustrate the efficiency of both approximations on a variety of numerical examples draw from the Chemical Engineering and Operations Research literature.