On Disjunction Convex Hulls by Lifting

We study the natural extended-variable formulation for the disjunction of n+1 polytopes in R-d. We demonstrate that the convex hull D in the natural extended-variable space Rd+n is given by full optimal big-M lifting (i) when d <= 2 (and that it is not generally true for d >= 3), and also (ii) when the polytopes are all axis-aligned hyper-rectangles. We give further results on the polyhedral structure of D, emphasizing the role of full optimal big-M lifting.