Gaining or Losing Perspective for Piecewise-Linear Under-Estimators of Convex Univariate Functions

We study mixed-integer nonlinear optimization (MINLO) formulations of the disjunction x is an element of {0} boolean OR [l, u], where z is a binary indicator for x is an element of [l, u] (0 <= l < u), and y "captures" f (x), which is assumed to be convex and positive on its domain [l, u], but otherwise y = 0 when x = 0. This model is very useful in nonlinear combinatorial optimization, where there is a fixed cost c for operating an activity at level x in the operating range [l, u], and then, there is a further (convex) variable cost f (x). So the overall cost is cz + f (x). In applied situations, there can be N 4-tuples (f , l, u, c), and associated (x, y, z), and so, the combinatorial nature of the problem is that for any of the 2(N) choices of the binary z-variables, the non-convexity associated with each of the (f , l, u) goes away. We study relaxations related to the perspective transformation of a natural piecewise-linear under-estimator of f, obtained by choosing linearization points for f . Using 3-d volume (in (x, y, z)) as a measure of the tightness of a convex relaxation, we investigate relaxation quality as a function of f, l, u, and the linearization points chosen. We make a detailed investigation for convex power functions f (x) := x(p), p > 1.