Summary: We study MINLO (mixed-integer non-linear optimization) formulations of the disjunction \(x \in \{0\} \cup [\ell,u]\), where \(z\) is a binary indicator of \(x \in [\ell,u]\) \((0 \leq \ell < u)\), and \(y\) “captures” \(f(x)\), which is assumed to be convex and positive on its domain \([\ell,u]\), but otherwise \(y = 0\) when \(x = 0\). This model is very useful in non-linear combinatorial optimization, where there is a fixed cost of operating an activity at level \(x\) in the operating range \([\ell,u]\), and then there is a further (convex) variable cost \(f(x)\). In particular, 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 \(f,\ell,u\), and the linearization points chosen. We make a careful investigation for convex power functions \(f(x) := x^p\), \(p > 1\).
For the entire collection see [Zbl 1465.05002].