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 ∈ { 0 } ∪ [ ℓ , u ] , where z is a binary indicator for x ∈ [ 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 c z + 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 .