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

TL;DR

This paper analyzes the effectiveness of piecewise-linear under-estimators for convex univariate functions within mixed-integer nonlinear optimization, focusing on how different linearization points affect relaxation tightness, especially for power functions.

Contribution

It provides a detailed study of relaxation quality for piecewise-linear under-estimators, introducing a volume-based measure and examining the impact of linearization choices for convex power functions.

Findings

Relaxation tightness varies with linearization points and function parameters.

Power functions with different exponents exhibit distinct relaxation behaviors.

Volume measure effectively quantifies relaxation quality.

Abstract

We study MINLO (mixed-integer nonlinear 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 nonlinear 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 detailed investigation for convex power functions $f(x):=x^p$, $p>1$.