A note on :“A Superior Representation Method for Piecewise Linear Functions” by Li, Lu, Huang and Hu

Two new Mixed Integer Linear Programming (MILP) formulations for modeling a univariate piecewise linear function f were introduced in Li et al. (2008). The first formulation (given by (1)–(3) in Li et al.) uses “Big-M” type constraints, so we denote it by LiBigM. The second formulation (given by (23)–(33) in Li et al.) uses a number of binary variables that is logarithmic in the number of segments in which f is affine, so we denote it by LiLog. Based on computational results that show that LiLog outperforms LiBigM, Li et al. declare LiLog to be superior to other MILP formulations for piecewise linear functions. In this paper we show that LiBigM and LiLog are both theoretically and computationally inferior to standard MILP formulations for piecewise linear functions. In Section 2 we show that both formulations from Li et al. are theoretically inferior to essentially every standard MILP formulation for piecewise linear functions. In Section 3 we present results of computational experiments that compare the formulations from Li et al. to two other standard formulations.