Building Formulations for Piecewise Linear Relaxations of Nonlinear Functions

Abstract: We study mixed-integer programming formulations for the piecewise linear lower and upper bounds (in other words, piecewise linear relaxations) of nonlinear functions that can be modeled by a new class of combinatorial disjunctive constraints (CDCs), generalized $n$D-ordered CDCs. We first introduce a general formulation technique to model piecewise linear lower and upper bounds of univariate nonlinear functions concurrently so that it uses fewer binary variables than modeling bounds separately. Next, we propose logarithmically sized ideal non-extended formulations to model the piecewise linear relaxations of univariate and higher-dimensional nonlinear functions under the CDC and independent branching frameworks. We also perform computational experiments for the approaches modeling the piecewise linear relaxations of univariate nonlinear functions and show significant speed-ups of our proposed formulations. Furthermore, we demonstrate that piecewise linear relaxations can provide strong dual bounds of the original problems with less computational time in order of magnitude.