Building Formulations for Piecewise Linear Relaxations of Nonlinear Functions
研究如何用混合整数规划公式为非线性函数构建分段线性下界和上界（即分段线性松弛），提出新公式减少二元变量并加速计算，实验显示能显著提升求解速度。
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 nD-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 nonextended 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 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 by an order of magnitude. Funding: This work was supported by the Office of Naval Research [Grant N000142412648] for Rice University. Supplemental Material: The online appendices are available at https://doi.org/10.1287/opre.2023.0187 .