AI 中文总结
本文针对序列耦合双线性规划，提出三种MILP公式，通过解析近似误差与数值实验，证明紧凑的“正方形”MILP公式在长序列SCBP实例上的计算性能优于QP求解器。
AI 中文摘要
数学建模中双线性项的存在通常会产生非凸二次规划（QP），这类问题在计算上仍难以求解至全局最优。虽然连续分段线性（CPWL）近似可将这些非线性项重构为混合整数线性规划（MILP），但域划分的几何构造极大地决定了求解器的效率。本文提出了用于近似双线性项的高效MILP公式，并与直接QP求解器严格评估其计算优势：首先，引入具有任意高精度的CPWL近似，明确考虑并利用单位双线性函数的固有对称性；其次，解析确定了CPWL近似的最大和平均近似误差；第三，构建并比较了三种不同的MILP公式（“三角形”、“正方形”和凸差“DC”公式）；第四，形式化了一类高度相关的优化模型——序列耦合双线性规划（SCBP），其中变量代表系统状态和状态变化；最后，通过大量数值实验证明，本文紧凑的“正方形”MILP公式在长序列时域的SCBP实例上实现了卓越的计算性能，使成熟的开源MILP求解器能够有效超越QP求解器。
英文摘要
The presence of bilinear terms in mathematical modeling generally yields nonconvex quadratic programs (QPs) that remain computationally challenging to solve to global optimality. While continuous piecewise linear (CPWL) approximations can reformulate these nonlinearities into mixed-integer linear programs (MILPs), the geometric construction of the domain partition heavily dictates the resulting solver efficiency. In this paper, we present efficient MILP formulations for approximating bilinear terms and rigorously evaluate their computational merits against direct QP solvers. First, we introduce CPWL approximations with arbitrary high degrees of accuracy that explicitly account for and exploit the inherent symmetries of the unit bilinear function. Second, we analytically establish the exact maximum and average approximation errors of the CPWL approximations. Third, we construct and compare three distinct MILP formulations (a ``Triangle'', a ``Square'', and a difference-of-convex ``DC'' formulation) of the CPWL functions. Fourth, we formalize a highly relevant class of optimization models, Sequentially Coupled Bilinear Programs (SCBP), where variables represent system states and state changes. Finally, through extensive numerical experiments, we demonstrate that our compact ``Square'' MILP formulation achieves superior computational performance on SCBP instances with long sequence horizons, allowing mature open-source MILP solvers to effectively outperform QP solvers.
