We present a new mixed-integer programming (MIP) approach for offline multiple change-point detection by casting the problem as a globally optimal piecewise linear (PWL) fitting problem. Our main contribution is a family of strengthened MIP formulations whose linear programming (LP) relaxations admit integral projections onto the segment assignment variables, which encode the segment membership of each data point. This property yields provably tighter relaxations than existing formulations for offline multiple change-point detection. We further extend the framework to two settings of active research interest: (i) multidimensional PWL models with shared change-points, and (ii) sparse change-point detection, where only a subset of dimensions undergo structural change. Extensive computational experiments on benchmark real-world datasets demonstrate that the proposed formulations achieve reductions in solution times under both $\ell_1$ and $\ell_2$ loss functions in comparison to the state-of-the-art.

翻译：本文提出了一种新的混合整数规划方法，用于离线多变点检测，该问题被建模为全局最优的分段线性拟合问题。我们的主要贡献在于提出了一系列强化的混合整数规划模型，其线性规划松弛在分段分配变量上具有整数投影特性，这些变量编码了每个数据点的分段隶属关系。该性质保证了所提模型相比现有离线多变点检测方法具有可证明的更紧松弛界。我们进一步将该框架扩展到两个当前活跃的研究方向：(i) 具有共享变点的多维分段线性模型；(ii) 稀疏变点检测，即仅部分维度发生结构变化。在真实世界基准数据集上的大量计算实验表明，相较于现有最优方法，所提模型在$\ell_1$和$\ell_2$损失函数下均实现了求解时间的显著缩减。