Mixed-Integer Programming for Change-point Detection

TL;DR

This work develops strengthened MIP formulations for offline change-point detection via piecewise-linear fitting, achieving globally optimal segmentations with tighter LP relaxations. The key advance is an extended, nested segment-assignment representation that yields an integral LP projection (polyhedron (C1)) and tighter relaxations than prior Basic/Alternate formulations, improving solver runtimes across univariate, multidimensional, and sparse change-point tasks. The framework naturally accommodates both continuity and discontinuity in univariate fits and extends to shared-breakpoint models across dimensions, plus a sparsity-enabled variant. Extensive experiments on real-world stock data demonstrate substantial runtime gains and robust performance advantages of the Extended formulations over existing benchmarks, validating the practical impact for large-scale change-point detection. Overall, the approach provides a unified, scalable optimization-based foundation for accurate, interpretable change-point detection in diverse settings.

Abstract

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.

Figures (4)

• Figure 1: Illustrative change-point detection scenarios on the same underlying data: (a) mean change detection, (b) piecewise linear without continuity, (c) piecewise linear with continuity, (d) piecewise polynomial without continuity.
• Figure 2: Illustration of how nested binary vectors $X_{1,\cdot}$ and $X_{2,\cdot}$ determine the segment-assignment variables $\delta_{1,\cdot}, \delta_{2,\cdot}, \delta_{3,\cdot}$ in a continuous three-segment piecewise-linear fit. The vectors $X_{1,\cdot}$ and $X_{2,\cdot}$ are temporally nonincreasing. This yields $\delta_{1,\cdot} = X_{1,\cdot}$, $\delta_{2,\cdot} = X_{2,\cdot} - X_{1,\cdot}$, and $\delta_{3,\cdot} = 1 - X_{2,\cdot}$, producing three contiguous segment blocks and the three piecewise linear segments $m_1x + c_1$, $m_2x + c_2$, and $m_3x + c_3$.
• Figure 3: Configuration that is feasible for the LP relaxations of the Basic and Alternate formulations but infeasible for their extended counterparts, as the sequences $\widetilde{\delta}_{2,\cdot}$ and $\widetilde{\delta}_{3,\cdot}$ violate the total-variation bound implied by $\widetilde{\delta}_{j,t} = \widetilde{X}_{j,t} - \widetilde{X}_{j-1,t}$.
• Figure 4: Illustration of piecewise-linear (PWL) fits obtained through MIP-based offline change-point detection under different modeling regimes. (a) Univariate PWL fit with continuity constraints, using $K=3$ linear segments to model a length-$T=100$ time series of daily closing prices for Amazon (AMZN). (b) Univariate PWL fit without continuity constraints for the same data and number of segments. (c) Multidimensional change-point detection with a shared set of change-points across $D=6$ dimensions, using $K=5$ linear segments to jointly fit all series. (d) Sparse change-point detection in a $D=20$-dimensional setting with a sparsity level of 20%, allowing 4 out of 20 dimensions to undergo a change in their fitted parameters.

Theorems & Definitions (18)

• Remark 1: Alternate Method of Contiguous Segment Assignment
• Remark 2: Alternate Method of Modeling Continuity
• Remark 3: Equivalence of the Basic and Extended Basic Formulations
• Remark 4: Feasible-Region of the Alternate formulation is strictly contained in that of the Extended Alternate Formulation
• Remark 5: Allowing Unused Segments and $\ell_0$-Regularized PWL Fitting
• Remark 6: Same Optimal Solution for the Basic, Alternate, and Extended Formulations
• Definition 3.1: Integral Polytope
• Definition 3.2: Projection of a Polyhedron
• Definition 3.3: Total Variation
• Lemma 3.4: Total Variation Bound for Differences