Piecewise Linear Function Fitting via Mixed-Integer Linear Programming
Steffen Rebennack, Vitaliy Krasko
INFORMS Journal on Computing · 2020 · Volume 32 · Issue 2
Abstract
Piecewise linear (PWL) functions are used in a variety of applications. Computing such continuous PWL functions, however, is a challenging task. Software packages and the literature on PWL function fitting are dominated by heuristic methods. This is true for both fitting discrete data points and continuous univariate functions. The only exact methods rely on nonconvex model formulations. Exact methods compute continuous PWL function for a fixed number of breakpoints minimizing some distance function between the original function and the PWL function. An optimal PWL function can only be computed if the breakpoints are allowed to be placed freely and are not fixed to a set of candidate breakpoints. In this paper, we propose the first convex model for optimal continuous univariate PWL function fitting. Dependent on the metrics chosen, the resulting formulations are either mixed-integer linear programming or mixed-integer quadratic programming problems. These models yield optimal continuous PWL functions for a set of discrete data. On the basis of these convex formulations, we further develop an exact algorithm to fit continuous univariate functions. Computational results for benchmark instances from the literature demonstrate the superiority of the proposed convex models compared with state-of-the-art nonconvex models.
Keywords
piecewise linear function, linear spline, splines of degree 1, polyhedral function, mixed-integer linear programming (MILP), mixed-integer quadratic programming (MIQP), function fitting, spline regression, global optimization
Enable JavaScript for citation tools and additional paper interactions.