Piecewise linear approximation using J1 compatible triangulations for efficient MILP representation

Abstract

For including piecewise linear (PWL) functions in MILP problems, the logarithmic convex combination (Log) formulation has been shown to yield very fast solving times. However, identifying approximations that can be used with Log is a big challenge since the approximation has to be compatible with a J1 triangulation. In this article, an algorithm is proposed that identifies approximations using J1 compatible triangulations. It seeks to satisfy the specified error tolerance with the minimum number of linear pieces, so that the MILP formulation is small. To evaluate the performance of the J1 approach it is applied to two sets of benchmark functions from literature and results are compared to state-of-the-art approaches. Overall the J1 approach is shown to efficiently approximate functions in up to 3 dimensions. Especially for tight error tolerances, these J1 approximations require fewer auxiliary variables in MILP compared to alternative approaches.