Optimization techniques for modeling with piecewise-linear functions

Abstract

In this paper we aim to construct piecewise-linear (PWL) approximations for functions of multiple variables and to build compact mixed-integer linear programming (MILP) formulations to represent the resulting PWL function. On the one hand, we describe a simple heuristic to iteratively construct a triangulation with a small number of triangles, while decreasing the error of the piecewise-linear approximation. On the other hand, we extend known techniques for modeling PWLs in MILPs more efficiently than state-of-the-art methods permit. The crux of our method is that the MILP model is a result of solving some hard combinatorial optimization problems, for which we present heuristic algorithms. The effectiveness of our techniques is demonstrated by a series of computational experiments including a short-term hydropower scheduling problem