Nonconvex Piecewise Linear Functions: Advanced Formulations and Simple Modeling Tools

Piecewise linear functions are deceptively simple structures that are nonetheless capable of approximating complex nonlinear behavior. As such, they have been adopted throughout operations research and engineering to approximate nonlinear structures in optimization problems which would otherwise render the problem extremely difficult to solve. In “Nonconvex Piecewise Linear Functions: Advanced Formulations and Simple Modeling Tools,” J. Huchette and J. P. Vielma derive new mixed-integer programming (MIP) formulations for embedding low-dimensional nonconvex piecewise linear functions in optimization models. These formulations computationally outperform the crowded field of existing approaches in a number of regimes of interest. As these formulations are derived using recently developed machinery that produce highly performant, but uninterpretable, formulations, the authors showcase the utility of high-level modeling tools by presenting PiecewiseLinearOpt.jl, an extension to the popular JuMP optimization modeling language that implements a host of MIP formulations for piecewise linear function in a single, easy-to-use interface. We present novel mixed-integer programming (MIP) formulations for optimization over nonconvex piecewise linear functions. We exploit recent advances in the systematic construction of MIP formulations to derive new formulations for univariate functions using a geometric approach and for bivariate functions using a combinatorial approach. All formulations are strong, small (so-called logarithmic formulations), and have other desirable computational properties. We present extensive experiments in which they exhibit substantial computational performance improvements over existing approaches. To accompany these advanced formulations, we present PiecewiseLinearOpt , an extension of the JuMP modeling language in Julia that implements our models (alongside other formulations from the literature) through a high-level interface, hiding the complexity of the formulations from the end user. Funding: This work was supported by the National Science Foundation [Grant CMMI-1351619].