PiecewiseLinearOpt.jl

PiecewiseLinearOpt.jl is a JuMP extension for modeling optimization problems containing piecewise linear functions, possibly nonconvex and of several variables.

The package accompanies the paper Nonconvex piecewise linear functions: Advanced formulations and simple modeling tools by Joey Huchette and Juan Pablo Vielma, and implements the formulations from it together with other formulations from the literature, see Formulation methods and References.

Version 0.5 is a rewrite of the code introduced with the paper, with a new interface. The original code can be found in the v0.4.2 release, the last of version 0.4.

Installation

import PkgPkg.add("PiecewiseLinearOpt")

Quick start

To minimize a piecewise linear approximation of $\sin(x)$ with breakpoints at 0:0.5:2π:

using JuMP, PiecewiseLinearOpt, HiGHSmodel = Model(HiGHS.Optimizer)set_silent(model)@variable(model, x)f = PiecewiseLinear(sin, 0:0.5:2π)@expression(model, y, f(x))@objective(model, Min, y)optimize!(model)value(x), value(y)

(4.5, -0.977530117665097)

The output y equals the piecewise linear function of x, which is formulated with auxiliary variables and constraints when f(x) is called. Functions of two variables work the same way, with breakpoints along each axis:

model = Model(HiGHS.Optimizer)set_silent(model)@variable(model, x)@variable(model, y)g = PiecewiseLinear(0:0.25:1, 0:0.25:1) do a, b    return (a - 0.3)^2 + (b - 0.6)^2end@expression(model, z, g(x, y))@objective(model, Min, z)optimize!(model)value(x), value(y), value(z)

(0.25000000000000033, 0.5000000000000006, 0.012499999999999994)

Contents

The manual:

• Piecewise linear functions describes how to construct functions.
• Adding functions to a model describes PiecewiseLinear, add_formulation and the formulations they add.
• Formulation methods gives an overview of the available methods and advice on choosing one.

The tutorials show the package at work on complete models:

• Transportation with economies of scale approximates concave and fixed-charge costs of a single variable, and compares with solving the nonlinear model directly.
• Hydropower scheduling approximates a production function of two variables in a model with on/off decisions.

Further pages:

• References lists the papers behind the formulations.
• Extending the package describes how to add a method.
• Release notes and Migrating from version 0.4 list the changes from previous versions.

Getting help

If you need help, please ask a question on the JuMP community forum. If you have a reproducible example of a bug, please open a GitHub issue.