# PiecewiseLinearOpt

A package for modeling optimization problems containing piecewise linear functions. Current support is for (the graphs of) continuous univariate functions.

This package is an accompaniment to a paper entitled *Nonconvex piecewise linear functions: Advanced formulations and simple modeling tools*, by Joey Huchette and Juan Pablo Vielma.

This package offers helper functions for the JuMP algebraic modeling language.

Consider a piecewise linear function. The function is described in terms of the breakpoints between pieces, and the function value at those breakpoints.

Consider a JuMP model

```
using JuMP, PiecewiseLinearOpt
m = Model()
@variable(m, x)
```

To model the graph of a piecewise linear function `f(x)`

, take `d`

as some set of breakpoints along the real line, and `fd = [f(x) for x in d]`

as the corresponding function values. You can model this function in JuMP using the following function:

```
z = piecewiselinear(m, x, d, fd)
@objective(m, Min, z) # minimize f(x)
```

For another example, think of a piecewise linear approximation for for the function $f(x,y) = exp(x+y)$:

```
using JuMP, PiecewiseLinearOpt
m = Model()
@variable(m, x)
@variable(m, y)
z = piecewiselinear(m, x, y, 0:0.1:1, 0:0.1:1, (u,v) -> exp(u+v))
@objective(m, Min, z)
```

Current support is limited to modeling the graph of a continuous piecewise linear function, either univariate or bivariate, with the goal of adding support for the epigraphs of lower semicontinuous piecewise linear functions.

Supported univariate formulations:

- Convex combination (
`:CC`

) - Multiple choice (
`:MC`

) - Native SOS2 branching (
`:SOS2`

) - Incremental (
`:Incremental`

) - Logarithmic (
`:Logarithmic`

; default) - Disaggregated Logarithmic (
`:DisaggLogarithmic`

) - Binary zig-zag (
`:ZigZag`

) - General integer zig-zag (
`:ZigZagInteger`

)

Supported bivariate formulations for entire constraint:

- Convex combination (
`:CC`

) - Multiple choice (
`:MC`

) - Dissaggregated Logarithmic (
`:DisaggLogarithmic`

)

Also, you can use any univariate formulation for bivariate functions as well. They will be used to impose two axis-aligned SOS2 constraints, along with the "6-stencil" formulation for the triangle selection portion of the constraint. See the associated paper for more details. In particular, the following are also acceptable bivariate formulation choices:

- Native SOS2 branching (
`:SOS2`

) - Incremental (
`:Incremental`

) - Logarithmic (
`:Logarithmic`

) - Binary zig-zag (
`:ZigZag`

) - General integer zig-zag (
`:ZigZagInteger`

)