Documentation


Model.setPWLObj()

setPWLObj ( var, x, y )

Set a piecewise-linear objective function for a variable.

The arguments to this method specify a list of points that define a piecewise-linear objective function for a single variable. Specifically, the <span>$</span>x<span>$</span> and <span>$</span>y<span>$</span> arguments give coordinates for the vertices of the function.

For example, suppose we want to define the function <span>$</span>f(x)<span>$</span> shown below:

Image pwl

The vertices of the function occur at the points <span>$</span>(1, 1)<span>$</span>, <span>$</span>(3,2)<span>$</span> and <span>$</span>(5,4)<span>$</span>, so <span>$</span>x<span>$</span> is [1, 3, 5] and <span>$</span>y<span>$</span> is [1, 2, 4]. With these arguments we define <span>$</span>f(1) = 1<span>$</span>, <span>$</span>f(3) = 2<span>$</span> and <span>$</span>f(5) = 4<span>$</span>. Other objective values are linearly interpolated between neighboring points. The first pair and last pair of points each define a ray, so values outside the specified <span>$</span>x<span>$</span> values are extrapolated from these points. Thus, in our example, <span>$</span>f(-1)=0<span>$</span> and <span>$</span>f(6)=5<span>$</span>.

More formally, a set of <span>$</span>n<span>$</span> points

\begin{displaymath}
\mathtt{x} = [x_1, \ldots, x_n], \quad \mathtt{y} = [y_1, \ldots, y_n]
\end{displaymath}

define the following piecewise-linear function:

\begin{displaymath}
f(v) =
\left\{
\begin{array}{ll}
y_1 + \frac{y_2-y_1}{x_2-x_...
...- x_n), & \mathrm{if}\; v \ge x_n. \ [7pt]
\end{array}\right.
\end{displaymath}

The <span>$</span>x<span>$</span> entries must appear in non-decreasing order. Two points can have the same <span>$</span>x<span>$</span> coordinate -- this can be useful for specifying a discrete jump in the objective function.

Note that a piecewise-linear objective can change the type of a model. Specifically, including a non-convex piecewise linear objective function in a continuous model will transform that model into a MIP. This can significantly increase the cost of solving the model.

Setting a piecewise-linear objective for a variable will set the Obj attribute on that variable to 0. Similarly, setting the Obj attribute will delete the piecewise-linear objective on that variable.

Each variable can have its own piecewise-linear objective function. They must be specified individually, even if multiple variables share the same function.

Arguments:

var: A Var object that gives the variable whose objective function is being set.

x: The <span>$</span>x<span>$</span> values for the points that define the piecewise-linear function. Must be in non-decreasing order.

y: The <span>$</span>y<span>$</span> values for the points that define the piecewise-linear function.

Example usage:

  model.setPWLObj(var, [1, 3, 5], [1, 2, 4])