## Thursday, May 18, 2017

### Simple piecewise linear problem, not so easy with binary variables

The following picture illustrates the problem:

The blue line is what we want to model:

 $\bbox[lightcyan,10px,border:3px solid darkblue]{ y = \begin{cases} 0 & \text{if 0 \le x \le a}\\ (x-a)\displaystyle\frac{H}{b-a} & \text{if a < x < b}\\ H & \text{if x\ge b}\end{cases}}$

Is there much we can exploit here, from this simple structure? I don’t believe so, and came up with:

 \begin{align}&x_1 \le a \delta_1\\&a \delta_2 \le x_2 \le b \delta_2\\&b \delta_3 \le x_3 \le U \delta_3\\&\delta_1+\delta_2+\delta_3 = 1\\&x = x_1+x_2+x_3\\&y = (x_2 - a \delta_2)\displaystyle\frac{H}{b-a} + H \delta_3 \\&\delta_k \in \{0,1\}\\&x_i \ge 0\\&0 \le x \le U\end{align}
##### Update

Added missing $$\sum_k \delta_k=1$$, see comments below.