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.

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