# Non-Linear objective function due to piecewise component (part 2)

This is a follow up to this question: Non-Linear objective function due to piecewise component

Consider a piecewise function but now with three segments but the objective remains the same as:

$$\sum_{n}(1-prob_{n})(1+x_n)$$

Where $$x_n$$ is the decision variable. However, $$prob_n$$ is now defined as:

$$prob_{n} =$$

$$\begin{cases} 0.25, & x_n \geq 2 \\ 0, & 0 < x_n < 2 \\ 0.05, & x \leq 0 \\ \end{cases}$$ I tried modifying the approach outlined by @Reinderien but I am unsure if I am on the right path.

First I re-write the objective as:

$$\sum_{n}(1-0.05 - (0.20 d_{1,n} - 0.05 d_{2,n}))(1+x_n)$$

Where $$d_{1,n}$$ and $$d_{2,n}$$ are binary variables.

Expanding this further I re-write it as:

$$\sum_{n}(0.95 + 0.95x_{n} - 0.2d_{1,n} - 0.2x_{n}d_{1,n} + 0.05d_{2,n} + 0.05x_{n}d_{2,n})$$

Now we have two non-linear terms so my guess is that I need two auxiliary variables, $$x_{n}^{'}$$ and $$x_{n}^{''}$$. But before introducing the auxiliary variables, my problem is that I am unsure how to re-write this using big-M. This is my initial attempt:

$$x_n \geq M_1d_{1,n}$$

$$x_n \geq 2 - M_2(1-d_{1,n})$$

$$x_{n} \leq 2 - M_{3}d_{2}$$

$$x_{n} \leq M_{4} (1 - d_{1,n} - d_{2,n})$$

But after this I am not sure how to continue. I was able to verify the constraints suggested by @Reinderien but I cannot extend it to the three segment case.

• I would suggest trying with 3 binary variables, one for each segment. Commented Nov 4, 2023 at 20:03