I am fairly new at linear programming/optimization and am currently working on implementing a linear program that is stated like this:
max $\sum_{i=1}^{k}{p(\vec \alpha \cdot \vec c_i)}$
$s.t. $
$|\alpha_j| \le 1$
Where p(x) = 2x if x < 0, x otherwise, and $ \vec c$ is a constant
The p(x) function is what's troubling me, since one can only determine x's sign after an assignment of $\vec \alpha$. How can I remove the function p from the objective and express this objective equivalently as a linear combination of the variables?
Thank you!
Best Answer
Sorry for making you wait 14 hours unnecessarily but you are partially guilty yourself: if you posted a correct and full version of the question from the beginning, you would get the answer in 5 minutes. Keep it in mind when you ask a question on a public forum next time.
Your problem is equivalent to maximizing the linear expression $\sum_i y_i$ under the linear restrictions $\alpha_j\ge -1$, $\alpha_j\le 1$, $y_i\le c_i\cdot\alpha$, $y_i\le 2c_i\cdot\alpha$. It is as simple as that but it is crucial that your $p$ is concave and that you maximize.