Sorry for the simple question, but my Google skills were not enough. I have a large optimization problem and I would like to add constraints of the following kind:
$$ x \in \{ 0 \} \cup [c, \infty) $$
with $c > 0$ a constant. Do these optimization problems have a name? Are there any algorithms to solve this?
Thanks in advance!
These are called semicontinuous variables. Because $x_i$ has an implied finite upper bound of $B$, you can model this situation by introducing a binary variable $y_i$ and linear constraints: $$c_i y_i \le x_i \le B y_i$$ If $y=0$ then $x_i=0$. If $y=1$ then $c_i \le x_i \le B$.