I'm reviewing a paper on Schedule Optimization of Electric Vehicles, and they're using a function that I am not familiar with.
They're proposing as one of the objective functions to minimize:
$$ \eta^{t-t_{dead},k_{e}} 1_{Q}\left(\sum^{t_{dead}}_{t=t_k} q^{'}_{k_{e},t}<Q_{k_{e},max}\right)$$
The expression in parentheses is the argument of the indicator function. The output of the indicator function ($1_{Q}$) is 1 if the expression inside the parentheses is true i.e., the output will be 1 if the net energy delivered to the battery of $k_e^{th}$ electric vehicle at deadline $t_{dead,k_{e}}$, is less than the requested energy $Q_{k_{e}, max}$ otherwise the output will be 0. $\eta$ is the penalty factor. Now, to transform the expression into a convex one, they suggest to transform the indicator function as:
$$- \min \left(0,\left(\sum_{t=t_{k}}^{t_{dead},k_{e}}q^{'}_{k_{e},t}-Q_{k_{e},max}\right)\right)$$.
Is it true? Why the first one is non-convex?