Triplet loss function for face recognition?
$\alpha$ is known as the margin. 

Not only that we want to minimize $D(A,P)$ and maximize $D(A,N)$, that is we want  $D(A,P)-D(A,N)$ to be small. Not only that we want it to be non-positive, we want it to be sufficiently negative.

That is not only that we want  $$D(A,P)-D(A,N) \le 0$$ 

We want $$D(A,P)-D(A,N) \le -\alpha.$$