When reading about the optimization problem for Support Vector Machine in Bishop's book (Pattern Recognition and Machine Learning) he wrote that:
The optimization problem then simply requires that we maximize $\| \mathbf w\|^{-1}$ which is equivalent to minimizing $\| \mathbf w\|^{2}$
Here $\| \mathbf w\|$ is the euclidian norm of the vector $\mathbf w$ so it is always a positive number.
I am wondering why do we not minimize $\| \mathbf w\|$ instead of $\| \mathbf w\|^2$ ?
Does using $\| \mathbf w\|^{2}$ simplifies the optimization problem?
At first I thought that taking the square made the problem convex, but the function $$f(x,y)=\sqrt{x^2+y^2}$$ looks convex to me too.
Minimizing $|w|$ is the same as minimizing $|w|^2$. As I understand it, the reason we choose the latter is purely for convenience. Would you rather deal with partials of $\sqrt{x^2+y^2}$ or $x^2+y^2$? The latter, obviously, because it's easier.