On the formulation of the Support Vector Machine problem, at some point, after the scaling of my constraints I arrive at the following formulation:
$$ \max_{\textbf{w},w_0} \|w\|^{-1}$$
subject to constraints $t_i (\textbf{w}^T\phi(x_i) +w_0) \ge 1$. Of course searching for the $max$ of the above quantity is the same as searching for the $\min_{\textbf{w},w_0} \|w\|$. But then, I read that it is equivalent again to minimzing $\|w\|^2$. I'm pretty new to optimization theory... so it is always true that searching for the minimum of something is equivalent to searching for the minimum of the same quantity squared? And what about the maximum? Thanks.
$\min\limits_{x} f(x)$ is equivalent to $\min\limits_{x}g(f(x))$ in the sense that their minimizers are the same if $g:\mathbb{R}\to\mathbb{R}$ is a strictly increasing function and $f:\mathbb{R}\to\mathbb{R}$ is convex.