Under the hood of Hard Margin SVM

You are right. You did not share the discussion that comes before (showing the definitions of these paramters), but it seems that in this problem the length of vector $w$ is minimized when the width of the street is maximized. However, since  
\[\frac{1}{2}||w||^2=\frac{1}{2}w^t w,\]
by minimizing $\frac{1}{2}||w||^2$ we also minimize $||w||$ which is the length of the vector $w$.

Note that for $w=(w_1, w_2, \dots, w_n)$. Then

\[||w||=\sqrt{w_1^2+w_2^2+\dots w_n^2},\]
and
\[||w||^2=w_1^2+w_2^2+\dots w_n^2.\]
We see that because of the square root $||w||$ is much more difficult to minimize. This is another reason to work with $||w||^2$ rather than $||w||$.