Is there any structured way to tackle $$\min_{\{x_n\},\{y_n\}}\frac{(\sum_{n=1}^{\infty}x_ny_n)^2}{\sum_{n=1}^{\infty}y_n^2\sum_{n=1}^{\infty}x_n^2}$$ when $x_n, y_n \neq 0$, $\sum_{n=1}^{\infty}x_n\neq 0$ and $\sum_{n=1}^{\infty}y_n\neq 0$?
It is clear to me that this positive ratio is at most $1$. Looking at the gradient I get that $x_n=Ay_n$ for some constant $A$. This however makes my objective function exactly equal to $1$. I hence should follow a different path, but what? ... I appreciate any hint.
The objective function is non-negative. If you take $\{x_n\}$ and $\{y_n\}$ to be orthogonal, it is clear that the minimum is equal to $0$. Take $a_n>0$ such that $\sum_{n=2}^\infty a_n^2=s<\infty$ and let $$ x_n=a_n,\quad y_n=-a_n,\quad n\ge2, $$ and $x_1$, $y_1$ such that $x_1y_1=s$. Then $\sum_{n=1}^\infty x_ny_n=s-\sum_{n=2}^\infty a_n^2=0$.