Ytukyg

Reading a research article, I came across the following statement:

The following function (where $x_i$ and $p_j$ are two vectors in $mathbb{R}^n$ and $mu_{i,j}$ is a constant) - it doesn't matter where it came from, is just an inner product actually:

$sum_{i} mu_{i,j} - left< sum_{i} mu_{i,j} frac{x_i}{||x_i||}, frac{p_j}{||p_j||} right>$

is minimized by using the Cauchy-Schwartz inequality iff

$p_j propto sum_{i} mu_{i,j} frac{x_i}{||x_i||}$.

I want to apply the Cauchy-Schwartz inequality on $left< sum_{i} mu_{i,j} frac{x_i}{||x_i||}, frac{p_j}{||p_j||} right>$ in order to understand where the proportionality thing came from.

So basically the Cauchy-Schwartz inequality states that:
$left< x, yright> le ||x|| cdot ||y||$. But how can I apply this formula on that monstrous term?

Beyond the inequality itself, the full Cauchy-Schwarz theorem states that $|langle u, vrangle|$ is maximized exactly when $u$ and $v$ are parallel or anti-parallel. I.e., when $v propto u$. For your inner product, this is

$$frac{p_j}{|p_j|} propto sum_{i} mu_{i,j} frac{x_i}{||x_i||}$$

Since $1/|p_j|$ is just a scalar multiplier, it can be absorbed into the constant of proportionality, leaving
$$p_j propto sum_{i} mu_{i,j} frac{x_i}{||x_i||}$$