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: $\left< x, y\right> \le ||x|| \cdot ||y||$. But how can I apply this formula on that monstrous term?
Beyond the inequality itself, the full Cauchy-Schwarz theorem states that $|\langle u, v\rangle|$ 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 $$p_j \propto \sum_{i} \mu_{i,j} \frac{x_i}{||x_i||}$$