Relation of $\max_P \; x^T \cdot Py$ and $\min_P \; \|x-Py\|_2$

Question

Consider a permutation matrix $P$ and two vectors $x$, $v$ with 2-norm = 1 and all positive entries.

Are the optimal solutions $P^\ast$ of $\max_P \; (x^T \cdot Py)$ and $\min_P \; \|x-Py\|_2$ the same?

Answer 1

Hint:

$$ \|x-Py\|_2^2=\|x\|_2^2+\|Py\|_2^2-2x^T\cdot Py=2(1-x^T\cdot Py) $$