This is straight from the book: Optimization Methods in Finance.
I'm trying to gain understanding of how the author derived the cone constraints from the the following quadratic constraint:
$$x^TQx + 2p^T x + γ ≤ 0$$
Assuming Matrix $Q$ is positive definite, there exists an invertible matrix, say $R$, satisfying $Q = RR^T$
This allows us to rearrange the equation to:
$$(R^Tx)^T(R^Tx) + 2p^Tx + γ ≤ 0 $$
This is fine and makes sense. However the next step I have had issues understanding:
Define $$ y = (y_1, . . . , y_k)^T = R^T x + R^{−1}p $$
Then we have:
$$ y^Ty = (R^Tx)^T(R^Tx) + 2p^Tx + p^TQ^{-1}p $$
Pictures from the text below show the entire derivation. Any suggestions or clarifications on how the last two lines are derived? Perhaps some linear algebra trick I am not aware of? Thanks!
If $y=R^Tx$ then
$$\|y\|_2^2=y^Ty=(R^Tx)^T(R^Tx)=x^TRR^Tx=x^TQx$$
so you can equivalently write your quadratic problem in conic form as
$$t+2p^Tx+\gamma\leq 0,\quad t\geq \|y\|_2^2.$$
The second constraint is a rotated quadratic cone.