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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!

part1 part2

user3547551
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    No tricks involved. Just the use of basic identities and the fact that $p^Tx=x^Tp$ (since that’s just the dot product of $x$ and $p$). – amd Nov 26 '18 at 00:04
  • Thank you, using the dot product rule of x and p did not come to mind! – user3547551 Nov 27 '18 at 15:48

1 Answers1

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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.

Michal Adamaszek
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