Suppose a QCQP problem : $$ \min_{x\in\mathbb{R}^{n}}f\left(x\right)=\frac{1}{2}x^{T}P_{0}x+q_{0}^{T}x$$ $$ s.t:\begin{cases} \frac{1}{2}x^{T}P_{i}x+q_{i}^{T}x+r_{i}\le0 & i=1,2\dots,m:m\le n \newline Ax=b \end{cases}$$ Where $P_i $ are all positive semi definite.
a SOCP problem : $$ \min_{y\in\mathbb{R}^{k}}f\left(x\right)=f^{T}y $$ $$ s.t:\begin{cases} \left|\left|A_{i}y-b_{i}\right|\right|\le c_{i}^{T}y+d_{i} & i=1,2\dots,l:l\le k\newline Fy=g \end{cases}$$
Show every QCQP problem can be written as a SOCP problem.
At first I tried converting the constraints :
$$ y^{T}A_{i}^{T}A_{i}y-y^{T}A_{i}^{T}b_{i}-b_{i}^{T}A_{i}y+b_{i}^{T}b_{i}\le\left(c_{i}^{T}y+d_{i}\right)^{2}=\left(y^{T}c_{i}+d_{i}\right)\left(c_{i}^{T}y+d_{i}\right)\iff $$ $$ y^{T}A_{i}^{T}A_{i}y-2b_{i}^{T}A_{i}y+b_{i}^{T}b_{i}\le y^{T}c_{i}c_{i}^{T}y+2d_{i}c_{i}^{T}y+d_{i}^{2} $$
Now to we want to show there exist $A_i,b_i,c_i,d_i$ such that: $$ y^{T}\left(A_{i}^{T}A_{i}-c_{i}c_{i}^{T}\right)y-2\left(b_{i}^{T}A_{i}+d_{i}c_{i}^{T}\right)y+b_{i}^{T}b_{i}-d_{i}^{2}\le 0\tag{SOCP}$$ $$ \frac{1}{2}x^{T}P_{i}x+q_{i}^{T}x+r_{i}\tag{QCQP}$$ $$ \begin{cases} A_{i}^{T}A_{i}-c_{i}c_{i}^{T}=\frac{1}{2}P_{i}\newline -2\left(b_{i}^{T}A_{i}+d_{i}c_{i}^{T}\right)=q_{i}^{T}\newline b_{i}^{T}b_{i}-d_{i}^{2}=r_{i}\newline y=x \end{cases} $$
How do I find $A_i$,$b_i$,$c_i$ and $d_i$ so those equations have solutions? And for the minimizing function, how do I find the vector $f$ that minimizing $$ \min_{x\in\mathbb{R}^{n}}f^{T}x\tag{SOCP} $$ will also minimize $$ \min_{x\in\mathbb{R}^{n}}\frac{1}{2}x^{T}P_{0}x+q_{0}^{T}x\tag{QCQP} $$
TLDR - how do I find the matrix $A_i$ and vectors $b_i,c_i,f$ and scalars $d_i $ so the reduction works given that I know $P_i,r_i,q_i$ and $P_i$ is positive semi definite.
According to https://web.stanford.edu/~boyd/papers/pdf/socp.pdf
It is enough to see $$ \min_{x\in\mathbb{R}^{n}}\frac{1}{2}x^{T}P_{0}x+q_{0}^{T}x=\min_{x\in\mathbb{R}^{n}}\left|\left|P^{\frac{1}{2}}x+P^{-\frac{1}{2}}q_{0}\right|\right|_{2}^{2}-q_{0}^{T}P^{-1}q_{0}$$ Can be reduced to : $$ \min_{x\in\mathbb{R}^{n},t\in\mathbb{R}}t $$ $$ s.t:\left\{ \left|\left|P^{\frac{1}{2}}x+P^{-\frac{1}{2}}q_{0}\right|\right|_{2}\le t\right\} $$ As for the other matrices : $$ c_{i}=\underline{0},d_{i}=\frac{1}{4}\sqrt{\left|\left|q_{i}\right|\right|^{2}+r_{i}},b_{i}=-\frac{1}{2}P_{i}^{\frac{1}{2}}q_{i},A_{i}=\frac{1}{\sqrt{2}}P_{i}^{\frac{1}{2}}$$ Those fulfill the equations I wrote above.