Linearize a CQ portfolio optimization problem

Hello,

I have a constraint quadratic (CQ) and linear objective portfolio optimization problem which I would like to reduce to a LP or QP standard form via some trick.

The original problem definition is to find the portfolio configuration x that maximizes the returns while keeping the cost to be at most C and the risk to be at most V where Sigma is the covariance matrix (SDP):

$\displaystyle \text{minimize}\ -r^Tx $

$\displaystyle \text{s.t.}\ c^Tx \leq C;\ x^T\Sigma x \leq V; x \ge 0$

I first thought of doing a variable transformation so the QC could be linearized i.e. define a variable vector:

$\displaystyle y_{ij}=x_ix_j; y'=(x_{1}^2, x_{2}^2, ... , x_{n}^2)$

and then my problem would become:

$\displaystyle \text{minimize}\ -r^Ty' $

$\displaystyle \text{s.t.}\ c^Tc y' \leq C^2;\ \Sigma y \leq V; y \ge 0$

and then my solution would be the squared root of the resulting $\displaystyle y_{ii}$ but I am afraid that the "equivalent" cost constraint is not exactly equivalent ...

Can anyone please advice?

Many thanks in advance,

Best regards,

Giovanni