Dealing with a constraint which is the square root of a quadratic form

I'm trying to maximize my portfolio, but don't know how to deal with the constraint which is on the form

max $2u^Tx-x^T \Sigma x$

Subject to

$e^Tx = 1$

$u^Tx - m (x^T \Sigma x)^{1/2} >= c$

Where $\Sigma$ is the covariance and psd matrix and $u$ is the expected return. $e^T$ is a vector consisting of ones (1,...,1). $m$ and $c$ are constants

I don't know how to deal with the square root. (I'm using R)

Cheers

$$u^Tx - m (x^T \Sigma x)^{1/2} \geq c$$ is the same as $$u^Tx-c \geq m (x^T \Sigma x)^{1/2}$$

which is the same as $$(u^Tx-c)^2 \geq m (x^T \Sigma x)$$

This has no square roots.

• After some thought I realized that you can't just square the both sides since $c$ and $u^T x$ can be both negative and positive and $m(x^T \Sigma x)$ is alway positive. I should have thought of that earlier... So back to square one. – user19265 Mar 17 '16 at 15:35

I thought of that at first but I was confused how to reformulate.

$(u^T x)^2 + 2 c u^Tx - c^2$

but it's the same as

$x^T (u \times u^T) x + 2 c u^Tx - c^2$

and now it's solvable. I'm using gurobi and so it needs the constraints on the form

$x^T Q x + u^Tx >= c$
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