Minimization problem convex set

Minimization problem convex set

I'm trying to minimize the function: $$f(w)=w^T\mu+k\sqrt{w^T\Sigma w}$$
where $w$ is a vector in $W=\{x \in \mathbb{R}^n|x_1+...+x_n=1 , x_i \geq
0 \forall i\}$.
The vector $\mu \in \mathbb{R}^n$, the constant $k\in \mathbb{R}$ and the
symmetric positive-semidefinite matrix $\Sigma\in \mathbb{R}^{n*n}$ are
given.
Does this problem have an unique solution? I am able to see that $W$ is
convex but I don't think $f(w)$ is convex. How can I numerically solve the
problem? I know some theory of linear and quadratic programming but I' not
able to use them in this case.