I have the following optimization problem in $x \in \mathbb{R}^n$
$$\begin{array}{ll} \text{maximize} & u^T x - c \sqrt{x^TAx}\\ \text{subject to} & \sum_{i} x_i = 1\\ & x_i \geq 0\end{array}$$
where positive semidefinite $n \times n$ matrix $A$, vector $u \in \mathbb{R}^n$ and scalar $c \in \mathbb{R_{+}}$ are given.
Is there a closed form solution? If so, can someone please show how to derive it?
I am trying to solve a RL research problem which results in the above-mentioned optimization problem. I am able to get the solution using gurobi solver which uses interior-point methods for socp but I was hoping there is a closed form solution to this problem.
Any lead would be appreciated.