Consider the standard second order cone programming problem:

\begin{equation} \begin{array}{ll} \operatorname{maximize} & \bar{p}^{T} x \\ \text { subject to } & \bar{p}^{T} x+\Phi^{-1}(\beta)\left\|\Sigma^{1 / 2} x\right\|_2 \geq \alpha. \end{array} \end{equation}

This is a problem appearing in portfolio optimization (here for more info). Here we assume $\bar{p},x \in\mathbb{R}^n$ and $\alpha, \beta\in[0,1]$, $\Sigma \in \mathbb{R}^{n\times n}$ is Positive Semi-Definite Matrix and $\Phi^{-1}(\cdot)$ is the inverse Gaussian CDF. The problem is convex for $\beta\leq0.5$.

Is there any closed-form solution for this problem? I tried solving the problem by imposing KKT but it did not lead to anything promising. Thanks.