First, you should restrict $x$ to be positive or use $|x|^\beta$ instead.

Then I think that the answer is **no**:

For $\beta\neq 1/2$ you can argue as follows:

The special case of diagonal $\Omega = \mathrm{diag}(w_1,\dots,w_n)$ ($w_i>0$) is simpler as in this case you problem is
$$
\min_x x^T\Lambda x - \alpha^T x + \sum_i w_i |x_i|^{2\beta}\quad\text{s.t. convex constraints}
$$
i.e. is is a convex quadratic problem with a weighted $\ell^{p}$ regularizer with $1\leq p=2\beta <2$.
As such it is a fairly simple convex problem.

I am fairly sure that even this special case can not be cast as SOCP (if found this claim in "Mixed norm FIR filter optimization using second-order cone programming" by Dan P. Scholnik (ICASSP 2002) and the report "Second-Order Cone Formulations of Mixed-Norm Error Constraints for FIR Filter Optimization" by Dan P. Scholnik and Jeffrey O. Coleman but no reference is given).

For $\beta=1/2$, i.e. $p=1$ one has to argue differently as the above case *can* be cast as an SOCP. The case including the spd matrices is equivalent to (neglecting the constraints)
$$
\min_x \|Ax-b\|_2^2 + \|Lx^{1/2}\|_2^2
$$
with some $A$, $b$ and $L$.

Here the penalty looks like
$$
\|Lx^{1/2}\|_2^2 = \sum_i (\sum_j l_{i,j}x_j^{1/2})^2.
$$
With $n=2$, and $L=\begin{bmatrix}1 & 0\\1 & 1\end{bmatrix}$ lead to
$$
\|Lx^{1/2}\|_2^2 = |x_1| + (\sqrt{x_1}+\sqrt{x_2})^2
$$
which is not a convex function (simply check that the level sets are not convex), so also here, the answer is **no**.

There is still a possibility that there may be a clever reformulation/substitution, but I doubt that. A prove that there is no such a reformulation seems very hard…

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