is the budget allocation right ?

[garaponp]

Was reconnecting with cvxpy and got this is very useful tutorials - Thanks!
One thing is puzzling me - in the risk budgeting allocation exercise
I don't find the objective function to make sense (maybe I am missing something)
I would say the first order condition for the below (non DCP):
$\log( \frac{1}{2} x^T \Sigma x ) - \sum_i \rho^{des}_i \log(x_i)$
yield the correct condition ?
I also did not find the optimum of $\frac{1}{2} x^T \Sigma x - \sum_i \rho^{des}_i \log(x_i)$ to yield the expected:
$\frac{x_i (\Sigma x)_i}{x^T \Sigma x } = \rho^{des}_i$
numerically.
Definitely could be missing something - but let me know if you get a chance