When solving for the minimum variance portfolio, we have the object:
$$ f(w) = \frac{1}{2} w^T \Sigma w $$ subject to a basic scaling constraint: $$ \sum_{i=1}^N w_i = 1 $$ or in matrix terms, $w^T \mathbf{1} = 1$ where $\mathbf{1}$ is the $n$ vector of all ones.
Forming the Lagrangian, we get: $$ \Sigma w - \lambda \mathbf{1} = 0 $$ From which we have: $$ w = \lambda \Sigma^{-1} \mathbf{1}$$ Using the constraint, we can solve for $\lambda$ which ends up being a normalizing constant.
I tried to solve for the "minimum standard deviation" portfolio in a similar way, subject to the same constraint. It has the objective function: $$ f(w) = \sqrt{w^T \Sigma w} $$ its solution should be the same as the minimum variance portfolio because the objective is simply a monotone transformation of of the minimum variance objective. Forming the lagrangian again, we get:
$$ \frac{\Sigma w}{\sqrt{w^T\Sigma w}} - \lambda I = 0 $$
I am however, unclear where to proceed from here, as I can't just invert $\Sigma$ to get a solution due to there being another function of $w$ in the denominator. Is there something wrong with apply Lagrange multipliers here, due to the non-differentiability of $\sqrt{.}$? Or am I missing something very obvious?