I would like to solve (as mathematically and formally as possible) that the following Markowitz problems are equivalent. The big point is: I want to show that it is equivalent to constrain the return of the portfolio to be greater than $m$ or equal to $m$.

Formulation I enter image description here

Formulation II enter image description here


I guess it amounts to saying that you want to exclude the case when the optimal portfolio $w_*$ is such that $\mu'w_{*}>m$. Notice that, given that $\Sigma$ is positive definite, you can choose another portfolio $w_{**}=w_{*}-1\epsilon$, with $\epsilon>0$ and small enough, such that $\mu'w_{**}=\mu'w_* - \mu'1\epsilon>m$, but clearly $w_{**} {'}\Sigma w_{**}=w_{*} {'}\Sigma w_{*} - 1{'}\Sigma 1 \epsilon <w_{*} {'}\Sigma w_{*}$, because $1{'}\Sigma 1>0$. This contradicts the optimality of $w_*$

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