i'd like to know if there is an analytical method to solve the following optimisation problem : $\forall i=1,..,n$ find $\omega_i^{}$ and $\alpha_i^{}$ such that:
$\dfrac{1}{n} \displaystyle \sum_{i=1}^{n} \Big(\omega_i^{} X_i - \alpha_i^{} Y_i \Big)= M$
where $\displaystyle\sum_{i=1}^{n} \omega_i^{}=\displaystyle\sum_{i=1}^{n} \alpha_i^{}=n$ and $X_i,Y_i \in \mathbb{R}_{+}$ and $M\in \mathbb{R}$