Let $A$ be a symetric and definite positive matrix of size $n \times n$. Let $\Sigma \subset \mathcal{P}( \{1, \dots, n\})$ a set of indices and consider the projection operator $P_{\Sigma} : \mathbb{R}^n \rightarrow \mathbb{R}^n$ defined by
$$\left[P_\Sigma (U) \right]_{i}= U_i \text{ if } i \in \Sigma \ \text{ and } \ 0 \text{ if } i \notin\Sigma.$$
(i.e all the components of $U$ outside of $\Sigma$ are put to zero). For simplification we will use the notation $U_\Sigma:= P_\Sigma(U)$.
I want to study the following minimization problem :
\begin{equation} \underset{U \in \mathbb{R}^n}{\min} \langle A(U-U_\Sigma) | U_\Sigma\rangle - \langle B_\Sigma | U_\Sigma\rangle \end{equation} where $\langle.|.\rangle$ is the usual scalar product on $\mathbb{R}^n$.
More precisely, I would like to find the Euler equation associated to this problem, but I'm not sure how to deal with the projection operator..
Any help or advices are welcomed !