Consider the optimization problem $$ \min_{ x_1, \ldots, x_N } \sum_{i=1}^{N} f_i( x_i ) \\ \text{s.t.: } \sum_{i=1}^{N} x_i \in X, \ x_i \in X_i \ \forall i \in \{1, \ldots, N\} $$
where $f_1, \ldots, f_N : \mathbb{R}^n \rightarrow \mathbb{R}$ are convex and continuous, $X, X_1, \ldots, X_N \subset \mathbb{R}^n$ are compact and convex.
I am looking for a decentralized scheme to solve the problem.
Comment. I tried with the standard augmented Lagrangian method and the ADMM method, with $x:= (x_1, \ldots, x_N) \in X_1 \times \cdots \times X_N$, an additional variable $y := \sum_{i=1}^{N} x_i \in X$, and augmented cost $f(x) + g(y) := \sum_{i=1}^{N} \{ f_i( x_i ) \} + I_{X}(y)$, where $I_X$ is the characteristic function of the set $X$. However, the primal update $x^{k+1}$ is not entirely decomposable, namely the updates $x_1^{k+1}, \ldots, x_N^{k+1}$ cannot be implemented in parallel.