Let $G=(V,E)$ be a connected network with $|V|=n$. Consider the following optimization problem I'm trying to know under which conditions the following minimization problem has solution : $${\sum _{i=1}^n}\underset{x\in \cap_{j=1}^n X_j(i)}{\min}{\sum _{j\in N(i)}}f_{i,j}(x),$$ where $N(i)$ is the neighborhood of the node $i$ on the network and the $X_i$ are a convex set of local constraints known only for the node $i$ and $f_{i,j}$ is a strictly convex function. I think the following:

First, we check have to make sure that $x$ is a feasible variale .i.e, $\cap_{i=1}^n\cap_{j=1}^n X_j(i) \neq \emptyset$.

Second, we solve $\underset{x\in \cap_{j=1}^n X_j(i)}{\min}{\sum _{j\in N(i)}}f_{i,j}(x)$ for each $i$ and finally we sum over $i$.

Does this will work or do I have to look for something else? Thank you in advance.