# How to solve the following optimization problem?

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.

• What is the equality you want to know about? Oct 5, 2022 at 22:34
• I want to know whether my strategy works. Thank you sir in advance .
– Goga
Oct 5, 2022 at 22:42
• @Goga: (paraphrasing Daniel) you say "equality" and there is no equality :) Oct 6, 2022 at 7:29
• Thank you Daniel and Geordie, I've corrected the post.
– Goga
Oct 6, 2022 at 10:17