# Has the following generalization of monotropic programming been studied in the literature?

I am interested in problems of the form

$$\min_{x \in C} \sum_{i=1}^n\sum_{j=1}^n f(x_i,x_j)$$

where $$C$$ is a convex subset of $$\mathbb{R}^{n}$$, and $$f \colon \mathbb{R}^{2} \to \mathbb{R}$$ is convex.

Question: Has this class of optimization problems being studied in some detail?

The Extended Monotropic Programming problem deals with a more general extension to all n variables at a time than the two variables at a time extension you are interested in, except that the literature I have seen deals only with the constraint set $$C$$ being a closed linear subspace, not a general convex set.

Specifically:

$$\min_{x \in \text{closed linear subsapce}} \Sigma_{i=1}^nf_i(x_i)$$ with respect to $$x$$, where $$x = (x_1,...,x_n)$$, $$x_i \in \mathbb{R}^{i}$$

2010 postprint available at https://www.mit.edu/~dimitrib/Extended_Mono.pdf