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.

Specifically:

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

See "Extended Monotropic Programming and Duality", Dimitri P. Bertsekas, Journal of Optimization Theory and Applications volume 139, pages209–225(2008)

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

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 $x = (x_1,...,x_n)$, $x_i \in \mathbb{R}^{i}$

See "Extended Monotropic Programming and Duality", Dimitri P. Bertsekas, Journal of Optimization Theory and Applications volume 139, pages209–225(2008)

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

Additional reference:

On a zero duality gap result in extended monotropic programming,Radu Ioan Bot, Ern ̈o Robert Csetnek"

Older published version: https://link.springer.com/article/10.1007/s10957-010-9733-y