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)][1]

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


  [1]: https://link.springer.com/article/10.1007/s10957-008-9393-3