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

Question

I am interested in problems of the form
$$min_{x in C} sum_{i=1}^nsum_{j=1}^n f(x_i,x_j)$$
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?
Thank you in advance for your help.

Mark L. Stone · Answer

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

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

One Answer

Add your own answers!

Ask a Question