Is the a numerical method (or even analytic solution!) for the following system of quations?
Let $x \in \mathbb R^J$ and $y \in \mathbb R^I$ be our unknowns, $A$ a known matrix of size $J\times I$ and $c \in \mathbb R^J$ and $b\in \mathbb R^I$ constant vectors, for which $\sum_{j=1}^J c_j = \sum_{i=1}^I b_i $. The system of equations is given by
$ D_x A y = c$ , $D_yA^Tx =b$
where $D_x=diag(x)$ and $D_y = diag(y)$. Note that, if $(x,y)$ form a solution to this system $(tx,\frac{1}{t} y)$ is also another solution, so for rulling out the trivial cases, we can always normalize $x_J=1$. In other words, we have $I+J-1$ equations and $I+J-1$ unknowns.