I'm attempting to find a matrix $X$ that minimizes some function $f(X)$ subject to the constraint that $$ X=W A Z $$ where $A$ is a given non-negative matrix with rows that sum to 1, and $W$ and $Z$ can be any non-negative matrices with rows that sum to 1.
Is it possible to express this in terms of linear constraints on $X$?
Some may wonder why I don't optimize over $W$ and $Z$ directly. The problem is that I may wish $X$ to satisfy several of the above constraints, e.g. $X=W A_i Z_i$ for $i=1..k$. Perhaps it might be possible to search over $W$ and the set of $Z_i$, then enforce $W A_i Z_i = W A_j Z_j$ for all $i\ne j$. However, this would involve (I believe) a quadratic equality constraint (assumin the $W$ and $Z_i$ were collapsed into a vector that was being optimized).