I'm wondering if there are any standard techniques for exploiting structure in multilinear equations. An example of what I have in mind is solving
$A_{ab} X_{bc} A_{cd} (B_{ad} B_{bc} + B_{ac} B_{bd}) = RHS_{ad}$
for X. Here, A, B, and X are n-by-n matrices and there is summation over b,c. Of, course, one can view this as a large equation system
$M vec(X) = vec(RHS)$
where M has dimension n^2 by n^2. But I'd like to somehow exploit the structure of M to reduce the rank.