to maximize a ratio of quadratic forms, $(u^\top Mu) / (u^\top Ku)$, or a canonical correlation analysis ratio $(u^\top Rv) / [(u^\top Ku)^{1/2} \times (v^\top Lv)^{1/2}]$, is done straightforwardly by a change of variables and then a singular value decomposition. Here $M$, $K$ and $L$ are symmetric positive definite matrices, and $R$ is any matrix ($u$ and $v$ are column vectors).
But, what if several norms are involved, like
(i) maximize w.r.t. $u$: $(u^\top Mu) / [(u^\top K_1u)^{1/2} \times (u^\top K_2 u)^{1/2}]$, or
(ii) maximize w.r.t. $(u,v)$: $$ \frac{u^\top R_1 v}{(u^\top K_1u)^{1/2}\times (v^\top L_1 v)^{1/2}} +\frac{u^\top R_2 v}{(u^\top K_2 u)^{1/2}\times (v^\top L_2 v)^{1/2}} $$ Are there any analytical or numerical good solutions for these problems? The objective functions in (i) and (ii) are both scale-invariant (multiplying $u$ and/or $v$ with a constant does not change the objective function), so there must be maximum. But can it be computed in an efficient way?
/Tobias