Consider this problem: $\max \limits_{C \in \mathbb{R}^{m*1},\theta \in \mathbb{R}^{n*1}} C^{T}\Sigma \theta$ ,s.t. $C^{T}C=1$ and $\theta^{T}H\theta=1$
where,$C \in \mathbb{R}^{m*1},\Sigma \in \mathbb{R}^{m*n},\theta \in \mathbb{R}^{n*1},H \in \mathbb{R}^{n*n}$,H is symmetric.
Before giving your advise , please take a look at my try:
Consider $f(C,\theta)=C^{T}\Sigma \theta -\lambda (C^{T}C-1)-\mu (\theta^{T}\theta-1)$,
we have,
$\frac{\partial f}{\partial C}=\Sigma \theta -2\lambda C=0$
$\frac{\partial f}{\partial \theta}=\Sigma C-\mu (H^{T}+H)\theta=0$
putting $C^{T}C=1$,$\theta^{T}H\theta=1$.
Hence, $\lambda=\frac{1}{2}C^{T}\Sigma \theta$ ,$\mu=\frac{1}{2}\theta^{T}\Sigma^{T}C$.
Then
$\Sigma \theta=C^{T}\Sigma \theta C$,$\Sigma^{T}C=\frac{1}{2}\theta^{T}\Sigma{T}C(H+H^{T})$.
But I don't know how to make $C$,$\theta$ out from the above two equations.
Thanks in advance!