I’m trying to understand what is the exact process to minimize the following constrained Frobenius norm equation: $$ ||A-CW||_{F}^2 $$ s.t. $$ WW^T\ge D $$ I understand one possible way is to use the lagrangian (i couldn't find the formal explanation for how to use it with matrices and norms). what are the steps to solve equations of such kind.
what will be the way to solve the same problem for different type of norm, for example $$ ||A-CW||_{2}^2 s.t. WW^T \ge D$$
i chose the lagrangian to be $$L(W,\mu)=Tr(AA^T-AW^TC^T-CWA^T+CWW^TC^T)+Tr(\mu^T(WW^T-D)$$ in such case $$\nabla_{W}L=2C^TCW-2C^TA2+\mu^TW=0$$ which means $$W=\frac{C^TA}{C^TC+\mu^T}$$ along with the constraint $$WW^T=D$$ we get $$\frac{C^TAA^TC-DC^TCC^TC}{D}=\mu^TC^TC+C^TC\mu+\mu^T\mu$$
how do i isolate $$\mu?$$ what did i do wrong? thank you.