Constrained matrix least squares with a set of quadratic form constraints

Question

WHAT I DON'T KNOW:

MATRIX CASE: I have a matrix equation $Y=AB$, where Y and A are given and we would like to find $B$. Here are more information: $Y \in R^{m \times n}$ with $m >n$, $A \in R^{m \times k}$ and $B \in R^{k \times n}$ with $k <n$. Obviously the problem is over-determined. Let $\{ \textbf{b}_i\}_{i=1}^{n}$ denote the column vectors of $B$. Now here are the constraints on these vectors. $\textbf{b}_i^TR_i \textbf{b}_i,i=0,1,2,\cdots,n$. How to formulate this problem as an optimization equation and solve it minimizing the least squares or Frobenius norm?

WHAT I KNOW

VECTOR CASE: I know for $\textbf{y}=A\textbf{b}~~$ ($\mathbf{y,b}$ are vectors) and with the constrain $\textbf{b}^TR \textbf{b}$, the optimization formulation is

\begin{equation} \underset{\textbf{b}}{\operatorname{min}}~ \| \textbf{y} - A \textbf{b} \|^{2} + \lambda (\textbf{b}^T R \textbf{b}) \end{equation}

with the least squares solution

\begin{equation} \widehat{\textbf{b}}=\left(A^T A +\lambda R \right)^{-1}A^T \textbf{y}~ \end{equation}

Could anyone help me with the matrix case?

Answer 1

If you want to have in a form you are familiar with, you can exploit the fact that $\text{vec}(Y) = (I^{k\times k} \otimes A) \text{vec}(B)$ where $I^{k\times k}$ denotes the identity matrix and $\otimes$ is the kronecker product.