I have the following equation:
$Y = Gh \ \circ \left[ D x \right] $
where $Y$ is a vector of size $L \times 1$, $G$ is a unitary matrix with size $L \times L$, $h$ is vector $L \times 1$ with first $M$ values are non-zeros and all others are zeros $(M \ll L)$, $\circ$ is point-wise multiplication operation, $D$ is a unitary matrix with size $L \times L$ and $x$ is vector $L \times 1$ with first $M$ values are known and all others are not known $(M \ll L)$.
In the above equations, $Y, G, D$ and first $M$ values of $x$ are known while $h$ and last $(L-M)$ values of $x$ to be calculated.
My question, can we calculate or (or using any optimization algorithm to calculate) the vector $h$ and last $(L-M)$ values of $x$ using the above equation? It easy to notice that above equation can be divided into two equations, the first one includes the first $M$ values of $Y$ which its corresponding values, and the second equation is the other values of $Y$, but I do not know if there is a way or algorithm to calculate $h$ and last $(L-M)$ values of $x$
NP: the first $M$ values of $x$ can be changed, but $h$ and $Y$ are random values.