under what conditions the following matrix optimization has a unique solution?

Question

So the problem is simple:

Consider the following matrix optimization problem on matrix D. What conditions on the matrix dimensions should apply so that the solution to the minimum is unique. please note that D $\in R^{n\times n}$ is square and $M\in R^{m\times n}$. and $a$ and $b$ are simply vectors of length $n$ and $m$ respectively.

Here is the optimization :

\begin{equation} \begin{aligned} & ~~~~~~~~ \underset{D}{\text{min}} & & \|b-MD a\|_2 \\ & \text{subject to:} & & \\ & & & \|D\|_2<1.\\ \end{aligned} \end{equation}

My comments: I think if $n>m^2$ then it has a unique solution (minimizer) because that means we have basically $m^2$ variables to solve with n equations. so

Answer 1

As stated, the problem might have no solutions at all. If the constraint would be replaced by $\|D\|\le 1$ then there is at least one solution.

If $n>1$ then the problem never has a unique solution.

Let $D_1$ be such at solution. Then there are plenty other matrices $D$ such that $Da = D_1a$ and $\|D\|\le1$, hence $D$ is also a solution.

Hence, the answer to the question is: The problem is uniquely solvable, if the constraint set is $\|D\|\le 1$, $n=1$, $M\ne 0$, $a\ne0$.