Given $\boldsymbol{a}\in\mathbb{C}^{n},\ \boldsymbol{b}\in\mathbb{C}^{m}$ and $\delta>0$. Show how to solve following problems: $$\min_{\|E\|_{F}\leq\delta}\|E\boldsymbol{a}-\boldsymbol{b}\|_{2} \quad \text{and} \quad \max_{\|E\|_{F}\leq\delta}\|E\boldsymbol{a}-\boldsymbol{b}\|_{2}$$ over all $E\in\mathbb{C}^{m\times n}$ where $\|\cdot\|_{F}$ is Frobenius norm and $\|\cdot\|_{2}$ is vector $2$-norm.
I tried to apply SVD and QR factorization of $E$ but these methods seem to be useless because the variable is matrix $E$ and vectors are fixed, which is different from the least squares problem we usually deal with. Are there some powerful tools to solve it or some tricks should be apllied here?
Any help is appreciated.
Squaring the norms, we have
$$\begin{array}{ll} \text{minimize} & \| \mathrm X \mathrm a - \mathrm b \|_2^2\\ \text{subject to} & \| \mathrm X \|_{\mathrm F}^2 \leq \delta^2\end{array}$$
Vectorizing, we recognize this as a quadratically constrained quadratic program (QCQP)
$$\begin{array}{ll} \text{minimize} & \| ( \mathrm a^{\top} \otimes \mathrm I ) \, \mbox{vec} (\mathrm X) - \mathrm b \|_2^2\\ \text{subject to} & \| \mbox{vec} (\mathrm X) \|_2^2 \leq \delta^2\end{array}$$
which is convex. If we maximize instead, then the problem ceases to be convex.