An Unconstrained Quadratic Optimization problem

Question

Let $\mathbf{A}_n,~ n=1,\ldots,N$ be a set of $N$ known $M\times M$ complex matrices and $\alpha_n,~ n=1,\ldots,N$ be a set of $N$ real numbers. What is the solution of the following optimization problem: \begin{align} \min_{\mathbf{B},\beta_1,\cdots,\beta_M}\sum_{n=1}^N \|\mathbf{A}_n-\mathbf{B}\cdot\mathrm{diag}\{\exp(j\alpha_n\beta_m)\}\|_F^2, \end{align} where $\beta_m$'s are real numbers and $\mathbf{B}$ is an $M\times M$ real matrix. Moreover, $\mathrm{diag}\{\exp(j\alpha_n\beta_m)\}$ is a diagonal matrix with $m$th diagonal entry equals to $\exp(j\alpha_n\beta_m)$.

Answer 1

I'm assuming $j=\sqrt{-1}$ in your problem? Let's expand this out into matrix elements. \begin{equation} F(\mathbf{B}, \{\beta\}) = \sum_{n,i,m} | (\mathbf{A}_n)_i^m - \mathbf{B}_i^m e^{j\alpha_n \beta_m} |^2 \end{equation} Now we take derivatives with respect to $\mathbf{B}_i^m$ and $\beta_m$ and set those derivatives to zero. We get \begin{equation} \sum_n [ (\mathbf{A}_n)_i^m - \mathbf{B}_i^m e^{j\alpha_n \beta_m} ]^* e^{j\alpha_n \beta_m} + c.c. = 0 \end{equation} \begin{equation} \sum_{n,i} [ (\mathbf{A}_n)_i^m - \mathbf{B}_i^m e^{j\alpha_n \beta_m} ]^* j\alpha_n \mathbf{B}_i^m e^{j\alpha_n \beta_m} + c.c. = 0 \end{equation} The first one gives \begin{equation} \mathbf{B}_i^m = Re(\sum_n (\mathbf{A_n})_i^m e^{-j\alpha_n \beta_m}) \end{equation} You can substitute that into the second equation to get constraints on the $\beta$s, but it's a bit complicated.