Supposing I have complex square matrices $B_i$ and $C_i$ ($i = 1,\dots,N$) of dimension $4 \times 4$.

- Is there an effective algorithm for solving the following problem?

$$\begin{align} A=&\underset{A\in\mathbb{C}^{4\times 4}}{\text{argmin}}\sum_{i=1}^{N}{\left\|AB_iA^{-1}-C_i\right\|}\\ \,\\ &\text{subject to }\left\{\begin{matrix}A_{12}=A_{13}=A_{42}=A_{43}=0\\A_{11}+A_{41}=1\\ A_{14}+A_{44}=1\end{matrix}\right. \end{align} $$ Here $\left\|\cdot\right\|$ is a matrix norm. I currently use $\left\|M\right\| = \sigma_{max}(M)$ where $\sigma_{max}(M)$ is the largest singular value of $M$. But if the other choice of the norm could simplify the problem that would be also very helpful.

- How can I make sure that there exists only one unique solution for $A$?

**Update**

I am pretty sure that the solution exists for the invertible $A$. Matrices $B_i$ and $C_i$ are pretty close for all $i$ (I have $N=15$) so I expect the solution to be somewhere nearby $A=I$.

Generally my case looks like follows. I know that there exists such $C_i^t$ that make objective function equal to zero. In this case I can transform the problem to a convex one by minimizing $\sum\left\|AB_i-C_iA\right\|$. This works just perfectly when I use CVX toolbox for MATLAB. However, I don't know the exact $C_i^t$ but have some rough estimations $C_i$.

The best results that I had so far were achieved with the help of genetic optimization algorithm. But the problems that:

- It takes a lot of time to find a minimum. The task looks simple at the first glance so I thought there could be a special solver for this.
- Every time I have a slightly different solution and I still can't understand why: because there are many local minimums nearby the global one or because the global minimum is degenerate. So the question (2) is also crucial for me as I am able to reduce or increase $N$.

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