Given $n$ trace-free hermitian mutually orthonormal matrices $M_1,...,M_n \in \mathbb{C}^{d \times d}$ and a matrix $A \in \mathbb{C}^{d \times d}.$
I want to solve the problem of finding the optimal $a_1,...,a_n \in \mathbb{C}$ such that $\left\lVert A - \sum_{i=1}^{n } a_i M_i \right\rVert$ is minimal under the constraint that $\langle \sum_{i=1}^{n}a_iM_i x,x \rangle \ge -1 $ for all $x \in \mathbb{C}^d.$
Apparently, this problem is convex, but I do not see how to implement the condition $\langle \sum_{i=1}^{n}a_iM_i x,x \rangle \ge -1 $ (diagonalising and checking eigenvalues in each step seems unreasonable) in an effective algorithm. Does anybody know how these things can be done?
Let $B(a) = - \sum_i a_i M_i$. The condition amounts to $$ -1 \le \min_{x :\; \|x\|_2 = 1}\langle -B(a)x ,x \rangle = - \max_{x :\; \|x\|_2 = 1}\langle B(a) x ,x \rangle = -\lambda_{\max}(B(a)). $$ That is the condition is $\lambda_\max(B(a)) \le 1$. I am going to assume that it is benign to also add the condition $\lambda_\max(B(-a)) \le 1$ (maybe this can be argued formally). The the two conditions amount to $\|B(a)\|_{op} \le 1$ for the operator 2-norm. The problem can be cast as an SDP: \begin{align} \min \| A - B(a)\| \quad \text{subject to} \\ \begin{pmatrix} I & B(a) \\ B^*(a) & I \end{pmatrix} \succeq 0 \end{align}