Let $H_1$ and $H_2$ be two given $n\times n$ matrices. Consider the following matrix-valued function, which maps an arbitrary $n\times n$ unitary matrix $U$ to \begin{equation} f(U)=U H_1 U^\dagger + i H_2 \end{equation} Given any $n\times n$ matrix $M$, define $\sigma_1(M)$ to be the greatest singular value of $M$.
Question: find $\underset{U}{\text{max}}\;\sigma_1(f(U))$.