Given some unitary matrices mit determinant one $U_1,\dots,U_n\in \mathrm{SU}(2)$, I would like to determine an/the element $U\in SU(2)$ such that $$f(U) = \sum_{k=1}^n \left(\mathrm{trace}(UU_k)\right)^2$$ is maximal.
My approach so far: any infinitesimal transformation of $U$ can be written as $U\mapsto \exp(i\epsilon H)U$ with $H\in \mathfrak{su}(2)$ a hermitian matrix with trace zero. So I can get the condition $$0 \stackrel{!}{=} \frac{\partial}{\partial\epsilon}f(\exp(i\epsilon H)U)\vert_{\epsilon=0} = 2i\sum_{k=1}^n\mathrm{trace}(UU_k)\mathrm{trace}(UU_kH)\hskip 1cm \forall H\in\mathfrak{su}(2)$$ $\mathfrak{su}(2)$ is a $3$-dimensional Lie-Algebra, so this condition consists of $3$ independant equations. They are only quadratic in (the matrix components of) $U$ so it should be possible to solve for $U$