Is there an elegant solution to the following minimization problem?
$$ \min_{B} \left( \sum_i a_i^\top C_i a_i \right) - \left( \sum_i C_i a_i \right)^\top \left( \sum_i C_i \right)^{-1} \left( \sum_i C_i a_i \right) $$ where the $a_i$ are given column vectors in $\mathbb{R}^n$ and each $C_i$ is a $\mathbb{R}^{n \times n}$ projection matrix defined as the projection onto the intersection of the linear subspace $B$ (defined as the column-space of $B$, a matrix in $\mathbb{R}^{n \times h}$) with the given linear subspace defined by $A_i$ (defined as the row-space of $A_i$, a matrix in $\mathbb{R}^{m \times n}$).
If it helps, $C_i$ can be written (via the Anderson-Duffin formula) as $2P_{A_i}(P_{A_i} + P_B)^{\dagger}P_B$. (I believe it is safe to assume that $P_{A_i} + P_B$ is full rank for my problem, so the inverse can be used rather than the pseudoinverse.)
This problem derives from the minimal distance between flats (affine subspaces): $| C( x_0 - y_0 ) |$, where $x_0$ is any point on one flat and $y_0$ is any point on the other flat and $C$ is the projection matrix onto the intersection of the two flats' orthogonal spaces. (See ``Distance and parallelism between flats in $R^n$'' [Arthur M. DuPré and Seymour Kass 1992] in Linear Algebra and Its Applications.) The sum of squared distances between a flat with linear subspace $B$ as above passing through a point $p$ and the flats with linear subspaces $A_i$ as above passing through points points $a_i$ is $\sum_i | C_i ( p - a_i ) |^2 = p^\top \left( \sum_i C_i \right) p + \left( \sum a_i^\top C_i a_i \right) - 2 p^\top \left( \sum C_i a_i \right)$. This is minimized (by setting the derivative with respect to $p$ to $0$) when $p = \left( \sum C_i \right)^{-1} \left( \sum_i C_i a_i \right)$. Substituting this expression for $p$ results in the expression at the top.