# How to minimize trace of the following matrix?

Let

$B=I_m+U{^\prime}U-\frac{1}{(m+3)}(1_m^{\prime}+1_2^{\prime} U){^\prime}(1_m^{\prime}+1_2^{\prime}U)$,

where $1_m$ is a column vector of all ones of size $m$ and $U=[u_1^{k_1}u_2^{k_2}u_3^{k_3}u_4^{k_4}]$ is size $2\times m$, where $u_1=(1\;1)^\prime$, $u_2=(1\;0)^\prime$, $u_3=(0\;1)^\prime$ and $u_4=(0\;0)^\prime$. Note that $k_1+k_2+k_3+k_4=m$, $m\ge3$ and $k_1, k_2, k_3, k_4$ are the multiplicity of each $u_1, u_2, u_3, u_4$, respectively.

I'm trying to find the values of $k_1, k_2, k_3$ and $k_4$ such that $minimize\{tr(B^{-1})\}$.