Let $\Sigma_0$, $\Sigma_1$ be known $p\times p$ symmetric positive semi-definite matrices, and $\Gamma_0$ and $\Gamma_1$ be $p\times p$ prespecified matrices. Define $W=\text{diag}(w_1,\ldots,w_p)$ as the a diagonal matrix, where $w_i\in[0,1]$ for $i=1,\ldots,p$.
Find the optimal diagonal weight matrix $W^*$ such that for any $W$,
\begin{equation} \left\{W\Gamma_0+(I-W)\Gamma_1\right\}^{-1}\left\{W\Sigma_0W+(I-W)\Sigma_1(I-W)\right\}\left\{W\Gamma_0+(I-W)\Gamma_1\right\}^{-1\top} -\left\{W^*\Gamma_0+(I-W^*)\Gamma_1\right\}^{-1}\left\{W^*\Sigma_0W^*+(I-W^*)\Sigma_1(I-W^*)\right\}\left\{W^*\Gamma_0+(I-W^*)\Gamma_1\right\}^{-1\top} \end{equation} is non-negative definite.
My first thought is to transform this problem to be an optimization problem \begin{array}{ll}\text{minimize} & \left\{W\Gamma_0+(I-W)\Gamma_1\right\}^{-1}\left\{W\Sigma_0W+(I-W)\Sigma_1(I-W)\right\}\left\{W\Gamma_0+(I-W)\Gamma_1\right\}^{-1\top} \\ \text{subject to} & W \succeq 0 \\ & I - W \succeq 0. \end{array}
This is applied to finding the most efficient estimator in statistics. Really appreciate any thoughts!