So you want $$\forall x,\\, x^t (\Lambda^{-1} - \Sigma^{-1}) x \geq 0$$
$(\Lambda^{-1} - \Sigma^{-1})$ is a symmetric matrix, and the condition expresses that it must be a positive matrix too. Let $T = \Sigma^{-1}$ and $M = \Lambda^{-1}$ we want to minimize $\det(M)$ with the constraint that $M-T$ is semi-definite positive.

This constraint can be expressed by calculating the Cholesky decomposition of $M-T$, giving $n$ inequality constraints. 

At this point, I would suggest resorting to quadratic programming to solve the KKT system. (See for instance [this][1]). The Cholesky algorithm can be adapted to compute derivatives in the constraints.

Edit: removed a system of equation that only relied on the (necessary) constraint $\det(M-T)\geq 0$


  [1]: http://homes.esat.kuleuven.be/~mdiehl/AALBORG/NLPinNutshell.pdf