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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). 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 $\det(M-T)\geq 0$

We can deduce a couple simple bounds. For instance, the non negativity of $M-T$ implies that all its element are themselves non negative, thus $\frac{1}{\lambda_i} \geq \Sigma^{-1}_{i,i}$ and $$\det(\Lambda^{-1}) \geq \prod_i^{n} \Sigma^{-1}_{i,i}$$

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). 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 $\det(M-T)\geq 0$

We can deduce a couple simple bounds. For instance, the non negativity of $M-T$ implies that all its element are themselves non negative, thus $\frac{1}{\lambda_i} \geq \Sigma^{-1}_{i,i}$ and $$\prod_i^{n} \Sigma^{-1}_{i,i} \leq \det(\Lambda^{-1}) \leq \hbox{Tr}(\Sigma^{-1})^n$$

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). 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 $\det(M-T)\geq 0$

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). 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 $\det(M-T)\geq 0$

We can deduce a couple simple bounds. For instance, the non negativity of $M-T$ implies that all its element are themselves non negative, thus $\frac{1}{\lambda_i} \geq \Sigma^{-1}_{i,i}$ and $$\det(\Lambda^{-1}) \geq \prod_i^{n} \Sigma^{-1}_{i,i}$$

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 maximize $\det(M)$ 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). 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 $\det(M-T)\geq 0$

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). 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 $\det(M-T)\geq 0$