I need to minimize the following L-infinity norm with respective to $\tau$

 $$ min_{\tau} \| I -S(S+\tau)^{-1}\| $$

$$ \text{subject to} \ S+\tau \ \text{is positive definite} $$ where $S$ is symmetric but not full rank, $\tau$ is non-singular diagonal matrix.

Any idea? Thanks.

I need to minimize the following L-infinity norm with respective to $\tau$. L-infinity norm of a matrix $A$ is defined as $\|A\| = max_{i,j}|a_{i,j}|$. $$ min_{\tau} \| I -S(S+\tau)^{-1}\| $$

$$ \text{subject to} \ S+\tau \ \text{is positive definite} $$ where $S$ is symmetric but not full rank, $\tau$ is non-singular diagonal matrix.

Any idea? Thanks.