$\underset{X}{\text{maximize}}$ $ \;\text{min}(\lambda_{\text{min}}(R-X),\; \lambda_{\text{min}}(S - X^{-1})) $
subject to $ \;R \succ0,\; S\succ 0,\; X \succ 0,\;R-X \succ 0,\; S-X^{-1} \succ 0$
where $R,S$ are positive definite matrix that are known and fixed, operator $\lambda_{\text{min}}$ takes the minimum eigenvalue of its target matrix.
I was wondering if the optimized value is taken when $R-X = S-X^{-1}$ just like the scalar case? Thank you.