Question on poset of positive semi-definite matrices

Question

Let $\Omega$ be a subset of the partially ordered set (poset) of $n\times n$ positive semi-definite matrices. I know that $\inf \Omega\in \bar{\Omega}$, where $\bar{\Omega}$ denotes the closure of $\Omega.$

If $\inf \Omega=X$, can I say that

\begin{array}{ll} \text{inf} & \operatorname{trace}(Y'\Omega Y)=\operatorname{trace}(Y'XY)?\\ \Omega\end{array}

I think trace is monotone increasing on the cone of positive definite matrices, it should hold?

Answer 1

I believe that you are trying to ask the following:

If there exists an $X \in \bar \Omega$ such that $X \preceq Z$ (Loewner order) for all $Z \in \Omega$, then does it necessarily hold that $$ \inf_{Z \in \Omega} \operatorname{trace}(Y'ZY) = \operatorname{trace}(Y'XY)? $$

The answer to this is yes. In particular, if such an $X$ exists, then for all $Z \in \Omega$, we have $$ \operatorname{trace}(Y'ZY) = \operatorname{trace}(Y'(Z - X)Y) + \operatorname{trace}(Y'XY) \geq \operatorname{trace}(Y'XY). $$