I have two real functions $f(X),g(X)$ where the argument $X$ is a real matrix. The solution $X^*$ for the problem of minimizing $f$ is ending up maximizing $g$ as well. I am looking for a way to prove or at least understand the condition/reasoning behind why this is happening for these two functions.
$f(X)=Tr(X^TAX)-Tr(X^TBX)$ and $g(X)=\frac{Tr(X^TBX)}{Tr(X^Th(X)X)}$ and
$h(X)=diag(XX^T\mathbb{1})-XX^T$. $A$ and $B$ are p.s.d matrices.
$B$ is a matrix that does not depend on $X$.
Coming to $A$, this is what I do, I first set $A=h(X)$ for some chosen $X$ and then I keep that matrix constant, in the optimization of $f(.)$, that is it is only optimized on the $X$ terms other than $A$. Let me know if you need more detail/clarification on how I use $A$ in this optimization.
Edit: Answered before new details were added. I'll revisit this.
This will be true generically when $g$ is a monotonic transformation of $f$ (transformation by a strictly increasing function). When this is the case: $g(x)\geq g(\tilde{x})\Leftrightarrow f(x)\geq f(\tilde{x}) $
Since $f(x^*)\geq f(x)$ for all $x$ in the domain, $g(x^*)\geq g(x)$ for all $x$ in the domain as well.