I need to solve the following problem.
Given $n \times n$ Hermitian matrices $A\geq 0$ and $B_1, ~ B_2$ (need not be positive semidefinite), with $Tr(AB_1)<0~,Tr(AB_2)<0$ construct a Hermitian matrix $B$ such that $B\geq B_1$ and $B\geq B_2$, for which $\min Tr(AB)$ is achieved.
For Hermitian matrices $X$ and $Y$, we write $X\geq Y$ if and only if $X-Y\geq 0$ (i.e. we define a partial order). There is no partial order relations between $B_1$ and $B_2$, otherwise the solution would have been easier.
Using some earlier programmes, I have constructed $A$, $B_1,~B_2$ (as well as $B_3,~B_4,...$, finite number of such $B$'s). Needless to say, there are all huge matrices and was time consuming to construct. I really do not want to extract the matrices as data and use it in some other software, unless it is absolutely necessary. Can someone please help me to pointing out some method for solving in Mathematica itself?
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