Consider the set of linear constraints $C = \{(x,y)\in \mathbb{R}_+^n\times \mathbb{R}_+^n: Bx \geq A y, e^{\top}x = 1\},$ where $e$ denotes the vector of ones, and the matrices $B$ and $A$ are assumed to be real positive definite (without requiring symmetry). I would like to know whether the vector $y$ over $C$ is bounded and how to estimate a bound of $y$ if the answer is yes. Otherwise, please provide a counterexample to show that $y$ is unbounded. I appreciate any help or ideas in advance.
What I've tried: I conducted several numerical tests with randomly generated real positive definite matrices $A$ and $B$ by solving the linear program $$\max\{ e^{\top} y: (x,y)\in C\}.$$ So far, I have not found any unbounded example yet. Hence, I suspect that $y$ might be bounded.
