There is a type of question which gets closed a lot which I find frustrating. An example is How to compute the volume of a region transformed by a matrix? , which I recently took the liberty of rewriting. If I understood the OP's question correctly, it is definitely a topic of current research. For example, Alex Barvinok and Michael Joswig think about the sort of polytope questions I raise. I am less familiar with the semidefinite programming literature, but Pablo Parrilo and Cynthia Vinzant are two people who think about variable elimination in nonpolyhedral convex geometry. It strikes me as unlikely that this question would get a good answer on math.SE ; it is not a textbook topic.
At the same time, there are a lot of clues in the OP's phrasing that suggests they are not at a level to understand this literature. Most substantively, they don't understand that the answer to a question about algorithms should be clear as to how the input data is specified and which parameters are growing and which are fixed. On a more superficial level, they don't emphasize the key point that $r < n$ and they don't use buzzwords like "linear programming" and/or "semidefinite programming" and "variable elimination".
I want us to help people in this situation. Is there a good way to help OP's in this situation connect with experts who can help them?