Let $P =$ {$ x \in \mathbb{R}^n : Ax \geq c$}. Assume that $P$ is a polytope (though maybe this is an unimportant assumption; perhaps the best algorithm for polytopes works for all polyhedra?).
Let $f : \mathbb{R}^n \to \mathbb{R}^m$ be a linear map. Then $Q = f(P)$ is a polytope (or more generally, a polyhedron), and hence also has a representations of the form $Q =$ {$ y \in \mathbb{R}^m : Bx \geq d$}.
My question is this:
Given the matrix $A$, the vector $c$, and matrix for $f$, what is the most efficient way to get some $B$ and $d$ as above?
Note: For my application, I really don't care whether the representation is minimal (in the sense of only including constraints corresponding to facets), I just care that the algorithm run quickly. (Thus, for example, converting the H-representation for $A$ to the V-representation, mapping the extreme points via $f$ to $\mathbb{R}^m$, then converting the V-representation for $B$ to the H-representation would not qualify, as the conversion from H-representation to V-representation for $A$ takes too long.)
PS: If you know any methods for doing this assuming some extra structure, I would also be happy to hear about these.
Computational geometry tools can define polytopes as well as perform affine transformations for you on low-dimensional polytopes. Here are some of the relevant tools:
See How to compute the image of a polyhedron under a linear transformation for the complexity of this problem, and a non-exhaustive list of approaches to perform this operation.