Well, there are not that many ways to define a facet, I guess. The only source of confusion I could imagine might be the concrete matrix representation: if you have a linear system A*x <= B, you have to specify INEQUALITIES and FACETS as a block matrix (B | -A)

because polymake's polyhedra are always defined as (FACETS)*x >= 0, but you surely know this already.

I don't know what do you call "affine hull". In polymake, AFFINE_HULL has a different meaning, it's the matrix B such that B*x=0 for all x in the polyhedron, aka linear span. It can't be taken "instead" of facets but they always have to come together. In the raw (possibly redundant) input they are called INEQUALITIES and EQUATIONS, later the non-redundant properties can be computed which are called FACETS and AFFINE_HULL. So you are probably using some CPLEX terminology?

Where do you obtain the "better" optimal value - with polymake (which algorithm?) or with some third-party software?