Instead of projecting (which requires some convention about bases to be 
able to work with coordinates) I would try to use the Minkowski sum with 
the linear space in the direction that you want to unconstrain.

E.g. the diagonal in the unit square:

sage: P = Polyhedron([(0,0), (1,1)])

Say you want to project along the x-axis. Instead, Minkowski-add the 
x-axis: 

sage: Q = P + Polyhedron(lines=[(1,0)])
sage: Q.Hrepresentation()
(An inequality (0, 1) x + 0 >= 0, An inequality (0, -1) x + 1 >= 0)

You can also intersect with some linear subspace not parallel to the x-axis 
(say, the y-axis) to realize the projection. 

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