  In pure integer linear programming it is often desirable to work with
polyhedra that are full-dimensional, and it is well known that it is possible
to reduce any polyhedron to a full-dimensional one in polynomial time. More
precisely, using the Hermite normal form, it is possible to map a non
full-dimensional polyhedron to a full-dimensional isomorphic one in a
lower-dimensional space, while preserving integer vectors. In this paper, we
extend the above result simultaneously in two directions. First, we consider
mixed integer vectors instead of integer vectors, by leveraging on the concept
of "integer reflexive generalized inverse." Second, we replace polyhedra with
convex quadratic sets, which are sets obtained from polyhedra by enforcing one
additional convex quadratic inequality. We study structural properties of
convex quadratic sets, and utilize them to obtain polynomial time algorithms to
recognize full-dimensional convex quadratic sets, and to find an affine
function that maps a non full-dimensional convex quadratic set to a
full-dimensional isomorphic one in a lower-dimensional space, while preserving
mixed integer vectors. We showcase the applicability and the potential impact
of these results by showing that they can be used to prove that mixed integer
convex quadratic programming is fixed parameter tractable with parameter the
number of integer variables. Our algorithm unifies and extends the known
polynomial time solvability of pure integer convex quadratic programming in
fixed dimension and of convex quadratic programming.
