  Many practical integer programming problems involve variables with one or
two-sided bounds. Dunkel and Schulz (2012) considered a strengthened version of
Chvatal-Gomory (CG) inequalities that use 0-1 bounds on variables, and showed
that the set of points in a rational polytope that satisfy all these
strengthened inequalities is a polytope. Recently, we generalized this result
by considering strengthened CG inequalities that use all variable bounds. In
this paper, we generalize further by considering not just variable bounds, but
general linear constraints on variables. We show that all points in a rational
polyhedron that satisfy such strengthened CG inequalities form a rational
polyhedron. We also consider mixed-integer sets defined by linear constraints,
to which we extend our polyhedrality result by defining the strengthened CG
inequalities on integer variables.
