The Frank–Wolfe theorem states that a quadratic function bounded from below on a polyhedron
attains its infimum. We study when this remains true in the presence of integer variables, for sets
defined by quadratic inequalities. Our main result is a complete classification for sets defined by convex
quadratic inequalities with rational coefficients and a rational affine hull, a condition that holds for all full-
dimensional sets. One integer variable never destroys attainment: if every rational quadratic objective that
is bounded from below attains its infimum over the set, even after adding rational linear constraints, the
same holds with one integer variable. In dimension at most 2, integer variables never destroy attainment.
In dimension at least 3 and with two or more integer variables, attainment holds for every such objective
if and only if the set has bounded curvature, that is, it is bounded except along directions in which all
constraints are linear. Without convexity, these results remain valid, except that bounded curvature is no
longer necessary. Moreover, integer variables can destroy attainment only if there are at least two of them
and at least one continuous variable. The result for one integer variable follows from a lifting theorem,
which also extends known results for cubic objectives and convex polynomial constraints. Finally, for
convex constraints and rational data, attainment holds when the objective is bounded from above, convex,
or quasi-convex over a full-dimensional convex set containing the feasible region.