  We consider in this paper a class of semi-continuous quadratic programming
problems which arises in many real-world applications such as production
planning, portfolio selection and subset selection in regression. We propose a
lift-and-convexification approach to derive an equivalent reformulation of the
original problem. This lift-and-convexification approach lifts the quadratic
term involving $x$ only in the original objective function $f(x,y)$ to a
quadratic function of both $x$ and $y$ and convexifies this equivalent
objective function. While the continuous relaxation of our new reformulation
attains the same tight bound as achieved by the continuous relaxation of the
well known perspective reformulation, the new reformulation also retains the
linearly constrained quadratic programming structure of the original
mix-integer problem. This prominent feature improves the performance of
branch-and-bound algorithms by providing the same tightness at the root node as
the state-of-the-art perspective reformulation and offering much faster
processing time at children nodes. We further combine the
lift-and-convexification approach and the quadratic convex reformulation
approach in the literature to form an even tighter reformulation. Promising
results from our computational tests in both portfolio selection and subset
selection problems numerically verify the benefits from these theoretical
features of our new reformulations.
