  We study robust convex quadratic programs where the uncertain problem
parameters can contain both continuous and integer components. Under the
natural boundedness assumption on the uncertainty set, we show that the generic
problems are amenable to exact copositive programming reformulations of
polynomial size. These convex optimization problems are NP-hard but admit a
conservative semidefinite programming (SDP) approximation that can be solved
efficiently. We prove that the popular approximate S-lemma method --- which is
valid only in the case of continuous uncertainty --- is weaker than our
approximation. We also show that all results can be extended to the two-stage
robust quadratic optimization setting if the problem has complete recourse. We
assess the effectiveness of our proposed SDP reformulations and demonstrate
their superiority over the state-of-the-art solution schemes on instances of
least squares, project management, and multi-item newsvendor problems.
