  In recent literature, many successful attempts have been made in
reformulating non-convex optimization problems as convex conic optimization
problems, or at least in giving powerful relaxations of this type. However, it
is often the case that these approaches suffer from poor computational
performance due to the size of the reformulation. In this paper we study QCQPs
whose special structure allows for a reduced conic relaxation where the
variables are lifted into a smaller space compared to traditional approaches.
We show that in special cases these relaxations are exact reformulations. We
contextualized these results in the field of matrix completion in two ways.
First we derive an exact reformulation of the QCQPs that is based on the
geometry of the cone of partial arrowhead matrix that have a set-completely
positive completion. However, we do not reference the completability in the
proof, but instead rely purely on geometric aspects. We also show, that our
reduced conic reformulation can be seen as relaxation of the latter exact
reformulation and describe the gap between those two in terms of the difference
of two subsets of a space that is isomorphic to the space of arrowhead matrices
if a certain type. Second we show that our partial exactness results imply a
new sufficient conditions for set-completely positive matrix completion.
