  In an effort to develop an alternative approach to traditional sparse
reformulations, we will provide a new type of convex reformulation of a large
class of stochastic quadratically constrained quadratic optimization problems
that is similar to Burer's reformulation, but lifts the variables into a
comparatively lower dimensional space. The reformulation rests on a
generalization of the set-completely positive matrix cone. This cone can then
be approximated via inner and outer approximations in order to obtain upper and
lower bounds, which potentially close the optimality gap, and hence can give a
certificate of exactness for these sparse reformulations outside of
traditional, known sufficient conditions. Finally, we provide some numerical
experiments, where we asses the quality of the inner and outer approximations,
thereby showing that the approximations may indeed close the optimality gap in
interesting cases.
