  Quadratically constrained quadratic programs (QCQPs) are an expressive family
of optimization problems that occur naturally in many applications. It is often
of interest to seek out sparse solutions, where many of the entries of the
solution are zero. This paper will consider QCQPs with a single linear
constraint, together with a sparsity constraint that requires that the set of
nonzero entries of a solution be small. This problem class includes many
fundamental problems of interest, such as sparse versions of linear regression
and principal component analysis, which are both known to be very hard to
approximate. We introduce a family of tractable approximations of such sparse
QCQPs using the roots of polynomials which can be expressed as linear
combinations of principal minors of a matrix. These polynomials arose naturally
from the study of hyperbolic polynomials. Our main contributions are
formulations of these approximations and computational methods for finding good
solutions to a sparse QCQP. We will also give numerical evidence that these
methods can be effective on practical problems.
