  We study sets defined as the intersection of a rank-1 constraint with
different choices of linear side constraints. We identify different conditions
on the linear side constraints, under which the convex hull of the rank-1 set
is polyhedral or second-order cone representable. In all these cases, we also
show that a linear objective can be optimized in polynomial time over these
sets. Towards the application side, we show how these sets relate to commonly
occurring substructures of a general quadratically constrained quadratic
program. To further illustrate the benefit of studying quadratically
constrained quadratic programs from a rank-1 perspective, we propose new rank-1
formulations for the generalized pooling problem and use our convexification
results to obtain several new convex relaxations for the pooling problem.
Finally, we run a comprehensive set of computational experiments and show that
our convexification results together with discretization significantly help in
improving dual bounds for the generalized pooling problem.
