  Quadratically constrained quadratic programs (QCQPs) are a fundamental class
of optimization problems well-known to be NP-hard in general. In this paper we
study conditions under which the standard semidefinite program (SDP) relaxation
of a QCQP is tight. We begin by outlining a general framework for proving such
sufficient conditions. Then using this framework, we show that the SDP
relaxation is tight whenever the quadratic eigenvalue multiplicity, a parameter
capturing the amount of symmetry present in a given problem, is large enough.
We present similar sufficient conditions under which the projected epigraph of
the SDP gives the convex hull of the epigraph in the original QCQP. Our results
also imply new sufficient conditions for the tightness (as well as convex hull
exactness) of a second order cone program relaxation of simultaneously
diagonalizable QCQPs.
