  In this paper, we study the convex quadratic optimization problem with
indicator variables. For the bivariate case, we describe the convex hull of the
epigraph in the original space of variables, and also give a conic quadratic
extended formulation. Then, using the convex hull description for the bivariate
case as a building block, we derive an extended SDP relaxation for the general
case. This new formulation is stronger than other SDP relaxations proposed in
the literature for the problem, including Shor's SDP relaxation, the optimal
perspective relaxation as well as the optimal rank-one relaxation.
Computational experiments indicate that the proposed formulations are quite
effective in reducing the integrality gap of the optimization problems.
