Convex relaxations for nonconvex quadratically constrained quadratic programming: matrix cone decomposition and polyhedral approximation

We present a decomposition-approximation method for generating convex relaxations for nonconvex quadratically constrained quadratic programming (QCQP). We first develop a general conic program relaxation for QCQP based on a matrix decomposition scheme and polyhedral (piecewise linear) underestimation. By employing suitable matrix cones, we then show that the convex conic relaxation can be reduced to a semidefinite programming (SDP) problem. In particular, we investigate polyhedral underestimations for several classes of matrix cones, including the cones of rank-1 and rank-2 matrices, the cone generated by the coefficient matrices, the cone of positive semidefinite matrices and the cones induced by rank-2 semidefinite inequalities. We demonstrate that in general the new SDP relaxations can generate lower bounds at least as tight as the best known SDP relaxations for QCQP. Moreover, we give examples for which tighter lower bounds can be generated by the new SDP relaxations. We also report comparison results of different convex relaxation schemes for nonconvex QCQP with convex quadratic/linear constraints, nonconvex quadratic constraints and 0–1 constraints.