  In this paper, we present new convex relaxations for nonconvex quadratically
constrained quadratic programming (QCQP) problems. While recent research has
focused on strengthening convex relaxations using reformulation-linearization
technique (RLT), the state-of-the-art methods lose their effectiveness when
dealing with (multiple) nonconvex quadratic constraints in QCQP. In this
research, we decompose and relax each nonconvex constraint to two second order
cone (SOC) constraints and then linearize the products of the SOC constraints
and linear constraints to construct some effective new valid constraints.
Moreover, we extend the reach of the RLT-like techniques for almost all
different types of constraint-pairs (including valid inequalities by
linearizing the product of a pair of SOC constraints, and the the Hadamard
product or the Kronecker product of two respective valid linear matrix
inequalities), examine dominance relationships among different valid
inequalities, and explore almost all possibilities of gaining benefits from
generating valid constraints. Especially, we successfully demonstrate that
applying RLT-like techniques to additional redundant linear constraints could
reduce the relaxation gap. We demonstrate the efficiency of our results with
numerical experiments.
