Quadratically constrained quadratic program  In mathematical optimization, a quadratically constrained quadratic program (QCQP) is an optimization problem in which both the objective function and the constraints are quadratic functions. It has the form  where "P"0, …, "P""m" are "n"-by-"n" matrices and "x" ∈ R"n" is the optimization variable. If "P"0, …, "P""m" are all positive semidefinite, then the problem is convex. If these matrices are neither positive nor negative semidefinite, the problem is non-convex. If "P"1, … ,"P""m" are all zero, then the constraints are in fact linear and the problem is a quadratic program. Solving the general case is an NP-hard problem. To see this, note that the two constraints "x"1("x"1 − 1) ≤ 0 and "x"1("x"1 − 1) ≥ 0 are equivalent to the constraint "x"1("x"1 − 1) = 0, which is in turn equivalent to the constraint "x"1 ∈ {0, 1}. Hence, any 0–1 integer program (in which all variables have to be either 0 or 1) can be formulated as a quadratically constrained quadratic program. Since 0–1 integer programming is NP-hard in general, QCQP is also NP-hard. There are two main relaxations of QCQP: using semidefinite programming (SDP), and using the reformulation-linearization technique (RLT). For some classes of QCQP problems (precisely, QCQPs with zero diagonal elements in the data matrices), second-order cone programming (SOCP) and linear programming (LP) relaxations providing the same objective value as the SDP relaxation are available. Nonconvex QCQPs with non-positive off-diagonal elements can be exactly solved by the SDP or SOCP relaxations, and there are polynomial-time-checkable sufficient conditions for SDP relaxations of general QCQPs to be exact. Moreover, it was shown that a class of random general QCQPs has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables.