SOCPs include QCQps as a Special Case

Theorem: QCQPs as SOCPs

The quadratically constrained quadratic programming problem

displaystylemin_x : a_0^Tx + x^TQ_0x ~:~ x^TQ_ix + a_i^Tx le b_i, ;; i=1,ldots, m ,

where $Q_i in mathbf{R}^{n times n}$ , Q_i succeq 0 , i=1,ldots,m , can be expressed as an SOCP with rotated second-order cone constraints:

$displaystylemin_{x,t,w_0,ldots,w_m} : a_0^Tx + t ~:~ begin{array}[t]{l} (w_0,t,1) in mathbf{K}_n, ;; w_0 = Q_0^{1/2}x, (w_i,b_i-a_i^Tx,1) in mathbf{K}_n , ;; w_i = Q_i^{1/2} x_i, ;; i=1,ldots, m . end{array}$

Proof: We first represent the problem in epigraph form:

$displaystylemin_{x,t} : a_0^Tx + t ~:~x^TQ_0x le t, ;; x^TQ_ix + a_i^Tx le b_i, ;; i=1,ldots, m .$

Now, a constraint of the form

is equivalent to the existence of w,y,z such that

$w^Tw le yz, ;; z = 1, ;; w = Q^{1/2}x, ;; y = b-a^Tx ,$

where $Q^{1/2}$ is the square-root of the PSD matrix .

Applyng this to the constraints of the above formulation, we obtain an equivalent representation of the original QCQP:

$displaystylemin_{x,t} : a_0^Tx + t ~:~ begin{array}[t]{l} w_0^Tw_0 le t, ;; w_0 = Q^{1/2}x, w_i^Tw_i le b_i-a_i^Tx, ;; w_i = Q_i^{1/2} x_i, ;; i=1,ldots, m . end{array}$

The above can be written as given in the theorem, as claimed.