SOCPs include QPs as Special Case

The quadratic program (QP)

where Q = Q^Tsucceq 0 , can be cast as an SOCP, as follows.

First we introduce a new variable and a new linear equality. With $w := Q^{1/2} x$ , we express the QP as

$min_{x,w} : c^Tx + w^Tw ~:~ w := Q^{1/2} x, ;; a_i^Tx le b_i, ;; i=1,ldots,m.$

Then, we add two new scalar variables, and a rotated cone constraint, to handle the second term in the objective. The QP bears the form

$min_{x,w} : c^Tx + y ~:~ yz ge w^Tw, ;; z = 1, ;; w := Q^{1/2} x, ;; a_i^Tx le b_i, ;; i=1,ldots,m.$

The above is indeed an SOCP: