{\displaystyle x\in \mathbb {R} ^{n}}
is the optimization variable.
‖
x
‖
2
{\displaystyle \lVert x\rVert _{2}}
is the Euclidean norm and
T
{\displaystyle ^{T}}
indicates transpose. The "second-order cone" in SOCP arises from the constraints, which are equivalent to requiring the affine function
(
A
x
+
b
,
c
T
x
+
d
)
{\displaystyle (Ax+b,c^{T}x+d)}
to lie in the second-order cone in
R
n
i
+
1
{\displaystyle \mathbb {R} ^{n_{i}+1}}
.SOCPs can be solved by interior point methods and in general, can be solved more efficiently than semidefinite programming (SDP) problems. Some engineering applications of SOCP include filter design, antenna array weight design, truss design, and grasping force optimization in robotics.