A **second-order cone program** (**SOCP**) is a convex optimization problem of the form

minimize

f
T
x
subject to

where the problem parameters are
f
∈
R
n
,
A
i
∈
R
n
i
×
n
,
b
i
∈
R
n
i
,
c
i
∈
R
n
,
d
i
∈
R
,
F
∈
R
p
×
n
, and
g
∈
R
p
. Here
x
∈
R
n
is the optimization variable. When
A
i
=
0
for
i
=
1
,
…
,
m
, the SOCP reduces to a linear program. When
c
i
=
0
for
i
=
1
,
…
,
m
, the SOCP is equivalent to a convex quadratically constrained linear program. Quadratically constrained quadratic programs can also be formulated as SOCPs by reformulating the objective function as a constraint. Semidefinite programming subsumes SOCPs as the SOCP constraints can be written as linear matrix inequalities (LMI) and can be reformulated as an instance of semi definite program. SOCPs can be solved with great efficiency by interior point methods.

Consider a quadratic constraint of the form

x
T
A
T
A
x
+
b
T
x
+
c
≤
0.
This is equivalent to the SOC constraint

∥
(
1
+
b
T
x
+
c
)
/
2
A
x
∥
2
≤
(
1
−
b
T
x
−
c
)
/
2.
Consider a stochastic linear program in inequality form

minimize

c
T
x
subject to

where the parameters
a
i
are independent Gaussian random vectors with mean
a
¯
i
and covariance
Σ
i
and
p
≥
0.5
. This problem can be expressed as the SOCP

minimize

c
T
x
subject to

where
Φ
−
1
is the inverse normal cumulative distribution function.

We refer to second-order cone programs as deterministic second-order cone programs since data deﬁning them are deterministic. Stochastic second-order cone programs is a class of optimization problems that deﬁned to handle uncertainty in data deﬁning deterministic second-order cone programs.