Second-order cone program

A second-order cone program (SOCP) is an optimization problem of the form

\[\begin{split}\begin{array}{ll} \mbox{minimize} & f^Tx\\ \mbox{subject to} & \|A_ix + b_i\|_2 \leq c_i^Tx + d_i, \quad i=1,\ldots,m \\ & Fx = g, \end{array}\end{split}\]

where \(x \in \mathcal{R}^{n}\) is the optimization variable and \(f \in \mathcal{R}^n\), \(A_i \in \mathcal{R}^{n_i \times n}\), \(b_i \in \mathcal{R}^{n_i}\), \(c_i \in \mathcal{R}^n\), \(d_i \in \mathcal{R}\), \(F \in \mathcal{R}^{p \times n}\), and \(g \in \mathcal{R}^p\) are problem data.

An example of an SOCP is the robust linear program

\[\begin{split}\begin{array}{ll} \mbox{minimize} & c^Tx\\ \mbox{subject to} & (a_i + u_i)^Tx \leq b_i \textrm{ for all } \|u_i\|_2 \leq 1, \quad i=1,\ldots,m, \end{array}\end{split}\]

where the problem data \(a_i\) are known within an \(\ell_2\)-norm ball of radius one. The robust linear program can be rewritten as the SOCP

\[\begin{split}\begin{array}{ll} \mbox{minimize} & c^Tx\\ \mbox{subject to} & a_i^Tx + \|x\|_2 \leq b_i, \quad i=1,\ldots,m, \end{array}\end{split}\]

When we solve a SOCP, in addition to a solution \(x^\star\), we obtain a dual solution \(\lambda_i^\star\) corresponding to each second-order cone constraint. A non-zero \(\lambda_i^\star\) indicates that the constraint \(\|A_ix + b_i\|_2 \leq c_i^Tx + d_i\) holds with equality for \(x^\star\) and suggests that changing \(d_i\) would change the optimal value.

Example

In the following code, we solve a SOCP with CVXPY.

# Import packages.
import cvxpy as cp
import numpy as np

# Generate a random feasible SOCP.
m = 3
n = 10
p = 5
n_i = 5
np.random.seed(2)
f = np.random.randn(n)
A = []
b = []
c = []
d = []
x0 = np.random.randn(n)
for i in range(m):
    A.append(np.random.randn(n_i, n))
    b.append(np.random.randn(n_i))
    c.append(np.random.randn(n))
    d.append(np.linalg.norm(A[i]@x0 + b, 2) - c[i].T@x0)
F = np.random.randn(p, n)
g = F@x0

# Define and solve the CVXPY problem.
x = cp.Variable(n)
# We use cp.SOC(t, x) to create the SOC constraint ||x||_2 <= t.
soc_constraints = [
      cp.SOC(c[i].T@x + d[i], A[i]@x + b[i]) for i in range(m)
]
prob = cp.Problem(cp.Minimize(f.T@x),
                  soc_constraints + [F@x == g])
prob.solve()

# Print result.
print("The optimal value is", prob.value)
print("A solution x is")
print(x.value)
for i in range(m):
    print("SOC constraint %i dual variable solution" % i)
    print(soc_constraints[i].dual_value)
The optimal value is -9.582695716265503
A solution x is
[ 1.40303325  2.4194569   1.69146656 -0.26922215  1.30825472 -0.70834842
  0.19313706  1.64153496  0.47698583  0.66581033]
SOC constraint 0 dual variable solution
[ 0.61662526  0.35370661 -0.02327185  0.04253095  0.06243588  0.49886837]
SOC constraint 1 dual variable solution
[ 0.35283078 -0.14301082  0.16539699 -0.22027817  0.15440264  0.06571645]
SOC constraint 2 dual variable solution
[ 0.86510445 -0.114638   -0.449291    0.37810251 -0.6144058  -0.11377797]