Semidefinite program

A semidefinite program (SDP) is an optimization problem of the form

\[\begin{split}\begin{array}{ll} \mbox{minimize} & \mathbf{tr}(CX) \\ \mbox{subject to} & \mathbf{tr}(A_iX) = b_i, \quad i=1,\ldots,p \\ & X \succeq 0, \end{array}\end{split}\]

where \(\mathbf{tr}\) is the trace function, \(X \in \mathcal{S}^{n}\) is the optimization variable and \(C, A_1, \ldots, A_p \in \mathcal{S}^{n}\), and \(b_1, \ldots, b_p \in \mathcal{R}\) are problem data, and \(X \succeq 0\) is a matrix inequality. Here \(\mathcal{S}^{n}\) denotes the set of \(n\)-by-\(n\) symmetric matrices.

An example of an SDP is to complete a covariance matrix \(\tilde \Sigma \in \mathcal{S}^{n}_+\) with missing entries \(M \subset \{1,\ldots,n\} \times \{1,\ldots,n\}\):

\[\begin{split}\begin{array}{ll} \mbox{minimize} & 0 \\ \mbox{subject to} & \Sigma_{ij} = \tilde \Sigma_{ij}, \quad (i,j) \notin M \\ & \Sigma \succeq 0, \end{array}\end{split}\]

Example

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

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

# Generate a random SDP.
n = 3
p = 3
np.random.seed(1)
C = np.random.randn(n, n)
A = []
b = []
for i in range(p):
    A.append(np.random.randn(n, n))
    b.append(np.random.randn())

# Define and solve the CVXPY problem.
# Create a symmetric matrix variable.
X = cp.Variable((n,n), symmetric=True)
# The operator >> denotes matrix inequality.
constraints = [X >> 0]
constraints += [
    cp.trace(A[i]@X) == b[i] for i in range(p)
]
prob = cp.Problem(cp.Minimize(cp.trace(C@X)),
                  constraints)
prob.solve()

# Print result.
print("The optimal value is", prob.value)
print("A solution X is")
print(X.value)
The optimal value is 2.654348003008652
A solution X is
[[ 1.6080571  -0.59770202 -0.69575904]
 [-0.59770202  0.22228637  0.24689205]
 [-0.69575904  0.24689205  1.39679396]]