如何优化 CVXPY 中的逆跟踪？

Question

How can I formulate an objective function that minimizes the trace of an inverse matrix using cvxpy?

Concretely the following problem:

subject to:

I have seen the trace_inv objective function in cvx but so far was not able to figure out how to translate this to cvxpy.

Answer 1

I figured it out by looking at the source code on git, https://github.com/cvxr/CVX/tree/master/functions .

The function matrix_frac() fits the problem.

Here is a minimal example:

import cvxpy as cp

n = 20
k = 5

A_list = []
for i in range(n):
    A = np.random.uniform(0,10,(6,6))
    A_list.append(A.T@A)

# Create two scalar optimization variables.
z = cp.Variable(n)

# Create two constraints.
constraints = [z >= 0,
               z <= 1,
              np.ones((n,))@z == k]

cost = sum([z[i] * A_list[i] for i in range(n)])

# Form objective.
obj = cp.Minimize(cp.matrix_frac(np.eye(6), cost))

# Form and solve problem.
prob = cp.Problem(obj, constraints)

prob.solve()  # Returns the optimal value.

print("status:", prob.status)
print("optimal value", prob.value)

# compute trace of inverse using z
S = np.zeros(A_list[0].shape)
for i in range(len(z.value)):
    S += z.value[i] * A_list[i]
value = np.trace(np.linalg.inv(S))

print("value", value)
print(prob.value == value)

Output:

status: optimal
optimal value 0.02061809722699777
value 0.02061809722699777
True