Block LMI in CVXPY

Question

I want to translate a LMI-constrained optimization problem from Matlab to Python. While reading the CVXPY documentation, I found that I can define an LMI-constrained problem by creating a matrix variable and adding the corresponding constraint. However, instead of constraining the problem by a simple LMI, I need to use the following block LMI:

Where P, L are matrix variables and gamma is a scalar. The other symbols are state space matrices of a dynamic system.

Is it possible to use this kind of LMI as a constraint in CVXPY? If not, is there any other tool that would allow me to solve this problem in Python?

Answer 1

I followed the example posted by Rodrigo de Azevedo and I managed to write the given LMI in cvxpy.

For reference, the code I wrote, It may be helpful for someone:

n = A.shape[0]

L = Variable((B2.shape[1], n))
P = Variable((n, n), PSD=True)
gamma2 = Variable()

LMI1 = bmat([
        [P, A*P + B2*L, B1, np.zeros((B1.shape[0], D11.shape[0]))],
        [P*A.T + L.T * B2.T, P, np.zeros((P.shape[0], B1.shape[1])), P*C1.T + L.T*D12.T],
        [B1.T, np.zeros((B1.shape[1], P.shape[1])), np.eye(B1.shape[1]), D11.T],
        [np.zeros((C1.shape[0], B1.shape[0])), C1*P + D12*L, D11, gamma2*np.eye(D11.shape[0])]
          ])

cons1 = LMI1 >> 0

cons2 = P == P.T
cons3 = gamma2 >= 0

And then, to solve the problem:

optprob = Problem(Minimize(gamma2), constraints=[cons1, cons2, cons3])
optprob.solve()

norm = np.sqrt(gamma2.value)
Pop = P.value
Lop = L.value