I am trying to optimize the following part of my code that involves solving a semi-definite programming (SDP) in CVXPY.

# optimization variables X and V

V = cp.Variable((p,p))
X = cp.Variable((num,p))

# constraints

constraints = [cp.bmat([[V, X.T @ R.conj().T], [R @ X, np.identity(num)]]) >> 0]
constraints += [X.T @ dersmatrix == np.identity(p)]

# minimization

prob = cp.Problem(cp.Minimize(cp.trace(W @ V)), constraints)
prob.solve(solver=cp.MOSEK,verbose=True)
v1=prob.value

The SDP is formulated as follows. The optimization variables V and X are matrices of size p x p and num x p respectively, where num = d^2 is typically very large i.e., > 6000 and p=2 (for my case).

The constraint for positive semi-definiteness in terms of X and V is given through the positive semi-definiteness of the block matrix (say M) M=cp.bmat([[V, X.T @ R.conj().T], [R @ X, np.identity(num)]]) >> 0. This is inferred from the positive semi-definiteness condition on the Schur complement of the identity block of M i.e., V - X.T @ R.conj().T @ R @ X. Another constraint reads: X.T @ dersmatrix == np.identity(p).

R and dersmatrix are matrices of size num x num and num x p respectively, which are defined previously in my code. W is a p x p weight matrix in the objective function cp.trace(W @ V).

For example, when d=81, num=6561, resulting in extremely large sizes for X, R, and dersmatrix, which significantly slows down the solution.

As one possible solution to obtain speed-up, I identified that R is quite sparse and tried to exploit its sparsity using R_spr=csr_matrix(R), which modifies the positive semi-definiteness constraint to cp.bmat([[V, X.T @ R_spr.conj().T], [R_spr @ X, np.identity(num)]]) >> 0. However, this still does not seem to yield faster results. I am also wondering if the block matrix form of the constraint slows down the code. Note that I am using the MOSEK solver.

Hence, I am looking for some suggestions to optimize this aspect of my code when dealing with very large matrices, especially for d>=81. Any ideas on this would be much helpful. Thanks.