CVXPY Square root of Singular Quadratic

Question

I need to model sqrt(x^T C x) for a singular positive semidefinite matrix C. Here, it is proposed to use norm(Q*x) where Q is obtained from the Cholesky decomposition of C.

How to take the square root of quad_form output in CVXPY?

But, np./scipy.linalg.cholskey does not work for singular matrices.

PS, using SVD or eigenvalue decomposition is too slow for my application.

PS2, this post Numpy Cholesky decomposition LinAlgError does not help as it does not offer a solution. Also, the matrix in the question seems to have negative eigenvalues (rather than being singular).

Possible duplicate of [Numpy Cholesky decomposition LinAlgError](https://stackoverflow.com/questions/21604498/numpy-cholesky-decomposition-linalgerror) — sophros, Nov 14 '18 at 12:39

Behrooz Ns · Accepted Answer · 2018-11-14T14:47:18.890

1

I found a solution using the ldl decomposition

    L,d,_ = scipy.linalg.ldl(C)
    d = np.diag(d).copy()
    inds = d >= d.max()*1e-8

    d = d[inds]
    d = np.sqrt(d)
    d.shape = (-1,1)
    Q = d * L.T[inds]


    loss = cp.norm(cp.matmul(Q, x))

The ldl decomposition needs scipy >= 1.1 though.

edited Nov 14 '18 at 14:47

answered Nov 14 '18 at 13:58

Behrooz Ns

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CVXPY Square root of Singular Quadratic

1 Answers1