def adjoint_derivative(dx, dy, ds, **kwargs):
"""Applies adjoint of derivative at (A, b, c) to perturbations dx, dy, ds
Args:
dx: NumPy array representing perturbation in `x`
dy: NumPy array representing perturbation in `y`
ds: NumPy array representing perturbation in `s`
Returns:
(`dA`, `db`, `dc`), the result of applying the adjoint to the
perturbations; the sparsity pattern of `dA` matches that of `A`.
"""
dw = -(x @ dx + y @ dy + s @ ds)
dz = np.concatenate(
[dx, D_proj_dual_cone.rmatvec(dy + ds) - ds,
np.array([dw])])
if np.allclose(dz, 0):
r = np.zeros(dz.shape)
elif mode == "dense":
r = _diffcp._solve_adjoint_derivative_dense(M, MT, dz)
else:
r = _diffcp.lsqr(MT, dz).solution
values = pi_z[cols] * r[rows + n] - pi_z[n + rows] * r[cols]
dA = sparse.csc_matrix((values, (rows, cols)), shape=A.shape)
db = pi_z[n:n + m] * r[-1] - pi_z[-1] * r[n:n + m]
dc = pi_z[:n] * r[-1] - pi_z[-1] * r[:n]
return dA, db, dc
def derivative(dA, db, dc, **kwargs):
"""Applies derivative at (A, b, c) to perturbations dA, db, dc
Args:
dA: SciPy sparse matrix in CSC format; must have same sparsity
pattern as the matrix `A` from the cone program
db: NumPy array representing perturbation in `b`
dc: NumPy array representing perturbation in `c`
Returns:
NumPy arrays dx, dy, ds, the result of applying the derivative
to the perturbations.
"""
dQ = sparse.bmat(
[[None, dA.T, np.expand_dims(dc, -1)],
[-dA, None, np.expand_dims(db, -1)],
[-np.expand_dims(dc, -1).T, -np.expand_dims(db, -1).T, None]])
rhs = dQ @ pi_z
if np.allclose(rhs, 0):
dz = np.zeros(rhs.size)
elif mode == "dense":
dz = _diffcp._solve_derivative_dense(M, MT, rhs)
else:
dz = _diffcp.lsqr(M, rhs).solution
du, dv, dw = np.split(dz, [n, n + m])
dx = du - x * dw
dy = D_proj_dual_cone.matvec(dv) - y * dw
ds = D_proj_dual_cone.matvec(dv) - dv - s * dw
return -dx, -dy, -ds