def test_power_cone_system(self):
n = 4
rng = np.random.default_rng(12345)
# Create w and z in one array, where the last one will be z
wz = Variable(shape=(n+1,), name='wz')
# Make the last element negative to indicate that that element is z in the wz variable
lamb = rng.random((n+1,))
lamb[-1] = -1*np.sum(lamb[:-1])
# Create simple constraints
constraints1 = [PowCone(wz, lamb)]
A, b, K, _, _, _ = compile_constrained_system(constraints1)
A = A.toarray()
assert np.allclose(A, np.identity(n+1))
assert np.allclose(b, np.zeros((n+1,)))
actual_annotations = {'weights': np.array(lamb[:-1]/np.sum(lamb[:-1])).tolist()}
assert K[0] == Cone('pow', n+1, annotations=actual_annotations)
# Increment z
offset = np.zeros((n+1,))
offset[-1] = 1
wz1 = wz + offset
constraints2 = [PowCone(wz1, lamb)]
A, b, K, _, _, _ = compile_constrained_system(constraints2)
A = A.toarray()
assert np.allclose(A, np.identity(n + 1))
expected_b = np.zeros((n + 1,))
expected_b[-1] = 1
assert np.allclose(b, expected_b)
actual_annotations = {'weights': np.array(lamb[:-1] / np.sum(lamb[:-1])).tolist()}
assert K[0] == Cone('pow', n + 1, annotations=actual_annotations)
#Increment w
r = rng.random((n+1,))
r[-1] = 0
wz2 = wz + r
constraints3 = [PowCone(wz2, lamb)]
A, b, K, _, _, _ = compile_constrained_system(constraints3)
A = A.toarray()
assert np.allclose(A, np.identity(n + 1))
expected_b = r
assert np.allclose(b[:-1], expected_b[:-1])
assert K[0] == Cone('pow', n+1, annotations=actual_annotations)
# Last test reconstructed from matrix creation
x = Variable(shape=(n+1,), name='x')
M = np.random.rand(n+1, n+1)
y = M @ x
constraints4 = [PowCone(y, lamb)]
A, b, K, _, _, _ = compile_constrained_system(constraints4)
A = A.toarray()
assert np.allclose(A, M)
assert K[0] == Cone('pow', n+1, annotations=actual_annotations)
def test_vector2norm_2(self):
x = Variable(shape=(3, ), name='x')
y = Variable(shape=(1, ), name='y')
nrm = vector2norm(x - np.array([0.1, 0.2, 0.3]))
con = [nrm <= y]
A_expect = np.array([
[0, 0, 0, 1,
-1], # linear inequality constraint in terms of epigraph variable
[0, 0, 0, 0,
1], # epigraph component in second order cone constraint
[1, 0, 0, 0,
0], # start of main block in second order cone constraint
[0, 1, 0, 0, 0],
[0, 0, 1, 0, 0]
]) # end of main block in second order cone constraint
b_expect = np.zeros(shape=(5, ))
b_expect[0] = 0
b_expect[2] = -0.1
b_expect[3] = -0.2
b_expect[4] = -0.3
A, b, K, _, _, _ = compile_constrained_system(con)
A = np.round(A.toarray(), decimals=1)
assert np.all(A == A_expect)
assert np.all(b == b_expect)
assert K == [Cone('+', 1), Cone('S', 4)]
def test_pos_1(self):
x = Variable(shape=(3, ), name='x')
con = [cl_pos(affine.sum(x) - 7) <= 5]
A_expect = np.array([[0, 0, 0, -1], [0, 0, 0, 1], [-1, -1, -1, 1]])
A, b, K, _1, _2, _3 = compile_constrained_system(con)
A = np.round(A.toarray(), decimals=1)
assert np.all(A == A_expect)
assert np.all(b == np.array([5, 0, 7]))
assert K == [Cone('+', 1), Cone('+', 2)]
# value propagation
x.value = np.array([4, 4, 4])
viol = con[0].violation()
assert viol == 0
x.value = np.array([4.2, 3.9, 4.0])
viol = con[0].violation()
assert abs(viol - 0.1) < 1e-7
def test_abs_1(self):
x = Variable(shape=(2, ), name='x')
one_norm = affine.sum(cl_abs(x))
con = [one_norm <= 5]
A_expect = np.array([[0, 0, -1, -1], [1, 0, 1, 0], [-1, 0, 1, 0],
[0, 1, 0, 1], [0, -1, 0, 1]])
A, b, K, _1, _2, _3 = compile_constrained_system(con)
A = np.round(A.toarray(), decimals=1)
assert np.all(A == A_expect)
assert np.all(b == np.array([5, 0, 0, 0, 0]))
assert K == [Cone('+', 1), Cone('+', 2), Cone('+', 2)]
# value propagation
x.value = np.array([1, -2])
viol = con[0].violation()
assert viol == 0
x.value = np.array([-3, 3])
viol = con[0].violation()
assert viol == 1
def test_relent_1(self):
# compilation and evaluation
x = Variable(shape=(2, ), name='x')
y = Variable(shape=(2, ), name='y')
re = relent(2 * x, np.exp(1) * y)
con = [re <= 10, 3 <= x, x <= 5]
# compilation
A, b, K, _, _, _ = compile_constrained_system(con)
A_expect = np.array([
[0., 0., 0., 0., -1.,
-1.], # linear inequality on epigraph for relent constr
[1., 0., 0., 0., 0., 0.], # bound constraints on x
[0., 1., 0., 0., 0., 0.], #
[-1., 0., 0., 0., 0., 0.], # more bound constraints on x
[0., -1., 0., 0., 0., 0.], #
[0., 0., 0., 0., -1., 0.], # first exponential cone
[0., 0., 2.72, 0., 0., 0.], #
[2., 0., 0., 0., 0., 0.], #
[0., 0., 0., 0., 0., -1.], # second exponential cone
[0., 0., 0., 2.72, 0., 0.], #
[0., 2., 0., 0., 0., 0.]
]) #
A = np.round(A.toarray(), decimals=2)
assert np.all(A == A_expect)
assert np.all(
b == np.array([10., -3., -3., 5., 5., 0., 0., 0., 0., 0., 0.]))
assert K == [
Cone('+', 1),
Cone('+', 2),
Cone('+', 2),
Cone('e', 3),
Cone('e', 3)
]
# value propagation
x0 = np.array([1, 2])
x.value = x0
y0 = np.array([3, 4])
y.value = y0
actual = re.value
expect = np.sum(rel_entr(2 * x0, np.exp(1) * y0))
assert abs(actual - expect) < 1e-7
def test_LP_systems(self):
n = 4
m = 5
x = Variable(shape=(n, 1), name='x')
G = np.random.randn(m, n)
h = G @ np.abs(np.random.randn(n, 1))
constraints = [G @ x == h,
x >= 0]
# Reference case : the constraints are over x, and we are interested in no variables other than x.
A0, b0, K0, var_mapping0, _, _ = compile_constrained_system(constraints)
A0 = A0.toarray()
# A0 should be the (m+n)-by-n matrix formed by stacking -G on top of the identity.
# b0 should be the (m+n)-length vector formed by concatenating h with the zero vector.
# Should see K0 == [('0',m), ('+',n)]
# var_mapping0 should be a length-1 dictionary with var_mapping0['x'] == np.arange(n).reshape((n,1)).
assert np.all(b0 == np.hstack([h.ravel(), np.zeros(n)]))
assert K0 == [Cone('0', m), Cone('+', n)]
assert var_maps_equal(var_mapping0, {'x': np.arange(0, n).reshape((n, 1))})
expected_A0 = np.vstack((-G, np.eye(n)))
assert np.all(A0 == expected_A0)
def test_relent_2(self):
# compilation with the elementwise option
x = Variable(shape=(2, ), name='x')
y = Variable(shape=(2, ), name='y')
re = affine.sum(relent(2 * x, np.exp(1) * y, elementwise=True))
con = [re <= 1]
A, b, K, _, _, _ = compile_constrained_system(con)
A_expect = np.array([
[0., 0., 0., 0., -1.,
-1.], # linear inequality on epigraph for relent constr
[0., 0., 0., 0., -1., 0.], # first exponential cone
[0., 0., 2.72, 0., 0., 0.], #
[2., 0., 0., 0., 0., 0.], #
[0., 0., 0., 0., 0., -1.], # second exponential cone
[0., 0., 0., 2.72, 0., 0.], #
[0., 2., 0., 0., 0., 0.]
]) #
A = np.round(A.toarray(), decimals=2)
assert np.all(A == A_expect)
assert np.all(b == np.array([1., 0., 0., 0., 0., 0., 0.]))
assert K == [Cone('+', 1), Cone('e', 3), Cone('e', 3)]
def test_vector2norm_1(self):
x = Variable(shape=(3, ), name='x')
nrm = vector2norm(x)
con = [nrm <= 1]
A_expect = np.array([
[0, 0, 0,
-1], # linear inequality constraint in terms of epigraph variable
[0, 0, 0, 1], # epigraph component in second order cone constraint
[1, 0, 0,
0], # start of main block in second order cone constraint
[0, 1, 0, 0],
[0, 0, 1, 0]
]) # end of main block in second order cone constraint
A, b, K, _1, _2, _3 = compile_constrained_system(con)
A = np.round(A.toarray(), decimals=1)
assert np.all(A == A_expect)
assert np.all(b == np.array([1, 0, 0, 0, 0]))
assert K == [Cone('+', 1), Cone('S', 4)]
# value propagation
x.value = np.zeros(3)
viol = con[0].violation()
assert viol == 0
def test_SDP_system_1(self):
x = Variable(shape=(2, 2), name='x', var_properties=['symmetric'])
D = np.random.randn(2, 2)
D += D.T
D /= 2.0
B = np.random.rand(1, 2)
C = np.diag([3, 0.5])
constraints = [affine.trace(D @ x) == 5,
B @ x @ B.T >= 1,
B @ x @ B.T >> 1, # a 1-by-1 LMI
C @ x @ C.T >> -2]
A, b, K, _, _, _ = compile_constrained_system(constraints)
A = A.toarray()
assert K == [Cone('0', 1), Cone('+', 1), Cone('P', 1), Cone('P', 3)]
expect_row_0 = -np.array([D[0, 0], 2 * D[1, 0], D[1, 1]])
assert np.allclose(expect_row_0, A[0, :])
temp = B.T @ B
expect_row_1and2 = np.array([temp[0, 0], 2 * temp[0, 1], temp[1, 1]])
assert np.allclose(expect_row_1and2, A[1, :])
assert np.allclose(expect_row_1and2, A[2, :])
expect_rows_3to6 = np.diag([C[0, 0] ** 2, C[0, 0] * C[1, 1], C[1, 1] ** 2])
assert np.allclose(expect_rows_3to6, A[3:, :])
assert np.all(b == np.array([5, -1, -1, 2, 2, 2]))