def test_conditional_sage_dual_1(self):
n, m = 2, 6
x = Variable(shape=(n, ), name='x')
cons = [1 >= vector2norm(x)]
gts = [lambda z: 1 - np.linalg.norm(z, 2)]
eqs = []
sigdom = SigDomain(n, coniclifts_cons=cons, gts=gts, eqs=eqs)
np.random.seed(0)
x0 = np.random.randn(n)
x0 /= 2 * np.linalg.norm(x0)
alpha = np.random.randn(m, n)
c = np.array([1, 2, 3, 4, -0.5, -0.1])
v0 = np.exp(alpha @ x0)
v = Variable(shape=(m, ), name='projected_v0')
t = Variable(shape=(1, ), name='epigraph_var')
sage_constraint = sage_cones.DualSageCone(v,
alpha,
name='test',
X=sigdom,
c=c)
epi_constraint = vector2norm(v - v0) <= t
constraints = [sage_constraint, epi_constraint]
prob = Problem(CL_MIN, t, constraints)
prob.solve(solver='ECOS')
v0 = sage_constraint.violation(norm_ord=1, rough=False)
assert v0 < 1e-6
v1 = sage_constraint.violation(norm_ord=np.inf, rough=True)
assert v1 < 1e-6
val = prob.value
assert val < 1e-7
def project(item, alpha):
"""
Calculates the shortest distance (the projection) of a vector
to a cone parametrized by :math:'\\alpha'
Parameters
----------
item - the point we are projecting
alpha - the :math:'\\alpha' parameter
for the Cone that we are projecting to
Returns
-------
The distance of the projection to the Cone
"""
from sageopt.coniclifts import MIN as CL_MIN
item = Expression(item).ravel()
w = Variable(shape=(item.size, ))
t = Variable(shape=(1, ))
cons = [vector2norm(item - w) <= t, PowCone(w, alpha)]
prob = Problem(CL_MIN, t, cons)
prob.solve(verbose=False)
return prob.value
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_ordinary_sage_dual_3(self):
# provide a vector "c" in the dual SAGE cone constructor.
# generate a point with zero distance from the dual SAGE cone
n, m = 2, 6
np.random.seed(0)
alpha = 10 * np.random.randn(m, n)
x0 = np.random.randn(n) / 10
v0 = np.exp(alpha @ x0)
dummy_vars = Variable(shape=(2, )).scalar_variables()
c = np.array([1, 2, 3, 4, dummy_vars[0], dummy_vars[1]])
c = Expression(c)
v = Variable(shape=(m, ), name='projected_v0')
t = Variable(shape=(1, ), name='epigraph_var')
sage_constraint = sage_cones.DualSageCone(v,
alpha,
X=None,
name='test_con',
c=c)
epi_constraint = vector2norm(v - v0) <= t
constraints = [sage_constraint, epi_constraint]
prob = Problem(CL_MIN, t, constraints)
prob.solve(solver='ECOS')
viol = sage_constraint.violation(norm_ord=1, rough=False)
assert viol < 1e-6
viol = sage_constraint.violation(norm_ord=np.inf, rough=True)
assert viol < 1e-6
val = prob.value
assert val < 1e-7
def test_separate_cone_constraints_2(self):
num_vars = 5
x = Variable(shape=(num_vars, ))
cons = [vector2norm(x) <= 1]
prob = Problem(CL_MIN, Expression([0]), cons)
A0, b0, K0 = prob.A, prob.b, prob.K
assert A0.shape == (num_vars + 2, num_vars + 1)
assert len(K0) == 2
assert K0[0].type == '+' and K0[0].len == 1
assert K0[1].type == 'S' and K0[1].len == num_vars + 1
A1, b1, K1, sepK1 = separate_cone_constraints(A0,
b0,
K0,
dont_sep={'+'})
A1 = A1.toarray()
assert A1.shape == (num_vars + 2, 2 * (num_vars + 1))
assert len(K1) == 2
assert K1[0].type == '+' and K1[0].len == 1
assert K1[1].type == '0' and K1[1].len == num_vars + 1
assert len(sepK1) == 1
assert sepK1[0].type == 'S' and sepK1[0].len == num_vars + 1
A0 = A0.toarray()
temp = np.vstack(
(np.zeros(shape=(1, num_vars + 1)), np.eye(num_vars + 1)))
expect_A1 = np.hstack((A0, -temp))
assert np.allclose(expect_A1, A1)
def project(item, K):
from sageopt.coniclifts import MIN as CL_MIN
item = Expression(item).ravel()
x = Variable(shape=(item.size, ))
t = Variable(shape=(1, ))
cons = [vector2norm(item - x) <= t, PrimalProductCone(x, K)]
prob = Problem(CL_MIN, t, cons)
prob.solve(verbose=False)
return prob.value
def project(item, alpha, X):
if np.all(item >= 0):
return 0
c = Variable(shape=(item.size,))
t = Variable(shape=(1,))
cons = [
vector2norm(item - c) <= t,
PrimalSageCone(c, alpha, X, 'temp_con')
]
prob = Problem(CL_MIN, t, cons)
prob.solve(verbose=False)
return prob.value
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_ordinary_sage_dual_2(self):
# generate a point with zero distance from the dual SAGE cone
n, m = 2, 6
np.random.seed(0)
alpha = 10 * np.random.randn(m, n)
x0 = np.random.randn(n) / 10
v0 = np.exp(alpha @ x0)
v = Variable(shape=(m, ), name='projected_v0')
t = Variable(shape=(1, ), name='epigraph_var')
sage_constraint = sage_cones.DualSageCone(v,
alpha,
X=None,
name='test_con')
epi_constraint = vector2norm(v - v0) <= t
constraints = [sage_constraint, epi_constraint]
prob = Problem(CL_MIN, t, constraints)
prob.solve(solver='ECOS')
viol = sage_constraint.violation(norm_ord=1, rough=False)
assert viol < 1e-6
viol = sage_constraint.violation(norm_ord=np.inf, rough=True)
assert viol < 1e-6
val = prob.value
assert val < 1e-7
def test_ordinary_sage_dual_1(self):
# generate a point which has positive distance from the dual SAGE cone
n, m = 2, 6
np.random.seed(0)
alpha = 10 * np.random.randn(m, n)
v0 = 10 * np.abs(np.random.randn(m)) + 0.01
v0[0] = -v0[0]
v = Variable(shape=(m, ), name='projected_v0')
t = Variable(shape=(1, ), name='epigraph_var')
sage_constraint = sage_cones.DualSageCone(v,
alpha,
X=None,
name='test_con')
epi_constraint = vector2norm(v - v0) <= t
constraints = [sage_constraint, epi_constraint]
prob = Problem(CL_MIN, t, constraints)
prob.solve(solver='ECOS')
viol = sage_constraint.violation(norm_ord=1, rough=False)
assert viol < 1e-6
viol = sage_constraint.violation(norm_ord=np.inf, rough=True)
assert viol < 1e-6
val = prob.value
assert val + 1e-6 >= abs(v0[0])
def test_ordinary_sage_primal_2(self):
n, m = 2, 6
np.random.seed(0)
alpha = 1 * np.random.randn(m - 1, n)
conv_comb = np.random.rand(m - 1)
conv_comb /= np.sum(conv_comb)
alpha_last = alpha.T @ conv_comb
alpha = np.row_stack([alpha, alpha_last])
c0 = np.array([1, 2, 3, 4, -0.5, -0.1])
c = Variable(shape=(m, ), name='projected_c0')
t = Variable(shape=(1, ), name='epigraph_var')
sage_constraint = sage_cones.PrimalSageCone(c,
alpha,
X=None,
name='test')
epi_constraint = vector2norm(c - c0) <= t
constraints = [sage_constraint, epi_constraint]
prob = Problem(CL_MIN, t, constraints)
prob.solve(solver='ECOS')
# constraint violations
v0 = sage_constraint.violation(norm_ord=1, rough=False)
assert v0 < 1e-6
v1 = sage_constraint.violation(norm_ord=np.inf, rough=True)
assert v1 < 1e-6
# certificates
w4 = sage_constraint.age_witnesses[4].value
c4 = sage_constraint.age_vectors[4].value
drop4 = np.array([True, True, True, True, False, True])
level4 = np.sum(rel_entr(w4[drop4], np.exp(1) * c4[drop4])) - c4[4]
assert level4 < 1e-6
w5 = sage_constraint.age_witnesses[5].value
c5 = sage_constraint.age_vectors[5].value
drop5 = np.array([True, True, True, True, True, False])
level5 = np.sum(rel_entr(w5[drop5], np.exp(1) * c5[drop5])) - c5[5]
assert level5 < 1e-6
