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_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 test_simple_sage_1(self):
"""
Solve a simple SAGE relaxation for a signomial minimization problem.
Do this without resorting to "Signomial" objects.
"""
alpha = np.array([[0, 0],
[1, 0],
[0, 1],
[1, 1],
[0.5, 0],
[0, 0.5]])
gamma = cl.Variable(shape=(), name='gamma')
c = cl.Expression([0 - gamma, 3, 2, 1, -4, -2])
expected_val = -1.8333331773244161
# with presolve
cl.presolve_trivial_age_cones(True)
con = cl.PrimalSageCone(c, alpha, None, 'test_con_name')
obj = gamma
prob = Problem(cl.MAX, obj, [con])
status, val = prob.solve(solver='ECOS', verbose=False)
assert abs(val - expected_val) < 1e-6
v = con.violation()
assert v < 1e-6
# without presolve
cl.presolve_trivial_age_cones(False)
con = cl.PrimalSageCone(c, alpha, None, 'test_con_name')
obj = gamma
prob = Problem(cl.MAX, obj, [con])
status, val = prob.solve(solver='ECOS', verbose=False)
assert abs(val - expected_val) < 1e-6
v = con.violation()
assert v < 1e-6
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 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 _geometric_program_1(self, solver, **kwargs):
"""
Solve a GP with a linear objective and single posynomial constraint.
The reference solution was computed by Wolfram Alpha.
"""
alpha = np.array([[1, 0],
[0, 1],
[1, 1],
[0.5, 0],
[0, 0.5]])
c = np.array([3, 2, 1, 4, 2])
x = cl.Variable(shape=(2,), name='x')
y = alpha @ x
expr = cl.weighted_sum_exp(c, y)
cons = [expr <= 1]
obj = - x[0] - 2 * x[1]
prob = Problem(cl.MIN, obj, cons)
status, val = prob.solve(solver=solver, **kwargs)
assert status == 'solved'
assert abs(val - 10.4075826) < 1e-6
x_star = x.value
expect = np.array([-4.93083, -2.73838])
assert np.allclose(x_star, expect, atol=1e-4)
return prob
def violation(self, norm_ord=np.inf, rough=False):
"""
Return a measure of violation for the constraint that ``self.v`` belongs to
:math:`C_{\\mathrm{SAGE}}(\\alpha, X)^{\\dagger}`.
Parameters
----------
norm_ord : int
The value of ``ord`` passed to numpy ``norm`` functions, when reducing
vector-valued residuals into a scalar residual.
rough : bool
Setting ``rough=False`` computes violation by solving an optimization
problem. Setting ``rough=True`` computes violation by taking norms of
residuals of appropriate elementwise equations and inequalities involving
``self.v`` and auxiliary variables.
Notes
-----
When ``rough=False``, the optimization-based violation is computed by projecting
the vector ``self.v`` onto a new copy of a dual SAGE constraint, and then returning
the L2-norm between ``self.v`` and that projection. This optimization step essentially
re-solves for all auxiliary variables used by this constraint.
"""
v = self.v.value
viols = []
for i in self.ech.U_I:
selector = self.ech.expcovers[i]
num_cover = self.ech.expcover_counts[i]
if num_cover > 0:
expr1 = np.tile(v[i], num_cover).ravel()
expr2 = v[selector].ravel()
lowerbounds = special_functions.rel_entr(expr1, expr2)
mat = -(self.alpha[selector, :] - self.alpha[i, :])
mu_i = self._lifted_mu_vars[i].value
# compute rough violation for this dual AGE cone
residual = mat @ mu_i[:self._n] - lowerbounds
residual[residual >= 0] = 0
curr_viol = np.linalg.norm(residual, ord=norm_ord)
if (self.X is not None) and (not np.isnan(curr_viol)):
AbK_val = self.X.A @ mu_i + v[i] * self.X.b
AbK_viol = PrimalProductCone.project(AbK_val, self.X.K)
curr_viol += AbK_viol
# as applicable, solve an optimization problem to compute the violation.
if (curr_viol > 0 or np.isnan(curr_viol)) and not rough:
temp_var = Variable(shape=(self._lifted_n,), name='temp_var')
cons = [mat @ temp_var[:self._n] >= lowerbounds]
if self.X is not None:
con = PrimalProductCone(self.X.A @ temp_var + v[i] * self.X.b, self.X.K)
cons.append(con)
prob = Problem(CL_MIN, Expression([0]), cons)
status, value = prob.solve(verbose=False)
if status in {CL_SOLVED, CL_INACCURATE} and abs(value) < 1e-7:
curr_viol = 0
viols.append(curr_viol)
else:
viols.append(0)
viol = max(viols)
return viol
def _presolve_trivial_ord_age(self, i, covers):
"""
Set covers[i][:] = False if the i-th (ordinary) AGE cone is trivial.
"""
mat = self.alpha[covers[i], :] - self.alpha[i, :]
x = Variable(shape=(mat.shape[1], ), name='temp_x')
objective = Expression([0])
cons = [mat @ x <= -1]
prob = Problem(CL_MIN, objective, cons)
# If prob is feasible, then there exists a sequence x_t where
# max(mat @ x_t) diverges to -\infty as t increases. Using this
# sequence we can satisfy the constraints for the i-th dual
# AGE cone no matter the value of the vector "v" that needs to belong
# to the dual AGE cone.
prob.solve(verbose=False, solver=self.settings['reduction_solver'])
if prob.status in {CL_SOLVED, CL_INACCURATE
} and abs(prob.value) < 1e-7:
covers[i][:] = False
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
def _presolve_trivial_cond_age(self, i, covers, threshold=-100):
"""
Overwrite covers[i][:] = False if it LOOKS LIKE the i-th X-AGE cone is trivial.
If X is a cone (which happens if self.AbK[1] is the zero vector), then this method
for checking triviality is necessary and sufficient.
If X is not a cone, then this method might incorrectly identify an AGE cone as trivial.
Setting "threshold" to a large negative number decreases the chance that we ignore
a useful AGE cone.
Note: If X is compact, then no X-AGE cone is trivial.
"""
mat = self.alpha[covers[i], :] - self.alpha[i, :]
x = Variable(shape=(mat.shape[1], ), name='temp_x')
t = Variable(shape=(1, ), name='temp_t')
objective = t
A, b, K = self.AbK
cons = [mat @ x <= t, PrimalProductCone(A @ x + b, K)]
prob = Problem(CL_MIN, objective, cons)
prob.solve(verbose=False, solver=self.settings['reduction_solver'])
if prob.status in {CL_SOLVED, CL_INACCURATE
} and prob.value < threshold:
covers[i][:] = False
def _default_exp_covers(self):
expcovers = dict()
for i in self.U_I:
cov = np.ones(shape=(self.m,), dtype=bool)
cov[self.N_I] = False
cov[i] = False
expcovers[i] = cov
if self.AbK is None or self.settings['heuristic_reduction']:
row_sums = np.sum(self.alpha, 1)
if np.all(self.alpha >= 0) and np.min(row_sums) == 0:
# Then apply the reduction.
zero_loc = np.nonzero(row_sums == 0)[0][0]
for i in self.U_I:
if i == zero_loc:
continue
curr_cover = expcovers[i]
curr_row = self.alpha[i, :]
for j in range(self.m):
if curr_cover[j] and j != zero_loc and curr_row @ self.alpha[j, :] == 0:
curr_cover[j] = False
"""
The above operation is without loss of generality for ordinary SAGE
constraints. For conditional SAGE constraints, the operation may or
may-not be without loss of generality. As a basic check, the above
operation is w.l.o.g. even for conditional SAGE constraints, as long
as the "conditioning" satisfies the following property:
Suppose "y" a geometric-form solution which is feasible w.r.t.
conditioning. Then "y" remains feasible (w.r.t. conditioning)
when we assign "y[k] = 0".
The comments below explain in greater detail.
Observation
-----------
By being in this part of the code, there must exist a "k" where
alpha[i,k] == 0 and alpha[j,k] > 0.
Also, we have alpha >= 0. These facts tell us that the expression
(alpha[j2,:] - alpha[i,:]) @ mu[:, i] (*)
is (1) non-decreasing in mu[k,i] for all 0 <= j2 < m, and (2)
strictly increasing in mu[k,i] when j2 == j. Therefore by
sending mu[i,k] to -\infty, we do not increase (*) for any
0 <= j2 < m, and in fact (*) goes to -\infty for j2 == j.
Consequence 1
-------------
If mu[:,i] is only subject to constraints of the form
v[i]*log(v[j2]/v[i]) >= (alpha[j2,:] - alpha[i,:]) @ mu[:, i]
with 0 <= j2 < m, then the particular constraint with j2 == j
is never active at any optimal solution. For ordinary SAGE cones,
this means the j-th term of alpha isn't used in the i-th AGE cone.
Consequence 2
-------------
For conditional SAGE cones, there is another constraint:
A @ mu[:, i] + v[i] * b \in K. (**)
However, as long as (**) allows us to send mu[k,i] to -\infty
without affecting feasibility, then the we arrive at the same
conclusion: the j-th term of alpha isn't used in the i-th AGE cone.
"""
if self.AbK is None:
for i in self.U_I:
if np.count_nonzero(expcovers[i]) == 1:
expcovers[i][:] = False
if self.settings['presolve_trivial_age_cones']:
if self.AbK is None:
for i in self.U_I:
if np.any(expcovers[i]):
mat = self.alpha[expcovers[i], :] - self.alpha[i, :]
x = Variable(shape=(mat.shape[1],), name='temp_x')
objective = Expression([0])
cons = [mat @ x <= -1]
prob = Problem(CL_MIN, objective, cons)
prob.solve(verbose=False,
solver=self.settings['reduction_solver'])
if prob.status in {CL_SOLVED, CL_INACCURATE} and abs(prob.value) < 1e-7:
expcovers[i][:] = False
else:
for i in self.U_I:
if np.any(expcovers[i]):
mat = self.alpha[expcovers[i], :] - self.alpha[i, :]
x = Variable(shape=(mat.shape[1],), name='temp_x')
t = Variable(shape=(1,), name='temp_t')
objective = t
A, b, K = self.AbK
cons = [mat @ x <= t, PrimalProductCone(A @ x + b, K)]
prob = Problem(CL_MIN, objective, cons)
prob.solve(verbose=False,
solver=self.settings['reduction_solver'])
if prob.status in {CL_SOLVED, CL_INACCURATE} and prob.value < -100:
expcovers[i][:] = False
return expcovers