def test_unconstrained_1(self):
# Background
#
# This example was constructed soley as a test case for sageopt.
#
# We consider two polynomial optimization problems, that are related
# to one another by a change of sign in one of the variables.
#
# Tests
#
# (1) Verify primal / dual consistency for the two problems, at level (0, 0).
#
# (2) Verify that the SAGE bound is the same for the two formulations.
#
alpha = np.array([[0, 0], [1, 1], [2, 2], [0, 2], [2, 0]])
# First formulation
p = Polynomial(alpha, np.array([1, -3, 1, 4, 4]))
res0 = primal_dual_unconstrained(p, 0, sigrep_ell=0)
assert abs(res0[0] - res0[1]) <= 1e-6
# Second formulation
p = Polynomial(alpha, np.array([1, 3, 1, 4, 4]))
res1 = primal_dual_unconstrained(p, 0, sigrep_ell=0)
assert abs(res1[0] - res1[1]) <= 1e-6
# Check for same results between the two formulations
expected = 1
assert abs(res0[0] - expected) <= 1e-5
assert abs(res1[0] - expected) <= 1e-5
def test_standard_monomials(self):
x = standard_poly_monomials(3)
y_actual = np.prod(x)
y_expect = Polynomial.from_dict({(1, 1, 1): 1})
assert TestPolynomials.are_equal(y_actual, y_expect)
x = standard_poly_monomials(2)
y_actual = np.sum(x)**2
y_expect = Polynomial.from_dict({(2, 0): 1, (1, 1): 2, (0, 2): 1})
assert TestPolynomials.are_equal(y_actual, y_expect)
def test_polynomial_exponentiation(self):
p = Polynomial.from_dict({(0,): -1, (1,): 1})
# square of (x-1)
res = p ** 2
expect = Polynomial.from_dict({(0,): 1, (1,): -2, (2,): 1})
assert res == expect
# cube of (2x+5)
p = Polynomial.from_dict({(0,): 5, (1,): 2})
expect = Polynomial.from_dict({(0,): 125, (1,): 150, (2,): 60, (3,): 8})
res = p ** 3
assert res == expect
self.assertRaises(RuntimeError, Polynomial.__pow__, p, p)
def test_addition_and_subtraction(self):
# data for tests
s0 = Polynomial(np.array([[0], [1], [2]]), np.array([1, 2, 3]))
t0 = Polynomial(np.array([[4]]), np.array([5]))
# tests
s = s0 - s0
s = s.without_zeros()
assert s.m == 1 and set(s.c) == {0}
s = -s0 + s0
s = s.without_zeros()
assert s.m == 1 and set(s.c) == {0}
s = s0 + t0
assert s.alpha_c == {(0, ): 1, (1, ): 2, (2, ): 3, (4, ): 5}
def test_composition(self):
p = Polynomial.from_dict({(2,): 1}) # represents lambda x: x ** 2
z = Polynomial.from_dict({(1,): 2, (0,): -1}) # represents lambda x: 2*x - 1
w = p(z) # represents lambda x: (2*x - 1) ** 2
assert w(0.5) == 0
assert w(1) == 1
assert w(0) == 1
x = standard_poly_monomials(3)
p = np.prod(x)
y = standard_poly_monomials(2)
expr = np.array([y[0], y[0]-y[1], y[1]])
w = p(expr)
assert w.n == 2
assert w(np.array([1, 1])) == 0
assert w(np.array([1, -2])) == -6
def test_polynomial_multiplication(self):
# data for tests
s0 = Polynomial(np.array([[0], [1], [2]]), np.array([1, 2, 3]))
t0 = Polynomial(np.array([[1]]), np.array([1]))
q0 = Polynomial(np.array([[5]]), np.array([0]))
# tests
s = s0 * t0
s = s.without_zeros()
assert s.alpha_c == {(1, ): 1, (2, ): 2, (3, ): 3}
s = t0 * s0
s = s.without_zeros()
assert s.alpha_c == {(1, ): 1, (2, ): 2, (3, ): 3}
s = s0 * q0
s = s.without_zeros()
assert s.alpha_c == {(0, ): 0}
def test_constant_location(self):
s0 = Polynomial(np.array([[0], [1], [2]]), np.array([1, 2, 3]))
t0 = Polynomial(np.array([[1]]), np.array([1]))
q0 = Polynomial(np.array([[5]]), np.array([0]))
# tests
loc = s0.constant_location()
assert (s0.alpha[loc, :] == 0).all()
loc = t0.constant_location()
assert loc == None
loc = q0.constant_location()
assert loc == None
def test_polynomial_division(self):
s0 = Polynomial(np.array([[0], [1], [2]]), np.array([1, 2, 3]))
t0 = Polynomial(np.array([[1]]), np.array([1]))
q0 = Polynomial(np.array([[5]]), np.array([0]))
#tests
self.assertRaises(ValueError, Polynomial.__truediv__, s0, t0)
s = s0 / 2
s = s.without_zeros()
assert s.alpha_c == {(0, ): 0.5, (1, ): 1, (2, ): 1.5}
s = t0 / 4
s = s.without_zeros()
assert s.alpha_c == {(1, ): 0.25}
s = q0 / 4
s = s.without_zeros()
assert s.alpha_c == {(5, ): 0}
def test_equal(self):
s0 = Polynomial(np.array([[0], [1], [2]]), np.array([1, 2, 3]))
assert s0 == s0
alpha = np.array([[1, 0], [0, 1], [1, 1]])
c = np.array([1, 2, 3])
f = Signomial(alpha, c)
assert f != s0
def poly_constrained_primal(f, gts, eqs, p=0, q=1, ell=0, X=None):
"""
Construct the primal SAGE-(p, q, ell) relaxation for the polynomial optimization problem
inf{ f(x) : g(x) >= 0 for g in gts,
g(x) == 0 for g in eqs,
and x in X }
where :math:`X = R^{\\texttt{f.n}}` by default.
"""
lagrangian, ineq_lag_mults, _, gamma = make_poly_lagrangian(f, gts, eqs, p=p, q=q)
metadata = {'lagrangian': lagrangian}
if ell > 0:
alpha_E_q = hierarchy_e_k([f] + list(gts) + list(eqs), k=1)
modulator = Polynomial(2 * alpha_E_q, np.ones(alpha_E_q.shape[0])) ** ell
lagrangian = lagrangian * modulator
metadata['modulator'] = modulator
# The Lagrangian (after possible multiplication, as above) must be a SAGE polynomial.
con_name = 'Lagrangian sage poly'
constrs = primal_sage_poly_cone(lagrangian, con_name, log_AbK=X)
# Lagrange multipliers (for inequality constraints) must be SAGE polynomials.
for s_h, _ in ineq_lag_mults:
con_name = str(s_h) + ' domain'
cons = primal_sage_poly_cone(s_h, con_name, log_AbK=X)
constrs += cons
# Construct the coniclifts problem.
prob = cl.Problem(cl.MAX, gamma, constrs)
prob.metadata = metadata
cl.clear_variable_indices()
return prob
def poly_constrained_dual(f, gts, eqs, p=0, q=1, ell=0, X=None, slacks=False):
"""
Construct the dual SAGE-(p, q, ell) relaxation for the polynomial optimization problem
inf{ f(x) : g(x) >= 0 for g in gts,
g(x) == 0 for g in eqs,
and x in X }
where :math:`X = R^{\\texttt{f.n}}` by default.
"""
lagrangian, ineq_lag_mults, eq_lag_mults, _ = make_poly_lagrangian(f, gts, eqs, p=p, q=q)
metadata = {'lagrangian': lagrangian, 'f': f, 'gts': gts, 'eqs': eqs, 'X': X}
if ell > 0:
alpha_E_1 = hierarchy_e_k([f, f.upcast_to_polynomial(1)] + gts + eqs, k=1)
modulator = Polynomial(2 * alpha_E_1, np.ones(alpha_E_1.shape[0])) ** ell
lagrangian = lagrangian * modulator
f = f * modulator
else:
modulator = f.upcast_to_polynomial(1)
metadata['modulator'] = modulator
# In primal form, the Lagrangian is constrained to be a SAGE polynomial.
# Introduce a dual variable "v" for this constraint.
v = cl.Variable(shape=(lagrangian.m, 1), name='v')
metadata['v_poly'] = v
constraints = relative_dual_sage_poly_cone(lagrangian, v, 'Lagrangian', log_AbK=X)
for s_g, g in ineq_lag_mults:
# These generalized Lagrange multipliers "s_g" are SAGE polynomials.
# For each such multiplier, introduce an appropriate dual variable "v_g", along
# with constraints over that dual variable.
g_m = g * modulator
c_g = sym_corr.moment_reduction_array(s_g, g_m, lagrangian)
name_base = 'v_' + str(g)
if slacks:
v_g = cl.Variable(name=name_base, shape=(s_g.m, 1))
con = c_g @ v == v_g
con.name += str(g) + ' >= 0'
constraints.append(con)
else:
v_g = c_g @ v
constraints += relative_dual_sage_poly_cone(s_g, v_g,
name_base=(name_base + ' domain'), log_AbK=X)
for z_g, g in eq_lag_mults:
# These generalized Lagrange multipliers "z_g" are arbitrary polynomials.
# They dualize to homogeneous equality constraints.
g_m = g * modulator
c_g = sym_corr.moment_reduction_array(z_g, g_m, lagrangian)
con = c_g @ v == 0
con.name += str(g) + ' == 0'
constraints.append(con)
# Equality constraint (for the Lagrangian to be bounded).
a = sym_corr.relative_coeff_vector(modulator, lagrangian.alpha)
constraints.append(a.T @ v == 1)
# Define the dual objective function.
obj_vec = sym_corr.relative_coeff_vector(f, lagrangian.alpha)
obj = obj_vec.T @ v
# Return the coniclifts Problem.
prob = cl.Problem(cl.MIN, obj, constraints)
prob.metadata = metadata
cl.clear_variable_indices()
return prob
def poly_dual(f, poly_ell=0, sigrep_ell=0, X=None):
if poly_ell == 0:
sr, _ = f.sig_rep
prob = sage_sigs.sig_dual(sr, sigrep_ell, X=X)
if AUTO_CLEAR_INDICES: # pragma:no cover
cl.clear_variable_indices()
return prob
elif sigrep_ell == 0:
modulator = f.standard_multiplier()**poly_ell
gamma = cl.Variable()
lagrangian = (f - gamma) * modulator
v = cl.Variable(shape=(lagrangian.m, 1), name='v')
con_base_name = v.name + ' domain'
constraints = relative_dual_sage_poly_cone(lagrangian,
v,
con_base_name,
log_AbK=X)
a = sym_corr.relative_coeff_vector(modulator, lagrangian.alpha)
constraints.append(a.T @ v == 1)
f_mod = Polynomial(f.alpha, f.c) * modulator
obj_vec = sym_corr.relative_coeff_vector(f_mod, lagrangian.alpha)
obj = obj_vec.T @ v
prob = cl.Problem(cl.MIN, obj, constraints)
if AUTO_CLEAR_INDICES: # pragma:no cover
cl.clear_variable_indices()
return prob
else: # pragma: no cover
raise NotImplementedError()
def test_unconstrained_3(self):
# Minimization of the six-hump camel back function.
p = Polynomial.from_dict({(0, 0): 0,
(2, 0): 4,
(1, 1): 1,
(0, 2): -4,
(4, 0): -2.1,
(0, 4): 4,
(6, 0): 1.0 / 3.0})
# sigrep_ell=0 has a decent bound, and sigrep_ell=1 is nearly optimal.
# ECOS is unable to solver sigrep_ell=2 due to conditioning problems.
# MOSEK easily solves sigrep_ell=2, and this is globally optimal
res00 = primal_dual_unconstrained(p, poly_ell=0, sigrep_ell=0)
expect00 = -1.18865
assert abs(res00[0] - res00[1]) <= 1e-6
assert abs(res00[0] - expect00) <= 1e-3
res10 = primal_dual_unconstrained(p, poly_ell=1, sigrep_ell=0)
expect10 = -1.03416
assert abs(res10[0] - res10[1]) < 1e-6
assert abs(res10[0] - expect10) <= 1e-3
if cl.Mosek.is_installed():
res01 = primal_dual_unconstrained(p, poly_ell=0, sigrep_ell=1, solver='MOSEK')
expect01 = -1.03221
assert abs(res01[0] - res01[1]) <= 1e-6
assert abs(res01[0] - expect01) <= 1e-3
res02 = primal_dual_unconstrained(p, poly_ell=0, sigrep_ell=2, solver='MOSEK')
expect02 = -1.0316
assert abs(res02[0] - res02[1]) < 1e-6
assert abs(res02[0] - expect02) <= 1e-3
def test_addition_and_subtraction(self):
# data for tests
s0 = Polynomial(np.array([[0], [1], [2]]), np.array([1, 2, 3]))
t0 = Polynomial(np.array([[4]]), np.array([5]))
# tests
s = s0 - s0
s = s.without_zeros()
assert s.m == 1 and set(s.c) == {0}
s = -s0 + s0
s = s.without_zeros()
assert s.m == 1 and set(s.c) == {0}
s = s0 + t0
assert s.alpha_c == {(0, ): 1, (1, ): 2, (2, ): 3, (4, ): 5}
alpha = np.array([[1, 0], [0, 1], [1, 1]])
c = np.array([1, 2, 3])
f = Signomial(alpha, c)
self.assertRaises(ValueError, Polynomial.__add__, s0, f)
def hierarchy_e_k(polys, k):
alphas = [s.alpha for s in polys]
alpha = np.vstack(alphas)
alpha = np.unique(alpha, axis=0)
c = np.ones(shape=(alpha.shape[0],))
s = Polynomial(alpha, c)
s = s ** k
return s.alpha
def test_sigrep_3(self):
alpha = np.random.randint(low=1, high=10, size=(10, 3))
alpha *= 2
c = np.random.randn(10)
p = Polynomial(alpha, c)
# The signomial representative has the same exponents and coeffs.
sr, sr_cons = p.sig_rep
assert len(sr_cons) == 0
assert p.alpha_c == sr.alpha_c
def as_polynomial(self):
"""
This function is only applicable if ``alpha`` is a matrix of nonnegative integers.
Returns
-------
f : Polynomial
For every vector ``x``, we have ``self(x) == f(np.exp(x))``.
"""
from sageopt.symbolic.polynomials import Polynomial
f = Polynomial(self.alpha, self.c)
return f
def test_sigrep_2(self):
c33 = cl.Variable(shape=(), name='c33')
alpha = np.array([[0, 0], [1, 1], [3, 3]])
c = cl.Expression([0, -1, c33])
p = Polynomial(alpha, c)
sr, sr_cons = p.sig_rep
assert len(sr_cons) == 2
var_names = set(v.name for v in sr_cons[0].variables())
var_names.union(set(v.name for v in sr_cons[1].variables()))
for v in var_names:
assert v == 'c33' or v == str(p) + ' variable sigrep coefficients'
assert sr.alpha_c[(1, 1)] == -1
def test_polynomial_multiplication(self):
# data for tests
s0 = Polynomial(np.array([[0], [1], [2]]), np.array([1, 2, 3]))
t0 = Polynomial(np.array([[1]]), np.array([1]))
q0 = Polynomial(np.array([[5]]), np.array([0]))
alpha = np.array([[1, 0], [0, 1], [1, 1]])
c = np.array([1, 2, 3])
f = Signomial(alpha, c)
# tests
s = s0 * t0
s = s.without_zeros()
assert s.alpha_c == {(1, ): 1, (2, ): 2, (3, ): 3}
s = t0 * s0
s = s.without_zeros()
assert s.alpha_c == {(1, ): 1, (2, ): 2, (3, ): 3}
s = s0 * q0
s = s.without_zeros()
assert s.alpha_c == {(0, ): 0}
self.assertRaises(ValueError, Polynomial.__sub__, s0, f)
self.assertRaises(ValueError, Polynomial.__mul__, s0, f)
def sage_multiplier_search(f, level=1, X=None):
"""
Constructs a coniclifts maximization Problem which is feasible if ``f`` can be certified as nonnegative
over ``X``, by using an appropriate X-SAGE modulating function.
Parameters
----------
f : Polynomial
We want to test if ``f`` is nonnegative over ``X``.
level : int
Controls the complexity of the X-SAGE modulating function. Must be a positive integer.
X : PolyDomain or None
If ``X`` is None, then we test nonnegativity of ``f`` over :math:`R^{\\texttt{f.n}}`.
Returns
-------
prob : sageopt.coniclifts.Problem
Notes
-----
This function provides an alternative to moving up the reference SAGE hierarchy, for the
goal of certifying nonnegativity of a polynomial ``f`` over some set ``X`` where ``|X|``
is log-convex. In general, the approach is to introduce a polynomial
``mult = Polynomial(alpha_hat, c_tilde)``
where the rows of alpha_hat are all "level"-wise sums of rows from ``f.alpha``, and ``c_tilde``
is a coniclifts Variable defining a nonzero SAGE polynomial. Then we can check if
``f_mod = f * mult`` is SAGE for any choice of ``c_tilde``.
"""
constraints = []
# Make the multiplier polynomial (and require that it be SAGE)
mult_alpha = hierarchy_e_k([f], k=level)
c_tilde = cl.Variable(shape=(mult_alpha.shape[0], ), name='c_tilde')
mult = Polynomial(mult_alpha, c_tilde)
temp_cons = primal_sage_poly_cone(mult,
name=(c_tilde.name + ' domain'),
log_AbK=X)
constraints += temp_cons
constraints.append(cl.sum(c_tilde) >= 1)
# Make "f_mod := f * mult", and require that it be SAGE.
f_mod = mult * f
temp_cons = primal_sage_poly_cone(f_mod, name='f_mod sage poly', log_AbK=X)
constraints += temp_cons
# noinspection PyTypeChecker
prob = cl.Problem(cl.MAX, 0, constraints)
if AUTO_CLEAR_INDICES: # pragma:no cover
cl.clear_variable_indices()
return prob
def test_unconstrained_2(self):
# Background
#
# Unconstrained minimization of a polynomial in 2 variables.
# This is Example 4.1 from a 2018 paper by Seidler and de Wolff
# (https://arxiv.org/abs/1808.08431).
#
# Tests
#
# (1) primal / dual consistency for (poly_ell, sigrep_ell) \in {(0, 0), (1, 0), (0, 1)}.
#
# (2) Show that the bound with (poly_ell=0, sigrep_ell=1) is strong than
# the bound with (poly_ell=1, sigrep_ell=0).
#
# Notes
#
# The global minimum of this polynomial (as verified by gloptipoly3) is 0.85018.
#
# The furthest we could progress up the hierarchy before encountering a solver failure
# was (poly_ell=0, sigrep_ell=5). In this case the SAGE bound was 0.8336.
#
p = Polynomial.from_dict({
(0, 0): 1,
(2, 6): 3,
(6, 2): 2,
(2, 2): 6,
(1, 2): -1,
(2, 1): 2,
(3, 3): -3
})
res00 = primal_dual_unconstrained(p, poly_ell=0, sigrep_ell=0)
expect00 = 0.6932
assert abs(res00[0] - res00[1]) <= 1e-6
assert abs(res00[0] - expect00) <= 1e-3
res10 = primal_dual_unconstrained(p, poly_ell=1, sigrep_ell=0)
expect10 = 0.7587
assert abs(res10[0] - res10[1]) <= 1e-5
assert abs(res10[0] - expect10) <= 1e-3
if cl.Mosek.is_installed():
# ECOS fails
res01 = primal_dual_unconstrained(p,
poly_ell=0,
sigrep_ell=1,
solver='MOSEK')
expect01 = 0.7876
assert abs(res01[0] - res01[1]) <= 1e-5
assert abs(res01[0] - expect01) <= 1e-3
def test_scalar_multiplication(self):
# data for tests
alpha0 = np.array([[0], [1], [2]])
c0 = np.array([1, 2, 3])
s0 = Polynomial(alpha0, c0)
# Tests
s = 2 * s0
# noinspection PyTypeChecker
assert set(s.c) == set(2 * s0.c)
s = s0 * 2
# noinspection PyTypeChecker
assert set(s.c) == set(2 * s0.c)
s = 1 * s0
assert s.alpha_c == s0.alpha_c
s = 0 * s0
s = s.without_zeros()
assert s.m == 1 and set(s.c) == {0}
def test_composition_sigs(self):
p = Polynomial.from_dict({
(1, ): 2,
(0, ): -1
}) # represents lambda x: 2*x - 1
s = Signomial.from_dict({(2, ): -1, (0, ): 1})
f = p(s) # lambda x: -2*exp(x) + 1
self.assertAlmostEqual(f(0.5), -2 * np.exp(1.0) + 1, places=4)
self.assertAlmostEqual(f(1), -2 * np.exp(2.0) + 1, places=4)
p = np.prod(standard_poly_monomials(3))
exp_x = standard_sig_monomials(2)
sig_vec = np.array([exp_x[0], exp_x[0] - exp_x[1], 1.0 / exp_x[1]])
f = p(sig_vec)
self.assertEqual(f.n, 2)
self.assertEqual(f(np.array([1, 1])), 0)
x_test = np.array([-3, 3])
self.assertAlmostEqual(f(x_test),
np.exp(-6) * (np.exp(-3) - np.exp(3)),
places=4)
def test_sigrep_1(self):
p = Polynomial.from_dict({(0, 0): -1, (1, 2): 1, (2, 2): 10})
gamma = cl.Variable(shape=(), name='gamma')
p -= gamma
sr, sr_cons = p.sig_rep
# Even though there is a Variable in p.c, no auxiliary
# variables should have been introduced by defining this
# signomial representative.
assert len(sr_cons) == 0
count_nonconstants = 0
for i, ci in enumerate(sr.c):
if isinstance(ci, cl.base.ScalarExpression):
if not ci.is_constant():
assert len(ci.variables()) == 1
count_nonconstants += 1
assert ci.variables()[0].name == 'gamma'
elif sr.alpha[i, 0] == 1 and sr.alpha[i, 1] == 2:
assert ci == -1
elif sr.alpha[i, 0] == 2 and sr.alpha[i, 1] == 2:
assert ci == 10
else:
assert False
assert count_nonconstants == 1
def poly_solrec(prob, ineq_tol=1e-8, eq_tol=1e-6, skip_ls=False, **kwargs):
"""
Recover a list of candidate solutions from a dual SAGE relaxation. Solutions are
guaranteed to be feasible up to specified tolerances, but not necessarily optimal.
Parameters
----------
prob : coniclifts.Problem
A dual-form SAGE relaxation, from ``poly_constrained_relaxation``.
ineq_tol : float
The amount by which recovered solutions can violate inequality constraints.
eq_tol : float
The amount by which recovered solutions can violate equality constraints.
skip_ls : bool
Whether or not to skip least-squares solution recovery.
Returns
-------
sols : list of ndarrays
A list of feasible solutions, sorted in increasing order of objective function value.
It is possible that this list is empty, in which case no feasible solutions were recovered.
Notes
-----
This function accepts the following keyword arguments:
zero_tol : float
Used in magnitude recovery. If a component of the Lagrangian's moment vector is smaller
than this (in absolute value), pretend it's zero in the least-squares step. Defaults to 1e-20.
heuristic_signs : bool
Used in sign recovery. If True, then attempts to infer variable signs from the Lagrangian's
moment vector even when a completely consistent set of signs does not exist. Defaults to True.
all_signs : bool
Used in sign recovery. If True, then consider returning solutions which differ only by sign.
Defaults to True.
This function is implemented only for poly_constrained_relaxation (not poly_relaxation).
"""
zero_tol = kwargs['zero_tol'] if 'zero_tol' in kwargs else 1e-20
heuristic = kwargs[
'heuristic_signs'] if 'heuristic_signs' in kwargs else True
all_signs = kwargs['all_signs'] if 'all_signs' in kwargs else True
metadata = prob.metadata
f = metadata['f']
lag_gts = metadata['gts']
lag_eqs = metadata['eqs']
lagrangian = _make_dummy_lagrangian(f, lag_gts, lag_eqs)
con = prob.constraints[0]
alpha = con.alpha
dummy_modulated_lagrangian = Polynomial(
alpha, np.ones(shape=(alpha.shape[0], ))) # coefficients dont matter
modulator = metadata['modulator']
v = metadata['v_poly'].value # possible that v_sig and v are the same
if np.any(np.isnan(v)):
return []
M = moment_reduction_array(lagrangian, modulator,
dummy_modulated_lagrangian)
v_reduced = M @ v
alpha_reduced = lagrangian.alpha
mags = variable_magnitudes(con, alpha_reduced, v_reduced, zero_tol,
skip_ls)
signs = variable_sign_patterns(alpha_reduced, v_reduced, heuristic,
all_signs)
# Now we need to build the candidate solutions, and check them for feasibility.
if con.X is not None:
gts = lag_gts + [g for g in con.X.gts]
eqs = lag_eqs + [g for g in con.X.eqs]
else:
gts = lag_gts
eqs = lag_eqs
solutions = []
for mag in mags:
for sign in signs:
x = mag * sign # elementwise
if is_feasible(x, gts, eqs, ineq_tol, eq_tol):
solutions.append(x)
solutions.sort(key=lambda xi: f(xi))
return solutions
def make_poly_lagrangian(f, gts, eqs, p, q):
"""
Given a problem
.. math::
\\begin{align*}
\min\{ f(x) :~& g(x) \geq 0 \\text{ for } g \\in \\text{gts}, \\\\
& g(x) = 0 \\text{ for } g \\in \\text{eqs}, \\\\
& \\text{and } x \\in X \}
\\end{align*}
construct the q-fold constraints ``q-gts`` and ``q-eqs``, by taking all products
of ``<= q`` elements from ``gts`` and ``eqs`` respectively. Then form the Lagrangian
.. math::
L = f - \\gamma
- \sum_{g \, \\in \, \\text{q-gts}} s_g \cdot g
- \sum_{g \, \\in \, \\text{q-eqs}} z_g \cdot g
where :math:`\\gamma` is a coniclifts Variable of dimension 1, and the coefficients
on Polynomials :math:`s_g` and :math:`z_g` are coniclifts Variables of a dimension
determined by ``p``.
Parameters
----------
f : Polynomial
The objective in a desired minimization problem.
gts : list of Polynomials
For every ``g in gts``, there is a desired constraint that variables ``x`` satisfy ``g(x) >= 0``.
eqs : list of Polynomials
For every ``g in eqs``, there is a desired constraint that variables ``x`` satisfy ``g(x) == 0``.
p : int
Controls the complexity of ``s_g`` and ``z_g``.
q : int
The number of folds of constraints ``gts`` and ``eqs``.
Returns
-------
L : Polynomial
``L.c`` is an affine expression of coniclifts Variables.
ineq_dual_polys : a list of pairs of Polynomials.
If the pair ``(s_g, g)`` is in this list, then ``s_g`` is a generalized Lagrange multiplier
to the constraint ``g(x) >= 0``.
eq_dual_polys : a list of pairs of Polynomials.
If the pair ``(z_g, g)`` is in this list, then ``z_g`` is a generalized Lagrange multiplier to the
constraint ``g(x) == 0``.
gamma : coniclifts.Variable.
In primal-form SAGE relaxations, we want to maximize ``gamma``. In dual form SAGE relaxations,
``gamma`` induces a normalizing equality constraint.
Notes
-----
The Lagrange multipliers ``s_g`` and ``z_g`` share a common matrix of exponent vectors,
which we call ``alpha_hat``.
When ``p = 0``, ``alpha_hat`` consists of a single row, of all zeros. In this case,
``s_g`` and ``z_g`` are constant Polynomials, and the coefficient vectors ``s_g.c``
and ``z_g.c`` are effectively scalars. When ``p > 0``, the rows of ``alpha_hat`` are
*initially* set set to all ``p``-wise sums of exponent vectors appearing in either ``f``,
or some ``g in gts``, or some ``g in eqs``. Then we replace ::
alpha_hat = np.vstack([2 * alpha_hat, alpha_hat])
alpha_multiplier = np.unique(alpha_hat, axis=0)
This has the effect of improving performance for problems where ``alpha_hat`` would otherwise
contain very few rows in the even integer lattice.
"""
folded_gt = con_gen.up_to_q_fold_cons(gts, q)
gamma = cl.Variable(name='gamma')
L = f - gamma
alpha_E_p = hierarchy_e_k([f, f.upcast_to_polynomial(1)] + gts + eqs, k=p)
alpha_multiplier = np.vstack([2 * alpha_E_p, alpha_E_p])
alpha_multiplier = np.unique(alpha_multiplier, axis=0)
ineq_dual_polys = []
for g in folded_gt:
s_g_coeff = cl.Variable(name='s_' + str(g), shape=(alpha_multiplier.shape[0],))
s_g = Polynomial(alpha_multiplier, s_g_coeff)
L -= s_g * g
ineq_dual_polys.append((s_g, g))
eq_dual_polys = []
folded_eq = con_gen.up_to_q_fold_cons(eqs, q)
for g in folded_eq:
z_g_coeff = cl.Variable(name='z_' + str(g), shape=(alpha_multiplier.shape[0],))
z_g = Polynomial(alpha_multiplier, z_g_coeff)
L -= z_g * g
eq_dual_polys.append((z_g, g))
return L, ineq_dual_polys, eq_dual_polys, gamma
def test_polynomial_hess_val(self):
f = Polynomial.from_dict({(3, ): 1, (0, ): -1})
actual = f.hess_val(np.array([0.1234]))
expect = 3 * 2 * 0.1234
assert abs(actual[0] - expect) < 1e-8
def test_sigrep_1(self):
p = Polynomial.from_dict({(0, 0): -1, (1, 2): 1, (2, 2): 10})
# One non-even lattice point (the only one) changes sign.
sr, sr_cons = p.sig_rep
assert len(sr_cons) == 0
assert sr.alpha_c == {(0, 0): -1, (1, 2): -1, (2, 2): 10}
def test_sigrep_2(self):
p = Polynomial.from_dict({(0, 0): 0, (1, 1): -1, (3, 3): 5})
# One non-even lattice point changes sign, another stays the same
sr, sr_cons = p.sig_rep
assert len(sr_cons) == 0
assert sr.alpha_c == {(0, 0): 0, (1, 1): -1, (3, 3): -5}
def test_as_signomial(self):
s0 = Polynomial(np.array([[0], [1], [2]]), np.array([1, 2, 3]))
f = s0.as_signomial()
assert s0.alpha_c == f.alpha_c