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_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_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_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_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 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_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_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_polynomial_grad_val(self):
f = Polynomial.from_dict({(3, ): 1, (0, ): -1})
actual = f.grad_val(np.array([0.5]))
expect = 3 * 0.5**2
assert abs(actual[0] - expect) < 1e-8