def test_construct_basis(self):
# 2 variables, 3rd degree. Even and odd.
x = Variable.multivariate('x', 2)
degree = 3
basis = BasisVector.construct_basis(x, degree)
basis_powers = [(0, 0), (1, 0), (0, 1), (2, 0), (0, 2), (1, 1), (3, 0),
(0, 3), (2, 1), (1, 2)]
self._test_basis_by_powers(x, basis, basis_powers)
# 2 variables, 3rd degree. Even only.
basis = BasisVector.construct_basis(x, degree, odd=False)
basis_powers = [(0, 0), (2, 0), (0, 2), (1, 1)]
self._test_basis_by_powers(x, basis, basis_powers)
# 2 variables, 3rd degree. Odd only.
basis = BasisVector.construct_basis(x, degree, even=False)
basis_powers = [(1, 0), (0, 1), (3, 0), (0, 3), (2, 1), (1, 2)]
self._test_basis_by_powers(x, basis, basis_powers)
# 3 variables, 2nd degree. Even and odd.
x = Variable.multivariate('x', 3)
degree = 2
basis = BasisVector.construct_basis(x, degree)
basis_powers = [(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), (2, 0, 0),
(0, 2, 0), (0, 0, 2), (1, 1, 0), (0, 1, 1), (1, 0, 1)]
self._test_basis_by_powers(x, basis, basis_powers)
# 3 variables, 2nd degree. Even only.
basis = BasisVector.construct_basis(x, degree, odd=False)
basis_powers = [(0, 0, 0), (2, 0, 0), (0, 2, 0), (0, 0, 2), (1, 1, 0),
(0, 1, 1), (1, 0, 1)]
self._test_basis_by_powers(x, basis, basis_powers)
# 3 variables, 2nd degree. Odd only.
basis = BasisVector.construct_basis(x, degree, even=False)
basis_powers = [(1, 0, 0), (0, 1, 0), (0, 0, 1)]
self._test_basis_by_powers(x, basis, basis_powers)
# 0 degree.
degree = 0
basis = BasisVector.construct_basis(x, degree)
basis_powers = [(0, 0, 0)]
self._test_basis_by_powers(x, basis, basis_powers)
basis = BasisVector.construct_basis(x, degree, odd=False)
self._test_basis_by_powers(x, basis, basis_powers)
basis = BasisVector.construct_basis(x, degree, even=False)
basis_powers = []
self._test_basis_by_powers(x, basis, basis_powers)
# Negative degree.
with self.assertRaises(ValueError):
basis = BasisVector.construct_basis(x, -3)
# Float degree.
with self.assertRaises(ValueError):
basis = BasisVector.construct_basis(x, 3.5)
def test_quadratic_form(self):
for Vector in Vectors:
x = Variable.multivariate('x', 2)
basis = Vector.construct_basis(x, 1)
Q = np.array([[1, 2, 3], [2, 4, 5], [3, 5, 6]])
p = Polynomial.quadratic_form(basis, Q)
p_target = basis[0] * basis[0] + (basis[0] * basis[1]) * 4 + \
(basis[0] * basis[2]) * 6 + (basis[1] * basis[1]) * 4 + \
(basis[1] * basis[2]) * 10 + (basis[2] * basis[2]) * 6
self.assertEqual(p, p_target)
def test_multivariate(self):
# Test the representation only.
x = Variable.multivariate('x', 5)
for i, xi in enumerate(x):
self.assertTrue(xi.__repr__() == f'x_{{{i+1}}}')
class TestSosProgram(unittest.TestCase):
x = Variable.multivariate('x', 2)
zero = {xi: 0 for xi in x}
one = {xi: 1 for xi in x}
two = {xi: 2 for xi in x}
def test_new_free_polynomial(self):
for Vector in Vectors:
# Fit free polynomial in 3 points.
prog = SosProgram()
basis = Vector.construct_basis(self.x, 3)
poly, coef = prog.add_polynomial(basis)
prog.add_linear_constraint(poly(self.zero) == 0)
prog.add_linear_constraint(poly(self.one) == 1)
prog.add_linear_constraint(poly(self.two) == 2)
prog.solve()
coef_opt = prog.substitute_minimizer(coef)
poly_opt = prog.substitute_minimizer(poly)
self.assertAlmostEqual(poly_opt(self.zero), 0, places=4)
self.assertAlmostEqual(poly_opt(self.one), 1, places=4)
self.assertAlmostEqual(poly_opt(self.two), 2, places=4)
def test_new_sos_polynomial(self):
# Fit free polynomial in 2 points, and minimize value at a third.
for Vector in Vectors:
# Normal polynomial.
prog = SosProgram()
basis = Vector.construct_basis(self.x, 3)
poly, gram, cons = prog.add_sos_polynomial(basis)
prog.add_linear_cost(poly(self.zero))
prog.add_linear_constraint(poly(self.one) == 1)
prog.add_linear_constraint(poly(self.two) == 2)
prog.solve()
poly_opt = prog.substitute_minimizer(poly)
self.assertAlmostEqual(prog.minimum(), 0, places=4)
self.assertAlmostEqual(poly_opt(self.zero), 0, places=4)
self.assertAlmostEqual(poly_opt(self.one), 1, places=4)
self.assertAlmostEqual(poly_opt(self.two), 2, places=4)
# Reconstruct polynomial from Gram matrix.
gram_opt = prog.substitute_minimizer(gram)
self.assertTrue(self._is_psd(gram_opt))
poly_opt_gram = Polynomial.quadratic_form(basis, gram_opt)
self.assertAlmostEqual(poly_opt, poly_opt_gram)
# Even polynomial.
prog = SosProgram()
poly, gram, cons = prog.add_even_sos_polynomial(basis)
prog.add_linear_cost(poly(self.zero))
prog.add_linear_constraint(poly(self.one) == 1)
prog.add_linear_constraint(poly(self.two) == 2)
prog.solve()
poly_opt = prog.substitute_minimizer(poly)
self.assertAlmostEqual(prog.minimum(), 0, places=4)
self.assertAlmostEqual(poly_opt(self.zero), 0, places=4)
self.assertAlmostEqual(poly_opt(self.one), 1, places=4)
self.assertAlmostEqual(poly_opt(self.two), 2, places=4)
# Reconstruct polynomial from Gram matrices.
gram_opt_e, gram_opt_o = [
prog.substitute_minimizer(gi) for gi in gram
]
self.assertTrue(self._is_psd(gram_opt_e))
self.assertTrue(self._is_psd(gram_opt_o))
basis_e = [v for v in basis if v.is_even()]
basis_o = [v for v in basis if v.is_odd()]
poly_opt_gram = Polynomial.quadratic_form(basis_e, gram_opt_e)
poly_opt_gram += Polynomial.quadratic_form(basis_o, gram_opt_o)
self.assertAlmostEqual(poly_opt, poly_opt_gram, places=4)
def test_add_sos_constraint(self):
# Fit free polynomial in 2 points, and minimize value at a third.
for Vector in Vectors:
# Normal polynomial.
prog = SosProgram()
basis = Vector.construct_basis(self.x, 6)
poly, coef = prog.add_polynomial(basis)
gram = prog.add_sos_constraint(poly)[1]
prog.add_linear_cost(poly(self.zero))
prog.add_linear_constraint(poly(self.one) == 1)
prog.add_linear_constraint(poly(self.two) == 2)
prog.solve()
poly_opt = prog.substitute_minimizer(poly)
self.assertAlmostEqual(prog.minimum(), 0, places=4)
self.assertAlmostEqual(poly_opt(self.zero), 0, places=4)
self.assertAlmostEqual(poly_opt(self.one), 1, places=4)
self.assertAlmostEqual(poly_opt(self.two), 2, places=4)
# Reconstruct polynomial from Gram matrix.
gram_opt = prog.substitute_minimizer(gram)
self.assertTrue(self._is_psd(gram_opt))
basis_half = Vector.construct_basis(self.x, 3)
poly_opt_gram = Polynomial.quadratic_form(basis_half, gram_opt)
self.assertAlmostEqual(poly_opt, poly_opt_gram)
# Even polynomial.
prog = SosProgram()
basis = Vector.construct_basis(self.x, 6, odd=False)
poly, coef = prog.add_polynomial(basis)
gram = prog.add_sos_constraint(poly)[1]
prog.add_linear_cost(poly(self.zero))
prog.add_linear_constraint(poly(self.one) == 1)
prog.add_linear_constraint(poly(self.two) == 2)
prog.solve()
poly_opt = prog.substitute_minimizer(poly)
self.assertAlmostEqual(prog.minimum(), 0, places=4)
self.assertAlmostEqual(poly_opt(self.zero), 0, places=4)
self.assertAlmostEqual(poly_opt(self.one), 1, places=4)
self.assertAlmostEqual(poly_opt(self.two), 2, places=4)
# Reconstruct polynomial from Gram matrices.
gram_opt_e, gram_opt_o = [
prog.substitute_minimizer(gi) for gi in gram
]
self.assertTrue(self._is_psd(gram_opt_e))
self.assertTrue(self._is_psd(gram_opt_o))
basis = Vector.construct_basis(self.x, 3)
basis_e = [v for v in basis if v.is_even()]
basis_o = [v for v in basis if v.is_odd()]
poly_opt_gram = Polynomial.quadratic_form(basis_e, gram_opt_e)
poly_opt_gram += Polynomial.quadratic_form(basis_o, gram_opt_o)
self.assertAlmostEqual(poly_opt, poly_opt_gram, places=4)
# Polynomial of odd degree.
prog = SosProgram()
basis = Vector.construct_basis(self.x, 3)
poly, c = prog.add_polynomial(basis)
with self.assertRaises(ValueError):
prog.add_sos_constraint(poly)
# Polynomial of length 0.
prog = SosProgram()
poly = Polynomial({})
with self.assertRaises(ValueError):
prog.add_sos_constraint(poly)
@staticmethod
def _is_psd(A, tol=1e-7):
return all(np.linalg.eig(A)[0] > -tol)