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_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 add_sos_polynomial(self, basis, name='Q'):
gram, cons = self.add_psd_variable(len(basis), name)
poly = Polynomial.quadratic_form(basis, gram)
return poly, gram, cons
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)