def sublevel_set():
if t.value > 1:
return s.INFEASIBLE
tsq = t.value**2
return ((1 - tsq**2) * atoms.sum_squares(x) -
atoms.matmul(2 * (a - tsq * b), x) + atoms.sum_squares(a) -
tsq * atoms.sum_squares(b)) <= 0
def quad_form_canon(expr, args):
scale, M1, M2 = decomp_quad(args[1].value)
if M1.size > 0:
expr = sum_squares(Constant(M1.T) * args[0])
if M2.size > 0:
scale = -scale
expr = sum_squares(Constant(M2.T) * args[0])
obj, constr = quad_over_lin_canon(expr, expr.args)
return scale * obj, constr
def quad_form_canon(expr, args):
# TODO this doesn't work with parameters!
scale, M1, M2 = decomp_quad(args[1].value)
if M1.size > 0:
expr = sum_squares(Constant(M1.T) @ args[0])
if M2.size > 0:
scale = -scale
expr = sum_squares(Constant(M2.T) @ args[0])
obj, constr = quad_over_lin_canon(expr, expr.args)
return scale * obj, constr
def smooth_ridge(self, solver):
np.random.seed(1)
n = 200
k = 50
eta = 1
A = np.ones((k, n))
b = np.ones((k))
obj = sum_squares(A*self.xsr - b) + \
eta*sum_squares(self.xsr[:-1]-self.xsr[1:])
p = Problem(Minimize(obj), [])
self.solve_QP(p, solver)
self.assertAlmostEqual(0, p.value, places=4)
def smooth_ridge(self, solver):
numpy.random.seed(1)
n = 500
k = 50
eta = 1
A = numpy.ones((k, n))
b = numpy.ones((k, 1))
obj = sum_squares(A * self.xsr -
b) + eta * sum_squares(self.xsr[:-1] - self.xsr[1:])
p = Problem(Minimize(obj), [])
s = self.solve_QP(p, solver)
self.assertAlmostEqual(0, s.opt_val)
def optimize(self):
num_points = len(self.graph.free_points)
point_dim = self.graph.point_dim
num_landmarks = len(self.graph.landmarks)
landmark_dim = self.graph.landmark_dim
Am, Ap, d, sigma_d = self.graph.observation_system()
Bp, t, sigma_t = self.graph.odometry_system()
S_d, S_t = sp.sparse.diags(
1 / _sanitized_noise_array(sigma_d)), sp.sparse.diags(
1 / _sanitized_noise_array(sigma_t))
if (num_points != 0) and (num_landmarks != 0):
M = cp.Variable((landmark_dim, num_landmarks))
P = cp.Variable((point_dim, num_points))
objective = cp.Minimize(
sum_squares(S_d * ((Am * vec(M)) + (Ap * vec(P)) - d)) +
sum_squares(S_t * ((Bp * vec(P)) - t)))
problem = cp.Problem(objective)
problem.solve(verbose=self._verbosity, solver=self._solver)
self.M = M.value
self.P = P.value
m = self.M.ravel(order='F')
p = self.P.ravel(order='F')
self.res_d = Am.dot(m) + Ap.dot(p) - d
self.res_t = Bp.dot(p) - t
elif (num_points != 0) and (num_landmarks == 0):
P = cp.Variable((point_dim, num_points))
objective = cp.Minimize(
sum_squares(sum_squares(S_t * ((Bp * vec(P)) - t))))
problem = cp.Problem(objective)
problem.solve(verbose=self._verbosity, solver=self._solver)
self.P = P.value
p = self.P.ravel(order='F')
self.res_t = Bp.dot(p) - t
else:
return
def control(self, solver):
# Some constraints on our motion
# The object should start from the origin, and end at rest
initial_velocity = numpy.matrix('-20; 100')
final_position = numpy.matrix('100; 100')
T = 100 # The number of timesteps
h = 0.1 # The time between time intervals
mass = 1 # Mass of object
drag = 0.1 # Drag on object
g = numpy.matrix('0; -9.8') # Gravity on object
# Create a problem instance
constraints = []
# Add constraints on our variables
for i in range(T - 1):
constraints += [
self.position[:, i + 1] == self.position[:, i] +
h * self.velocity[:, i]
]
acceleration = self.force[:,
i] / mass + g - drag * self.velocity[:,
i]
constraints += [
self.velocity[:,
i + 1] == self.velocity[:, i] + h * acceleration
]
# Add position constraints
constraints += [self.position[:, 0] == 0]
constraints += [self.position[:, -1] == final_position]
# Add velocity constraints
constraints += [self.velocity[:, 0] == initial_velocity]
constraints += [self.velocity[:, -1] == 0]
# Solve the problem
p = Problem(Minimize(sum_squares(self.force)), constraints)
s = self.solve_QP(p, solver)
self.assertAlmostEqual(17850.0, s.opt_val)
def control(self, solver):
# Some constraints on our motion
# The object should start from the origin, and end at rest
initial_velocity = np.array([-20, 100])
final_position = np.array([100, 100])
T = 100 # The number of timesteps
h = 0.1 # The time between time intervals
mass = 1 # Mass of object
drag = 0.1 # Drag on object
g = np.array([0, -9.8]) # Gravity on object
# Create a problem instance
constraints = []
# Add constraints on our variables
for i in range(T - 1):
constraints += [
self.position[:, i + 1] == self.position[:, i] +
h * self.velocity[:, i]
]
acceleration = self.force[:, i]/mass + g - \
drag * self.velocity[:, i]
constraints += [
self.velocity[:,
i + 1] == self.velocity[:, i] + h * acceleration
]
# Add position constraints
constraints += [self.position[:, 0] == 0]
constraints += [self.position[:, -1] == final_position]
# Add velocity constraints
constraints += [self.velocity[:, 0] == initial_velocity]
constraints += [self.velocity[:, -1] == 0]
# Solve the problem
p = Problem(Minimize(.01 * sum_squares(self.force)), constraints)
self.solve_QP(p, solver)
self.assertAlmostEqual(178.500, p.value, places=1)
def test_sum_squares(self):
X = Variable(5, 4)
P = np.asmatrix(np.random.randn(3, 5))
Q = np.asmatrix(np.random.randn(4, 7))
M = np.asmatrix(np.random.randn(3, 7))
y = P*X*Q + M
self.assertFalse(y.is_constant())
self.assertTrue(y.is_affine())
self.assertTrue(y.is_quadratic())
self.assertTrue(y.is_dcp())
s = sum_squares(y)
self.assertFalse(s.is_constant())
self.assertFalse(s.is_affine())
self.assertTrue(s.is_quadratic())
self.assertTrue(s.is_dcp())
# Frobenius norm squared is indeed quadratic
# but can't show quadraticity using recursive rules
t = norm(y, 'fro')**2
self.assertFalse(t.is_constant())
self.assertFalse(t.is_affine())
self.assertFalse(t.is_quadratic())
self.assertTrue(t.is_dcp())
def test_sum_squares(self):
X = Variable(5, 4)
P = np.asmatrix(np.random.randn(3, 5))
Q = np.asmatrix(np.random.randn(4, 7))
M = np.asmatrix(np.random.randn(3, 7))
y = P*X*Q + M
self.assertFalse(y.is_constant())
self.assertTrue(y.is_affine())
self.assertTrue(y.is_quadratic())
self.assertTrue(y.is_dcp())
s = sum_squares(y)
self.assertFalse(s.is_constant())
self.assertFalse(s.is_affine())
self.assertTrue(s.is_quadratic())
self.assertTrue(s.is_dcp())
# Frobenius norm squared is indeed quadratic
# but can't show quadraticity using recursive rules
t = norm(y, 'fro')**2
self.assertFalse(t.is_constant())
self.assertFalse(t.is_affine())
self.assertFalse(t.is_quadratic())
self.assertTrue(t.is_dcp())
def optimize(self):
num_points = len(self.graph.free_points)
point_dim = self.graph.point_dim
num_landmarks = len(self.graph.landmarks)
landmark_dim = self.graph.landmark_dim
Am, Ap, d, _ = self.graph.observation_system()
Bp, t, _ = self.graph.odometry_system()
if (num_points != 0) and (num_landmarks != 0):
M = cp.Variable((landmark_dim, num_landmarks))
P = cp.Variable((point_dim, num_points))
objective = cp.Minimize(
sum_squares(Am * vec(M) + Ap * vec(P) - d) +
sum_squares(Bp * vec(P) - t))
problem = cp.Problem(objective)
problem.solve(verbose=self._verbosity, solver=self._solver)
self.M = M.value
self.P = P.value
m = self.M.ravel(order='F')
p = self.P.ravel(order='F')
self.res_d = Am.dot(m) + Ap.dot(p) - d
self.res_t = Bp.dot(p) - t
elif (num_points != 0) and (num_landmarks == 0):
P = cp.Variable((point_dim, num_points))
objective = cp.Minimize(sum_squares(Bp * vec(P) - t))
problem = cp.Problem(objective)
problem.solve(verbose=self._verbosity, solver=self._solver)
self.P = P.value
p = self.P.ravel(order='F')
self.res_t = Bp.dot(p) - t
else:
return
def square_affine(self, solver):
A = np.random.randn(10, 2)
b = np.random.randn(10)
p = Problem(Minimize(sum_squares(A * self.x - b)))
self.solve_QP(p, solver)
for var in p.variables():
self.assertItemsAlmostEqual(lstsq(A, b)[0].flatten(),
var.value,
places=1)
def abs(self, solver):
u = Variable(2)
constr = []
constr += [abs(u[1] - u[0]) <= 100]
prob = Problem(Minimize(sum_squares(u)), constr)
print(("The problem is QP: ", prob.is_qp()))
self.assertEqual(prob.is_qp(), True)
result = prob.solve(solver=solver)
self.assertAlmostEqual(result, 0)
def square_affine(self, solver):
A = numpy.random.randn(10, 2)
b = numpy.random.randn(10, 1)
p = Problem(Minimize(sum_squares(A * self.x - b)))
s = self.solve_QP(p, solver)
for var in p.variables():
self.assertItemsAlmostEqual(lstsq(A, b)[0].flatten(),
s.primal_vars[var.id],
places=1)
def test_qp_maximization_reduction_path_ecos(self):
qp_maximization = Problem(Maximize(-sum_squares(self.x)),
[self.x <= -1])
self.assertTrue(qp_maximization.is_dcp())
path = PathFinder().reduction_path(ProblemType(qp_maximization),
[ECOS])
self.assertEquals(4, len(path))
self.assertEquals(path[1], ConeMatrixStuffing)
self.assertEquals(path[2], Dcp2Cone)
self.assertEquals(path[3], FlipObjective)
def sparse_system(self, solver):
m = 100
n = 80
np.random.seed(1)
density = 0.4
A = sp.rand(m, n, density)
b = np.random.randn(m)
p = Problem(Minimize(sum_squares(A * self.xs - b)), [self.xs == 0])
self.solve_QP(p, solver)
self.assertAlmostEqual(b.T.dot(b), p.value, places=4)
def sparse_system(self, solver):
m = 1000
n = 800
numpy.random.seed(1)
density = 0.2
A = sp.rand(m, n, density)
b = numpy.random.randn(m, 1)
p = Problem(Minimize(sum_squares(A * self.xs - b)), [self.xs == 0])
s = self.solve_QP(p, solver)
self.assertAlmostEqual(b.T.dot(b), s.opt_val)
def _m_step(self, W, Am, Ap, d, sigma_d):
num_points = len(self.graph.free_points)
point_dim = self.graph.point_dim
num_landmarks = len(self.graph.landmarks)
landmark_dim = self.graph.landmark_dim
Bp, t, sigma_t = self.graph.odometry_system()
S_t = sp.sparse.diags(1 / _sanitized_noise_array(sigma_t))
sigma_d = np.tile(_sanitized_noise_array(sigma_d), num_landmarks)
S_d = sp.sparse.diags(1 / sigma_d)
W = sp.sparse.diags(W.flatten('F'))
Am = [
np.zeros((Am.shape[0], Am.shape[1])) for _ in range(num_landmarks)
]
for j in range(num_landmarks):
Am[j][:, j] = 1
Am = sp.sparse.csr_matrix(np.concatenate(Am, axis=0))
Ap = sp.sparse.vstack([Ap for _ in range(num_landmarks)])
d = np.tile(d, num_landmarks)
M = cp.Variable((landmark_dim, num_landmarks))
P = cp.Variable((point_dim, num_points))
objective = cp.Minimize(
sum_squares(W * S_d * ((Am * vec(M)) + (Ap * vec(P)) - d)) +
sum_squares(S_t * ((Bp * vec(P)) - t)))
problem = cp.Problem(objective)
problem.solve(verbose=self._verbosity, solver=self._solver)
self.M = M.value
self.P = P.value
m = self.M.ravel(order='F')
p = self.P.ravel(order='F')
self.res_d = Am.dot(m) + Ap.dot(p) - d
self.res_t = Bp.dot(p) - t
def prox_step(prob, rho_init, scaled=False, spectral=False):
"""Formulates the proximal operator for a given objective, constraints, and step size.
Parikh, Boyd. "Proximal Algorithms."
Parameters
----------
prob : Problem
The objective and constraints associated with the proximal operator.
The sign of the objective function is flipped if `prob` is a maximization problem.
rho_init : float
The initial step size.
scaled : logical, optional
Should the dual variable be scaled?
spectral : logical, optional
Will spectral step sizes be used?
Returns
----------
prox : Problem
The proximal step problem.
vmap : dict
A map of each proximal variable id to a dictionary containing that variable `x`,
the mean variable parameter `xbar`, the associated dual parameter `y`, and the
step size parameter `rho`. If `spectral = True`, the estimated dual parameter
`yhat` is also included.
"""
vmap = {} # Store consensus variables
f = flip_obj(prob).args[0]
# Add penalty for each variable.
for xvar in prob.variables():
xid = xvar.id
shape = xvar.shape
vmap[xid] = {
"x": xvar,
"xbar": Parameter(shape, value=np.zeros(shape)),
"y": Parameter(shape, value=np.zeros(shape)),
"rho": Parameter(value=rho_init[xid], nonneg=True)
}
if spectral:
vmap[xid]["yhat"] = Parameter(shape, value=np.zeros(shape))
dual = vmap[xid]["y"] if scaled else vmap[xid]["y"] / vmap[xid]["rho"]
f += (vmap[xid]["rho"] / 2.0) * sum_squares(xvar - vmap[xid]["xbar"] +
dual)
prox = Problem(Minimize(f), prob.constraints)
return prox, vmap
def regression_2(self, solver):
np.random.seed(1)
# Number of examples to use
n = 100
# Specify the true value of the variable
true_coeffs = np.array([2, -2, 0.5])
# Generate data
x_data = np.random.rand(n) * 5
x_data_expanded = np.vstack([np.power(x_data, i) for i in range(1, 4)])
print((x_data_expanded.shape, true_coeffs.shape))
y_data = x_data_expanded.T.dot(true_coeffs) + 0.5 * np.random.rand(n)
quadratic = self.offset + x_data*self.slope + \
self.quadratic_coeff*np.power(x_data, 2)
residuals = quadratic.T - y_data
fit_error = sum_squares(residuals)
p = Problem(Minimize(fit_error), [])
self.solve_QP(p, solver)
self.assertAlmostEqual(139.225660756, p.value, places=4)
def regression_1(self, solver):
np.random.seed(1)
# Number of examples to use
n = 100
# Specify the true value of the variable
true_coeffs = np.array([[2, -2, 0.5]]).T
# Generate data
x_data = np.random.rand(n) * 5
x_data = np.atleast_2d(x_data)
x_data_expanded = np.vstack([np.power(x_data, i) for i in range(1, 4)])
x_data_expanded = np.atleast_2d(x_data_expanded)
y_data = x_data_expanded.T.dot(true_coeffs) + 0.5 * np.random.rand(
n, 1)
y_data = np.atleast_2d(y_data)
line = self.offset + x_data * self.slope
residuals = line.T - y_data
fit_error = sum_squares(residuals)
p = Problem(Minimize(fit_error), [])
self.solve_QP(p, solver)
self.assertAlmostEqual(1171.60037715, p.value, places=4)
def regression_1(self, solver):
numpy.random.seed(1)
# Number of examples to use
n = 100
# Specify the true value of the variable
true_coeffs = numpy.matrix('2; -2; 0.5')
# Generate data
x_data = numpy.random.rand(n) * 5
x_data = numpy.asmatrix(x_data)
x_data_expanded = numpy.vstack([numpy.power(x_data, i)
for i in range(1, 4)])
x_data_expanded = numpy.asmatrix(x_data_expanded)
y_data = x_data_expanded.T * true_coeffs + 0.5 * numpy.random.rand(n,1)
y_data = numpy.asmatrix(y_data)
line = self.offset + x_data * self.slope
residuals = line.T - y_data
fit_error = sum_squares(residuals)
p = Problem(Minimize(fit_error), [])
s = self.solve_QP(p, solver)
self.assertAlmostEqual(1171.60037715, p.value, places=4)
def regression_2(self, solver):
numpy.random.seed(1)
# Number of examples to use
n = 100
# Specify the true value of the variable
true_coeffs = numpy.matrix('2; -2; 0.5')
# Generate data
x_data = numpy.random.rand(n, 1) * 5
x_data = numpy.asmatrix(x_data)
x_data_expanded = numpy.hstack(
[numpy.power(x_data, i) for i in range(1, 4)])
x_data_expanded = numpy.asmatrix(x_data_expanded)
y_data = x_data_expanded * true_coeffs + 0.5 * numpy.random.rand(n, 1)
y_data = numpy.asmatrix(y_data)
quadratic = self.offset + x_data * self.slope + self.quadratic_coeff * numpy.power(
x_data, 2)
residuals = quadratic - y_data
fit_error = sum_squares(residuals)
p = Problem(Minimize(fit_error), [])
s = self.solve_QP(p, solver)
self.assertAlmostEqual(139.225660756, s.opt_val)
def test_warm_start(self):
"""Test warm start.
"""
m = 200
n = 100
np.random.seed(1)
A = np.random.randn(m, n)
b = Parameter(m)
# Construct the problem.
x = Variable(n)
prob = Problem(Minimize(sum_squares(A * x - b)))
b.value = np.random.randn(m)
result = prob.solve(warm_start=False)
result2 = prob.solve(warm_start=True)
self.assertAlmostEqual(result, result2)
b.value = np.random.randn(m)
result = prob.solve(warm_start=True)
result2 = prob.solve(warm_start=False)
self.assertAlmostEqual(result, result2)
pass
def optimize(self):
self._pre_optimizer.optimize()
self._pre_optimizer.update()
points = self.graph.free_points
landmarks = self.graph.landmarks
num_points = len(points)
point_dim = self.graph.point_dim
num_landmarks = len(landmarks)
landmark_dim = self.graph.landmark_dim
transforms = [
equivalence.SumMass(self.graph.correspondence_map.set_map()),
equivalence.ExpDistance(self._sigma),
equivalence.Facing()
]
E, W = equivalence.equivalence_matrix(landmarks, transforms=transforms)
if E.shape[0] == 0:
self.M = self._pre_optimizer.M
self.P = self._pre_optimizer.P
self.res_d = self._pre_optimizer.res_d
self.res_t = self._pre_optimizer.res_t
self.equivalence_pairs = []
return
Am, Ap, d, sigma_d = self.graph.observation_system()
Bp, t, sigma_t = self.graph.odometry_system()
S_d, S_t = sp.sparse.diags(
1 / _sanitized_noise_array(sigma_d)), sp.sparse.diags(
1 / _sanitized_noise_array(sigma_t))
M = cp.Variable((landmark_dim, num_landmarks))
P = cp.Variable((point_dim, num_points))
M.value = self._pre_optimizer.M
P.value = self._pre_optimizer.P
objective = cp.Minimize(mixed_norm(W * E * M.T))
constraints = [
norm((Am * vec(M)) +
(Ap * vec(P)) - d) <= 2 * np.linalg.norm(sigma_d + 1e-6),
norm((Bp * vec(P)) - t) <= 2 * np.linalg.norm(sigma_t + 1e-6)
]
problem = cp.Problem(objective, constraints)
problem.solve(verbose=self._verbosity,
solver=self._solver,
warm_start=True)
if problem.solution.status == 'infeasible':
self.M = self._pre_optimizer.M
self.P = self._pre_optimizer.P
self.res_d = self._pre_optimizer.res_d
self.res_t = self._pre_optimizer.res_t
self.equivalence_pairs = []
return
E_ = E[np.abs(np.linalg.norm(E * M.value.T, axis=1)) < 0.001, :]
objective = cp.Minimize(
sum_squares(S_d * ((Am * vec(M)) + (Ap * vec(P)) - d)) +
sum_squares(S_t * ((Bp * vec(P)) - t)))
constraints = [E_ * M.T == 0] if E_.shape[0] > 0 else []
problem = cp.Problem(objective, constraints)
problem.solve(verbose=self._verbosity,
solver=self._solver,
warm_start=True)
self.M = M.value
self.P = P.value
m = self.M.ravel(order='F')
p = self.P.ravel(order='F')
self.res_d = Am.dot(m) + Ap.dot(p) - d
self.res_t = Bp.dot(p) - t
self.equivalence_pairs = [(landmarks[i], landmarks[j])
for (i, j) in E_.tolil().rows]
def reconstruct(self, measurement_results):
from cvxpy import Variable, atoms, abs, reshape, Minimize, Problem, CVXOPT
from traceback import print_exc
reconstruction_operator_names = []
reconstruction_operators = []
basis_axes_names = self.reconstruction_basis.keys()
basis_vector_norms = np.asarray([
np.linalg.norm(self.reconstruction_basis[r]['operator'])
for r in basis_axes_names
])
for reconstruction_operator_name, reconstruction_operator in self.reconstruction_basis.items(
):
reconstruction_operator_names.append(reconstruction_operator_name)
reconstruction_operators.append(
reconstruction_operator['operator'])
reconstruction_matrix = []
for rot, projection in self.proj_seq.items():
for measurement_name, projection_operator in projection[
'operators'].items():
reconstruction_matrix.append([
np.sum(projection_operator *
np.conj(reconstruction_operator)) /
np.sum(np.abs(reconstruction_operator)**2)
for reconstruction_operator in reconstruction_operators
])
reconstruction_matrix_pinv = np.linalg.pinv(reconstruction_matrix)
reconstruction_matrix = np.asarray(reconstruction_matrix)
self.reconstruction_matrix = reconstruction_matrix
self.reconstruction_matrix_pinv = reconstruction_matrix_pinv
projections = np.dot(reconstruction_matrix_pinv, measurement_results)
reconstruction = {
str(k): v
for k, v in zip(basis_axes_names, projections)
}
if self.reconstruction_type == 'cvxopt':
#x = cvxpy.Variable(len(projections), complex=True)
x = Variable(len(projections), complex=True)
rmat_normalized = np.asarray(reconstruction_matrix /
np.mean(np.abs(measurement_results)),
dtype=complex)
meas_normalized = np.asarray(measurement_results).ravel(
) / np.mean(np.abs(measurement_results))
#lstsq_objective = cvxpy.atoms.sum_squares(cvxpy.abs(rmat_normalized @ x - meas_normalized))
lstsq_objective = atoms.sum_squares(
abs(rmat_normalized @ x - meas_normalized))
matrix_size = int(np.round(np.sqrt(len(projections))))
#x_reshaped = cvxpy.reshape(x, (matrix_size, matrix_size))
x_reshaped = reshape(x, (matrix_size, matrix_size))
psd_constraint = x_reshaped >> 0
hermitian_constraint = x_reshaped.H == x_reshaped
# Create two constraints.
constraints = [psd_constraint, hermitian_constraint]
# Form objective.
#obj = cvxpy.Minimize(lstsq_objective)
obj = Minimize(lstsq_objective)
# Form and solve problem.
prob = cvxpy.Problem(obj, constraints)
try:
prob.solve(solver=cvxpy.CVXOPT, verbose=True)
reconstruction = {
str(k): v
for k, v in zip(basis_axes_names, np.asarray(x.value))
}
except ValueError as e:
print_exc()
if self.reconstruction_output_mode == 'array':
it = np.nditer([self.reconstruction_output_array, None],
flags=['refs_ok'],
op_dtypes=(object, complex))
with it:
for x, z in it:
z[...] = reconstruction[str(x)]
reconstruction = it.operands[1]
return reconstruction