def sub_graph_finder(A_tau, A0):
m, n = A0.shape
A = cvx.Symmetric(n)
#objective function
objective = cvx.Maximize(cvx.trace(A * A0))
# A_i,j = 0 if A0_i,j = 0 (and i != j)
edge_constraints = EdgeLocationConstraint(n, A0, A, 0)
# γ =eigenvalue corresponding to the largest eigenspace of the corresponding graph (eigenvalue with highest multiplicity)
gamma = most_common_eigenvalue(A_tau)
clique_size, same_clique_size = A_tau.shape
#TODO SCHUR-HORN CONSTRAINT
schur_horn_constraints = CliqueSchurHornOrbitopeConstraint(
transform_matrix(n, A_tau, gamma), A,
clique_size) # currently will only work for finding cliques
constraints = edge_constraints.constraint_list + schur_horn_constraints.constraint_list
problem = cvx.Problem(objective, constraints)
problem.solve(kktsolver='robust')
if problem.status == 'optimal':
return problem.status, problem.value, A.value
else:
return problem.status, np.nan, np.nan
def _restrict(self, matrix):
w, V = np.linalg.eigh(matrix)
w_sorted_idxs = np.argsort(-w)
pos_w = w[w_sorted_idxs[:self.k]]
pos_V = V[:, w_sorted_idxs[:self.k]]
Sigma = cvx.Symmetric(self.k, self.k)
return [self == pos_V * Sigma * pos_V.T]
def ntf_fir_from_digested(Qs, A, C, H_inf, **opts):
"""
Synthesize FIR NTF from predigested specification
Version for the cvxpy modeler.
"""
verbose = opts['show_progress']
if opts['cvxpy_opts']['solver'] == 'cvxopt':
opts['cvxpy_opts']['solver'] = cvxpy.CVXOPT
elif opts['cvxpy_opts']['solver'] == 'scs':
opts['cvxpy_opts']['solver'] = cvxpy.SCS
order = np.size(Qs, 0) - 1
br = cvxpy.Variable(order, 1, name='br')
b = cvxpy.vstack(1, br)
X = cvxpy.Symmetric(order, name='X')
target = cvxpy.Minimize(cvxpy.norm2(Qs * b))
B = np.vstack((np.zeros((order - 1, 1)), 1.))
C = C + br[::-1].T
D = np.matrix(1.)
M1 = A.T * X
M2 = M1 * B
M = cvxpy.bmat([[M1 * A - X, M2, C.T], [M2.T, B.T * X * B - H_inf**2, D],
[C, D, np.matrix(-1.)]])
constraints = [M << 0, X >> 0]
p = cvxpy.Problem(target, constraints)
p.solve(verbose=verbose, **opts['cvxpy_opts'])
return np.hstack((1, np.asarray(br.value.T)[0]))
def deconvolve(self, A, epsilon_vector):
'''
Recover the precise labelling of A1 and A2, A = A1+A2.
Args:
A (matrix) : Composition of A1 and A2
Returns:
problem.status (str) : Optimal/
problem_correct (bool) : Correctness of solution
problem.value (float) : Value of objective function (Euclidean norm of (A-A1*-A2*))
A1* (matrix) : Precise labelling of A1
A2* (matrix) : Precise labelling of A2
'''
# Realisations of the precise labelling of A1 and A2
A1_labelled = cvx.Symmetric(self.n)
A2_labelled = cvx.Symmetric(self.n)
# objective function
objective = cvx.Minimize(cvx.pnorm((A - A1_labelled - A2_labelled), 2))
# Convex hull constraints
A1_hull = SpectralHullConstraint(self.A1, A1_labelled, epsilon_vector)
A2_hull = SpectralHullConstraint(self.A2, A2_labelled, epsilon_vector)
# all values between 0 and 1
A1_limits = NodeLimitConstraint(A1_labelled,
lower_limit=0,
upper_limit=1)
A2_limits = NodeLimitConstraint(A2_labelled,
lower_limit=0,
upper_limit=1)
constraints = A1_hull.constraint_list + A2_hull.constraint_list + A1_limits.constraint_list + A2_limits.constraint_list
problem = cvx.Problem(objective, constraints)
problem.solve(solver=cvx.MOSEK)
if problem.status == 'optimal':
return problem.status, problem.value, A1_labelled.value, A2_labelled.value
else:
return problem.status, np.nan, np.nan, np.nan
def denoise(self, A_noisy, epsilon_vector):
A_recovered= cvx.Symmetric(self.n)
spectral_hull = SpectralHullConstraint(self.A, A_recovered, epsilon_vector)
objective = cvx.Minimize(spectral_hull.constraint_list)
node_limits = NodeLimitConstraint(A_recovered,lower_limit=0, upper_limit=1)
constraints = spectral_hull.constraint_list + node_limits.constraint_list
problem = cvx.Problem(objective,constraints)
problem.solve(solver=cvx.MOSEK)
if problem.status=='optimal':
return problem.status,problem.value,A_recovered.value
else:
return problem.status,np.nan,np.nan
def ntf_fir_from_digested(order, osrs, H_inf, f0s, zf, **opts):
"""
Synthesize FIR NTF with minmax approach from predigested specification
Version for the cvxpy_tinoco modeler.
"""
verbose = opts['show_progress']
if opts['cvxpy_opts']['solver'] == 'cvxopt':
opts['cvxpy_opts']['solver'] = cvxpy.CVXOPT
elif opts['cvxpy_opts']['solver'] == 'scs':
opts['cvxpy_opts']['solver'] = cvxpy.SCS
# State space representation of NTF
A = np.matrix(np.eye(order, order, 1))
B = np.matrix(np.vstack((np.zeros((order-1, 1)), 1.)))
# C contains the NTF coefficients
D = np.matrix(1)
# Set up the problem
bands = len(f0s)
c = cvxpy.Variable(1, order)
F = []
gg = cvxpy.Variable(bands, 1)
for idx in range(bands):
f0 = f0s[idx]
osr = osrs[idx]
omega0 = 2*f0*np.pi
Omega = 1./osr*np.pi
P = cvxpy.Symmetric(order)
Q = cvxpy.Semidef(order)
if f0 == 0:
# Lowpass modulator
M1 = A.T*P*A+Q*A+A.T*Q-P-2*Q*np.cos(Omega)
M2 = A.T*P*B + Q*B
M3 = B.T*P*B - gg[idx, 0]
M = cvxpy.bmat([[M1, M2, c.T],
[M2.T, M3, D],
[c, D, -1]])
F += [M << 0]
if zf:
# Force a zero at DC
F += [cvxpy.sum_entries(c) == -1]
else:
# Bandpass modulator
M1r = (A.T*P*A + Q*A*np.cos(omega0) + A.T*Q*np.cos(omega0) -
P - 2*Q*np.cos(Omega))
M2r = A.T*P*B + Q*B*np.cos(omega0)
M3r = B.T*P*B - gg[idx, 0]
M1i = A.T*Q*np.sin(omega0) - Q*A*np.sin(omega0)
M21i = -Q*B*np.sin(omega0)
M22i = B.T*Q*np.sin(omega0)
Mr = cvxpy.bmat([[M1r, M2r, c.T],
[M2r.T, M3r, D],
[c, D, -1]])
Mi = cvxpy.bmat([[M1i, M21i, np.zeros((order, 1))],
[M22i, 0, 0],
[np.zeros((1, order)), 0, 0]])
M = cvxpy.bmat([[Mr, Mi],
[-Mi, Mr]])
F += [M << 0]
if zf:
# Force a zero at z=np.exp(1j*omega0)
nn = np.arange(order).reshape((order, 1))
vr = np.matrix(np.cos(omega0*nn))
vi = np.matrix(np.sin(omega0*nn))
vn = np.matrix(
[-np.cos(omega0*order), -np.sin(omega0*order)])
F += [c*cvxpy.hstack(vr, vi) == vn]
if H_inf < np.inf:
# Enforce the Lee constraint
R = cvxpy.Semidef(order)
MM = cvxpy.bmat([[A.T*R*A-R, A.T*R*B, c.T],
[B.T*R*A, -H_inf**2+B.T*R*B, D],
[c, D, -1]])
F += [MM << 0]
target = cvxpy.Minimize(cvxpy.max_entries(gg))
p = cvxpy.Problem(target, F)
p.solve(verbose=verbose, **opts['cvxpy_opts'])
return np.hstack((1, np.asarray(c.value)[0, ::-1]))
def latent_variable_gmm_cvx(X_o,
alpha=1,
lambda_s=1,
S_init=None,
verbose=False):
'''
A cvx implementation of the Latent Variable Gaussian Graphical Model
see review of "Venkat Chandrasekaran, Pablo A Parrilo, and Alan S Willsky. Latent variable graphical model selection via convex optimization. The Annals of Statistics, 40(4):1935–1967, 2012."
min_{S, L} -log det (S-L) + trace(emp_Cov*(S-L)) + alpha*lambda_s*\|S\|_{1} + alpha*\|L\|_{*}
s.t. S-L \succeq 0
L \succeq 0
return S, L
'''
n, m = X_o.shape
emp_cov = np.cov(X_o)
if alpha == 0:
if return_costs:
precision = np.linalg.pinv(emp_cov)
cost = -2. * log_likelihood(emp_cov, precision)
cost += n_features * np.log(2 * np.pi)
d_gap = np.sum(emp_cov * precision) - n
return emp_cov, precision, (cost, d_gap)
else:
return emp_cov, np.linalg.pinv(emp_cov)
costs = list()
if S_init is None:
covariance_o = emp_cov.copy()
else:
covariance_o = S_init.copy()
# As a trivial regularization (Tikhonov like), we scale down the
# off-diagonal coefficients of our starting point: This is needed, as
# in the cross-validation the cov_init can easily be
# ill-conditioned, and the CV loop blows. Beside, this takes
# conservative stand-point on the initial conditions, and it tends to
# make the convergence go faster.
covariance_o *= 0.95
diagonal_o = emp_cov.flat[::n + 1]
covariance_o.flat[::n + 1] = diagonal_o
# define the low-rank term L and sparse term S
L = cvx.Semidef(n)
S = cvx.Symmetric(n)
# define the SDP problem
objective = cvx.Minimize(-cvx.log_det(S - L) + cvx.trace(covariance_o *
(S - L)) +
alpha * lambda_s * cvx.norm(S, 1) +
alpha * cvx.norm(L, "nuc"))
constraints = [S - L >> 0]
# solve the problem
problem = cvx.Problem(objective, constraints)
problem.solve(verbose=verbose)
return (S.value, L.value)
# sparse well-connected graphs on 16 nodes
# all values between 0 and 1
limit_constraints = NodeLimitConstraint(M, lower_limit=0, upper_limit=1)
# diag(M) == 0
diagonal_constraints = DiagonalConstraint(n, M, 0)
# (A*ones)_i <= 2.5
degree_constraints = MaxWeightedDegreeConstraint(n, M, 2.5)
# 2nd smalled eigenvalue of the laplacian >= 1.1
laplacian_constraints = LaplacianLambdaSecondMinConstraint(M, 1.1)
return limit_constraints.constraint_list + diagonal_constraints.constraint_list + degree_constraints.constraint_list + laplacian_constraints.constraint_list
if __name__ == '__main__':
# a 16 node path (cycle with vertex removed
n = 16
A = ((0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8),
(8, 9), (9, 10), (10, 11), (11, 12), (12, 13), (13, 14), (14, 15))
A_matrix = Graph.create_adjacency_matrix(n, A)
M = cvx.Symmetric(n)
family_1_constraints = generate_cycle_family_constraints(n, M)
family_2_constraints = generate_sparse_well_connected_constraints(n, M)
test_families(A_matrix, M, family_1_constraints, family_2_constraints)
import numpy as np
import cvxpy as cp
import scipy as sp
import scipy.sparse as sps
import scipy.linalg as la
np.random.seed(1)
n = 100
A = np.diag(-np.logspace(-0.5, 1, n))
U = la.orth(np.random.randn(n, n))
A = U.T.dot(A.dot(U))
P = cp.Symmetric(n, n)
f = cp.trace(P)
C = [A.T * P + P * A << np.eye(n), P >> np.eye(n)]
prob = cp.Problem(cp.Minimize(f), C)
problemDict = {"problemID": "trace", "problem": prob, "opt_val": None}
problems = [problemDict]
# For debugging individual problems:
if __name__ == "__main__":
def printResults(problemID="", problem=None, opt_val=None):
print(problemID)
problem.solve()
def sls_common_lyapunov(A, B, Q, R, eps_A, eps_B, tau, logger=None):
"""
Solves the common Lyapunov relaxation to the robust
synthesis problem.
Taken from
lstd-lqr/blob/master/code/policy_iteration.ipynb
learning-lqr/experiments/matlab/sls_synth_yalmip/common_lyap_synth_var2_alpha.m
"""
if logger is None:
logger = logging.getLogger(__name__)
d, p = B.shape
X = cvx.Symmetric(d) # inverse Lyapunov function
Z = cvx.Variable(p, d) # -K*X
W_11 = cvx.Symmetric(d)
W_12 = cvx.Variable(d, p)
W_22 = cvx.Symmetric(p)
alph = cvx.Variable() # scalar for tuning the H_inf constraint
constraints = []
# H2 cost: trace(W)=H2 cost
mat1 = cvx.bmat([[X, X, Z.T], [X, W_11, W_12], [Z, W_12.T, W_22]])
constraints.append(mat1 == cvx.Semidef(2 * d + p))
# H_infinity constraint
mat2 = cvx.bmat(
[[X - np.eye(d), (A * X + B * Z),
np.zeros((d, d)),
np.zeros((d, p))], [(X * A.T + Z.T * B.T), X, eps_A * X,
eps_B * Z.T],
[
np.zeros((d, d)), eps_A * X, alph * (tau**2) * np.eye(d),
np.zeros((d, p))
],
[
np.zeros((p, d)), eps_B * Z,
np.zeros((p, d)), (1 - alph) * (tau**2) * np.eye(p)
]])
constraints.append(mat2 == cvx.Semidef(3 * d + p))
# constrain alpha to be in [0,1]:
constraints.append(alph >= 0)
constraints.append(alph <= 1)
# Solve!
objective = cvx.Minimize(cvx.trace(Q * W_11) + cvx.trace(R * W_22))
prob = cvx.Problem(objective, constraints)
try:
obj = prob.solve(solver=cvx.MOSEK)
except cvx.SolverError:
logger.warn("SolverError encountered")
return (False, None, None, None)
if prob.status == cvx.OPTIMAL:
logging.debug("common_lyapunov: found optimal solution")
X_value = np.array(X.value)
P_value = scipy.linalg.solve(X_value, np.eye(d), sym_pos=True)
# NOTE: the K returned here is meant to be used
# as A + BK **NOT** A - BK
K_value = np.array(Z.value).dot(P_value)
return (True, obj, P_value, K_value)
else:
logging.debug("common_lyapunov: could not solve (status={})".format(
prob.status))
return (False, None, None, None)
def __init__(self, n):
self.n = n
self.A = cvx.Symmetric(n)
np.random.seed(1) # move out of class?
self.M = np.random.normal(size=(n, n))
