def test_cvx_sdp():
"""
Test that the CVX SDP optimization works.
"""
x1, x2, x3 = cp.Variable(), cp.Variable(), cp.Variable(),
A1 = np.matrix([[-7, -11], [-11, 3]])
A2 = np.matrix([[7, -18], [-18, 8]])
A3 = np.matrix([[-2, -8], [-8, 1]])
A0 = np.matrix([[33, -9], [-9, 26]])
B1 = np.matrix([
[-21, -11, 0],
[-11, 10, 8],
[0, 8, 5],
])
B2 = np.matrix([
[0, 10, 16],
[10, -10, -10],
[16, -10, 3],
])
B3 = np.matrix([
[-5, 2, -17],
[2, -6, -7],
[-17, 8, 6],
])
B0 = np.matrix([
[14, 9, 40],
[9, 91, 10],
[40, 10, 15],
])
prob = cp.Problem(cp.Minimize(x1 - x2 + x3), [
A0 - x1 * A1 - x2 * A2 - x3 * A3 == cp.semidefinite(2),
B0 - x1 * B1 - x2 * B2 - x3 * B3 == cp.semidefinite(3),
])
prob.solve()
print prob.status
x = np.matrix([
x1.value,
x2.value,
x3.value,
]).T
x_ = np.matrix([
[-3.68e-01],
[1.90e+00],
[-8.88e-01],
])
print x.T, x_.T
assert np.allclose(x, x_, atol=1e-2)
def test_cvx_sdp():
"""
Test that the CVX SDP optimization works.
"""
x1, x2, x3 = cp.Variable(), cp.Variable(), cp.Variable(),
A1 = np.matrix([[-7, -11],
[-11, 3]])
A2 = np.matrix([[7, -18],
[-18, 8]])
A3 = np.matrix([[-2, -8],
[-8, 1]])
A0 = np.matrix([[33, -9],
[-9, 26]])
B1 = np.matrix([[-21, -11, 0],
[-11, 10, 8],
[0, 8, 5], ])
B2 = np.matrix([[ 0, 10, 16],
[10, -10, -10],
[16, -10, 3], ])
B3 = np.matrix([[-5, 2, -17],
[2, -6, -7],
[-17, 8, 6], ])
B0 = np.matrix([[14, 9, 40],
[9, 91, 10],
[40, 10, 15], ])
prob = cp.Problem(
cp.Minimize( x1 - x2 + x3 ), [
A0 - x1 * A1 - x2 * A2 - x3 * A3 == cp.semidefinite(2),
B0 - x1 * B1 - x2 * B2 - x3 * B3 == cp.semidefinite(3),
])
prob.solve()
print prob.status
x = np.matrix([
x1.value,
x2.value,
x3.value,
]).T
x_ = np.matrix([
[-3.68e-01],
[ 1.90e+00],
[-8.88e-01],
])
print x.T, x_.T
assert np.allclose(x, x_, atol=1e-2)
def test_key_error(self):
"""Test examples that caused key error.
"""
import cvxpy as cvx
x = cvx.Variable()
u = -cvx.exp(x)
prob = cvx.Problem(cvx.Maximize(u), [x == 1])
prob.solve(verbose=True, solver=cvx.CVXOPT)
prob.solve(verbose=True, solver=cvx.CVXOPT)
###########################################
import numpy as np
import cvxopt
import cvxpy as cp
kD=2
Sk=cp.semidefinite(kD)
Rsk=cp.Parameter(kD,kD)
mk=cp.Variable(kD,1)
musk=cp.Parameter(kD,1)
logpart=-0.5*cp.log_det(Sk)+0.5*cp.matrix_frac(mk,Sk)+(kD/2.)*np.log(2*np.pi)
linpart=mk.T*musk-0.5*cp.trace(Sk*Rsk)
obj=logpart-linpart
prob=cp.Problem(cp.Minimize(obj))
musk.value=np.ones((2,1))
covsk=np.diag([0.3,0.5])
Rsk.value=covsk+(musk.value*musk.value.T)
prob.solve(verbose=True,solver=cp.CVXOPT)
print "second solve"
prob.solve(verbose=False, solver=cp.CVXOPT)
def omega_solve(self, snk, alpha):
# Create a variable that is constrained to the positive semidefinite cone.
n = len(snk[0])
S = cvx.semidefinite(n)
# Form the logdet(S) - tr(SY) objective. Note the use of a set
# comprehension to form a set of the diagonal elements of S*Y, and the
# native sum function, which is compatible with cvxpy, to compute the trace.
# TODO: If a cvxpy trace operator becomes available, use it!
obj = cvx.Minimize(-cvx.log_det(S) + sum([(S * np.array(snk))[i, i]
for i in range(n)]))
# Set constraint.
constraints = [cvx.sum_entries(cvx.abs(S)) <= alpha]
# Form and solve optimization problem
prob = cvx.Problem(obj, constraints)
prob.solve()
if prob.status != cvx.OPTIMAL:
raise Exception('CVXPY Error')
# If the covariance matrix R is desired, here is how it to create it.
# Threshold S element values to enforce exact zeros:
S = S.value
S[abs(S) <= 1e-4] = 0
return np.array(S).tolist()
def test_cvx_polynomial():
"""
Test that the SDP relaxation for polynomial optimization works on simple examples.
In particular, consider min (x-2)^2.
We get this to be x^2 - 2x + 1, which is
min [1 -2 1]' y,
subj to yy' >= 0
"""
a = 4.0
A = cp.Variable(2,2)
# No equality constraint
prob = cp.Problem(
cp.Minimize( a**2 * A[0,0] - 2 * a * A[0,1] + A[1,1] ), [
A == cp.semidefinite(2),
A[0,0] == 1.
])
prob.solve()
print prob.status
print A.value
assert False
def cvx_solve(self, obj, verbose=False):
"""Solve an objective function with cvx.
obj: vector representing the linear objective to be minimized
verbose: whether cvx should print verbose output to stdout
Returns a pair representing the optimum of the psd and nsd
components (None if that component is either infeasible or
unbounded along the optimization direction). Also sets
psd_spec, nsd_spec, and fully_(un)bounded_directions.
"""
import cvxpy as cvx
x = cvx.Variable(name="x")
y = cvx.Variable(name="y")
z = cvx.Variable(name="z")
# dummy variable to code semidefinite constraint
T = cvx.semidefinite(5, name="T")
spec = self.A * x + self.B * y + self.C * z + self.D
objective = cvx.Minimize(obj[0] * x + obj[1] * y + obj[2] * z)
out_psd = out_nsd = None
# check PSD component
if self.psd_spec:
psd = cvx.Problem(objective, [T == spec])
psd.solve(verbose=verbose)
if psd.status == cvx.OPTIMAL:
out_psd = [x.value, y.value, z.value]
elif psd.status == cvx.INFEASIBLE:
self.psd_spec = False
# check NSD component
if self.nsd_spec:
nsd = cvx.Problem(objective, [T == -spec])
nsd.solve(verbose=verbose)
if nsd.status == cvx.OPTIMAL:
out_nsd = [x.value, y.value, z.value]
if self.psd_spec and psd.status == cvx.OPTIMAL:
self.fully_bounded_directions += 1
elif nsd.status == cvx.UNBOUNDED and self.psd_spec and psd.status == cvx.UNBOUNDED:
self.fully_unbounded_directions += 1
elif nsd.status == cvx.INFEASIBLE:
self.nsd_spec = False
return out_psd, out_nsd
def __setup(self):
M = self.__M
Rreal = cp.semidefinite(2*M, 'Rreal')
# Rreal = [B1 B3
# B2 B4 ]
# = [ReR -ImR
# ImR ReR]
B1 = Rreal[0:M, 0:M]
B2 = Rreal[M:(2*M), 0:M]
B3 = Rreal[0:M, M:(2*M)]
B4 = Rreal[M:(2*M), M:(2*M)]
rho = cp.Variable(1+M**2, 1, 'rho')
constraints = [] # the list of the constraint equations and inequations
constraints.append(0 <= rho[0])
k = 1
for q in range(M):
for p in range(M):
if p == q:
constraints.append(rho[k] == B1[p, q])
k += 1
elif p > q:
constraints.append(rho[k] == B1[p, q])
k += 1
constraints.append(rho[k] == B2[p, q])
k += 1
constraints.append(B1 == B4)
constraints.append(B1 == B1.T)
constraints.append(B3 == -B2)
constraints.append(B3 == -B3.T)
constraints.extend([B1[k,k]==1 for k in range(M)])
objective = cp.Minimize(cp.quad_form(rho, self.__Gamma))
problem = cp.Problem(objective, constraints)
self.__problem = problem
self.__ReR = B1
self.__ImR = B2
def __setup(self):
M = self.__M
Rreal = cp.semidefinite(2 * M, 'Rreal')
# Rreal = [B1 B3
# B2 B4 ]
# = [ReR -ImR
# ImR ReR]
B1 = Rreal[0:M, 0:M]
B2 = Rreal[M:(2 * M), 0:M]
B3 = Rreal[0:M, M:(2 * M)]
B4 = Rreal[M:(2 * M), M:(2 * M)]
rho = cp.Variable(1 + M**2, 1, 'rho')
constraints = [
] # the list of the constraint equations and inequations
constraints.append(0 <= rho[0])
k = 1
for q in range(M):
for p in range(M):
if p == q:
constraints.append(rho[k] == B1[p, q])
k += 1
elif p > q:
constraints.append(rho[k] == B1[p, q])
k += 1
constraints.append(rho[k] == B2[p, q])
k += 1
constraints.append(B1 == B4)
constraints.append(B1 == B1.T)
constraints.append(B3 == -B2)
constraints.append(B3 == -B3.T)
constraints.extend([B1[k, k] == 1 for k in range(M)])
objective = cp.Minimize(cp.quad_form(rho, self.__Gamma))
problem = cp.Problem(objective, constraints)
self.__problem = problem
self.__ReR = B1
self.__ImR = B2
def graphlasso(X, tauk, muk, lambd): # tauk nxk matrix, muk: pxk
Ss = []
for k in range(len(muk)):
Xc = X - muk[k]
cluster_cov = np.array([
np.array(np.mat(vect).T * np.mat(vect)) for vect in Xc
]) #check perf with np.dot(X.T, X) and take diag
Snk = np.dot(cluster_cov.T, tauk[:, k])
S = cvx.semidefinite(len(muk[0]))
obj = cvx.Maximize(
cvx.log_det(S) - sum([(S * Snk)[i, i] for i in range(len(muk))]))
constraints = [cvx.sum_entries(cvx.abs(S)) <= lambd[k]]
# Form and solve optimization problem
prob = cvx.Problem(obj, constraints)
prob.solve()
if prob.status != cvx.OPTIMAL:
print "error"
raise Exception('CVXPY Error')
S = S.value
# S[abs(S) <= 1e-4] = 0 #force exact zeros
Ss += [S]
return Ss
def test_cvx_polynomial():
"""
Test that the SDP relaxation for polynomial optimization works on simple examples.
In particular, consider min (x-2)^2.
We get this to be x^2 - 2x + 1, which is
min [1 -2 1]' y,
subj to yy' >= 0
"""
a = 4.0
A = cp.Variable(2, 2)
# No equality constraint
prob = cp.Problem(cp.Minimize(a**2 * A[0, 0] - 2 * a * A[0, 1] + A[1, 1]),
[A == cp.semidefinite(2), A[0, 0] == 1.])
prob.solve()
print prob.status
print A.value
assert False
# -*- coding: utf-8 -*- """ Created on Mon Jul 14 13:45:05 2014 @author: cyrusl """ import cvxpy as cvx S = cvx.semidefinite(4) x = cvx.vstack(0.1, 0.2, -0.05, 0.1) obj = cvx.Maximize(x.T*S*x) constraints = [S[0,0] == 0.2] constraints += [S[1,1] == 0.1] constraints += [ S[2,2] == 0.3] constraints += [ S[3,3] == 0.1 ] constraints += [ S[0,1] >= 0 ] constraints += [ S[0,2] >= 0 ] constraints += [ S[1,2] <= 0 ] constraints += [ S[1,3] <= 0 ] constraints += [ S[2,3] >= 0 ] #constraints += [ S[0,3] == 0 ] prob = cvx.Problem(obj, constraints) prob.solve() print "status:", prob.status print "optimal value:", sqrt(prob.value) print "optimal covariance matrix:", S.value
# # create data P
# P = cvxopt.matrix(np.matrix('4 1 3; 1 3.5 0.8; 3 0.8 1'))
# Z = cvx.Variable(3,3)
# objective = cvx.Minimize( sum(cvx.square(P - Z)) )
# constr = [cvx.constraints.semi_definite.SDP(P)]
# prob = cvx.Problem(objective, constr)
# prob.solve()
import cvxpy as cp
import numpy as np
import cvxopt
# create data P
P = cp.Parameter(3,3)
Z = cp.semidefinite(3)
objective = cp.Minimize( cp.lambda_max(P) - cp.lambda_min(P - Z) )
prob = cp.Problem(objective, [Z >= 0])
P.value = cvxopt.matrix(np.matrix('4 1 3; 1 3.5 0.8; 3 0.8 1'))
prob.solve()
print "optimal value =", prob.value
# [ 4, 1+2*j, 3-j ; ...
# 1-2*j, 3.5, 0.8+2.3*j ; ...
# 3+j, 0.8-2.3*j, 4 ];
#
# % Construct and solve the model
# n = size( P, 1 );
# cvx_begin sdp
def TVGL(data,
lengthOfSlice,
lamb,
beta,
indexOfPenalty,
verbose=False,
eps=3e-3,
epsAbs=1e-3,
epsRel=1e-3):
if indexOfPenalty == 1:
print('Use l-1 penalty function')
penalty = inferGraphL1
elif indexOfPenalty == 2:
print('Use l-2 penalty function')
penalty = inferGraphL2
elif indexOfPenalty == 3:
print('Use laplacian penalty function')
penalty = inferGraphLaplacian
elif indexOfPenalty == 4:
print('Use l-inf penalty function')
penalty = inferGraphLinf
else:
print('Use perturbation node penalty function')
penalty = inferGraphPN
numberOfTotalSamples = data.shape[0]
timestamps = int(numberOfTotalSamples / lengthOfSlice)
size = data.shape[1]
# Generate empirical covariance matrices
sampleSet = [] # list of array
k = 0
empCovSet = [] # list of array
for i in range(timestamps):
# Generate the slice of samples for each timestamp from data
k_next = min(k + lengthOfSlice, numberOfTotalSamples)
samples = data[k:k_next, :]
k = k_next
sampleSet.append(samples)
empCov = GenEmpCov(sampleSet[i].T)
empCovSet.append(empCov)
# delete: for checking
print((sampleSet.__len__())) #
# print empCovSet
print(('lambda = %s, beta = %s' % (lamb, beta)))
# Define a graph representation to solve
gvx = penalty.TGraphVX()
for i in range(timestamps):
n_id = i
S = cvxpy.semidefinite(size, name='S')
obj = -cvxpy.log_det(S) + cvxpy.trace(
empCovSet[i] * S) #+ alpha*norm(S,1)
gvx.AddNode(n_id, obj)
if (i > 0): #Add edge to previous timestamp
prev_Nid = n_id - 1
currVar = gvx.GetNodeVariables(n_id)
prevVar = gvx.GetNodeVariables(prev_Nid)
edge_obj = beta * cvxpy.norm(currVar['S'] - prevVar['S'],
indexOfPenalty)
gvx.AddEdge(n_id, prev_Nid, Objective=edge_obj)
#Add rake nodes, edges
gvx.AddNode(n_id + timestamps)
gvx.AddEdge(n_id, n_id + timestamps, Objective=lamb * cvxpy.norm(S, 1))
# need to write the parameters of ADMM
# gvx.Solve()
gvx.Solve(EpsAbs=epsAbs, EpsRel=epsRel, Verbose=verbose)
# gvx.Solve(MaxIters = 700, Verbose = True, EpsAbs=eps_abs, EpsRel=eps_rel)
# gvx.Solve( NumProcessors = 1, MaxIters = 3)
# Extract the set of estimated theta
thetaSet = []
for nodeID in range(timestamps):
val = gvx.GetNodeValue(nodeID, 'S')
thetaEst = upper2FullTVGL(val, eps)
thetaSet.append(thetaEst)
return thetaSet, empCovSet, gvx.status, gvx
def _generate_matrix_variables(self):
self._true_cov_matrix = cv.semidefinite(self._n,
name=u"true_cov_matrix")
self._error_cov_matrix = cv.Variable(self._n, self._n,
name=u"error_cov_matrix")
# # create data P
# P = cvxopt.matrix(np.matrix('4 1 3; 1 3.5 0.8; 3 0.8 1'))
# Z = cvx.Variable(3,3)
# objective = cvx.Minimize( sum(cvx.square(P - Z)) )
# constr = [cvx.constraints.semi_definite.SDP(P)]
# prob = cvx.Problem(objective, constr)
# prob.solve()
import cvxpy as cp
import numpy as np
import cvxopt
# create data P
P = cp.Parameter(3,3)
Z = cp.semidefinite(3)
objective = cp.Minimize( cp.lambda_max(P) - cp.lambda_min(P - Z) )
prob = cp.Problem(objective, 10*[Z >= 0])
P.value = cvxopt.matrix(np.matrix('4 1 3; 1 3.5 0.8; 3 0.8 1'))
prob.solve()
# [ 4, 1+2*j, 3-j ; ...
# 1-2*j, 3.5, 0.8+2.3*j ; ...
# 3+j, 0.8-2.3*j, 4 ];
#
# % Construct and solve the model
# n = size( P, 1 );
# cvx_begin sdp
