def embed_SDP(P,order="AMD",cholmod=False):
if not isinstance(P,SDP): raise ValueError, "not an SDP object"
if order=='AMD':
from cvxopt.amd import order
elif order=='METIS':
from cvxopt.metis import order
else: raise ValueError, "unknown ordering: %s " %(order)
p = order(P.V)
if cholmod:
from cvxopt import cholmod
V = +P.V + spmatrix([float(i+1) for i in xrange(P.n)],xrange(P.n),xrange(P.n))
F = cholmod.symbolic(V,p=p)
cholmod.numeric(V,F)
f = cholmod.getfactor(F)
fd = [(j,i) for i,j in enumerate(f[:P.n**2:P.n+1])]
fd.sort()
ip = matrix([j for _,j in fd])
Ve = chompack.tril(chompack.perm(chompack.symmetrize(f),ip))
Ie = misc.sub2ind((P.n,P.n),Ve.I,Ve.J)
else:
#Vc,n = chompack.embed(P.V,p)
symb = chompack.symbolic(P.V,p)
#Ve = chompack.sparse(Vc)
Ve = symb.sparsity_pattern(reordered=False)
Ie = misc.sub2ind((P.n,P.n),Ve.I,Ve.J)
Pe = SDP()
Pe._A = +P.A; Pe._b = +P.b
Pe._A[:,0] += spmatrix(0.0,Ie,[0 for i in range(len(Ie))],(Pe._A.size[0],1))
Pe._agg_sparsity()
Pe._pname = P._pname + "_embed"
Pe._ischordal = True; Pe._blockstruct = P._blockstruct
return Pe
def test_merge_nc(self):
symb = cp.symbolic(self.A_nc, p = None, merge_function = cp.merge_size_fill(0,0))
self.assertEqual(symb.n, 23)
self.assertEqual(symb.nnz, 150)
self.assertEqual(symb.clique_number, 12)
self.assertEqual(symb.Nsn, 10)
self.assertEqual(symb.fill,(73,0))
p = amd.order(self.A_nc)
symb = cp.symbolic(self.A_nc, p = p, merge_function = cp.merge_size_fill(4,4))
self.assertEqual(symb.n, 23)
self.assertTrue(symb.nnz > 150)
self.assertTrue(symb.Nsn < 10)
self.assertTrue(symb.fill[0] >= 36)
self.assertTrue(symb.fill[1] > 0)
def mk_rand(V, cone='posdef', seed=0):
"""
Generates random matrix U with sparsity pattern V.
- U is positive definite if cone is 'posdef'.
- U is completable if cone is 'completable'.
"""
if not (cone == 'posdef' or cone == 'completable'):
raise ValueError, "cone must be 'posdef' (default) or 'completable' "
from cvxopt import amd
setseed(seed)
n = V.size[0]
U = +V
U.V *= 0.0
for i in xrange(n):
u = normal(n, 1) / sqrt(n)
base.syrk(u, U, beta=1.0, partial=True)
# test if U is in cone: if not, add multiple of identity
t = 0.1
Ut = +U
p = amd.order(Ut)
# Vc, NF = chompack.embed(U,p)
symb = chompack.symbolic(U, p)
Vc = chompack.cspmatrix(symb) + U
while True:
# Uc = chompack.project(Vc,Ut)
Uc = chompack.cspmatrix(symb) + Ut
try:
if cone == 'posdef':
# positive definite?
# Y = chompack.cholesky(Uc)
Y = Uc.copy()
chompack.cholesky(Y)
elif cone == 'completable':
# completable?
# Y = chompack.completion(Uc)
Y = Uc.copy()
chompack.completion(Y)
# Success: Ut is in cone
U = +Ut
break
except:
Ut = U + spmatrix(t, xrange(n), xrange(n), (n, n))
t *= 2.0
return U
def test_symbolic_nc(self):
symb = cp.symbolic(self.A_nc, p=None)
self.assertEqual(symb.n, 23)
self.assertEqual(symb.nnz, 150)
self.assertEqual(symb.clique_number, 12)
self.assertEqual(symb.Nsn, 10)
self.assertEqual(symb.fill, (73, 0))
#self.assertEqual(symb.p, None)
#self.assertEqual(symb.ip, None)
p = amd.order(self.A_nc)
symb = cp.symbolic(self.A_nc, p=p)
self.assertEqual(symb.n, 23)
self.assertEqual(symb.nnz, 113)
self.assertEqual(symb.clique_number, 9)
self.assertEqual(symb.Nsn, 15)
self.assertEqual(symb.fill, (36, 0))
def test_symbolic_nc(self):
symb = cp.symbolic(self.A_nc, p = None)
self.assertEqual(symb.n, 23)
self.assertEqual(symb.nnz, 150)
self.assertEqual(symb.clique_number, 12)
self.assertEqual(symb.Nsn, 10)
self.assertEqual(symb.fill,(73,0))
#self.assertEqual(symb.p, None)
#self.assertEqual(symb.ip, None)
p = amd.order(self.A_nc)
symb = cp.symbolic(self.A_nc, p = p)
self.assertEqual(symb.n, 23)
self.assertEqual(symb.nnz, 113)
self.assertEqual(symb.clique_number, 9)
self.assertEqual(symb.Nsn, 15)
self.assertEqual(symb.fill,(36,0))
def mk_rand(V,cone='posdef',seed=0):
"""
Generates random matrix U with sparsity pattern V.
- U is positive definite if cone is 'posdef'.
- U is completable if cone is 'completable'.
"""
if not (cone=='posdef' or cone=='completable'):
raise ValueError("cone must be 'posdef' (default) or 'completable' ")
from cvxopt import amd
setseed(seed)
n = V.size[0]
U = +V
U.V *= 0.0
for i in range(n):
u = normal(n,1)/sqrt(n)
base.syrk(u,U,beta=1.0,partial=True)
# test if U is in cone: if not, add multiple of identity
t = 0.1; Ut = +U
p = amd.order(Ut)
# Vc, NF = chompack.embed(U,p)
symb = chompack.symbolic(U,p)
Vc = chompack.cspmatrix(symb) + U
while True:
# Uc = chompack.project(Vc,Ut)
Uc = chompack.cspmatrix(symb) + Ut
try:
if cone=='posdef':
# positive definite?
# Y = chompack.cholesky(Uc)
Y = Uc.copy()
chompack.cholesky(Y)
elif cone=='completable':
# completable?
# Y = chompack.completion(Uc)
Y = Uc.copy()
chompack.completion(Y)
# Success: Ut is in cone
U = +Ut
break
except:
Ut = U + spmatrix(t,range(n),range(n),(n,n))
t*=2.0
return U
def test_merge_nc(self):
symb = cp.symbolic(self.A_nc,
p=None,
merge_function=cp.merge_size_fill(0, 0))
self.assertEqual(symb.n, 23)
self.assertEqual(symb.nnz, 150)
self.assertEqual(symb.clique_number, 12)
self.assertEqual(symb.Nsn, 10)
self.assertEqual(symb.fill, (73, 0))
p = amd.order(self.A_nc)
symb = cp.symbolic(self.A_nc,
p=p,
merge_function=cp.merge_size_fill(4, 4))
self.assertEqual(symb.n, 23)
self.assertTrue(symb.nnz > 150)
self.assertTrue(symb.Nsn < 10)
self.assertTrue(symb.fill[0] >= 36)
self.assertTrue(symb.fill[1] > 0)
def multstuff(self, A, AT, m, n):
"""Multiplies and stuffs matrices."""
# Make a sparse mask of rn. Use A*A'.
Am = zeros(m, n)
for k in A.keys():
Am[k] = 1
Am = Am * tp(Am)
p = order(Am)
Ao = Am[p, p]
# Design a simple storage scheme for Ao.
k = 0
ss = {}
for (i, j) in zip(Ao.I, Ao.J):
if i >= j:
ss[i, j] = k
k += 1
# Find reduced Newton system matrix ASAT.
d = {}
ASAT = {}
# Write some code.
h = param('CM_h', m, 1)
for i in range(m):
ip = p[i]
for j in range(i + 1):
jp = p[j]
for k in range(n):
if (ip, k) in A and (k, jp) in AT:
if ss[i, j] not in d:
d[ss[i, j]] = exprtoC(A[ip, k] * h[k] * AT[k, jp])
else:
d[ss[i, j]] += ' + ' + exprtoC(
A[ip, k] * h[k] * AT[k, jp])
s = ''
for k in sorted(d.keys()):
s += 'CM_ASAT[%d] = %s;\n' % (k, d[k])
return (Ao, s, ss, p)
def multstuff(self, A, AT, m, n):
"""Multiplies and stuffs matrices."""
# Make a sparse mask of rn. Use A*A'.
Am = zeros(m, n)
for k in A.keys():
Am[k] = 1
Am = Am*tp(Am)
p = order(Am)
Ao = Am[p,p]
# Design a simple storage scheme for Ao.
k = 0
ss = {}
for (i, j) in zip(Ao.I, Ao.J):
if i >= j:
ss[i,j] = k
k += 1
# Find reduced Newton system matrix ASAT.
d = {}
ASAT = {}
# Write some code.
h = param('CM_h', m, 1)
for i in range(m):
ip = p[i]
for j in range(i + 1):
jp = p[j]
for k in range(n):
if (ip,k) in A and (k,jp) in AT:
if ss[i,j] not in d:
d[ss[i,j]] = exprtoC(A[ip,k] * h[k] * AT[k,jp])
else:
d[ss[i,j]] += ' + ' + exprtoC(A[ip,k] * h[k] * AT[k,jp])
s = ''
for k in sorted(d.keys()):
s += 'CM_ASAT[%d] = %s;\n' % (k, d[k])
return (Ao, s, ss, p)
def completion(X):
"""
Returns maximum-determinant positive definite completion of X
if it exists, and otherwise an exception is raised.
"""
from cvxopt.amd import order
n = X.size[0]
Xt = chompack.tril(X)
p = order(Xt)
# Xc,N = chompack.embed(Xt,p)
# L = chompack.completion(Xc)
symb = chompack.symbolic(Xt,p)
L = chompack.cspmatrix(symb) + Xt
chompack.completion(L)
Xt = matrix(0.,(n,n))
Xt[::n+1] = 1.
# chompack.solve(L, Xt, mode=0)
# chompack.solve(L, Xt, mode=1)
chompack.trsm(L, Xt)
chompack.trsm(L, Xt, trans = 'T')
return Xt
def completion(X):
"""
Returns maximum-determinant positive definite completion of X
if it exists, and otherwise an exception is raised.
"""
from cvxopt.amd import order
n = X.size[0]
Xt = chompack.tril(X)
p = order(Xt)
# Xc,N = chompack.embed(Xt,p)
# L = chompack.completion(Xc)
symb = chompack.symbolic(Xt, p)
L = chompack.cspmatrix(symb) + Xt
chompack.completion(L)
Xt = matrix(0., (n, n))
Xt[::n + 1] = 1.
# chompack.solve(L, Xt, mode=0)
# chompack.solve(L, Xt, mode=1)
chompack.trsm(L, Xt)
chompack.trsm(L, Xt, trans='T')
return Xt
def solve_phase1(self,kktsolver='chol',MM = 1e5):
"""
Solves primal Phase I problem using the feasible
start solver.
Returns primal feasible X.
"""
from cvxopt import cholmod, amd
k = 1e-3
# compute Schur complement matrix
Id = [i*(self.n+1) for i in range(self.n)]
As = self._A[:,1:]
As[Id,:] /= sqrt(2.0)
M = spmatrix([],[],[],(self.m,self.m))
base.syrk(As,M,trans='T')
u = +self.b
# compute least-norm solution
F = cholmod.symbolic(M)
cholmod.numeric(M,F)
cholmod.solve(F,u)
x = 0.5*self._A[:,1:]*u
X0 = spmatrix(x[self.V[:].I],self.V.I,self.V.J,(self.n,self.n))
# test feasibility
p = amd.order(self.V)
#Xc,Nf = chompack.embed(X0,p)
#E = chompack.project(Xc,spmatrix(1.0,range(self.n),range(self.n)))
symb = chompack.symbolic(self.V,p)
Xc = chompack.cspmatrix(symb) + X0
try:
# L = chompack.completion(Xc)
L = Xc.copy()
chompack.completion(L)
# least-norm solution is feasible
return X0,None
except:
pass
# create Phase I SDP object
trA = matrix(0.0,(self.m+1,1))
e = matrix(1.0,(self.n,1))
Aa = self._A[Id,1:]
base.gemv(Aa,e,trA,trans='T')
trA[-1] = MM
P1 = SDP()
P1._A = misc.phase1_sdp(self._A,trA)
P1._b = matrix([self.b-k*trA[:self.m],MM])
P1._agg_sparsity()
# find feasible starting point for Phase I problem
tMIN = 0.0
tMAX = 1.0
while True:
t = (tMIN+tMAX)/2.0
#Xt = chompack.copy(Xc)
#chompack.axpy(E,Xt,t)
Xt = Xc.copy() + spmatrix(t,list(range(self.n)),list(range(self.n)))
try:
# L = chompack.completion(Xt)
L = Xt.copy()
chompack.completion(L)
tMAX = t
if tMAX - tMIN < 1e-1:
break
except:
tMAX *= 2.0
tMIN = t
tt = t + 1.0
U = X0 + spmatrix(tt,list(range(self.n,)),list(range(self.n)))
trU = sum(U[:][Id])
Z0 = spdiag([U,spmatrix([tt+k,MM-trU],[0,1],[0,1],(2,2))])
sol = P1.solve_feas(primalstart = {'x':Z0}, kktsolver = kktsolver)
s = sol['x'][-2,-2] - k
if s > 0:
return None,P1
else:
sol.pop('y')
sol.pop('s')
X0 = sol.pop('x')[:self.n,:self.n]\
- spmatrix(s,list(range(self.n)),list(range(self.n)))
return X0,sol
# Read in X
X = pandas.read_csv('X.csv', header=None).values
# Read in b
b = pandas.read_csv('true_b.csv', header=None).values
# Read in Y
Y = pandas.read_csv('Y.csv', header=None).values
# Read in ffx variance
sigma2 = pandas.read_csv('true_ffxvar.csv', header=None).values
ZtZ = cvxopt.spmatrix.trans(Z2) * Z2
# Use minimum degree ordering
P = amd.order(ZtZ)
# Set the factorisation to use LL' instead of LDL'
cholmod.options['supernodal'] = 2
# Make an expression for the factorisation
F = cholmod.symbolic(ZtZ, p=P)
# Calculate the factorisation
cholmod.numeric(ZtZ, F)
# Get the sparse cholesky factorisation
L = cholmod.getfactor(F)
LLt = L * cvxopt.spmatrix.trans(L)
# Create initial lambda
def solve(A, b, C, L, dims, proxqp=None, sigma=1.0, rho=1.0, **kwargs):
"""
Solves the SDP
min. < c, x >
s.t. A(x) = b
x >= 0
and its dual
max. -< b, y >
s.t. s >= 0.
c + A'(y) = s
Input arguments.
A is an N x M sparse matrix where N = sum_i ns[i]**2 and M = sum_j ms[j]
and ns and ms are the SDP variable sizes and constraint block lengths respectively.
The expression A(x) = b can be written as A.T*xtilde = b, where
xtilde is a stacked vector of vectorized versions of xi.
b is a stacked vector containing constraint vectors of
size m_i x 1.
C is a stacked vector containing vectorized 'd' matrices
c_k of size n_k**2 x 1, representing symmetric matrices.
L is an N X P sparse matrix, where L.T*X = 0 represents the consistency
constraints. If an index k appears in different cliques i,j, and
in converted form are indexed by it, jt, then L[it,l] = 1,
L[jt,l] = -1 for some l.
dims is a dictionary containing conic dimensions.
dims['l'] contains number of linear variables under nonnegativity constrant
dims['q'] contains a list of quadratic cone orders (not implemented!)
dims['s'] contains a list of semidefinite cone matrix orders
proxqp is either a function pointer to a prox implementation, or, if
the problem has block-diagonal correlative sparsity, a pointer
to the prox implementation of a single clique. The choices are:
proxqp_general : solves prox for general sparsity pattern
proxqp_clique : solves prox for a single dense clique with
only semidefinite variables.
proxqp_clique_SNL : solves prox for sensor network localization
problem
sigma is a nonnegative constant (step size)
rho is a nonnegative constaint between 0 and 2 (overrelaxation parameter)
In addition, the following paramters are optional:
maxiter : maximum number of iterations (default 100)
reltol : relative tolerance (default 0.01).
If rp < reltol and rd < reltol and iteration < maxiter,
solver breaks and returns current value.
adaptive : boolean toggle on whether adaptive step size should be
used. (default False)
mu, tau, tauscale : parameters for adaptive step size (see paper)
multiprocess : number of parallel processes (default 1).
if multiprocess = 1, no parallelization is used.
blockdiagonal : boolean toggle on whether problem has block diagonal
correlative sparsity. Note that even if the problem
does have block-diagonal correlative sparsity, if
this parameter is set to False, then general mode
is used. (default False)
verbose : toggle printout (default True)
log_cputime : toggle whether cputime should be logged.
The output is returned in a dictionary with the following files:
x : primal variable in stacked form (X = [x0, ..., x_{N-1}]) where
xk is the vectorized form of the nk x nk submatrix variable.
y, z : iterates in Spingarn's method
cputime, walltime : total cputime and walltime, respectively, spent in
main loop. If log_cputime is False, then cputime is
returned as 0.
primal, rprimal, rdual : evolution of primal optimal value, primal
residual, and dual residual (resp.)
sigma : evolution of step size sigma (changes if adaptive step size is used.)
"""
solvers.options['show_progress'] = False
maxiter = kwargs.get('maxiter', 100)
reltol = kwargs.get('reltol', 0.01)
adaptive = kwargs.get('adaptive', False)
mu = kwargs.get('mu', 2.0)
tau = kwargs.get('tau', 1.5)
multiprocess = kwargs.get('multiprocess', 1)
tauscale = kwargs.get('tauscale', 0.9)
blockdiagonal = kwargs.get('blockdiagonal', False)
verbose = kwargs.get('verbose', True)
log_cputime = kwargs.get('log_cputime', True)
if log_cputime:
try:
import psutil
except (ImportError):
assert False, "Python package psutil required to log cputime. Package can be downloaded at http://code.google.com/p/psutil/"
#format variables
nl, ns = dims['l'], dims['s']
C = C[nl:]
L = L[nl:, :]
As, bs = [], []
cons = []
offset = 0
for k in xrange(len(ns)):
Atmp = sparse(A[nl + offset:nl + offset + ns[k]**2, :])
J = list(set(list(Atmp.J)))
Atmp = Atmp[:, J]
if len(sparse(Atmp).V) == Atmp[:].size[0]: Atmp = matrix(Atmp)
else: Atmp = sparse(Atmp)
As.append(Atmp)
bs.append(b[J])
cons.append(J)
offset += ns[k]**2
if blockdiagonal:
if sum([len(c) for c in cons]) > len(b):
print "Problem does not have block-diagonal correlative sparsity. Switching to general mode."
blockdiagonal = False
#If not block-diagonal correlative sprasity, represent A as a list of lists:
# A[i][j] is a matrix (or spmatrix) if ith clique involves jth constraint block
#Otherwise, A is a list of matrices, where A[i] involves the ith clique and
#ith constraint block only.
if not blockdiagonal:
while sum([len(c) for c in cons]) > len(b):
tobreak = False
for i in xrange(len(cons)):
for j in xrange(i):
ci, cj = set(cons[i]), set(cons[j])
s1 = ci.intersection(cj)
if len(s1) > 0:
s2 = ci.difference(cj)
s3 = cj.difference(ci)
cons.append(list(s1))
if len(s2) > 0:
s2 = list(s2)
if not (s2 in cons): cons.append(s2)
if len(s3) > 0:
s3 = list(s3)
if not (s3 in cons): cons.append(s3)
cons.pop(i)
cons.pop(j)
tobreak = True
break
if tobreak: break
As, bs = [], []
for i in xrange(len(cons)):
J = cons[i]
bs.append(b[J])
Acol = []
offset = 0
for k in xrange(len(ns)):
Atmp = sparse(A[nl + offset:nl + offset + ns[k]**2, J])
if len(Atmp.V) == 0:
Acol.append(0)
elif len(Atmp.V) == Atmp[:].size[0]:
Acol.append(matrix(Atmp))
else:
Acol.append(Atmp)
offset += ns[k]**2
As.append(Acol)
ms = [len(i) for i in bs]
bs = matrix(bs)
meq = L.size[1]
if (not blockdiagonal) and multiprocess > 1:
print "Multiprocessing mode can only be used if correlative sparsity is block diagonal. Switching to sequential mode."
multiprocess = 1
assert rho > 0 and rho < 2, 'Overrelaxaton parameter (rho) must be (strictly) between 0 and 2'
# create routine for projecting on { x | L*x = 0 }
#{ x | L*x = 0 } -> P = I - L*(L.T*L)i *L.T
LTL = spmatrix([], [], [], (meq, meq))
offset = 0
for k in ns:
Lk = L[offset:offset + k**2, :]
base.syrk(Lk, LTL, trans='T', beta=1.0)
offset += k**2
LTLi = cholmod.symbolic(LTL, amd.order(LTL))
cholmod.numeric(LTL, LTLi)
#y = y - L*LTLi*L.T*y
nssq = sum(matrix([nsk**2 for nsk in ns]))
def proj(y, ip=True):
if not ip: y = +y
tmp = matrix(0.0, size=(meq, 1))
ypre = +y
base.gemv(L,y,tmp,trans='T',\
m = nssq, n = meq, beta = 1)
cholmod.solve(LTLi, tmp)
base.gemv(L,tmp,y,beta=1.0,alpha=-1.0,trans='N',\
m = nssq, n = meq)
if not ip: return y
time_to_solve = 0
#initialize variables
X = C * 0.0
Y = +X
Z = +X
dualS = +X
dualy = +b
PXZ = +X
proxargs = {
'C': C,
'A': As,
'b': bs,
'Z': Z,
'X': X,
'sigma': sigma,
'dualS': dualS,
'dualy': dualy,
'ns': ns,
'ms': ms,
'multiprocess': multiprocess
}
if blockdiagonal: proxqp = proxqp_blockdiagonal(proxargs, proxqp)
else: proxqp = proxqp_general
if log_cputime: utime = psutil.cpu_times()[0]
wtime = time.time()
primal = []
rpvec, rdvec = [], []
sigmavec = []
for it in xrange(maxiter):
pv, gap = proxqp(proxargs)
blas.copy(Z, Y)
blas.axpy(X, Y, alpha=-2.0)
proj(Y, ip=True)
#PXZ = sigma*(X-Z)
blas.copy(X, PXZ)
blas.scal(sigma, PXZ)
blas.axpy(Z, PXZ, alpha=-sigma)
#z = z + rho*(y-x)
blas.axpy(X, Y, alpha=1.0)
blas.axpy(Y, Z, alpha=-rho)
xzn = blas.nrm2(PXZ)
xn = blas.nrm2(X)
xyn = blas.nrm2(Y)
proj(PXZ, ip=True)
rdual = blas.nrm2(PXZ)
rpri = sqrt(abs(xyn**2 - rdual**2)) / sigma
if log_cputime: cputime = psutil.cpu_times()[0] - utime
else: cputime = 0
walltime = time.time() - wtime
if rpri / max(xn, 1.0) < reltol and rdual / max(1.0, xzn) < reltol:
break
rpvec.append(rpri / max(xn, 1.0))
rdvec.append(rdual / max(1.0, xzn))
primal.append(pv)
if adaptive:
if (rdual / xzn * mu < rpri / xn):
sigmanew = sigma * tau
elif (rpri / xn * mu < rdual / xzn):
sigmanew = sigma / tau
else:
sigmanew = sigma
if it % 10 == 0 and it > 0 and tau > 1.0:
tauscale *= 0.9
tau = 1 + (tau - 1) * tauscale
sigma = max(min(sigmanew, 10.0), 0.1)
sigmavec.append(sigma)
if verbose:
if log_cputime:
print "%d: primal = %e, gap = %e, (rp,rd) = (%e,%e), sigma = %f, (cputime,walltime) = (%f, %f)" % (
it, pv, gap, rpri / max(xn, 1.0), rdual / max(1.0, xzn),
sigma, cputime, walltime)
else:
print "%d: primal = %e, gap = %e, (rp,rd) = (%e,%e), sigma = %f, walltime = %f" % (
it, pv, gap, rpri / max(xn, 1.0), rdual / max(1.0, xzn),
sigma, walltime)
sol = {}
sol['x'] = X
sol['y'] = Y
sol['z'] = Z
sol['cputime'] = cputime
sol['walltime'] = walltime
sol['primal'] = primal
sol['rprimal'] = rpvec
sol['rdual'] = rdvec
sol['sigma'] = sigmavec
return sol
def SW_lmerTest(theta3D,L,nlevels,nparams,ZtX,ZtY,XtX,ZtZ,XtY,YtX,YtZ,XtZ,YtY,n,beta):# TODO inputs
#================================================================================
# Initial theta
#================================================================================
theta0 = np.array([])
r = np.amax(nlevels.shape)
for i in np.arange(r):
theta0 = np.hstack((theta0, mat2vech2D(np.eye(nparams[i])).reshape(np.int64(nparams[i]*(nparams[i]+1)/2))))
#================================================================================
# Sparse Permutation, P
#================================================================================
tinds,rinds,cinds=get_mapping2D(nlevels, nparams)
tmp = np.random.randn(theta0.shape[0])
Lam=mapping2D(tmp,tinds,rinds,cinds)
# Obtain Lambda'Z'ZLambda
LamtZtZLam = spmatrix.trans(Lam)*cvxopt.sparse(matrix(ZtZ[0,:,:]))*Lam
# Obtaining permutation for PLS
cholmod.options['supernodal']=2
P=amd.order(LamtZtZLam)
# Identity
I = spmatrix(1.0, range(Lam.size[0]), range(Lam.size[0]))
# These are not spatially varying
XtX_current = cvxopt.matrix(XtX[0,:,:])
XtZ_current = cvxopt.matrix(XtZ[0,:,:])
ZtX_current = cvxopt.matrix(ZtX[0,:,:])
ZtZ_current = cvxopt.sparse(cvxopt.matrix(ZtZ[0,:,:]))
df = np.zeros(YtY.shape[0])
# Get the sigma^2 and D estimates.
for i in np.arange(theta3D.shape[0]):
# Get current theta
theta = theta3D[i,:]
# Convert product matrices to CVXopt form
XtY_current = cvxopt.matrix(XtY[i,:,:])
YtX_current = cvxopt.matrix(YtX[i,:,:])
YtY_current = cvxopt.matrix(YtY[i,:,:])
YtZ_current = cvxopt.matrix(YtZ[i,:,:])
ZtY_current = cvxopt.matrix(ZtY[i,:,:])
# Convert to gamma form
gamma = theta2gamma(theta, ZtX_current, ZtY_current, XtX_current, ZtZ_current, XtY_current, YtX_current, YtZ_current, XtZ_current, YtY_current, n, P, I, tinds, rinds, cinds)
# Estimate hessian
H = nd.Hessian(llh_gamma)(gamma, beta[i,:,:], np.array(ZtX_current), np.array(ZtY_current), np.array(XtX_current), np.array(matrix(ZtZ_current)), np.array(XtY_current), np.array(YtX_current), np.array(YtZ_current), np.array(XtZ_current), np.array(YtY_current), nlevels, nparams, n, P, tinds, rinds, cinds)
# Estimate Jacobian
J = nd.Jacobian(S2_gammavec)(gamma, L, np.array(ZtX_current), np.array(ZtY_current), np.array(XtX_current), np.array(matrix(ZtZ_current)), np.array(XtY_current), np.array(YtX_current), np.array(YtZ_current), np.array(XtZ_current), np.array(YtY_current), nparams, nlevels)
# print('J shape')
# print(J.shape)
# Calulcate S^2
S2 = S2_gamma(gamma, L, ZtX_current, ZtY_current, XtX_current, ZtZ_current, XtY_current, YtX_current,
YtZ_current, XtZ_current, YtY_current, n, P, I, tinds, rinds, cinds)
# Calculate the degrees of freedom
df[i] = 2*(S2**2)/(J @ np.linalg.pinv(H) @ J.transpose())
return(df)
0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 9, 9, 9, 10,
10, 10, 11, 11, 11, 11, 12, 12, 12, 13, 13, 14, 14, 15
]
A = spmatrix(1.0, I, J, (17, 17))
# Test if A is chordal
p = cp.maxcardsearch(A)
print("\nMaximum cardinality search")
print(" -- perfect elimination order:"), cp.peo(A, p)
# Test if natural ordering 0,1,2,...,17 is a perfect elimination order
p = range(17)
print("\nNatural ordering")
print(" -- perfect elimination order:"), cp.peo(A, p)
p = amd.order(A)
print("\nAMD ordering")
print(" -- perfect elimination order:"), cp.peo(A, p)
# Compute a symbolic factorization
symb = cp.symbolic(A, p)
print("\nSymbolic factorization:")
print("Fill :"), sum(symb.fill)
print("Number of cliques :"), symb.Nsn
print(symb)
# Compute a symbolic factorization with clique merging
symb2 = cp.symbolic(A, p, merge_function=cp.merge_size_fill(3, 3))
print("Symbolic factorization with clique merging:")
print("Fill (fact.+merging) :"), sum(symb2.fill)
print("Number of cliques :"), symb2.Nsn
# Define sparse matrix
I = range(17) + [2,2,3,3,4,14,4,14,8,14,15,8,15,7,8,14,8,14,14,15,10,12,13,16,12,13,16,12,13,15,16,13,15,16,15,16,15,16,16]
J = range(17) + [0,1,1,2,2,2,3,3,4,4,4,5,5,6,6,6,7,7,8,8,9,9,9,9,10,10,10,11,11,11,11,12,12,12,13,13,14,14,15]
A = spmatrix(1.0,I,J,(17,17))
# Test if A is chordal
p = cp.maxcardsearch(A)
print("\nMaximum cardinality search")
print(" -- perfect elimination order:"), cp.peo(A,p)
# Test if natural ordering 0,1,2,...,17 is a perfect elimination order
p = range(17)
print("\nNatural ordering")
print(" -- perfect elimination order:"), cp.peo(A,p)
p = amd.order(A)
print("\nAMD ordering")
print(" -- perfect elimination order:"), cp.peo(A,p)
# Compute a symbolic factorization
symb = cp.symbolic(A, p)
print("\nSymbolic factorization:")
print("Fill :"), sum(symb.fill)
print("Number of cliques :"), symb.Nsn
print(symb)
# Compute a symbolic factorization with clique merging
symb2 = cp.symbolic(A, p, merge_function = cp.merge_size_fill(3,3))
print("Symbolic factorization with clique merging:")
print("Fill (fact.+merging) :"), sum(symb2.fill)
print("Number of cliques :"), symb2.Nsn
def solve_phase1(self, kktsolver='chol', MM=1e5):
"""
Solves primal Phase I problem using the feasible
start solver.
Returns primal feasible X.
"""
from cvxopt import cholmod, amd
k = 1e-3
# compute Schur complement matrix
Id = [i * (self.n + 1) for i in range(self.n)]
As = self._A[:, 1:]
As[Id, :] /= sqrt(2.0)
M = spmatrix([], [], [], (self.m, self.m))
base.syrk(As, M, trans='T')
u = +self.b
# compute least-norm solution
F = cholmod.symbolic(M)
cholmod.numeric(M, F)
cholmod.solve(F, u)
x = 0.5 * self._A[:, 1:] * u
X0 = spmatrix(x[self.V[:].I], self.V.I, self.V.J, (self.n, self.n))
# test feasibility
p = amd.order(self.V)
#Xc,Nf = chompack.embed(X0,p)
#E = chompack.project(Xc,spmatrix(1.0,range(self.n),range(self.n)))
symb = chompack.symbolic(self.V, p)
Xc = chompack.cspmatrix(symb) + X0
try:
# L = chompack.completion(Xc)
L = Xc.copy()
chompack.completion(L)
# least-norm solution is feasible
return X0
except:
pass
# create Phase I SDP object
trA = matrix(0.0, (self.m + 1, 1))
e = matrix(1.0, (self.n, 1))
Aa = self._A[Id, 1:]
base.gemv(Aa, e, trA, trans='T')
trA[-1] = MM
P1 = SDP()
P1._A = misc.phase1_sdp(self._A, trA)
P1._b = matrix([self.b - k * trA[:self.m], MM])
P1._agg_sparsity()
# find feasible starting point for Phase I problem
tMIN = 0.0
tMAX = 1.0
while True:
t = (tMIN + tMAX) / 2.0
#Xt = chompack.copy(Xc)
#chompack.axpy(E,Xt,t)
Xt = Xc.copy() + spmatrix(t, range(self.n), range(self.n))
try:
# L = chompack.completion(Xt)
L = Xt.copy()
chompack.completion(L)
tMAX = t
if tMAX - tMIN < 1e-1:
break
except:
tMAX *= 2.0
tMIN = t
tt = t + 1.0
U = X0 + spmatrix(tt, range(self.n, ), range(self.n))
trU = sum(U[:][Id])
Z0 = spdiag([U, spmatrix([tt + k, MM - trU], [0, 1], [0, 1], (2, 2))])
sol = P1.solve_feas(primalstart={'x': Z0}, kktsolver=kktsolver)
s = sol['x'][-2, -2] - k
if s > 0:
return None, P1
else:
sol.pop('y')
sol.pop('s')
X0 = sol.pop('x')[:self.n,:self.n]\
- spmatrix(s,range(self.n),range(self.n))
return X0, sol