def _gen_bandsdp(self,n,m,bw,seed):
"""Random data generator for SDP with band structure"""
setseed(seed)
I = matrix([ i for j in range(n) for i in range(j,min(j+bw+1,n))])
J = matrix([ j for j in range(n) for i in range(j,min(j+bw+1,n))])
V = spmatrix(0.,I,J,(n,n))
Il = misc.sub2ind((n,n),I,J)
Id = matrix([i for i in range(len(Il)) if I[i]==J[i]])
# generate random y with norm 1
y0 = normal(m,1)
y0 /= blas.nrm2(y0)
# generate random S0
S0 = mk_rand(V,cone='posdef',seed=seed)
X0 = mk_rand(V,cone='completable',seed=seed)
# generate random A1,...,Am and set A0 = sum Ai*yi + S0
A_ = normal(len(I),m+1,std=1./len(I))
u = +S0.V
blas.gemv(A_[:,1:],y0,u,beta=1.0)
A_[:,0] = u
# compute b
x = +X0.V
x[Id] *= 0.5
self._b = matrix(0.,(m,1))
blas.gemv(A_[:,1:],x,self._b,trans='T',alpha=2.0)
# form A
self._A = spmatrix(A_[:],
[i for j in range(m+1) for i in Il],
[j for j in range(m+1) for i in Il],(n**2,m+1))
self._X0 = X0; self._y0 = y0; self._S0 = S0
def _gen_mtxnormsdp(self,p,q,r,d,seed):
"""
Random data generator for matrix norm minimization SDP.
"""
setseed(seed)
n = p + q;
I = matrix([ i for j in range(q) for i in range(q,n)])
J = matrix([ j for j in range(q) for i in range(q,n)])
Il = list(misc.sub2ind((n,n),I,J))
if type(d) is float:
nz = min(max(1, int(round(d*p*q))), p*q)
A = sparse([[spmatrix(normal(p*q,1),Il,[0 for i in range(p*q)],(n**2,1))],
[spmatrix(normal(nz*r,1),
[i for j in range(r) for i in random.sample(Il,nz)],
[j for j in range(r) for i in range(nz)],(n**2,r))]])
elif type(d) is list:
if len(d) == r:
nz = [min(max(1, int(round(v*p*q))), p*q) for v in d]
nnz = sum(nz)
A = sparse([[spmatrix(normal(p*q,1),Il,[0 for i in range(p*q)],(n**2,1))],
[spmatrix(normal(nnz,1),
[i for j in range(r) for i in random.sample(Il,nz[j])],
[j for j in range(r) for i in range(nz[j])],(n**2,r))]])
else: raise TypeError
self._A = sparse([[A],[spmatrix(-1.,list(range(0,n**2,n+1)),[0 for i in range(n)],(n**2,1))]])
self._b = matrix(0.,(r+1,1))
self._b[-1] = -1.
def robustLS_toep(q, r, delta=None, seed=0):
"""
Random data generator for matrix norm minimization SDP.
Al,b = robustLS_toep(q,r,delta=0.1,seed=0])
PURPOSE
Generates random data for a robust least squares problem
with Toeplitz structure:
minimize e_wc(x)^2
where
e_wc(x)=sup_{norm(u)<=1} norm((Ab+A1*u1+...+Ar*ur)*x - b),
Ab,A1,...,Ar \in \reals^{p \times q}
The parameter r determines the number of nonzero diagonals
in Ab (1 <= r <= p).
ARGUMENTS
q integer
r integer
delta float
RETURNS
Al list of CVXOPT matrices, [Ab,A1,...,Ar]
b CVXOPT matrix
"""
setseed(seed)
p = q + r - 1
Alist = [
sparse(
misc.toeplitz(
matrix([normal(r, 1), matrix(0., (p - r, 1))], (p, 1)),
matrix(0., (q, 1))))
]
x = normal(q, 1)
b = normal(p, 1, std=0.1)
b += Alist[0] * x
if delta is None:
delta = 1. / r
for i in xrange(r):
Alist.append(
spmatrix(delta, xrange(i, min(p, q + i)), xrange(min(q, p - i)),
(p, q)))
return robustLS_SDP(Alist, b)
def _gen_randsdp(self,V,m,d,seed):
"""
Random data generator
"""
setseed(seed)
n = self._n
V = chompack.tril(V)
N = len(V)
I = V.I; J = V.J
Il = misc.sub2ind((n,n),I,J)
Id = matrix([i for i in range(len(Il)) if I[i]==J[i]])
# generate random y with norm 1
y0 = normal(m,1)
y0 /= blas.nrm2(y0)
# generate random S0, X0
S0 = mk_rand(V,cone='posdef',seed=seed)
X0 = mk_rand(V,cone='completable',seed=seed)
# generate random A1,...,Am
if type(d) is float:
nz = min(max(1, int(round(d*N))), N)
A = sparse([[spmatrix(normal(N,1),Il,[0 for i in range(N)],(n**2,1))],
[spmatrix(normal(nz*m,1),
[i for j in range(m) for i in random.sample(Il,nz)],
[j for j in range(m) for i in range(nz)],(n**2,m))]])
elif type(d) is list:
if len(d) == m:
nz = [min(max(1, int(round(v*N))), N) for v in d]
nnz = sum(nz)
A = sparse([[spmatrix(normal(N,1),Il,[0 for i in range(N)],(n**2,1))],
[spmatrix(normal(nnz,1),
[i for j in range(m) for i in random.sample(Il,nz[j])],
[j for j in range(m) for i in range(nz[j])],(n**2,m))]])
else: raise TypeError
# compute A0
u = +S0.V
for k in range(m):
base.gemv(A[:,k+1][Il],matrix(y0[k]),u,beta=1.0,trans='N')
A[Il,0] = u
self._A = A
# compute b
X0[Il[Id]] *= 0.5
self._b = matrix(0.,(m,1))
u = matrix(0.)
for k in range(m):
base.gemv(A[:,k+1][Il],X0.V,u,trans='T',alpha=2.0)
self._b[k] = u
def robustLS_toep(q,r,delta=None,seed=0):
"""
Random data generator for matrix norm minimization SDP.
Al,b = robustLS_toep(q,r,delta=0.1,seed=0])
PURPOSE
Generates random data for a robust least squares problem
with Toeplitz structure:
minimize e_wc(x)^2
where
e_wc(x)=sup_{norm(u)<=1} norm((Ab+A1*u1+...+Ar*ur)*x - b),
Ab,A1,...,Ar \in \reals^{p \times q}
The parameter r determines the number of nonzero diagonals
in Ab (1 <= r <= p).
ARGUMENTS
q integer
r integer
delta float
RETURNS
Al list of CVXOPT matrices, [Ab,A1,...,Ar]
b CVXOPT matrix
"""
setseed(seed)
p = q+r-1
Alist = [sparse(misc.toeplitz(matrix([normal(r,1),matrix(0.,(p-r,1))],(p,1)),matrix(0.,(q,1))))]
x = normal(q,1)
b = normal(p,1,std=0.1)
b += Alist[0]*x
if delta is None:
delta = 1./r
for i in range(r):
Alist.append(spmatrix(delta,range(i,min(p,q+i)),range(min(q,p-i)),(p,q)))
return robustLS_SDP(Alist, b)
def _gen_mtxnormsdp(self, p, q, r, d, seed):
"""
Random data generator for matrix norm minimization SDP.
"""
setseed(seed)
n = p + q
I = matrix([i for j in range(q) for i in range(q, n)])
J = matrix([j for j in range(q) for i in range(q, n)])
Il = misc.sub2ind((n, n), I, J)
if type(d) is float:
nz = min(max(1, int(round(d * p * q))), p * q)
A = sparse(
[[
spmatrix(normal(p * q, 1), Il, [0 for i in range(p * q)],
(n**2, 1))
],
[
spmatrix(
normal(nz * r, 1),
[i for j in range(r) for i in random.sample(Il, nz)],
[j for j in range(r) for i in range(nz)], (n**2, r))
]])
elif type(d) is list:
if len(d) == r:
nz = [min(max(1, int(round(v * p * q))), p * q) for v in d]
nnz = sum(nz)
A = sparse([[
spmatrix(normal(p * q, 1), Il, [0 for i in range(p * q)],
(n**2, 1))
],
[
spmatrix(normal(nnz, 1), [
i for j in range(r)
for i in random.sample(Il, nz[j])
], [j for j in range(r) for i in range(nz[j])],
(n**2, r))
]])
else:
raise TypeError
self._A = sparse([[A],
[
spmatrix(-1., range(0, n**2, n + 1),
[0 for i in range(n)], (n**2, 1))
]])
self._b = matrix(0., (r + 1, 1))
self._b[-1] = -1.
def main(args):
# Generate an analytic centering problem
#
# -b1 <= Ar*x <= b2
#
# with random mxn Ar and random b1, b2.
if len(args[0]) > 0 and args[0] == "reftest":
m, n = 10, 5
# matrix in column order
A0 = matrix([[-7.44e-01, 4.59e-01, -2.95e-02, -7.75e-01, -1.80e+00],
[1.11e-01, 7.06e-01, -2.22e-01, 1.03e-01, 1.24e+00],
[1.29e+00, 3.16e-01, -2.07e-01, -1.22e+00, -2.61e+00],
[2.62e+00, -1.06e-01, -9.11e-01, -5.74e-01, -9.31e-01],
[-1.82e+00, 7.80e-01, -3.92e-01, -3.32e-01, -6.38e-01]])
b0 = matrix([
8.38e-01, 9.92e-01, 9.56e-01, 6.14e-01, 6.56e-01, 3.57e-01,
6.36e-01, 5.08e-01, 8.81e-03, 7.08e-02
])
A = matrix([A0, -A0])
b = b0
else:
m, n = 500, 500
Ar = base.normal(m, n)
A = matrix([Ar, -Ar])
b = base.uniform(2 * m, 1)
x, ntdecrs = acent(A, b)
print "solution:\n", helpers.str2(x, "%.17f")
print "ntdecrs :\n", ntdecrs
def _gen_bandsdp(self, n, m, bw, seed):
"""Random data generator for SDP with band structure"""
setseed(seed)
I = matrix(
[i for j in xrange(n) for i in xrange(j, min(j + bw + 1, n))])
J = matrix(
[j for j in xrange(n) for i in xrange(j, min(j + bw + 1, n))])
V = spmatrix(0., I, J, (n, n))
Il = misc.sub2ind((n, n), I, J)
Id = matrix([i for i in xrange(len(Il)) if I[i] == J[i]])
# generate random y with norm 1
y0 = normal(m, 1)
y0 /= blas.nrm2(y0)
# generate random S0
S0 = mk_rand(V, cone='posdef', seed=seed)
X0 = mk_rand(V, cone='completable', seed=seed)
# generate random A1,...,Am and set A0 = sum Ai*yi + S0
A_ = normal(len(I), m + 1, std=1. / len(I))
u = +S0.V
blas.gemv(A_[:, 1:], y0, u, beta=1.0)
A_[:, 0] = u
# compute b
x = +X0.V
x[Id] *= 0.5
self._b = matrix(0., (m, 1))
blas.gemv(A_[:, 1:], x, self._b, trans='T', alpha=2.0)
# form A
self._A = spmatrix(A_[:], [i for j in xrange(m + 1) for i in Il],
[j for j in xrange(m + 1) for i in Il],
(n**2, m + 1))
self._X0 = X0
self._y0 = y0
self._S0 = S0
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 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 main(args):
# Generate an analytic centering problem
#
# -b1 <= Ar*x <= b2
#
# with random mxn Ar and random b1, b2.
if len(args[0]) > 0 and args[0] == "reftest":
m, n = 10, 5
# matrix in column order
A0 = matrix([[-7.44e-01, 4.59e-01, -2.95e-02, -7.75e-01, -1.80e+00],
[ 1.11e-01, 7.06e-01, -2.22e-01, 1.03e-01, 1.24e+00],
[ 1.29e+00, 3.16e-01, -2.07e-01, -1.22e+00, -2.61e+00],
[ 2.62e+00, -1.06e-01, -9.11e-01, -5.74e-01, -9.31e-01],
[-1.82e+00, 7.80e-01, -3.92e-01, -3.32e-01, -6.38e-01]])
b0 = matrix([8.38e-01,
9.92e-01,
9.56e-01,
6.14e-01,
6.56e-01,
3.57e-01,
6.36e-01,
5.08e-01,
8.81e-03,
7.08e-02])
A = matrix([A0, -A0])
b = b0
else:
m, n = 500, 500
Ar = base.normal(m,n);
A = matrix([Ar, -Ar])
b = base.uniform(2*m,1)
x, ntdecrs = acent(A, b)
print "solution:\n", helpers.str2(x, "%.17f")
print "ntdecrs :\n", ntdecrs
import pylab, math, pickle, nucnrm, sysid
from cvxopt import base, blas, lapack, solvers
from cvxopt.base import normal, matrix, spmatrix, mul
maxiter = 8
data = "random"
#data = "sysid"
solvers.options['show_progress'] = False
if data is "random":
# Generate random problem data
base.setseed(0)
p, q, n = 20, 20, 40
A = normal(p*q, n)
B = normal(p, q)
x0 = normal(n, 1)
elif data is "sysid":
iddata = pickle.load(open("CD_player_arm.bin","r"))
N = 100
u = iddata['u'][:,:N]
y = iddata['y'][:,:N]
m, N, p = u.size[0], u.size[1], y.size[0]
r = min(int(30/p),int((N+1.0)/(p+m+1)+1.0))
a = r*p
c = r*m
b = N-r+1
d = b-c
U = sysid.Hankel(u,r,b,p=m,q=1)
import pylab, math, pickle, nucnrm, sysid
from cvxopt import base, blas, lapack, solvers
from cvxopt.base import normal, matrix, spmatrix, mul
maxiter = 8
data = "random"
#data = "sysid"
solvers.options['show_progress'] = False
if data is "random":
# Generate random problem data
base.setseed(0)
p, q, n = 20, 20, 40
A = normal(p * q, n)
B = normal(p, q)
x0 = normal(n, 1)
elif data is "sysid":
iddata = pickle.load(open("CD_player_arm.bin", "r"))
N = 100
u = iddata['u'][:, :N]
y = iddata['y'][:, :N]
m, N, p = u.size[0], u.size[1], y.size[0]
r = min(int(30 / p), int((N + 1.0) / (p + m + 1) + 1.0))
a = r * p
c = r * m
b = N - r + 1
d = b - c
U = sysid.Hankel(u, r, b, p=m, q=1)
def _gen_randsdp(self, V, m, d, seed):
"""
Random data generator
"""
setseed(seed)
n = self._n
V = chompack.tril(V)
N = len(V)
I = V.I
J = V.J
Il = misc.sub2ind((n, n), I, J)
Id = matrix([i for i in xrange(len(Il)) if I[i] == J[i]])
# generate random y with norm 1
y0 = normal(m, 1)
y0 /= blas.nrm2(y0)
# generate random S0, X0
S0 = mk_rand(V, cone='posdef', seed=seed)
X0 = mk_rand(V, cone='completable', seed=seed)
# generate random A1,...,Am
if type(d) is float:
nz = min(max(1, int(round(d * N))), N)
A = sparse(
[[
spmatrix(normal(N, 1), Il, [0 for i in xrange(N)],
(n**2, 1))
],
[
spmatrix(
normal(nz * m, 1),
[i for j in xrange(m) for i in random.sample(Il, nz)],
[j for j in xrange(m) for i in xrange(nz)], (n**2, m))
]])
elif type(d) is list:
if len(d) == m:
nz = [min(max(1, int(round(v * N))), N) for v in d]
nnz = sum(nz)
A = sparse(
[[
spmatrix(normal(N, 1), Il, [0 for i in xrange(N)],
(n**2, 1))
],
[
spmatrix(normal(nnz, 1), [
i for j in xrange(m)
for i in random.sample(Il, nz[j])
], [j for j in xrange(m) for i in xrange(nz[j])],
(n**2, m))
]])
else:
raise TypeError
# compute A0
u = +S0.V
for k in xrange(m):
base.gemv(A[:, k + 1][Il], matrix(y0[k]), u, beta=1.0, trans='N')
A[Il, 0] = u
self._A = A
# compute b
X0[Il[Id]] *= 0.5
self._b = matrix(0., (m, 1))
u = matrix(0.)
for k in xrange(m):
base.gemv(A[:, k + 1][Il], X0.V, u, trans='T', alpha=2.0)
self._b[k] = u
if trans=='N':
gx[1::3,:]=-x[ : n,:]
gx[2::3,:]=-x[ n:2*n,:]
gx[ ::3,:]=-x[-n: ,:]
elif trans=='T':
gx[ : n,:]=-x[1::3,:]
gx[ n:2*n,:]=-x[2::3,:]
gx[-n: ,:]=-x[ ::3,:]
y[:,:]=alpha*gx+beta*y
h = matrix(0.0, (3*n, 1))
dims = {'l': 0, 'q': n*[3], 's': []}
factor=mykktchol(P)
sol = solvers.coneqp(P, q, Gfun, h, dims,kktsolver=factor)
return sol['x'][:n] + 1j*sol['x'][n:2*n]
if __name__ == '__main__':
from cvxopt.base import normal
from numpy.random import choice
m, n = 1500, 900
A = normal(m,n) + 1j*normal(m,n)
b = normal(n,1) + 1j*normal(n,1)
b = array(b)
b[choice(range(n),3*n/4,False)]=0
b=matrix(b)
y=A*b
gamma = 1.0
x = l1regls(A, y, gamma)
while 1 - step * max(y) < 0:
step *= BETA
while True:
if -sum(log(1 - step * y)) < ALPHA * step * lam: break
step *= BETA
x += step * v
# Generate an analytic centering problem
#
# -b1 <= Ar*x <= b2
#
# with random mxn Ar and random b1, b2.
m, n = 500, 500
Ar = base.normal(m, n)
A = matrix([Ar, -Ar])
b = base.uniform(2 * m, 1)
x, ntdecrs = acent(A, b)
try:
import pylab
except ImportError:
pass
else:
pylab.semilogy(range(len(ntdecrs)), ntdecrs, 'o', range(len(ntdecrs)),
ntdecrs, '-')
pylab.xlabel('Iteration number')
pylab.ylabel('Newton decrement')
pylab.show()
while 1 - step * max(y) < 0:
step *= BETA
while True:
if -sum(log(1 - step * y)) < ALPHA * step * lam:
break
step *= BETA
x += step * v
# Generate an analytic centering problem
#
# -b1 <= Ar*x <= b2
#
# with random mxn Ar and random b1, b2.
m, n = 500, 500
Ar = base.normal(m, n)
A = matrix([Ar, -Ar])
b = base.uniform(2 * m, 1)
x, ntdecrs = acent(A, b)
try:
import pylab
except ImportError:
pass
else:
pylab.semilogy(range(len(ntdecrs)), ntdecrs, "o", range(len(ntdecrs)), ntdecrs, "-")
pylab.xlabel("Iteration number")
pylab.ylabel("Newton decrement")
pylab.show()
