def spy(P,i=None,file=None,scale=None):
"""Generates sparsity plot using Pylab"""
if type(P) is spmatrix:
V = chompack.symmetrize(chompack.tril(P))
n = V.size[0]
else:
if not P._A: raise AttributeError, "SDP data missing"
n = +P.n;
if i == None: V = chompack.symmetrize(P.V)
elif i>=0 and i<=P.m and P._A:
r = P._A.CCS[1][P._A.CCS[0][i]:P._A.CCS[0][i+1]]
if type(r) is int: I = [r%n]; J = [r/n]
else: I,J = misc.ind2sub(n,r)
V = chompack.symmetrize(spmatrix(0.,I,J,(n,n)))
else: raise ValueError, "index out of range"
from math import floor
msize = max(1,int(floor(100./n)))
if file==None: pylab.ion()
else: pylab.ioff()
f = pylab.figure(figsize=(6,6)); f.clf()
if scale is None: scale = 1.0
I = V.I*scale+1; J = (n-V.J)*scale
p = pylab.plot(I,J, 's', linewidth = 1, hold = 'False')
pylab.setp(p, markersize = msize, markerfacecolor = 'k')
g = pylab.gca()
pylab.axis([0.5,n*scale+0.5,0.5,n*scale+0.5])
g.set_aspect('equal')
locs,labels = pylab.yticks()
locs = locs[locs<=n*scale]; locs = locs[locs>=1]
pylab.yticks(locs[::-1]-(locs[-1]-n*scale-1)-locs[0],
[str(int(loc)) for loc in locs])
if file: pylab.savefig(file)
def setUp(self):
I = list(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 = list(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
]
self.A = spmatrix(matrix(range(len(I)), tc='d'), I, J, (17, 17))
n = 23
nz = [
1, 2, 10, 12, 13, 14, 15, 16, 19, 24, 48, 58, 71, 76, 81, 85, 91,
103, 108, 109, 111, 114, 116, 117, 118, 134, 143, 145, 161, 174,
178, 180, 183, 192, 194, 202, 203, 205, 212, 214, 219, 224, 226,
227, 228, 229, 235, 240, 241, 243, 247, 255, 256, 260, 269, 273,
275, 279, 280, 314, 315, 320, 321, 328, 340, 342, 344, 345, 349,
350, 357, 359, 370, 372, 375, 384, 385, 394, 399, 402, 411, 412,
419, 420, 422, 433, 439, 441, 447, 452, 454, 458, 460, 474, 476,
479, 481, 483, 485, 497, 506, 517, 519, 523, 526, 527
]
self.A_nc = cp.tril(spmatrix(matrix(range(1,len(nz)+1),tc='d')/len(nz),[ni%n for ni in nz],[int(ii/n) for ii in nz],(n,n)))\
+ spmatrix(10.0,range(n),range(n))
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_cspmatrix(self):
Ac = cp.cspmatrix(self.symb) + self.A
self.assertTrue(Ac.is_factor is False)
self.assertTrue(Ac.size[0] == Ac.size[1])
self.assertTrue(Ac.size[0] == 17)
diff = list((self.A - Ac.spmatrix(reordered=False,symmetric=False)).V)
self.assertAlmostEqualLists(diff, len(diff)*[0.0])
diff = list(cp.symmetrize(self.A) - Ac.spmatrix(reordered=False,symmetric=True))
self.assertAlmostEqualLists(diff, len(diff)*[0.0])
diff = list(cp.symmetrize(self.A)[self.symb.p,self.symb.p] - Ac.spmatrix(reordered=True,symmetric=True))
self.assertAlmostEqualLists(diff, len(diff)*[0.0])
diff = list(cp.tril(cp.symmetrize(self.A)[self.symb.p,self.symb.p]) - Ac.spmatrix(reordered=True,symmetric=False))
self.assertAlmostEqualLists(diff, len(diff)*[0.0])
diff = list((2*self.A - (2*Ac).spmatrix(reordered=False,symmetric=False)).V)
self.assertAlmostEqualLists(diff, len(diff)*[0.0])
diff = list((2*self.A - (Ac+Ac).spmatrix(reordered=False,symmetric=False)).V)
self.assertAlmostEqualLists(diff, len(diff)*[0.0])
Ac2 = Ac.copy()
Ac2 += Ac
diff = list((2*self.A - Ac2.spmatrix(reordered=False,symmetric=False)).V)
self.assertAlmostEqualLists(diff, len(diff)*[0.0])
self.assertAlmostEqualLists(list(Ac.diag(reordered=False)), list(self.A[::18]))
self.assertAlmostEqualLists(list(Ac.diag(reordered=True)), list(self.A[self.symb.p,self.symb.p][::18]))
def setUp(self):
n = 31
nnz = int(round(0.15*n**2))
random.seed(1)
nz = matrix(random.sample(range(n**2), nnz), tc='i')
self.A = cp.tril(spmatrix(matrix(range(1,nnz+1),tc='d')/nnz, nz%n, matrix([int(ii) for ii in nz/n]), (n,n)))\
+ spmatrix(10.0,range(n),range(n))
self.symb = cp.symbolic(self.A, p = amd.order)
def setUp(self):
n = 31
nnz = int(round(0.15 * n**2))
random.seed(1)
nz = matrix(random.sample(range(n**2), nnz), tc='i')
self.A = cp.tril(spmatrix(matrix(range(1,nnz+1),tc='d')/nnz, nz%n, matrix([int(ii) for ii in nz/n]), (n,n)))\
+ spmatrix(10.0,range(n),range(n))
self.symb = cp.symbolic(self.A, p=amd.order)
def setUp(self):
I = list(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 = list(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]
self.A = spmatrix(matrix(range(len(I)),tc='d'),I,J,(17,17))
n = 23
nz = [1, 2, 10, 12, 13, 14, 15, 16, 19, 24, 48, 58, 71, 76, 81, 85, 91, 103, 108, 109, 111, 114, 116, 117, 118, 134, 143, 145, 161, 174, 178, 180, 183, 192, 194, 202, 203, 205, 212, 214, 219, 224, 226, 227, 228, 229, 235, 240, 241, 243, 247, 255, 256, 260, 269, 273, 275, 279, 280, 314, 315, 320, 321, 328, 340, 342, 344, 345, 349, 350, 357, 359, 370, 372, 375, 384, 385, 394, 399, 402, 411, 412, 419, 420, 422, 433, 439, 441, 447, 452, 454, 458, 460, 474, 476, 479, 481, 483, 485, 497, 506, 517, 519, 523, 526, 527]
self.A_nc = cp.tril(spmatrix(matrix(range(1,len(nz)+1),tc='d')/len(nz),[ni%n for ni in nz],[int(ii/n) for ii in nz],(n,n)))\
+ spmatrix(10.0,range(n),range(n))
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 test_cspmatrix(self):
Ac = cp.cspmatrix(self.symb) + self.A
self.assertTrue(Ac.is_factor is False)
self.assertTrue(Ac.size[0] == Ac.size[1])
self.assertTrue(Ac.size[0] == 17)
diff = list((self.A - Ac.spmatrix(reordered=False, symmetric=False)).V)
self.assertAlmostEqualLists(diff, len(diff) * [0.0])
diff = list(
cp.symmetrize(self.A) -
Ac.spmatrix(reordered=False, symmetric=True))
self.assertAlmostEqualLists(diff, len(diff) * [0.0])
diff = list(
cp.symmetrize(self.A)[self.symb.p, self.symb.p] -
Ac.spmatrix(reordered=True, symmetric=True))
self.assertAlmostEqualLists(diff, len(diff) * [0.0])
diff = list(
cp.tril(cp.symmetrize(self.A)[self.symb.p, self.symb.p]) -
Ac.spmatrix(reordered=True, symmetric=False))
self.assertAlmostEqualLists(diff, len(diff) * [0.0])
diff = list((2 * self.A -
(2 * Ac).spmatrix(reordered=False, symmetric=False)).V)
self.assertAlmostEqualLists(diff, len(diff) * [0.0])
diff = list((2 * self.A -
(Ac + Ac).spmatrix(reordered=False, symmetric=False)).V)
self.assertAlmostEqualLists(diff, len(diff) * [0.0])
Ac2 = Ac.copy()
Ac2 += Ac
diff = list(
(2 * self.A - Ac2.spmatrix(reordered=False, symmetric=False)).V)
self.assertAlmostEqualLists(diff, len(diff) * [0.0])
self.assertAlmostEqualLists(list(Ac.diag(reordered=False)),
list(self.A[::18]))
self.assertAlmostEqualLists(
list(Ac.diag(reordered=True)),
list(self.A[self.symb.p, self.symb.p][::18]))
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 test_cholesky(self):
L = cp.cspmatrix(self.symb) + self.A
cp.cholesky(L)
Lm = L.spmatrix(reordered=True)
diff = list( (cp.tril(cp.perm(Lm*Lm.T,self.symb.ip)) - self.A).V )
self.assertAlmostEqualLists(diff, len(diff)*[0.0])
# generate random sparse matrix
def sp_rand(m,n,a):
"""
Generates an m-by-n sparse 'd' matrix with round(a*m*n) nonzeros.
"""
if m == 0 or n == 0: return spmatrix([], [], [], (m,n))
nnz = min(max(0, int(round(a*m*n))), m*n)
nz = matrix(random.sample(range(m*n), nnz), tc='i')
return spmatrix(normal(nnz,1), nz%m, nz/m, (m,n))
# generate random sparsity pattern and sparse SDP problem data
random.seed(1)
m, n = 50, 200
print("Generating random sparse SDP (n=%i, m=%i constraints).."%(n,m))
A = sp_rand(n,n,0.015) + spmatrix(1.0,range(n),range(n))
I = cp.tril(A)[:].I
N = len(I)/50 # each data matrix has 1/50 of total nonzeros in pattern
Ig = []; Jg = []
for j in range(m):
Ig += sorted(random.sample(I,N))
Jg += N*[j]
G = spmatrix(normal(len(Ig),1),Ig,Jg,(n**2,m))
h = G*normal(m,1) + spmatrix(1.0,range(n),range(n))[:]
c = normal(m,1)
dims = {'l':0, 'q':[], 's': [n]};
# solve SDP with CVXOPT
print("Solving SDP with CVXOPT..")
prob = (c, G, matrix(h), dims)
sol = solvers.conelp(*prob)
Z1 = matrix(sol['z'], (n,n))
print("Projected inverse/completion (upd) : err = %.3e" % (blas.nrm2(tmp)))
# Compute norm of error: At - U
tmp = (At-U).spmatrix().V
print("Hessian factors NN/TN/TI/NI : err = %.3e" % (blas.nrm2(tmp)))
# Test triangular matrix products and solve
p = L.symb.p
B = eye(n)
trsm(L, B)
print("trsm, trans = 'N' : err = %.3e" % (blas.nrm2(L.spmatrix(reordered = True)*B[p,p] - eye(n))))
B = eye(n)
trsm(L, B, trans = 'T')
print("trsm, trans = 'T' : err = %.3e" % (blas.nrm2(L.spmatrix(reordered = True).T*B[p,p] - eye(n))))
B = eye(n)
trmm(L, B)
print("trmm, trans = 'N' : err = %.3e" % (blas.nrm2(L.spmatrix(reordered = True) - B[p,p])))
B = eye(n)
trmm(L, B, trans = 'T')
print("trmm, trans = 'T' : err = %.3e" % (blas.nrm2(L.spmatrix(reordered = True).T - B[p,p])))
B = eye(n)
trmm(L,B,trans='T')
trmm(L,B)
print("llt(L) - trmm N/T : err = %.3e" % (blas.nrm2(tril(B - As))))
def test_cholesky(self):
L = cp.cspmatrix(self.symb) + self.A
cp.cholesky(L)
Lm = L.spmatrix(reordered=True)
diff = list((cp.tril(cp.perm(Lm * Lm.T, self.symb.ip)) - self.A).V)
self.assertAlmostEqualLists(diff, len(diff) * [0.0])
def solve(self, solver = "mosek"):
if self.to_real == False: raise ValueError("Solvers do not support complex-valued data.")
sol = {}
c,G,h,dims = self.problem_data
if solver == "mosek":
if self.__verbose:
msk.options[msk.mosek.iparam.log] = 1
else:
msk.options[msk.mosek.iparam.log] = 0
solsta,mu,zz = msk.conelp(c,G,matrix(h),dims)
sol['status'] = str(solsta).split('.')[-1]
elif solver == "cvxopt":
if self.__verbose:
options = {'show_progress':True}
else:
options = {'show_progress':False}
csol = solvers.conelp(c,G,matrix(h),dims,options=options)
zz = csol['z']
mu = csol['x']
sol['status'] = csol['status']
else:
raise ValueError("Unknown solver %s" % (solver))
if zz is None: return sol
sol['mu'] = mu
offset = self.offset
sol['t'] = zz[:offset['wpl']]
sol['wpl'] = zz[offset['wpl']:offset['wpu']]
sol['wpu'] = zz[offset['wpu']:offset['wql']]
sol['wql'] = zz[offset['wql']:offset['wqu']]
sol['wqu'] = zz[offset['wqu']:offset['ul']]
sol['ul'] = zz[offset['ul']:offset['uu']]
sol['uu'] = zz[offset['uu']:offset['z']]
sol['z'] = zz[offset['z']:offset['w']]
sol['w'] = zz[offset['w']:offset['X']]
if self.conversion:
dims = self.problem_data[3]
offset = dims['l'] + sum(dims['q'])
self.Xc = []
sol['eigratio'] = []
for k,sk in enumerate(dims['s']):
zk = matrix(zz[offset:offset+sk**2],(sk,sk))
offset += sk**2
zkr = 0.5*(zk[:sk//2,:sk//2] + zk[sk//2:,sk//2:])
zki = 0.5*(zk[sk//2:,:sk//2] - zk[:sk//2,sk//2:])
zki[::sk+1] = 0.0
zk = zkr + complex(0,1.0)*zki
self.Xc.append(zk)
ev = matrix(0.0,(zk.size[0],1),tc='d')
lapack.heev(+zk,ev)
ev = sorted(list(ev),reverse=True)
sol['eigratio'].append(ev[0]/ev[1])
# Build partial Hermitian matrix
z = matrix([zk[:] for zk in self.Xc])
blki,I,J,bn = self.blocks_to_sparse[0]
X = spmatrix(z[blki],I,J)
idx = [i for i,ij in enumerate(zip(X.I,X.J)) if ij[0] > ij[1]]
sol['X'] = chompack.tril(X) + spmatrix(X.V[idx].H, X.J[idx], X.I[idx], X.size)
else:
X = matrix(zz[offset['X']:],(2*self.nbus,2*self.nbus))
Xr = X[:self.nbus,:self.nbus]
Xi = X[self.nbus:,:self.nbus]
Xi[::self.nbus+1] = 0.0
X = Xr + complex(0.0,1.0)*Xi
sol['X'] = +X
V = matrix(0.0,(self.nbus,5),tc='z')
w = matrix(0.0,(self.nbus,1))
lapack.heevr(X, w, Z = V, jobz='V', range='I', il = self.nbus-4, iu = self.nbus)
sol['eigratio'] = [w[4]/w[3]]
if w[4]/w[3] < self.eigtol and self.__verbose:
print("Eigenvalue ratio smaller than %e."%(self.eigtol))
sol['eigs'] = w[:5]
V = V[:,-1]*sqrt(w[4])
sol['V'] = V
# Branch injections
sol['Sf'] = self.baseMVA*(sol['z'][1::6] + complex(0.0,1.0)*sol['z'][2::6])
sol['St'] = self.baseMVA*(sol['z'][4::6] + complex(0.0,1.0)*sol['z'][5::6])
# Generation
sol['Sg'] = (matrix([gen['Pmin'] for gen in self.generators]) +\
matrix([0.0 if gen['pslack'] is None else sol['wpl'][gen['pslack']] for gen in self.generators])) +\
complex(0.0,1.0)*(matrix([gen['Qmin'] for gen in self.generators]) +\
matrix([0.0 if gen['qslack'] is None else sol['wql'][gen['qslack']] for gen in self.generators]))
Pg = sol['Sg'].real()
Qg = sol['Sg'].imag()
for k,gen in enumerate(self.generators):
gen['Pg'] = Pg[k]
gen['Qg'] = Qg[k]
sol['Sg'] *= self.baseMVA
sol['cost'] = 0.0
for ii,gen in enumerate(self.generators):
for nk in range(gen['Pcost']['ncoef']):
sol['cost'] += gen['Pcost']['coef'][-1-nk]*(Pg[ii]*self.baseMVA)**nk
sol['cost_objective'] = -(self.problem_data[0].T*mu)[0]*self.cost_scale + self.const_cost
sol['Vm'] = sqrt(matrix(sol['X'][::self.nbus+1]).real())
return sol
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