def Fkkt(W):
# Factor
#
# S = A*D^-1*A' + I
#
# where D = 2*D1*D2*(D1+D2)^-1, D1 = d[:n]**2, D2 = d[n:]**2.
d1, d2 = W['di'][:n]**2, W['di'][n:]**2
# ds is square root of diagonal of D
ds = sqrt(2.0) * div( mul( W['di'][:n], W['di'][n:]), sqrt(d1+d2) )
d3 = div(d2 - d1, d1 + d2)
# Asc = A*diag(d)^-1/2
blas.copy(A, Asc)
for k in range(m):
blas.tbsv(ds, Asc, n=n, k=0, ldA=1, incx=m, offsetx=k)
# S = I + A * D^-1 * A'
blas.syrk(Asc, S)
S[::m+1] += 1.0
lapack.potrf(S)
def g(x, y, z):
x[:n] = 0.5 * ( x[:n] - mul(d3, x[n:]) + \
mul(d1, z[:n] + mul(d3, z[:n])) - \
mul(d2, z[n:] - mul(d3, z[n:])) )
x[:n] = div( x[:n], ds)
# Solve
#
# S * v = 0.5 * A * D^-1 * ( bx[:n]
# - (D2-D1)*(D1+D2)^-1 * bx[n:]
# + D1 * ( I + (D2-D1)*(D1+D2)^-1 ) * bz[:n]
# - D2 * ( I - (D2-D1)*(D1+D2)^-1 ) * bz[n:] )
blas.gemv(Asc, x, v)
lapack.potrs(S, v)
# x[:n] = D^-1 * ( rhs - A'*v ).
blas.gemv(Asc, v, x, alpha=-1.0, beta=1.0, trans='T')
x[:n] = div(x[:n], ds)
# x[n:] = (D1+D2)^-1 * ( bx[n:] - D1*bz[:n] - D2*bz[n:] )
# - (D2-D1)*(D1+D2)^-1 * x[:n]
x[n:] = div( x[n:] - mul(d1, z[:n]) - mul(d2, z[n:]), d1+d2 )\
- mul( d3, x[:n] )
# z[:n] = D1^1/2 * ( x[:n] - x[n:] - bz[:n] )
# z[n:] = D2^1/2 * ( -x[:n] - x[n:] - bz[n:] ).
z[:n] = mul( W['di'][:n], x[:n] - x[n:] - z[:n] )
z[n:] = mul( W['di'][n:], -x[:n] - x[n:] - z[n:] )
return g
def Fkkt(W):
# Factor
#
# S = A*D^-1*A' + I
#
# where D = 2*D1*D2*(D1+D2)^-1, D1 = d[:n]**2, D2 = d[n:]**2.
d1, d2 = W['di'][:n]**2, W['di'][n:]**2
# ds is square root of diagonal of D
ds = sqrt(2.0) * div(mul(W['di'][:n], W['di'][n:]), sqrt(d1 + d2))
d3 = div(d2 - d1, d1 + d2)
# Asc = A*diag(d)^-1/2
blas.copy(A, Asc)
for k in range(m):
blas.tbsv(ds, Asc, n=n, k=0, ldA=1, incx=m, offsetx=k)
# S = I + A * D^-1 * A'
blas.syrk(Asc, S)
S[::m + 1] += 1.0
lapack.potrf(S)
def g(x, y, z):
x[:n] = 0.5 * ( x[:n] - mul(d3, x[n:]) + \
mul(d1, z[:n] + mul(d3, z[:n])) - \
mul(d2, z[n:] - mul(d3, z[n:])) )
x[:n] = div(x[:n], ds)
# Solve
#
# S * v = 0.5 * A * D^-1 * ( bx[:n]
# - (D2-D1)*(D1+D2)^-1 * bx[n:]
# + D1 * ( I + (D2-D1)*(D1+D2)^-1 ) * bz[:n]
# - D2 * ( I - (D2-D1)*(D1+D2)^-1 ) * bz[n:] )
blas.gemv(Asc, x, v)
lapack.potrs(S, v)
# x[:n] = D^-1 * ( rhs - A'*v ).
blas.gemv(Asc, v, x, alpha=-1.0, beta=1.0, trans='T')
x[:n] = div(x[:n], ds)
# x[n:] = (D1+D2)^-1 * ( bx[n:] - D1*bz[:n] - D2*bz[n:] )
# - (D2-D1)*(D1+D2)^-1 * x[:n]
x[n:] = div( x[n:] - mul(d1, z[:n]) - mul(d2, z[n:]), d1+d2 )\
- mul( d3, x[:n] )
# z[:n] = D1^1/2 * ( x[:n] - x[n:] - bz[:n] )
# z[n:] = D2^1/2 * ( -x[:n] - x[n:] - bz[n:] ).
z[:n] = mul(W['di'][:n], x[:n] - x[n:] - z[:n])
z[n:] = mul(W['di'][n:], -x[:n] - x[n:] - z[n:])
return g
def d_kernel(R, k, norm=True):
m = R.size[0]
n = R.size[1]
x_choose_k = [0]*(n+1)
x_choose_k[0] = 0
for i in range(1, n+1):
x_choose_k[i] = spc.binom(i,k)
nCk = x_choose_k[n]
X = R*R.T
K = co.matrix(0.0, (X.size[0], X.size[1]))
for i in range(m):
for j in range(i, m):
n_niCk = x_choose_k[n - int(X[i,i])]
n_njCk = x_choose_k[n - int(X[j,j])]
n_ni_nj_nijCk = x_choose_k[n - int(X[i,i]) - int(X[j,j]) + int(X[i,j])]
K[i,j] = K[j,i] = nCk - n_niCk - n_njCk + n_ni_nj_nijCk
if norm:
YY = co.matrix([K[i,i] for i in range(K.size[0])])
YY = co.sqrt(YY)**(-1)
K = co.mul(K, YY*YY.T)
return K
def F(x=None, z=None):
if x is None: return 0, matrix(0.0, (n, 1))
y = A * x - b
w = sqrt(rho + y**2)
f = sum(w)
Df = div(y, w).T * A
if z is None: return f, Df
H = A.T * spdiag(z[0] * rho * (w**-3)) * A
return f, Df, H
def normalize_rows(X): """ @param X: the matrix @type X: cvxopt dense matrix @return: the row-normalized matrix @rtype: cvxopt dense matrix """ d = diagonal_vec(X*X.T) N = co.sqrt(d * ones_vec(X.size[1]).T) return co.div(X,N)
def normalize_cols(X):
"""
@param X: the matrix
@type X: cvxopt dense matrix
@return: the col-normalized matrix
@rtype: cvxopt dense matrix
"""
d = diagonal_vec(X.T * X)
N = co.sqrt(ones_vec(X.size[0]) * d.T)
return co.div(X, N)
def normalize_cols(X): """ @param X: the matrix @type X: cvxopt dense matrix @return: the col-normalized matrix @rtype: cvxopt dense matrix """ d = diagonal_vec(X.T*X) N = co.sqrt(ones_vec(X.size[0])*d.T) return co.div(X,N)
def normalize_rows(X):
"""
@param X: the matrix
@type X: cvxopt dense matrix
@return: the row-normalized matrix
@rtype: cvxopt dense matrix
"""
d = diagonal_vec(X * X.T)
N = co.sqrt(d * ones_vec(X.size[1]).T)
return co.div(X, N)
def Fkkt(W):
# Factor
#
# S = A*D^-1*A' + I
#
# where D = 2*D1*D2*(D1+D2)^-1, D1 = d[:n]**-2, D2 = d[n:]**-2.
d1, d2 = W['di'][:n]**2, W['di'][n:]**2
print 'printing: ', W['di']
# ds is square root of diagonal of D
ds = math.sqrt(2.0) * div(mul(W['di'][:n], W['di'][n:]), sqrt(d1 + d2))
d3 = div(d2 - d1, d1 + d2)
Asc = matrix(0.0, (m, n))
# Asc = A*diag(d)^-1/2
Asc = A * spdiag(ds**-1)
# S = I + A * D^-1 * A'
blas.syrk(Asc, S)
S[::m + 1] += 1.0
lapack.potrf(S)
def g(x, y, z):
x[:n] = 0.5 * (x[:n] - mul(d3, x[n:]) + mul(
d1, z[:n] + mul(d3, z[:n])) - mul(d2, z[n:] - mul(d3, z[n:])))
x[:n] = div(x[:n], ds)
# Solve
#
# S * v = 0.5 * A * D^-1 * ( bx[:n] -
# (D2-D1)*(D1+D2)^-1 * bx[n:] +
# D1 * ( I + (D2-D1)*(D1+D2)^-1 ) * bzl[:n] -
# D2 * ( I - (D2-D1)*(D1+D2)^-1 ) * bzl[n:] )
blas.gemv(Asc, x, v)
lapack.potrs(S, v)
# x[:n] = D^-1 * ( rhs - A'*v ).
blas.gemv(Asc, v, x, alpha=-1.0, beta=1.0, trans='T')
x[:n] = div(x[:n], ds)
# x[n:] = (D1+D2)^-1 * ( bx[n:] - D1*bzl[:n] - D2*bzl[n:] )
# - (D2-D1)*(D1+D2)^-1 * x[:n]
x[n:] = div( x[n:] - mul(d1, z[:n]) - mul(d2, z[n:]), d1+d2 )\
- mul( d3, x[:n] )
# zl[:n] = D1^1/2 * ( x[:n] - x[n:] - bzl[:n] )
# zl[n:] = D2^1/2 * ( -x[:n] - x[n:] - bzl[n:] ).
z[:n] = mul(W['di'][:n], x[:n] - x[n:] - z[:n])
z[n:] = mul(W['di'][n:], -x[:n] - x[n:] - z[n:])
return g
def Fkkt(W):
# Factor
#
# S = A*D^-1*A' + I
#
# where D = 2*D1*D2*(D1+D2)^-1, D1 = d[:n]**-2, D2 = d[n:]**-2.
d1, d2 = W['di'][:n]**2, W['di'][n:]**2
# ds is square root of diagonal of D
ds = math.sqrt(2.0) * div( mul( W['di'][:n], W['di'][n:]),
sqrt(d1+d2) )
d3 = div(d2 - d1, d1 + d2)
# Asc = A*diag(d)^-1/2
Asc = A * spdiag(ds**-1)
# S = I + A * D^-1 * A'
blas.syrk(Asc, S)
S[::m+1] += 1.0
lapack.potrf(S)
def g(x, y, z):
x[:n] = 0.5 * ( x[:n] - mul(d3, x[n:]) +
mul(d1, z[:n] + mul(d3, z[:n])) - mul(d2, z[n:] -
mul(d3, z[n:])) )
x[:n] = div( x[:n], ds)
# Solve
#
# S * v = 0.5 * A * D^-1 * ( bx[:n] -
# (D2-D1)*(D1+D2)^-1 * bx[n:] +
# D1 * ( I + (D2-D1)*(D1+D2)^-1 ) * bzl[:n] -
# D2 * ( I - (D2-D1)*(D1+D2)^-1 ) * bzl[n:] )
blas.gemv(Asc, x, v)
lapack.potrs(S, v)
# x[:n] = D^-1 * ( rhs - A'*v ).
blas.gemv(Asc, v, x, alpha=-1.0, beta=1.0, trans='T')
x[:n] = div(x[:n], ds)
# x[n:] = (D1+D2)^-1 * ( bx[n:] - D1*bzl[:n] - D2*bzl[n:] )
# - (D2-D1)*(D1+D2)^-1 * x[:n]
x[n:] = div( x[n:] - mul(d1, z[:n]) - mul(d2, z[n:]), d1+d2 )\
- mul( d3, x[:n] )
# zl[:n] = D1^1/2 * ( x[:n] - x[n:] - bzl[:n] )
# zl[n:] = D2^1/2 * ( -x[:n] - x[n:] - bzl[n:] ).
z[:n] = mul( W['di'][:n], x[:n] - x[n:] - z[:n] )
z[n:] = mul( W['di'][n:], -x[:n] - x[n:] - z[n:] )
return g
def Fkkt(W):
# Returns a function f(x, y, z) that solves
#
# [ 0 0 P' -P' ] [ x[:n] ] [ bx[:n] ]
# [ 0 0 -I -I ] [ x[n:] ] [ bx[n:] ]
# [ P -I -D1^{-1} 0 ] [ z[:m] ] = [ bz[:m] ]
# [-P -I 0 -D2^{-1} ] [ z[m:] ] [ bz[m:] ]
#
# where D1 = diag(di[:m])^2, D2 = diag(di[m:])^2 and di = W['di'].
#
# On entry bx, bz are stored in x, z.
# On exit x, z contain the solution, with z scaled (di .* z is
# returned instead of z).
# Factor A = 4*P'*D*P where D = d1.*d2 ./(d1+d2) and
# d1 = d[:m].^2, d2 = d[m:].^2.
di = W['di']
d1, d2 = di[:m]**2, di[m:]**2
D = div(mul(d1, d2), d1 + d2)
Ds = spdiag(2 * sqrt(D))
base.gemm(Ds, P, Ps)
blas.syrk(Ps, A, trans='T')
lapack.potrf(A)
def f(x, y, z):
# Solve for x[:n]:
#
# A*x[:n] = bx[:n] + P' * ( ((D1-D2)*(D1+D2)^{-1})*bx[n:]
# + (2*D1*D2*(D1+D2)^{-1}) * (bz[:m] - bz[m:]) ).
blas.copy((mul(div(d1 - d2, d1 + d2), x[n:]) +
mul(2 * D, z[:m] - z[m:])), u)
blas.gemv(P, u, x, beta=1.0, trans='T')
lapack.potrs(A, x)
# x[n:] := (D1+D2)^{-1} * (bx[n:] - D1*bz[:m] - D2*bz[m:]
# + (D1-D2)*P*x[:n])
base.gemv(P, x, u)
x[n:] = div(
x[n:] - mul(d1, z[:m]) - mul(d2, z[m:]) + mul(d1 - d2, u),
d1 + d2)
# z[:m] := d1[:m] .* ( P*x[:n] - x[n:] - bz[:m])
# z[m:] := d2[m:] .* (-P*x[:n] - x[n:] - bz[m:])
z[:m] = mul(di[:m], u - x[n:] - z[:m])
z[m:] = mul(di[m:], -u - x[n:] - z[m:])
return f
def Fkkt(W):
# Returns a function f(x, y, z) that solves
#
# [ 0 0 P' -P' ] [ x[:n] ] [ bx[:n] ]
# [ 0 0 -I -I ] [ x[n:] ] [ bx[n:] ]
# [ P -I -D1^{-1} 0 ] [ z[:m] ] = [ bz[:m] ]
# [-P -I 0 -D2^{-1} ] [ z[m:] ] [ bz[m:] ]
#
# where D1 = diag(di[:m])^2, D2 = diag(di[m:])^2 and di = W['di'].
#
# On entry bx, bz are stored in x, z.
# On exit x, z contain the solution, with z scaled (di .* z is
# returned instead of z).
# Factor A = 4*P'*D*P where D = d1.*d2 ./(d1+d2) and
# d1 = d[:m].^2, d2 = d[m:].^2.
di = W['di']
d1, d2 = di[:m]**2, di[m:]**2
D = div( mul(d1,d2), d1+d2 )
Ds = spdiag(2 * sqrt(D))
base.gemm(Ds, P, Ps)
blas.syrk(Ps, A, trans = 'T')
lapack.potrf(A)
def f(x, y, z):
# Solve for x[:n]:
#
# A*x[:n] = bx[:n] + P' * ( ((D1-D2)*(D1+D2)^{-1})*bx[n:]
# + (2*D1*D2*(D1+D2)^{-1}) * (bz[:m] - bz[m:]) ).
blas.copy(( mul( div(d1-d2, d1+d2), x[n:]) +
mul( 2*D, z[:m]-z[m:] ) ), u)
blas.gemv(P, u, x, beta = 1.0, trans = 'T')
lapack.potrs(A, x)
# x[n:] := (D1+D2)^{-1} * (bx[n:] - D1*bz[:m] - D2*bz[m:]
# + (D1-D2)*P*x[:n])
base.gemv(P, x, u)
x[n:] = div( x[n:] - mul(d1, z[:m]) - mul(d2, z[m:]) +
mul(d1-d2, u), d1+d2 )
# z[:m] := d1[:m] .* ( P*x[:n] - x[n:] - bz[:m])
# z[m:] := d2[m:] .* (-P*x[:n] - x[n:] - bz[m:])
z[:m] = mul(di[:m], u-x[n:]-z[:m])
z[m:] = mul(di[m:], -u-x[n:]-z[m:])
return f
def normalize(K):
"""
@param K: the matrix
@type K: cvxopt dense matrix or numpy array
@return: the row-normalized matrix
@rtype: cvxopt dense matrix
"""
if type(K) is np.ndarray:
d = np.array([[K[i, i] for i in range(K.shape[0])]])
return K / np.sqrt(np.dot(d.T, d))
else:
YY = cvx.diagonal_vec(K)
YY = co.sqrt(YY)**(-1)
return co.mul(K, YY * YY.T)
def normalize_cols_sparse(X):
"""
@param X: the matrix
@type X: cvxopt matrix
@return: the col-normalized matrix
@rtype: cvxopt sparse matrix
"""
d = co.matrix([sum(X[:, i].V) for i in xrange(X.size[1])])
N = co.sqrt(d.T)
I, J = X.I, X.J
V = []
for j in X.J:
V += [(1.0 / N[j]) if N[j] != 0.0 else 0.0]
return co.spmatrix(V, I, J)
def normalize(K): """ @param K: the matrix @type K: cvxopt dense matrix or numpy array @return: the row-normalized matrix @rtype: cvxopt dense matrix """ if type(K) is np.ndarray: d = np.array([[K[i,i] for i in range(K.shape[0])]]) return K / np.sqrt(np.dot(d.T,d)) else: YY = cvx.diagonal_vec(K) YY = co.sqrt(YY)**(-1) return co.mul(K, YY*YY.T)
def normalize_cols_sparse(X): """ @param X: the matrix @type X: cvxopt matrix @return: the col-normalized matrix @rtype: cvxopt sparse matrix """ d = co.matrix([sum(X[:,i].V) for i in xrange(X.size[1])]) N = co.sqrt(d.T) I, J = X.I, X.J V = [] for j in X.J: V += [(1.0 / N[j]) if N[j] != 0.0 else 0.0] return co.spmatrix(V, I, J)
# least squares solution: minimize || A*x - b ||_2^2 xls = +b lapack.gels(+A, xls) xls = xls[:n] # Tikhonov solution: minimize || A*x - b ||_2^2 + 0.1*||x||^2_2 xtik = A.T*b S = A.T*A S[::n+1] += 0.1 lapack.posv(S, xtik) # Worst case solution xwc = wcls(A, Ap, b) notrials = 100000 r = sqrt(uniform(1,notrials)) theta = 2.0 * pi * uniform(1,notrials) u = matrix(0.0, (2,notrials)) u[0,:] = mul(r, cos(theta)) u[1,:] = mul(r, sin(theta)) # LS solution q = A*xls - b P = matrix(0.0, (m,2)) P[:,0], P[:,1] = Ap[0]*xls, Ap[1]*xls r = P*u + q[:,notrials*[0]] resls = sqrt( matrix(1.0, (1,m)) * mul(r,r) ) q = A*xtik - b P[:,0], P[:,1] = Ap[0]*xtik, Ap[1]*xtik r = P*u + q[:,notrials*[0]]
# Figures for linear placement.
pylab.figure(1, figsize=(10, 4), facecolor='w')
pylab.subplot(121)
X = matrix(0.0, (N + M, 2))
X[:N, :], X[N:, :] = X1, Xf
pylab.plot(Xf[:, 0], Xf[:, 1], 'sw', mec='k')
pylab.plot(X1[:, 0], X1[:, 1], 'or', ms=10)
for s, t in E:
pylab.plot([X[s, 0], X[t, 0]], [X[s, 1], X[t, 1]], 'b:')
pylab.axis([-1.1, 1.1, -1.1, 1.1])
pylab.axis('equal')
pylab.title('Linear placement')
pylab.subplot(122)
lngths = sqrt((A1 * X1 + B)**2 * matrix(1.0, (2, 1)))
pylab.hist(list(lngths), [.1 * k for k in range(15)])
x = pylab.arange(0, 1.6, 1.6 / 500)
pylab.plot(x, 5.0 / 1.6 * x, '--k')
pylab.axis([0, 1.6, 0, 5.5])
pylab.title('Length distribution')
# Figures for quadratic placement.
pylab.figure(2, figsize=(10, 4), facecolor='w')
pylab.subplot(121)
X[:N, :], X[N:, :] = X2, Xf
pylab.plot(Xf[:, 0], Xf[:, 1], 'sw', mec='k')
pylab.plot(X2[:, 0], X2[:, 1], 'or', ms=10)
for s, t in E:
pylab.plot([X[s, 0], X[t, 0]], [X[s, 1], X[t, 1]], 'b:')
def F(W):
"""
Generate a solver for
A'(uz0) = bx[0]
-uz0 - uz1 = bx[1]
A(ux[0]) - ux[1] - r0*r0' * uz0 * r0*r0' = bz0
- ux[1] - r1*r1' * uz1 * r1*r1' = bz1.
uz0, uz1, bz0, bz1 are symmetric m x m-matrices.
ux[0], bx[0] are n-vectors.
ux[1], bx[1] are symmetric m x m-matrices.
We first calculate a congruence that diagonalizes r0*r0' and r1*r1':
U' * r0 * r0' * U = I, U' * r1 * r1' * U = S.
We then make a change of variables
usx[0] = ux[0],
usx[1] = U' * ux[1] * U
usz0 = U^-1 * uz0 * U^-T
usz1 = U^-1 * uz1 * U^-T
and define
As() = U' * A() * U'
bsx[1] = U^-1 * bx[1] * U^-T
bsz0 = U' * bz0 * U
bsz1 = U' * bz1 * U.
This gives
As'(usz0) = bx[0]
-usz0 - usz1 = bsx[1]
As(usx[0]) - usx[1] - usz0 = bsz0
-usx[1] - S * usz1 * S = bsz1.
1. Eliminate usz0, usz1 using equations 3 and 4,
usz0 = As(usx[0]) - usx[1] - bsz0
usz1 = -S^-1 * (usx[1] + bsz1) * S^-1.
This gives two equations in usx[0] an usx[1].
As'(As(usx[0]) - usx[1]) = bx[0] + As'(bsz0)
-As(usx[0]) + usx[1] + S^-1 * usx[1] * S^-1
= bsx[1] - bsz0 - S^-1 * bsz1 * S^-1.
2. Eliminate usx[1] using equation 2:
usx[1] + S * usx[1] * S
= S * ( As(usx[0]) + bsx[1] - bsz0 ) * S - bsz1
i.e., with Gamma[i,j] = 1.0 + S[i,i] * S[j,j],
usx[1] = ( S * As(usx[0]) * S ) ./ Gamma
+ ( S * ( bsx[1] - bsz0 ) * S - bsz1 ) ./ Gamma.
This gives an equation in usx[0].
As'( As(usx[0]) ./ Gamma )
= bx0 + As'(bsz0) +
As'( (S * ( bsx[1] - bsz0 ) * S - bsz1) ./ Gamma )
= bx0 + As'( ( bsz0 - bsz1 + S * bsx[1] * S ) ./ Gamma ).
"""
# Calculate U s.t.
#
# U' * r0*r0' * U = I, U' * r1*r1' * U = diag(s).
# Cholesky factorization r0 * r0' = L * L'
blas.syrk(W['r'][0], L)
lapack.potrf(L)
# SVD L^-1 * r1 = U * diag(s) * V'
blas.copy(W['r'][1], U)
blas.trsm(L, U)
lapack.gesvd(U, s, jobu='O')
# s := s**2
s[:] = s**2
# Uti := U
blas.copy(U, Uti)
# U := L^-T * U
blas.trsm(L, U, transA='T')
# Uti := L * Uti = U^-T
blas.trmm(L, Uti)
# Us := U * diag(s)^-1
blas.copy(U, Us)
for i in range(m):
blas.tbsv(s, Us, n=m, k=0, ldA=1, incx=m, offsetx=i)
# S is m x m with lower triangular entries s[i] * s[j]
# sqrtG is m x m with lower triangular entries sqrt(1.0 + s[i]*s[j])
# Upper triangular entries are undefined but nonzero.
blas.scal(0.0, S)
blas.syrk(s, S)
Gamma = 1.0 + S
sqrtG = sqrt(Gamma)
# Asc[i] = (U' * Ai * * U ) ./ sqrtG, for i = 1, ..., n
# = Asi ./ sqrt(Gamma)
blas.copy(A, Asc)
misc.scale(
Asc, # only 'r' part of the dictionary is used
{
'dnl': matrix(0.0, (0, 1)),
'dnli': matrix(0.0, (0, 1)),
'd': matrix(0.0, (0, 1)),
'di': matrix(0.0, (0, 1)),
'v': [],
'beta': [],
'r': [U],
'rti': [U]
})
for i in range(n):
blas.tbsv(sqrtG, Asc, n=msq, k=0, ldA=1, offsetx=i * msq)
# Convert columns of Asc to packed storage
misc.pack2(Asc, {'l': 0, 'q': [], 's': [m]})
# Cholesky factorization of Asc' * Asc.
H = matrix(0.0, (n, n))
blas.syrk(Asc, H, trans='T', k=mpckd)
lapack.potrf(H)
def solve(x, y, z):
"""
1. Solve for usx[0]:
Asc'(Asc(usx[0]))
= bx0 + Asc'( ( bsz0 - bsz1 + S * bsx[1] * S ) ./ sqrtG)
= bx0 + Asc'( ( bsz0 + S * ( bsx[1] - bssz1) S )
./ sqrtG)
where bsx[1] = U^-1 * bx[1] * U^-T, bsz0 = U' * bz0 * U,
bsz1 = U' * bz1 * U, bssz1 = S^-1 * bsz1 * S^-1
2. Solve for usx[1]:
usx[1] + S * usx[1] * S
= S * ( As(usx[0]) + bsx[1] - bsz0 ) * S - bsz1
usx[1]
= ( S * (As(usx[0]) + bsx[1] - bsz0) * S - bsz1) ./ Gamma
= -bsz0 + (S * As(usx[0]) * S) ./ Gamma
+ (bsz0 - bsz1 + S * bsx[1] * S ) . / Gamma
= -bsz0 + (S * As(usx[0]) * S) ./ Gamma
+ (bsz0 + S * ( bsx[1] - bssz1 ) * S ) . / Gamma
Unscale ux[1] = Uti * usx[1] * Uti'
3. Compute usz0, usz1
r0' * uz0 * r0 = r0^-1 * ( A(ux[0]) - ux[1] - bz0 ) * r0^-T
r1' * uz1 * r1 = r1^-1 * ( -ux[1] - bz1 ) * r1^-T
"""
# z0 := U' * z0 * U
# = bsz0
__cngrnc(U, z, trans='T')
# z1 := Us' * bz1 * Us
# = S^-1 * U' * bz1 * U * S^-1
# = S^-1 * bsz1 * S^-1
__cngrnc(Us, z, trans='T', offsetx=msq)
# x[1] := Uti' * x[1] * Uti
# = bsx[1]
__cngrnc(Uti, x[1], trans='T')
# x[1] := x[1] - z[msq:]
# = bsx[1] - S^-1 * bsz1 * S^-1
blas.axpy(z, x[1], alpha=-1.0, offsetx=msq)
# x1 = (S * x[1] * S + z[:msq] ) ./ sqrtG
# = (S * ( bsx[1] - S^-1 * bsz1 * S^-1) * S + bsz0 ) ./ sqrtG
# = (S * bsx[1] * S - bsz1 + bsz0 ) ./ sqrtG
# in packed storage
blas.copy(x[1], x1)
blas.tbmv(S, x1, n=msq, k=0, ldA=1)
blas.axpy(z, x1, n=msq)
blas.tbsv(sqrtG, x1, n=msq, k=0, ldA=1)
misc.pack2(x1, {'l': 0, 'q': [], 's': [m]})
# x[0] := x[0] + Asc'*x1
# = bx0 + Asc'( ( bsz0 - bsz1 + S * bsx[1] * S ) ./ sqrtG)
# = bx0 + As'( ( bz0 - bz1 + S * bx[1] * S ) ./ Gamma )
blas.gemv(Asc, x1, x[0], m=mpckd, trans='T', beta=1.0)
# x[0] := H^-1 * x[0]
# = ux[0]
lapack.potrs(H, x[0])
# x1 = Asc(x[0]) .* sqrtG (unpacked)
# = As(x[0])
blas.gemv(Asc, x[0], tmp, m=mpckd)
misc.unpack(tmp, x1, {'l': 0, 'q': [], 's': [m]})
blas.tbmv(sqrtG, x1, n=msq, k=0, ldA=1)
# usx[1] = (x1 + (x[1] - z[:msq])) ./ sqrtG**2
# = (As(ux[0]) + bsx[1] - bsz0 - S^-1 * bsz1 * S^-1)
# ./ Gamma
# x[1] := x[1] - z[:msq]
# = bsx[1] - bsz0 - S^-1 * bsz1 * S^-1
blas.axpy(z, x[1], -1.0, n=msq)
# x[1] := x[1] + x1
# = As(ux) + bsx[1] - bsz0 - S^-1 * bsz1 * S^-1
blas.axpy(x1, x[1])
# x[1] := x[1] / Gammma
# = (As(ux) + bsx[1] - bsz0 + S^-1 * bsz1 * S^-1 ) / Gamma
# = S^-1 * usx[1] * S^-1
blas.tbsv(Gamma, x[1], n=msq, k=0, ldA=1)
# z[msq:] := r1' * U * (-z[msq:] - x[1]) * U * r1
# := -r1' * U * S^-1 * (bsz1 + ux[1]) * S^-1 * U * r1
# := -r1' * uz1 * r1
blas.axpy(x[1], z, n=msq, offsety=msq)
blas.scal(-1.0, z, offset=msq)
__cngrnc(U, z, offsetx=msq)
__cngrnc(W['r'][1], z, trans='T', offsetx=msq)
# x[1] := S * x[1] * S
# = usx1
blas.tbmv(S, x[1], n=msq, k=0, ldA=1)
# z[:msq] = r0' * U' * ( x1 - x[1] - z[:msq] ) * U * r0
# = r0' * U' * ( As(ux) - usx1 - bsz0 ) * U * r0
# = r0' * U' * usz0 * U * r0
# = r0' * uz0 * r0
blas.axpy(x1, z, -1.0, n=msq)
blas.scal(-1.0, z, n=msq)
blas.axpy(x[1], z, -1.0, n=msq)
__cngrnc(U, z)
__cngrnc(W['r'][0], z, trans='T')
# x[1] := Uti * x[1] * Uti'
# = ux[1]
__cngrnc(Uti, x[1])
return solve
def F(W):
"""
Create a solver for the linear equations
C * ux + G' * uzl - 2*A'(uzs21) = bx
-uzs11 = bX1
-uzs22 = bX2
G * ux - Dl^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ uzs11 uzs21' ]
[ ] - r*r' * [ ] * r*r' = bzs
[ -A(ux) -uX2 ] [ uzs21 uzs22 ]
where Dl = diag(W['l']), r = W['r'][0].
On entry, x = (bx, bX1, bX2) and z = [ bzl; bzs[:] ].
On exit, x = (ux, uX1, uX2) and z = [ Dl*uzl; (r'*uzs*r)[:] ].
1. Compute matrices V1, V2 such that (with T = r*r')
[ V1 0 ] [ T11 T21' ] [ V1' 0 ] [ I S' ]
[ ] [ ] [ ] = [ ]
[ 0 V2' ] [ T21 T22 ] [ 0 V2 ] [ S I ]
and S = [ diag(s); 0 ], s a positive q-vector.
2. Factor the mapping X -> X + S * X' * S:
X + S * X' * S = L( L'( X )).
3. Compute scaled mappings: a matrix As with as its columns the
coefficients of the scaled mapping
L^-1( V2' * A() * V1' )
and the matrix Gs = Dl^-1 * G.
4. Cholesky factorization of H = C + Gs'*Gs + 2*As'*As.
"""
# 1. Compute V1, V2, s.
r = W['r'][0]
# LQ factorization R[:q, :] = L1 * Q1.
lapack.lacpy(r, Q1, m = q)
lapack.gelqf(Q1, tau1)
lapack.lacpy(Q1, L1, n = q, uplo = 'L')
lapack.orglq(Q1, tau1)
# LQ factorization R[q:, :] = L2 * Q2.
lapack.lacpy(r, Q2, m = p, offsetA = q)
lapack.gelqf(Q2, tau2)
lapack.lacpy(Q2, L2, n = p, uplo = 'L')
lapack.orglq(Q2, tau2)
# V2, V1, s are computed from an SVD: if
#
# Q2 * Q1' = U * diag(s) * V',
#
# then V1 = V' * L1^-1 and V2 = L2^-T * U.
# T21 = Q2 * Q1.T
blas.gemm(Q2, Q1, T21, transB = 'T')
# SVD T21 = U * diag(s) * V'. Store U in V2 and V' in V1.
lapack.gesvd(T21, s, jobu = 'A', jobvt = 'A', U = V2, Vt = V1)
# # Q2 := Q2 * Q1' without extracting Q1; store T21 in Q2
# this will requires lapack.ormlq or lapack.unmlq
# V2 = L2^-T * U
blas.trsm(L2, V2, transA = 'T')
# V1 = V' * L1^-1
blas.trsm(L1, V1, side = 'R')
# 2. Factorization X + S * X' * S = L( L'( X )).
#
# The factor L is stored as a diagonal matrix D and a sparse lower
# triangular matrix P, such that
#
# L(X)[:] = D**-1 * (I + P) * X[:]
# L^-1(X)[:] = D * (I - P) * X[:].
# SS is q x q with SS[i,j] = si*sj.
blas.scal(0.0, SS)
blas.syr(s, SS)
# For a p x q matrix X, P*X[:] is Y[:] where
#
# Yij = si * sj * Xji if i < j
# = 0 otherwise.
#
P.V = SS[Itril2]
# For a p x q matrix X, D*X[:] is Y[:] where
#
# Yij = Xij / sqrt( 1 - si^2 * sj^2 ) if i < j
# = Xii / sqrt( 1 + si^2 ) if i = j
# = Xij otherwise.
#
DV[Idiag] = sqrt(1.0 + SS[::q+1])
DV[Itriu] = sqrt(1.0 - SS[Itril3]**2)
D.V = DV**-1
# 3. Scaled linear mappings
# Ask := V2' * Ask * V1'
blas.scal(0.0, As)
base.axpy(A, As)
for i in xrange(n):
# tmp := V2' * As[i, :]
blas.gemm(V2, As, tmp, transA = 'T', m = p, n = q, k = p,
ldB = p, offsetB = i*p*q)
# As[:,i] := tmp * V1'
blas.gemm(tmp, V1, As, transB = 'T', m = p, n = q, k = q,
ldC = p, offsetC = i*p*q)
# As := D * (I - P) * As
# = L^-1 * As.
blas.copy(As, As2)
base.gemm(P, As, As2, alpha = -1.0, beta = 1.0)
base.gemm(D, As2, As)
# Gs := Dl^-1 * G
blas.scal(0.0, Gs)
base.axpy(G, Gs)
for k in xrange(n):
blas.tbmv(W['di'], Gs, n = m, k = 0, ldA = 1, offsetx = k*m)
# 4. Cholesky factorization of H = C + Gs' * Gs + 2 * As' * As.
blas.syrk(As, H, trans = 'T', alpha = 2.0)
blas.syrk(Gs, H, trans = 'T', beta = 1.0)
base.axpy(C, H)
lapack.potrf(H)
def f(x, y, z):
"""
Solve
C * ux + G' * uzl - 2*A'(uzs21) = bx
-uzs11 = bX1
-uzs22 = bX2
G * ux - D^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ uzs11 uzs21' ]
[ ] - T * [ ] * T = bzs.
[ -A(ux) -uX2 ] [ uzs21 uzs22 ]
On entry, x = (bx, bX1, bX2) and z = [ bzl; bzs[:] ].
On exit, x = (ux, uX1, uX2) and z = [ D*uzl; (r'*uzs*r)[:] ].
Define X = uzs21, Z = T * uzs * T:
C * ux + G' * uzl - 2*A'(X) = bx
[ 0 X' ] [ bX1 0 ]
T * [ ] * T - Z = T * [ ] * T
[ X 0 ] [ 0 bX2 ]
G * ux - D^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ Z11 Z21' ]
[ ] - [ ] = bzs
[ -A(ux) -uX2 ] [ Z21 Z22 ]
Return x = (ux, uX1, uX2), z = [ D*uzl; (rti'*Z*rti)[:] ].
We use the congruence transformation
[ V1 0 ] [ T11 T21' ] [ V1' 0 ] [ I S' ]
[ ] [ ] [ ] = [ ]
[ 0 V2' ] [ T21 T22 ] [ 0 V2 ] [ S I ]
and the factorization
X + S * X' * S = L( L'(X) )
to write this as
C * ux + G' * uzl - 2*A'(X) = bx
L'(V2^-1 * X * V1^-1) - L^-1(V2' * Z21 * V1') = bX
G * ux - D^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ Z11 Z21' ]
[ ] - [ ] = bzs,
[ -A(ux) -uX2 ] [ Z21 Z22 ]
or
C * ux + Gs' * uuzl - 2*As'(XX) = bx
XX - ZZ21 = bX
Gs * ux - uuzl = D^-1 * bzl
-As(ux) - ZZ21 = bbzs_21
-uX1 - Z11 = bzs_11
-uX2 - Z22 = bzs_22
if we introduce scaled variables
uuzl = D * uzl
XX = L'(V2^-1 * X * V1^-1)
= L'(V2^-1 * uzs21 * V1^-1)
ZZ21 = L^-1(V2' * Z21 * V1')
and define
bbzs_21 = L^-1(V2' * bzs_21 * V1')
[ bX1 0 ]
bX = L^-1( V2' * (T * [ ] * T)_21 * V1').
[ 0 bX2 ]
Eliminating Z21 gives
C * ux + Gs' * uuzl - 2*As'(XX) = bx
Gs * ux - uuzl = D^-1 * bzl
-As(ux) - XX = bbzs_21 - bX
-uX1 - Z11 = bzs_11
-uX2 - Z22 = bzs_22
and eliminating uuzl and XX gives
H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bX - bbzs_21)
Gs * ux - uuzl = D^-1 * bzl
-As(ux) - XX = bbzs_21 - bX
-uX1 - Z11 = bzs_11
-uX2 - Z22 = bzs_22.
In summary, we can use the following algorithm:
1. bXX := bX - bbzs21
[ bX1 0 ]
= L^-1( V2' * ((T * [ ] * T)_21 - bzs_21) * V1')
[ 0 bX2 ]
2. Solve H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bXX).
3. From ux, compute
uuzl = Gs*ux - D^-1 * bzl and
X = V2 * L^-T(-As(ux) + bXX) * V1.
4. Return ux, uuzl,
rti' * Z * rti = r' * [ -bX1, X'; X, -bX2 ] * r
and uX1 = -Z11 - bzs_11, uX2 = -Z22 - bzs_22.
"""
# Save bzs_11, bzs_22, bzs_21.
lapack.lacpy(z, bz11, uplo = 'L', m = q, n = q, ldA = p+q,
offsetA = m)
lapack.lacpy(z, bz21, m = p, n = q, ldA = p+q, offsetA = m+q)
lapack.lacpy(z, bz22, uplo = 'L', m = p, n = p, ldA = p+q,
offsetA = m + (p+q+1)*q)
# zl := D^-1 * zl
# = D^-1 * bzl
blas.tbmv(W['di'], z, n = m, k = 0, ldA = 1)
# zs := r' * [ bX1, 0; 0, bX2 ] * r.
# zs := [ bX1, 0; 0, bX2 ]
blas.scal(0.0, z, offset = m)
lapack.lacpy(x[1], z, uplo = 'L', m = q, n = q, ldB = p+q,
offsetB = m)
lapack.lacpy(x[2], z, uplo = 'L', m = p, n = p, ldB = p+q,
offsetB = m + (p+q+1)*q)
# scale diagonal of zs by 1/2
blas.scal(0.5, z, inc = p+q+1, offset = m)
# a := tril(zs)*r
blas.copy(r, a)
blas.trmm(z, a, side = 'L', m = p+q, n = p+q, ldA = p+q, ldB =
p+q, offsetA = m)
# zs := a'*r + r'*a
blas.syr2k(r, a, z, trans = 'T', n = p+q, k = p+q, ldB = p+q,
ldC = p+q, offsetC = m)
# bz21 := L^-1( V2' * ((r * zs * r')_21 - bz21) * V1')
#
# [ bX1 0 ]
# = L^-1( V2' * ((T * [ ] * T)_21 - bz21) * V1').
# [ 0 bX2 ]
# a = [ r21 r22 ] * z
# = [ r21 r22 ] * r' * [ bX1, 0; 0, bX2 ] * r
# = [ T21 T22 ] * [ bX1, 0; 0, bX2 ] * r
blas.symm(z, r, a, side = 'R', m = p, n = p+q, ldA = p+q,
ldC = p+q, offsetB = q)
# bz21 := -bz21 + a * [ r11, r12 ]'
# = -bz21 + (T * [ bX1, 0; 0, bX2 ] * T)_21
blas.gemm(a, r, bz21, transB = 'T', m = p, n = q, k = p+q,
beta = -1.0, ldA = p+q, ldC = p)
# bz21 := V2' * bz21 * V1'
# = V2' * (-bz21 + (T*[bX1, 0; 0, bX2]*T)_21) * V1'
blas.gemm(V2, bz21, tmp, transA = 'T', m = p, n = q, k = p,
ldB = p)
blas.gemm(tmp, V1, bz21, transB = 'T', m = p, n = q, k = q,
ldC = p)
# bz21[:] := D * (I-P) * bz21[:]
# = L^-1 * bz21[:]
# = bXX[:]
blas.copy(bz21, tmp)
base.gemv(P, bz21, tmp, alpha = -1.0, beta = 1.0)
base.gemv(D, tmp, bz21)
# Solve H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bXX).
# x[0] := x[0] + Gs'*zl + 2*As'(bz21)
# = bx + G' * D^-1 * bzl + 2 * As'(bXX)
blas.gemv(Gs, z, x[0], trans = 'T', alpha = 1.0, beta = 1.0)
blas.gemv(As, bz21, x[0], trans = 'T', alpha = 2.0, beta = 1.0)
# x[0] := H \ x[0]
# = ux
lapack.potrs(H, x[0])
# uuzl = Gs*ux - D^-1 * bzl
blas.gemv(Gs, x[0], z, alpha = 1.0, beta = -1.0)
# bz21 := V2 * L^-T(-As(ux) + bz21) * V1
# = X
blas.gemv(As, x[0], bz21, alpha = -1.0, beta = 1.0)
blas.tbsv(DV, bz21, n = p*q, k = 0, ldA = 1)
blas.copy(bz21, tmp)
base.gemv(P, tmp, bz21, alpha = -1.0, beta = 1.0, trans = 'T')
blas.gemm(V2, bz21, tmp)
blas.gemm(tmp, V1, bz21)
# zs := -zs + r' * [ 0, X'; X, 0 ] * r
# = r' * [ -bX1, X'; X, -bX2 ] * r.
# a := bz21 * [ r11, r12 ]
# = X * [ r11, r12 ]
blas.gemm(bz21, r, a, m = p, n = p+q, k = q, ldA = p, ldC = p+q)
# z := -z + [ r21, r22 ]' * a + a' * [ r21, r22 ]
# = rti' * uzs * rti
blas.syr2k(r, a, z, trans = 'T', beta = -1.0, n = p+q, k = p,
offsetA = q, offsetC = m, ldB = p+q, ldC = p+q)
# uX1 = -Z11 - bzs_11
# = -(r*zs*r')_11 - bzs_11
# uX2 = -Z22 - bzs_22
# = -(r*zs*r')_22 - bzs_22
blas.copy(bz11, x[1])
blas.copy(bz22, x[2])
# scale diagonal of zs by 1/2
blas.scal(0.5, z, inc = p+q+1, offset = m)
# a := r*tril(zs)
blas.copy(r, a)
blas.trmm(z, a, side = 'R', m = p+q, n = p+q, ldA = p+q, ldB =
p+q, offsetA = m)
# x[1] := -x[1] - a[:q,:] * r[:q, :]' - r[:q,:] * a[:q,:]'
# = -bzs_11 - (r*zs*r')_11
blas.syr2k(a, r, x[1], n = q, alpha = -1.0, beta = -1.0)
# x[2] := -x[2] - a[q:,:] * r[q:, :]' - r[q:,:] * a[q:,:]'
# = -bzs_22 - (r*zs*r')_22
blas.syr2k(a, r, x[2], n = p, alpha = -1.0, beta = -1.0,
offsetA = q, offsetB = q)
# scale diagonal of zs by 1/2
blas.scal(2.0, z, inc = p+q+1, offset = m)
return f
def F(W):
"""
Create a solver for the linear equations
C * ux + G' * uzl - 2*A'(uzs21) = bx
-uzs11 = bX1
-uzs22 = bX2
G * ux - Dl^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ uzs11 uzs21' ]
[ ] - r*r' * [ ] * r*r' = bzs
[ -A(ux) -uX2 ] [ uzs21 uzs22 ]
where Dl = diag(W['l']), r = W['r'][0].
On entry, x = (bx, bX1, bX2) and z = [ bzl; bzs[:] ].
On exit, x = (ux, uX1, uX2) and z = [ Dl*uzl; (r'*uzs*r)[:] ].
1. Compute matrices V1, V2 such that (with T = r*r')
[ V1 0 ] [ T11 T21' ] [ V1' 0 ] [ I S' ]
[ ] [ ] [ ] = [ ]
[ 0 V2' ] [ T21 T22 ] [ 0 V2 ] [ S I ]
and S = [ diag(s); 0 ], s a positive q-vector.
2. Factor the mapping X -> X + S * X' * S:
X + S * X' * S = L( L'( X )).
3. Compute scaled mappings: a matrix As with as its columns the
coefficients of the scaled mapping
L^-1( V2' * A() * V1' )
and the matrix Gs = Dl^-1 * G.
4. Cholesky factorization of H = C + Gs'*Gs + 2*As'*As.
"""
# 1. Compute V1, V2, s.
r = W['r'][0]
# LQ factorization R[:q, :] = L1 * Q1.
lapack.lacpy(r, Q1, m=q)
lapack.gelqf(Q1, tau1)
lapack.lacpy(Q1, L1, n=q, uplo='L')
lapack.orglq(Q1, tau1)
# LQ factorization R[q:, :] = L2 * Q2.
lapack.lacpy(r, Q2, m=p, offsetA=q)
lapack.gelqf(Q2, tau2)
lapack.lacpy(Q2, L2, n=p, uplo='L')
lapack.orglq(Q2, tau2)
# V2, V1, s are computed from an SVD: if
#
# Q2 * Q1' = U * diag(s) * V',
#
# then V1 = V' * L1^-1 and V2 = L2^-T * U.
# T21 = Q2 * Q1.T
blas.gemm(Q2, Q1, T21, transB='T')
# SVD T21 = U * diag(s) * V'. Store U in V2 and V' in V1.
lapack.gesvd(T21, s, jobu='A', jobvt='A', U=V2, Vt=V1)
# # Q2 := Q2 * Q1' without extracting Q1; store T21 in Q2
# this will requires lapack.ormlq or lapack.unmlq
# V2 = L2^-T * U
blas.trsm(L2, V2, transA='T')
# V1 = V' * L1^-1
blas.trsm(L1, V1, side='R')
# 2. Factorization X + S * X' * S = L( L'( X )).
#
# The factor L is stored as a diagonal matrix D and a sparse lower
# triangular matrix P, such that
#
# L(X)[:] = D**-1 * (I + P) * X[:]
# L^-1(X)[:] = D * (I - P) * X[:].
# SS is q x q with SS[i,j] = si*sj.
blas.scal(0.0, SS)
blas.syr(s, SS)
# For a p x q matrix X, P*X[:] is Y[:] where
#
# Yij = si * sj * Xji if i < j
# = 0 otherwise.
#
P.V = SS[Itril2]
# For a p x q matrix X, D*X[:] is Y[:] where
#
# Yij = Xij / sqrt( 1 - si^2 * sj^2 ) if i < j
# = Xii / sqrt( 1 + si^2 ) if i = j
# = Xij otherwise.
#
DV[Idiag] = sqrt(1.0 + SS[::q + 1])
DV[Itriu] = sqrt(1.0 - SS[Itril3]**2)
D.V = DV**-1
# 3. Scaled linear mappings
# Ask := V2' * Ask * V1'
blas.scal(0.0, As)
base.axpy(A, As)
for i in xrange(n):
# tmp := V2' * As[i, :]
blas.gemm(V2,
As,
tmp,
transA='T',
m=p,
n=q,
k=p,
ldB=p,
offsetB=i * p * q)
# As[:,i] := tmp * V1'
blas.gemm(tmp,
V1,
As,
transB='T',
m=p,
n=q,
k=q,
ldC=p,
offsetC=i * p * q)
# As := D * (I - P) * As
# = L^-1 * As.
blas.copy(As, As2)
base.gemm(P, As, As2, alpha=-1.0, beta=1.0)
base.gemm(D, As2, As)
# Gs := Dl^-1 * G
blas.scal(0.0, Gs)
base.axpy(G, Gs)
for k in xrange(n):
blas.tbmv(W['di'], Gs, n=m, k=0, ldA=1, offsetx=k * m)
# 4. Cholesky factorization of H = C + Gs' * Gs + 2 * As' * As.
blas.syrk(As, H, trans='T', alpha=2.0)
blas.syrk(Gs, H, trans='T', beta=1.0)
base.axpy(C, H)
lapack.potrf(H)
def f(x, y, z):
"""
Solve
C * ux + G' * uzl - 2*A'(uzs21) = bx
-uzs11 = bX1
-uzs22 = bX2
G * ux - D^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ uzs11 uzs21' ]
[ ] - T * [ ] * T = bzs.
[ -A(ux) -uX2 ] [ uzs21 uzs22 ]
On entry, x = (bx, bX1, bX2) and z = [ bzl; bzs[:] ].
On exit, x = (ux, uX1, uX2) and z = [ D*uzl; (r'*uzs*r)[:] ].
Define X = uzs21, Z = T * uzs * T:
C * ux + G' * uzl - 2*A'(X) = bx
[ 0 X' ] [ bX1 0 ]
T * [ ] * T - Z = T * [ ] * T
[ X 0 ] [ 0 bX2 ]
G * ux - D^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ Z11 Z21' ]
[ ] - [ ] = bzs
[ -A(ux) -uX2 ] [ Z21 Z22 ]
Return x = (ux, uX1, uX2), z = [ D*uzl; (rti'*Z*rti)[:] ].
We use the congruence transformation
[ V1 0 ] [ T11 T21' ] [ V1' 0 ] [ I S' ]
[ ] [ ] [ ] = [ ]
[ 0 V2' ] [ T21 T22 ] [ 0 V2 ] [ S I ]
and the factorization
X + S * X' * S = L( L'(X) )
to write this as
C * ux + G' * uzl - 2*A'(X) = bx
L'(V2^-1 * X * V1^-1) - L^-1(V2' * Z21 * V1') = bX
G * ux - D^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ Z11 Z21' ]
[ ] - [ ] = bzs,
[ -A(ux) -uX2 ] [ Z21 Z22 ]
or
C * ux + Gs' * uuzl - 2*As'(XX) = bx
XX - ZZ21 = bX
Gs * ux - uuzl = D^-1 * bzl
-As(ux) - ZZ21 = bbzs_21
-uX1 - Z11 = bzs_11
-uX2 - Z22 = bzs_22
if we introduce scaled variables
uuzl = D * uzl
XX = L'(V2^-1 * X * V1^-1)
= L'(V2^-1 * uzs21 * V1^-1)
ZZ21 = L^-1(V2' * Z21 * V1')
and define
bbzs_21 = L^-1(V2' * bzs_21 * V1')
[ bX1 0 ]
bX = L^-1( V2' * (T * [ ] * T)_21 * V1').
[ 0 bX2 ]
Eliminating Z21 gives
C * ux + Gs' * uuzl - 2*As'(XX) = bx
Gs * ux - uuzl = D^-1 * bzl
-As(ux) - XX = bbzs_21 - bX
-uX1 - Z11 = bzs_11
-uX2 - Z22 = bzs_22
and eliminating uuzl and XX gives
H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bX - bbzs_21)
Gs * ux - uuzl = D^-1 * bzl
-As(ux) - XX = bbzs_21 - bX
-uX1 - Z11 = bzs_11
-uX2 - Z22 = bzs_22.
In summary, we can use the following algorithm:
1. bXX := bX - bbzs21
[ bX1 0 ]
= L^-1( V2' * ((T * [ ] * T)_21 - bzs_21) * V1')
[ 0 bX2 ]
2. Solve H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bXX).
3. From ux, compute
uuzl = Gs*ux - D^-1 * bzl and
X = V2 * L^-T(-As(ux) + bXX) * V1.
4. Return ux, uuzl,
rti' * Z * rti = r' * [ -bX1, X'; X, -bX2 ] * r
and uX1 = -Z11 - bzs_11, uX2 = -Z22 - bzs_22.
"""
# Save bzs_11, bzs_22, bzs_21.
lapack.lacpy(z, bz11, uplo='L', m=q, n=q, ldA=p + q, offsetA=m)
lapack.lacpy(z, bz21, m=p, n=q, ldA=p + q, offsetA=m + q)
lapack.lacpy(z,
bz22,
uplo='L',
m=p,
n=p,
ldA=p + q,
offsetA=m + (p + q + 1) * q)
# zl := D^-1 * zl
# = D^-1 * bzl
blas.tbmv(W['di'], z, n=m, k=0, ldA=1)
# zs := r' * [ bX1, 0; 0, bX2 ] * r.
# zs := [ bX1, 0; 0, bX2 ]
blas.scal(0.0, z, offset=m)
lapack.lacpy(x[1], z, uplo='L', m=q, n=q, ldB=p + q, offsetB=m)
lapack.lacpy(x[2],
z,
uplo='L',
m=p,
n=p,
ldB=p + q,
offsetB=m + (p + q + 1) * q)
# scale diagonal of zs by 1/2
blas.scal(0.5, z, inc=p + q + 1, offset=m)
# a := tril(zs)*r
blas.copy(r, a)
blas.trmm(z,
a,
side='L',
m=p + q,
n=p + q,
ldA=p + q,
ldB=p + q,
offsetA=m)
# zs := a'*r + r'*a
blas.syr2k(r,
a,
z,
trans='T',
n=p + q,
k=p + q,
ldB=p + q,
ldC=p + q,
offsetC=m)
# bz21 := L^-1( V2' * ((r * zs * r')_21 - bz21) * V1')
#
# [ bX1 0 ]
# = L^-1( V2' * ((T * [ ] * T)_21 - bz21) * V1').
# [ 0 bX2 ]
# a = [ r21 r22 ] * z
# = [ r21 r22 ] * r' * [ bX1, 0; 0, bX2 ] * r
# = [ T21 T22 ] * [ bX1, 0; 0, bX2 ] * r
blas.symm(z,
r,
a,
side='R',
m=p,
n=p + q,
ldA=p + q,
ldC=p + q,
offsetB=q)
# bz21 := -bz21 + a * [ r11, r12 ]'
# = -bz21 + (T * [ bX1, 0; 0, bX2 ] * T)_21
blas.gemm(a,
r,
bz21,
transB='T',
m=p,
n=q,
k=p + q,
beta=-1.0,
ldA=p + q,
ldC=p)
# bz21 := V2' * bz21 * V1'
# = V2' * (-bz21 + (T*[bX1, 0; 0, bX2]*T)_21) * V1'
blas.gemm(V2, bz21, tmp, transA='T', m=p, n=q, k=p, ldB=p)
blas.gemm(tmp, V1, bz21, transB='T', m=p, n=q, k=q, ldC=p)
# bz21[:] := D * (I-P) * bz21[:]
# = L^-1 * bz21[:]
# = bXX[:]
blas.copy(bz21, tmp)
base.gemv(P, bz21, tmp, alpha=-1.0, beta=1.0)
base.gemv(D, tmp, bz21)
# Solve H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bXX).
# x[0] := x[0] + Gs'*zl + 2*As'(bz21)
# = bx + G' * D^-1 * bzl + 2 * As'(bXX)
blas.gemv(Gs, z, x[0], trans='T', alpha=1.0, beta=1.0)
blas.gemv(As, bz21, x[0], trans='T', alpha=2.0, beta=1.0)
# x[0] := H \ x[0]
# = ux
lapack.potrs(H, x[0])
# uuzl = Gs*ux - D^-1 * bzl
blas.gemv(Gs, x[0], z, alpha=1.0, beta=-1.0)
# bz21 := V2 * L^-T(-As(ux) + bz21) * V1
# = X
blas.gemv(As, x[0], bz21, alpha=-1.0, beta=1.0)
blas.tbsv(DV, bz21, n=p * q, k=0, ldA=1)
blas.copy(bz21, tmp)
base.gemv(P, tmp, bz21, alpha=-1.0, beta=1.0, trans='T')
blas.gemm(V2, bz21, tmp)
blas.gemm(tmp, V1, bz21)
# zs := -zs + r' * [ 0, X'; X, 0 ] * r
# = r' * [ -bX1, X'; X, -bX2 ] * r.
# a := bz21 * [ r11, r12 ]
# = X * [ r11, r12 ]
blas.gemm(bz21, r, a, m=p, n=p + q, k=q, ldA=p, ldC=p + q)
# z := -z + [ r21, r22 ]' * a + a' * [ r21, r22 ]
# = rti' * uzs * rti
blas.syr2k(r,
a,
z,
trans='T',
beta=-1.0,
n=p + q,
k=p,
offsetA=q,
offsetC=m,
ldB=p + q,
ldC=p + q)
# uX1 = -Z11 - bzs_11
# = -(r*zs*r')_11 - bzs_11
# uX2 = -Z22 - bzs_22
# = -(r*zs*r')_22 - bzs_22
blas.copy(bz11, x[1])
blas.copy(bz22, x[2])
# scale diagonal of zs by 1/2
blas.scal(0.5, z, inc=p + q + 1, offset=m)
# a := r*tril(zs)
blas.copy(r, a)
blas.trmm(z,
a,
side='R',
m=p + q,
n=p + q,
ldA=p + q,
ldB=p + q,
offsetA=m)
# x[1] := -x[1] - a[:q,:] * r[:q, :]' - r[:q,:] * a[:q,:]'
# = -bzs_11 - (r*zs*r')_11
blas.syr2k(a, r, x[1], n=q, alpha=-1.0, beta=-1.0)
# x[2] := -x[2] - a[q:,:] * r[q:, :]' - r[q:,:] * a[q:,:]'
# = -bzs_22 - (r*zs*r')_22
blas.syr2k(a,
r,
x[2],
n=p,
alpha=-1.0,
beta=-1.0,
offsetA=q,
offsetB=q)
# scale diagonal of zs by 1/2
blas.scal(2.0, z, inc=p + q + 1, offset=m)
return f
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
if pylab_installed:
# Figures for linear placement.
pylab.figure(1, figsize=(10,4), facecolor='w')
pylab.subplot(121)
X = matrix(0.0, (N+M,2))
X[:N,:], X[N:,:] = X1, Xf
pylab.plot(Xf[:,0], Xf[:,1], 'sw', mec = 'k')
pylab.plot(X1[:,0], X1[:,1], 'or', ms=10)
for s, t in E: pylab.plot([X[s,0], X[t,0]], [X[s,1],X[t,1]], 'b:')
pylab.axis([-1.1, 1.1, -1.1, 1.1])
pylab.axis('equal')
pylab.title('Linear placement')
pylab.subplot(122)
lngths = sqrt((A1*X1 + B)**2 * matrix(1.0, (2,1)))
pylab.hist(list(lngths), [.1*k for k in range(15)])
x = pylab.arange(0, 1.6, 1.6/500)
pylab.plot( x, 5.0/1.6*x, '--k')
pylab.axis([0, 1.6, 0, 5.5])
pylab.title('Length distribution')
# Figures for quadratic placement.
pylab.figure(2, figsize=(10,4), facecolor='w')
pylab.subplot(121)
X[:N,:], X[N:,:] = X2, Xf
pylab.plot(Xf[:,0], Xf[:,1], 'sw', mec = 'k')
pylab.plot(X2[:,0], X2[:,1], 'or', ms=10)
for s, t in E: pylab.plot([X[s,0], X[t,0]], [X[s,1],X[t,1]], 'b:')
def Fkkt(W):
"""
Custom solver:
v := alpha * 2*A'*A * u + beta * v
"""
global mmS
mmS = matrix(0.0, (iR, iR))
global vvV
vvV = matrix(0.0, (iR, 1))
# Factor
#
# S = A*D^-1*A' + I
#
# where D = 2*D1*D2*(D1+D2)^-1, D1 = d[:n]**2, D2 = d[n:]**2.
mmAsc = matrix(0.0, (iR, iC))
d1, d2 = W['di'][:iC]**2, W['di'][iC:]**2
# ds is square root of diagonal of D
ds = sqrt(2.0) * div(mul(W['di'][:iC], W['di'][iC:]), sqrt(d1 + d2))
d3 = div(d2 - d1, d1 + d2)
# Asc = A*diag(d)^-1/2
blas.copy(mmTh, mmAsc)
for k in range(iR):
blas.tbsv(ds, mmAsc, n=iC, k=0, ldA=1, incx=iR, offsetx=k)
# S = I + A * D^-1 * A'
blas.syrk(mmAsc, mmS)
mmS[::iR + 1] += 1.0
lapack.potrf(mmS)
def g(x, y, z):
x[:iC] = 0.5 * ( x[:iC] - mul(d3, x[iC:]) + \
mul(d1, z[:iC] + mul(d3, z[:iC])) - \
mul(d2, z[iC:] - mul(d3, z[iC:])) )
x[:iC] = div(x[:iC], ds)
# Solve
#
# S * v = 0.5 * A * D^-1 * ( bx[:n]
# - (D2-D1)*(D1+D2)^-1 * bx[n:]
# + D1 * ( I + (D2-D1)*(D1+D2)^-1 ) * bz[:n]
# - D2 * ( I - (D2-D1)*(D1+D2)^-1 ) * bz[n:] )
blas.gemv(mmAsc, x, vvV)
lapack.potrs(mmS, vvV)
# x[:n] = D^-1 * ( rhs - A'*v ).
blas.gemv(mmAsc, vvV, x, alpha=-1.0, beta=1.0, trans='T')
x[:iC] = div(x[:iC], ds)
# x[n:] = (D1+D2)^-1 * ( bx[n:] - D1*bz[:n] - D2*bz[n:] )
# - (D2-D1)*(D1+D2)^-1 * x[:n]
x[iC:] = div( x[iC:] - mul(d1, z[:iC]) - mul(d2, z[iC:]), d1+d2 )\
- mul( d3, x[:iC] )
# z[:n] = D1^1/2 * ( x[:n] - x[n:] - bz[:n] )
# z[n:] = D2^1/2 * ( -x[:n] - x[n:] - bz[n:] ).
z[:iC] = mul(W['di'][:iC], x[:iC] - x[iC:] - z[:iC])
z[iC:] = mul(W['di'][iC:], -x[:iC] - x[iC:] - z[iC:])
return g
def F(W):
"""
Custom solver for the system
[ It 0 0 Xt' 0 At1' ... Atk' ][ dwt ] [ rwt ]
[ 0 0 0 -d' 0 0 ... 0 ][ db ] [ rb ]
[ 0 0 0 -I -I 0 ... 0 ][ dv ] [ rv ]
[ Xt -d -I -Wl1^-2 ][ dzl1 ] [ rl1 ]
[ 0 0 -I -Wl2^-2 ][ dzl2 ] = [ rl2 ]
[ At1 0 0 -W1^-2 ][ dz1 ] [ r1 ]
[ | | | . ][ | ] [ | ]
[ Atk 0 0 -Wk^-2 ][ dzk ] [ rk ]
where
It = [ I 0 ] Xt = [ -D*X E ] Ati = [ 0 -e_i' ]
[ 0 0 ] [ -Pi 0 ]
dwt = [ dw ] rwt = [ rw ]
[ dt ] [ rt ].
"""
# scalings and 'intermediate' vectors
# db = inv(Wl1)^2 + inv(Wl2)^2
db = W['di'][:m]**2 + W['di'][m:2*m]**2
dbi = div(1.0,db)
# dt = I - inv(Wl1)*Dbi*inv(Wl1)
dt = 1.0 - mul(W['di'][:m]**2,dbi)
dtsqrt = sqrt(dt)
# lam = Dt*inv(Wl1)*d
lam = mul(dt,mul(W['di'][:m],d))
# lt = E'*inv(Wl1)*lam
lt = matrix(0.0,(k,1))
base.gemv(E, mul(W['di'][:m],lam), lt, trans = 'T')
# Xs = sqrt(Dt)*inv(Wl1)*X
tmp = mul(dtsqrt,W['di'][:m])
Xs = spmatrix(tmp,range(m),range(m))*X
# Es = D*sqrt(Dt)*inv(Wl1)*E
Es = spmatrix(mul(d,tmp),range(m),range(m))*E
# form Ab = I + sum((1/bi)^2*(Pi'*Pi + 4*(v'*v + 1)*Pi'*y*y'*Pi)) + Xs'*Xs
# and Bb = -sum((1/bi)^2*(4*ui*v'*v*Pi'*y*ei')) - Xs'*Es
# and D2 = Es'*Es + sum((1/bi)^2*(1+4*ui^2*(v'*v - 1))
Ab = matrix(0.0,(n,n))
Ab[::n+1] = 1.0
base.syrk(Xs,Ab,trans = 'T', beta = 1.0)
Bb = matrix(0.0,(n,k))
Bb = -Xs.T*Es # inefficient!?
D2 = spmatrix(0.0,range(k),range(k))
base.syrk(Es,D2,trans = 'T', partial = True)
d2 = +D2.V
del D2
py = matrix(0.0,(n,1))
for i in range(k):
binvsq = (1.0/W['beta'][i])**2
Ab += binvsq*Pt[i]
dvv = blas.dot(W['v'][i],W['v'][i])
blas.gemv(P[i], W['v'][i][1:], py, trans = 'T', alpha = 1.0, beta = 0.0)
blas.syrk(py, Ab, alpha = 4*binvsq*(dvv+1), beta = 1.0)
Bb[:,i] -= 4*binvsq*W['v'][i][0]*dvv*py
d2[i] += binvsq*(1+4*(W['v'][i][0]**2)*(dvv-1))
d2i = div(1.0,d2)
d2isqrt = sqrt(d2i)
# compute a = alpha - lam'*inv(Wl1)*E*inv(D2)*E'*inv(Wl1)*lam
alpha = blas.dot(lam,mul(W['di'][:m],d))
tmp = matrix(0.0,(k,1))
base.gemv(E,mul(W['di'][:m],lam), tmp, trans = 'T')
tmp = mul(tmp, d2isqrt) #tmp = inv(D2)^(1/2)*E'*inv(Wl1)*lam
a = alpha - blas.dot(tmp,tmp)
# compute M12 = X'*D*inv(Wl1)*lam + Bb*inv(D2)*E'*inv(Wl1)*lam
tmp = mul(tmp, d2isqrt)
M12 = matrix(0.0,(n,1))
blas.gemv(Bb,tmp,M12, alpha = 1.0)
tmp = mul(d,mul(W['di'][:m],lam))
blas.gemv(X,tmp,M12, trans = 'T', alpha = 1.0, beta = 1.0)
# form and factor M
sBb = Bb * spmatrix(d2isqrt,range(k), range(k))
base.syrk(sBb, Ab, alpha = -1.0, beta = 1.0)
M = matrix([[Ab, M12.T],[M12, a]])
lapack.potrf(M)
def f(x,y,z):
# residuals
rwt = x[:n+k]
rb = x[n+k]
rv = x[n+k+1:n+k+1+m]
iw_rl1 = mul(W['di'][:m],z[:m])
iw_rl2 = mul(W['di'][m:2*m],z[m:2*m])
ri = [z[2*m+i*(n+1):2*m+(i+1)*(n+1)] for i in range(k)]
# compute 'derived' residuals
# rbwt = rwt + sum(Ai'*inv(Wi)^2*ri) + [-X'*D; E']*inv(Wl1)^2*rl1
rbwt = +rwt
for i in range(k):
tmp = +ri[i]
qscal(tmp,W['beta'][i],W['v'][i],inv=True)
qscal(tmp,W['beta'][i],W['v'][i],inv=True)
rbwt[n+i] -= tmp[0]
blas.gemv(P[i], tmp[1:], rbwt, trans = 'T', alpha = -1.0, beta = 1.0)
tmp = mul(W['di'][:m],iw_rl1)
tmp2 = matrix(0.0,(k,1))
base.gemv(E,tmp,tmp2,trans='T')
rbwt[n:] += tmp2
tmp = mul(d,tmp) # tmp = D*inv(Wl1)^2*rl1
blas.gemv(X,tmp,rbwt,trans='T', alpha = -1.0, beta = 1.0)
# rbb = rb - d'*inv(Wl1)^2*rl1
rbb = rb - sum(tmp)
# rbv = rv - inv(Wl2)*rl2 - inv(Wl1)^2*rl1
rbv = rv - mul(W['di'][m:2*m],iw_rl2) - mul(W['di'][:m],iw_rl1)
# [rtw;rtt] = rbwt + [-X'*D; E']*inv(Wl1)^2*inv(Db)*rbv
tmp = mul(W['di'][:m]**2, mul(dbi,rbv))
rtt = +rbwt[n:]
base.gemv(E, tmp, rtt, trans = 'T', alpha = 1.0, beta = 1.0)
rtw = +rbwt[:n]
tmp = mul(d,tmp)
blas.gemv(X, tmp, rtw, trans = 'T', alpha = -1.0, beta = 1.0)
# rtb = rbb - d'*inv(Wl1)^2*inv(Db)*rbv
rtb = rbb - sum(tmp)
# solve M*[dw;db] = [rtw - Bb*inv(D2)*rtt; rtb + lt'*inv(D2)*rtt]
tmp = mul(d2i,rtt)
tmp2 = matrix(0.0,(n,1))
blas.gemv(Bb,tmp,tmp2)
dwdb = matrix([rtw - tmp2,rtb + blas.dot(mul(d2i,lt),rtt)])
lapack.potrs(M,dwdb)
# compute dt = inv(D2)*(rtt - Bb'*dw + lt*db)
tmp2 = matrix(0.0,(k,1))
blas.gemv(Bb, dwdb[:n], tmp2, trans='T')
dt = mul(d2i, rtt - tmp2 + lt*dwdb[-1])
# compute dv = inv(Db)*(rbv + inv(Wl1)^2*(E*dt - D*X*dw - d*db))
dv = matrix(0.0,(m,1))
blas.gemv(X,dwdb[:n],dv,alpha = -1.0)
dv = mul(d,dv) - d*dwdb[-1]
base.gemv(E, dt, dv, beta = 1.0)
tmp = +dv # tmp = E*dt - D*X*dw - d*db
dv = mul(dbi, rbv + mul(W['di'][:m]**2,dv))
# compute wdz1 = inv(Wl1)*(E*dt - D*X*dw - d*db - dv - rl1)
wdz1 = mul(W['di'][:m], tmp - dv) - iw_rl1
# compute wdz2 = - inv(Wl2)*(dv + rl2)
wdz2 = - mul(W['di'][m:2*m],dv) - iw_rl2
# compute wdzi = inv(Wi)*([-ei'*dt; -Pi*dw] - ri)
wdzi = []
tmp = matrix(0.0,(n,1))
for i in range(k):
blas.gemv(P[i],dwdb[:n],tmp, alpha = -1.0, beta = 0.0)
tmp1 = matrix([-dt[i],tmp])
blas.axpy(ri[i],tmp1,alpha = -1.0)
qscal(tmp1,W['beta'][i],W['v'][i],inv=True)
wdzi.append(tmp1)
# solution
x[:n] = dwdb[:n]
x[n:n+k] = dt
x[n+k] = dwdb[-1]
x[n+k+1:] = dv
z[:m] = wdz1
z[m:2*m] = wdz2
for i in range(k):
z[2*m+i*(n+1):2*m+(i+1)*(n+1)] = wdzi[i]
return f
#INSERT CODE HERE TO MODIFY FEATURES
# features is a n_examplesxn_features sparse matrix
#binarize features
print "Binarizing features"
binarizer = preprocessing.Binarizer(threshold=0.0)
binfeatures=binarizer.transform(features)
#print binfeatures
#cALCULATE DOT PRODUCT BETWEEN FEATURE REPRESENTATION OF EXAMPLES
GMo=np.array(features.dot(features.T).todense())
#normalize GM matrix
GMo=co.matrix(GMo)
YY = co.matrix([GMo[i,i] for i in range(GMo.size[0])])
YY = co.sqrt(YY)**(-1)
GMo = co.mul(GMo, YY*YY.T)
#print GMo
# START MIRKO
print "Calculating D-kernel..."
R = co.matrix(binfeatures.todense())
K = d_kernel(R, d)
#GM = np.array(K)+ GMo#.tolist()
GM = K+ GMo#.tolist()
#GM=co.matrix(GM)
YY = co.matrix([GM[i,i] for i in range(GM.size[0])])
YY = co.sqrt(YY)**(-1)
GM = np.array(co.mul(GM, YY*YY.T))
# END MIRKO
def proxqp_clique_SNL(c, A, b, z, rho):
"""
Solves the 1-norm regularized conic LP
min. < c, x > + || A(x) - b ||_1 + (rho/2) || x - z ||^2
s.t. x >= 0
for a single dense clique .
The method is used in this package to solve the
sensor node localization problem
Input arguments.
c is a 'd' matrix of size n_k**2 x 1
A is a 'd' matrix. with size n_k**2 times m_k. Each of its columns
represents a symmetric matrix of order n_k in unpacked column-major
order. The term A ( x ) in the primal constraint is given by
A(x) = A' * vec(x).
The adjoint A'( y ) in the dual constraint is given by
A'(y) = mat( A * y ).
Only the entries of A corresponding to lower-triangular
positions are accessed.
b is a 'd' matrix of size m_k x 1.
z is a 'd' matrix of size n_k**2 x 1
rho is a positive scalar.
Output arguments.
sol : Solution dictionary for quadratic optimization problem.
primal : objective for optimization problem without prox term (trace C*X)
"""
ns2, ms = A.size
nl, msl = len(b) * 2, len(b)
ns = int(sqrt(ns2))
dims = {'l': nl, 'q': [], 's': [ns]}
c = matrix([matrix(1.0, (nl, 1)), c])
z = matrix([matrix(0.0, (nl, 1)), z])
q = +c
blas.axpy(z, q, alpha=-rho, offsetx=nl, offsety=nl)
symmetrize(q, ns, offset=nl)
q = q[:]
h = matrix(0.0, (nl + ns2, 1))
bz = +q
xp = +q
def P(u, v, alpha=1.0, beta=0.0):
# v := alpha * rho * u + beta * v
blas.scal(beta, v)
blas.axpy(u, v, alpha=alpha * rho, offsetx=nl, offsety=nl)
def xdot(x, y):
misc.trisc(x, dims)
adot = blas.dot(x, y)
misc.triusc(x, dims)
return adot
def Gf(u, v, alpha=1.0, beta=0.0, trans='N'):
# v = -alpha*u + beta * v
blas.scal(beta, v)
blas.axpy(u, v, alpha=-alpha)
def Af(u, v, alpha=1.0, beta=0.0, trans="N"):
# v := alpha * A(u) + beta * v if trans is 'N'
# v := alpha * A'(u) + beta * v if trans is 'T'
blas.scal(beta, v)
if trans == "N":
blas.axpy(u, v, alpha=alpha, n=nl / 2)
blas.axpy(u, v, alpha=-alpha, offsetx=nl / 2, n=nl / 2)
sgemv(A,
u,
v,
n=ns,
m=ms,
alpha=alpha,
beta=1.0,
trans="T",
offsetx=nl)
elif trans == "T":
blas.axpy(u, v, alpha=alpha, n=nl / 2)
blas.axpy(u, v, alpha=-alpha, offsety=nl / 2, n=nl / 2)
sgemv(A,
u,
v,
n=ns,
m=ms,
alpha=alpha,
beta=1.0,
trans="N",
offsety=nl)
U = matrix(0.0, (ns, ns))
Vt = matrix(0.0, (ns, ns))
sv = matrix(0.0, (ns, 1))
Gamma = matrix(0.0, (ns, ns))
if type(A) is spmatrix:
VecAIndex = +A[:].I
As = matrix(A)
Aspkd = matrix(0.0, ((ns + 1) * ns / 2, ms))
tmp = matrix(0.0, (ms, 1))
def F(W):
# SVD R[j] = U[j] * diag(sig[j]) * Vt[j]
lapack.gesvd(+W['r'][0], sv, jobu='A', jobvt='A', U=U, Vt=Vt)
W2 = mul(+W['d'], +W['d'])
# Vt[j] := diag(sig[j])^-1 * Vt[j]
for k in xrange(ns):
blas.tbsv(sv, Vt, n=ns, k=0, ldA=1, offsetx=k * ns)
# Gamma[j] is an ns[j] x ns[j] symmetric matrix
# (sig[j] * sig[j]') ./ sqrt(1 + rho * (sig[j] * sig[j]').^2)
# S = sig[j] * sig[j]'
S = matrix(0.0, (ns, ns))
blas.syrk(sv, S)
Gamma = div(S, sqrt(1.0 + rho * S**2))
symmetrize(Gamma, ns)
# As represents the scaled mapping
#
# As(x) = A(u * (Gamma .* x) * u')
# As'(y) = Gamma .* (u' * A'(y) * u)
#
# stored in a similar format as A, except that we use packed
# storage for the columns of As[i][j].
if type(A) is spmatrix:
blas.scal(0.0, As)
As[VecAIndex] = +A[VecAIndex]
else:
blas.copy(A, As)
# As[i][j][:,k] = diag( diag(Gamma[j]))*As[i][j][:,k]
# As[i][j][l,:] = Gamma[j][l,l]*As[i][j][l,:]
for k in xrange(ms):
cngrnc(U, As, trans='T', offsetx=k * (ns2))
blas.tbmv(Gamma, As, n=ns2, k=0, ldA=1, offsetx=k * (ns2))
misc.pack(As, Aspkd, {'l': 0, 'q': [], 's': [ns] * ms})
# H is an m times m block matrix with i, k block
#
# Hik = sum_j As[i,j]' * As[k,j]
#
# of size ms[i] x ms[k]. Hik = 0 if As[i,j] or As[k,j]
# are zero for all j
H = matrix(0.0, (ms, ms))
blas.syrk(Aspkd, H, trans='T', beta=1.0, k=ns * (ns + 1) / 2)
#H = H + spmatrix(W2[:nl/2] + W2[nl/2:] ,range(nl/2),range(nl/2))
blas.axpy(W2, H, n=ms, incy=ms + 1, alpha=1.0)
blas.axpy(W2, H, offsetx=ms, n=ms, incy=ms + 1, alpha=1.0)
lapack.potrf(H)
def solve(x, y, z):
"""
Returns solution of
rho * ux + A'(uy) - r^-T * uz * r^-1 = bx
A(ux) = by
-ux - r * uz * r' = bz.
On entry, x = bx, y = by, z = bz.
On exit, x = ux, y = uy, z = uz.
"""
# bz is a copy of z in the format of x
blas.copy(z, bz)
blas.axpy(bz, x, alpha=rho, offsetx=nl, offsety=nl)
# x := Gamma .* (u' * x * u)
# = Gamma .* (u' * (bx + rho * bz) * u)
cngrnc(U, x, trans='T', offsetx=nl)
blas.tbmv(Gamma, x, n=ns2, k=0, ldA=1, offsetx=nl)
blas.tbmv(+W['d'], x, n=nl, k=0, ldA=1)
# y := y - As(x)
# := by - As( Gamma .* u' * (bx + rho * bz) * u)
misc.pack(x, xp, dims)
blas.gemv(Aspkd, xp, y, trans = 'T',alpha = -1.0, beta = 1.0, \
m = ns*(ns+1)/2, n = ms,offsetx = nl)
#y = y - mul(+W['d'][:nl/2],xp[:nl/2])+ mul(+W['d'][nl/2:nl],xp[nl/2:nl])
blas.tbmv(+W['d'], xp, n=nl, k=0, ldA=1)
blas.axpy(xp, y, alpha=-1, n=ms)
blas.axpy(xp, y, alpha=1, n=ms, offsetx=nl / 2)
# y := -y - A(bz)
# = -by - A(bz) + As(Gamma .* (u' * (bx + rho * bz) * u)
Af(bz, y, alpha=-1.0, beta=-1.0)
# y := H^-1 * y
# = H^-1 ( -by - A(bz) + As(Gamma.* u'*(bx + rho*bz)*u) )
# = uy
blas.trsv(H, y)
blas.trsv(H, y, trans='T')
# bz = Vt' * vz * Vt
# = uz where
# vz := Gamma .* ( As'(uy) - x )
# = Gamma .* ( As'(uy) - Gamma .* (u'*(bx + rho *bz)*u) )
# = Gamma.^2 .* ( u' * (A'(uy) - bx - rho * bz) * u ).
misc.pack(x, xp, dims)
blas.scal(-1.0, xp)
blas.gemv(Aspkd,
y,
xp,
alpha=1.0,
beta=1.0,
m=ns * (ns + 1) / 2,
n=ms,
offsety=nl)
#xp[:nl/2] = xp[:nl/2] + mul(+W['d'][:nl/2],y)
#xp[nl/2:nl] = xp[nl/2:nl] - mul(+W['d'][nl/2:nl],y)
blas.copy(y, tmp)
blas.tbmv(+W['d'], tmp, n=nl / 2, k=0, ldA=1)
blas.axpy(tmp, xp, n=nl / 2)
blas.copy(y, tmp)
blas.tbmv(+W['d'], tmp, n=nl / 2, k=0, ldA=1, offsetA=nl / 2)
blas.axpy(tmp, xp, alpha=-1, n=nl / 2, offsety=nl / 2)
# bz[j] is xp unpacked and multiplied with Gamma
blas.copy(xp, bz) #,n = nl)
misc.unpack(xp, bz, dims)
blas.tbmv(Gamma, bz, n=ns2, k=0, ldA=1, offsetx=nl)
# bz = Vt' * bz * Vt
# = uz
cngrnc(Vt, bz, trans='T', offsetx=nl)
symmetrize(bz, ns, offset=nl)
# x = -bz - r * uz * r'
# z contains r.h.s. bz; copy to x
#so far, z = bzc (untouched)
blas.copy(z, x)
blas.copy(bz, z)
cngrnc(W['r'][0], bz, offsetx=nl)
blas.tbmv(W['d'], bz, n=nl, k=0, ldA=1)
blas.axpy(bz, x)
blas.scal(-1.0, x)
return solve
sol = solvers.coneqp(P, q, Gf, h, dims, Af, b, None, F, xdot=xdot)
primal = blas.dot(sol['s'], c)
sol['s'] = sol['s'][nl:]
sol['z'] = sol['z'][nl:]
return sol, primal
def mrcompletion(A, reordered=True):
"""
Minimum rank positive semidefinite completion. The routine takes a
positive semidefinite cspmatrix :math:`A` and returns a dense
matrix :math:`Y` with :math:`r` columns that satisfies
.. math::
P( YY^T ) = A
where
.. math::
r = \max_{i} |\gamma_i|
is the clique number (the size of the largest clique).
:param A: :py:class:`cspmatrix`
:param reordered: boolean
"""
assert isinstance(A, cspmatrix) and A.is_factor is False, "A must be a cspmatrix"
symb = A.symb
n = symb.n
snpost = symb.snpost
snptr = symb.snptr
chptr = symb.chptr
chidx = symb.chidx
relptr = symb.relptr
relidx = symb.relidx
blkptr = symb.blkptr
blkval = A.blkval
stack = []
r = 0
maxr = symb.clique_number
Y = matrix(0.0,(n,maxr)) # storage for factorization
Z = matrix(0.0,(maxr,maxr)) # storage for EVD of cliques
w = matrix(0.0,(maxr,1)) # storage for EVD of cliques
P = matrix(0.0,(maxr,maxr)) # storage for left singular vectors
Q1t = matrix(0.0,(maxr,maxr)) # storage for right singular vectors (1)
Q2t = matrix(0.0,(maxr,maxr)) # storage for right singular vectors (2)
S = matrix(0.0,(maxr,1)) # storage for singular values
V = matrix(0.0,(maxr,maxr))
Ya = matrix(0.0,(maxr,maxr))
# visit supernodes in reverse topological order
for k in range(symb.Nsn-1,-1,-1):
nn = snptr[k+1]-snptr[k] # |Nk|
na = relptr[k+1]-relptr[k] # |Ak|
nj = na + nn
# allocate F and copy X_{Jk,Nk} to leading columns of F
F = matrix(0.0, (nj,nj))
lapack.lacpy(blkval, F, offsetA = blkptr[k], ldA = nj, m = nj, n = nn, uplo = 'L')
# if supernode k is not a root node:
if na > 0:
# copy Vk to 2,2 block of F
Vk = stack.pop()
lapack.lacpy(Vk, F, offsetB = nn*nj+nn, m = na, n = na, uplo = 'L')
# if supernode k has any children:
for ii in range(chptr[k],chptr[k+1]):
stack.append(frontal_get_update(F,relidx,relptr,chidx[ii]))
# Compute factorization of F
lapack.syevr(F, w, jobz='V', range='A', uplo='L', Z=Z, n=nj,ldZ=maxr)
rk = sum([1 for wi in w[:nj] if wi > 1e-14*w[nj-1]]) # determine rank of clique k
r = max(rk,r) # update rank
# Scale last rk cols of Z and copy parts to Yn
for j in range(nj-rk,nj):
Z[:nj,j] *= sqrt(w[j])
In = symb.snrowidx[symb.sncolptr[k]:symb.sncolptr[k]+nn]
Y[In,:rk] = Z[:nn,nj-rk:nj]
# if supernode k is not a root node:
if na > 0:
# Extract data
Ia = symb.snrowidx[symb.sncolptr[k]+nn:symb.sncolptr[k+1]]
Ya[:na,:r] = Y[Ia,:r]
V[:na,:rk] = Z[nn:nj,nj-rk:nj]
V[:na,rk:r] *= 0.0
# Compute SVDs: V = P*S*Q1t and Ya = P*S*Q2t
lapack.gesvd(V,S,jobu='A',jobvt='A',U=P,Vt=Q1t,ldU=maxr,ldVt=maxr,m=na,n=r,ldA=maxr)
lapack.gesvd(Ya,S,jobu='N',jobvt='A',Vt=Q2t,ldVt=maxr,m=na,n=r,ldA=maxr)
# Scale Q2t
for i in range(min(na,rk)):
if S[i] > 1e-14*S[0]: Q2t[i,:r] = P[:na,i].T*Y[Ia,:r]/S[i]
# Scale Yn
Y[In,:r] = Y[In,:r]*Q1t[:r,:r].T*Q2t[:r,:r]
if reordered:
return Y[:,:r]
else:
return Y[symb.ip,:r]
def proxqp_clique(c, A, b, z, rho):
"""
Solves the conic QP
min. < c, x > + (rho/2) || x - z ||^2
s.t. A(x) = b
x >= 0
and its dual
max. -< b, y > - 1/(2*rho) * || c + A'(y) - rho * z - s ||^2
s.t. s >= 0.
for a single dense clique.
If the problem has block-arrow correlative sparsity, then the previous
function
X = proxqp(c,A,b,z,rho,**kwargs)
is equivalent to
for k in xrange(ncliques):
X[k] = proxqp_clique(c[k],A[k][k],b[k],z[k],rho,**kwargs)
and each call can be implemented in parallel.
Input arguments.
c is a 'd' matrix of size n_k**2 x 1
A is a 'd' matrix. with size n_k**2 times m_k. Each of its columns
represents a symmetric matrix of order n_k in unpacked column-major
order. The term A ( x ) in the primal constraint is given by
A(x) = A' * vec(x).
The adjoint A'( y ) in the dual constraint is given by
A'(y) = mat( A * y ).
b is a 'd' matrix of size m_k x 1.
z is a 'd' matrix of size n_k**2 x 1
rho is a positive scalar.
Output arguments.
sol : Solution dictionary for quadratic optimization problem.
primal : objective for optimization problem without prox term (trace C*X)
"""
ns2, ms = A.size
ns = int(sqrt(ns2))
dims = {'l': 0, 'q': [], 's': [ns]}
q = +c
blas.axpy(z, q, alpha=-rho)
symmetrize(q, ns, offset=0)
q = q[:]
h = matrix(0.0, (ns2, 1))
bz = +q
xp = +q
def P(u, v, alpha=1.0, beta=0.0):
# v := alpha * rho * u + beta * v
#if not (beta==0.0):
blas.scal(beta, v)
blas.axpy(u, v, alpha=alpha * rho)
def xdot(x, y):
misc.trisc(x, {'l': 0, 'q': [], 's': [ns]})
adot = blas.dot(x, y)
misc.triusc(x, {'l': 0, 'q': [], 's': [ns]})
return adot
def Gf(u, v, alpha=1.0, beta=0.0, trans='N'):
# v = -alpha*u + beta * v
# u and v are vectors representing N symmetric matrices in the
# cvxopt format.
blas.scal(beta, v)
blas.axpy(u, v, alpha=-alpha)
def Af(u, v, alpha=1.0, beta=0.0, trans="N"):
# v := alpha * A(u) + beta * v if trans is 'N'
# v := alpha * A'(u) + beta * v if trans is 'T'
blas.scal(beta, v)
if trans == "N":
sgemv(A,
u,
v,
n=ns,
m=ms,
alpha=alpha,
beta=1.0,
trans="T",
offsetx=0)
elif trans == "T":
sgemv(A,
u,
v,
n=ns,
m=ms,
alpha=alpha,
beta=1.0,
trans="N",
offsetx=0)
U = matrix(0.0, (ns, ns))
Vt = matrix(0.0, (ns, ns))
sv = matrix(0.0, (ns, 1))
Gamma = matrix(0.0, (ns, ns))
if type(A) is spmatrix:
VecAIndex = +A[:].I
Aspkd = matrix(0.0, ((ns + 1) * ns / 2, ms))
As = matrix(A)
def F(W):
# SVD R[j] = U[j] * diag(sig[j]) * Vt[j]
lapack.gesvd(+W['r'][0], sv, jobu='A', jobvt='A', U=U, Vt=Vt)
# Vt[j] := diag(sig[j])^-1 * Vt[j]
for k in xrange(ns):
blas.tbsv(sv, Vt, n=ns, k=0, ldA=1, offsetx=k * ns)
# Gamma[j] is an ns[j] x ns[j] symmetric matrix
#
# (sig[j] * sig[j]') ./ sqrt(1 + rho * (sig[j] * sig[j]').^2)
# S = sig[j] * sig[j]'
S = matrix(0.0, (ns, ns))
blas.syrk(sv, S)
Gamma = div(S, sqrt(1.0 + rho * S**2))
symmetrize(Gamma, ns)
# As represents the scaled mapping
#
# As(x) = A(u * (Gamma .* x) * u')
# As'(y) = Gamma .* (u' * A'(y) * u)
#
# stored in a similar format as A, except that we use packed
# storage for the columns of As[i][j].
if type(A) is spmatrix:
blas.scal(0.0, As)
try:
As[VecAIndex] = +A['s'][VecAIndex]
except:
As[VecAIndex] = +A[VecAIndex]
else:
blas.copy(A, As)
# As[i][j][:,k] = diag( diag(Gamma[j]))*As[i][j][:,k]
# As[i][j][l,:] = Gamma[j][l,l]*As[i][j][l,:]
for k in xrange(ms):
cngrnc(U, As, trans='T', offsetx=k * (ns2))
blas.tbmv(Gamma, As, n=ns2, k=0, ldA=1, offsetx=k * (ns2))
misc.pack(As, Aspkd, {'l': 0, 'q': [], 's': [ns] * ms})
# H is an m times m block matrix with i, k block
#
# Hik = sum_j As[i,j]' * As[k,j]
#
# of size ms[i] x ms[k]. Hik = 0 if As[i,j] or As[k,j]
# are zero for all j
H = matrix(0.0, (ms, ms))
blas.syrk(Aspkd, H, trans='T', beta=1.0, k=ns * (ns + 1) / 2)
lapack.potrf(H)
def solve(x, y, z):
"""
Returns solution of
rho * ux + A'(uy) - r^-T * uz * r^-1 = bx
A(ux) = by
-ux - r * uz * r' = bz.
On entry, x = bx, y = by, z = bz.
On exit, x = ux, y = uy, z = uz.
"""
# bz is a copy of z in the format of x
blas.copy(z, bz)
blas.axpy(bz, x, alpha=rho)
# x := Gamma .* (u' * x * u)
# = Gamma .* (u' * (bx + rho * bz) * u)
cngrnc(U, x, trans='T', offsetx=0)
blas.tbmv(Gamma, x, n=ns2, k=0, ldA=1, offsetx=0)
# y := y - As(x)
# := by - As( Gamma .* u' * (bx + rho * bz) * u)
#blas.copy(x,xp)
#pack_ip(xp,n = ns,m=1,nl=nl)
misc.pack(x, xp, {'l': 0, 'q': [], 's': [ns]})
blas.gemv(Aspkd, xp, y, trans = 'T',alpha = -1.0, beta = 1.0, \
m = ns*(ns+1)/2, n = ms,offsetx = 0)
# y := -y - A(bz)
# = -by - A(bz) + As(Gamma .* (u' * (bx + rho * bz) * u)
Af(bz, y, alpha=-1.0, beta=-1.0)
# y := H^-1 * y
# = H^-1 ( -by - A(bz) + As(Gamma.* u'*(bx + rho*bz)*u) )
# = uy
blas.trsv(H, y)
blas.trsv(H, y, trans='T')
# bz = Vt' * vz * Vt
# = uz where
# vz := Gamma .* ( As'(uy) - x )
# = Gamma .* ( As'(uy) - Gamma .* (u'*(bx + rho *bz)*u) )
# = Gamma.^2 .* ( u' * (A'(uy) - bx - rho * bz) * u ).
#blas.copy(x,xp)
#pack_ip(xp,n=ns,m=1,nl=nl)
misc.pack(x, xp, {'l': 0, 'q': [], 's': [ns]})
blas.scal(-1.0, xp)
blas.gemv(Aspkd,
y,
xp,
alpha=1.0,
beta=1.0,
m=ns * (ns + 1) / 2,
n=ms,
offsety=0)
# bz[j] is xp unpacked and multiplied with Gamma
misc.unpack(xp, bz, {'l': 0, 'q': [], 's': [ns]})
blas.tbmv(Gamma, bz, n=ns2, k=0, ldA=1, offsetx=0)
# bz = Vt' * bz * Vt
# = uz
cngrnc(Vt, bz, trans='T', offsetx=0)
symmetrize(bz, ns, offset=0)
# x = -bz - r * uz * r'
# z contains r.h.s. bz; copy to x
blas.copy(z, x)
blas.copy(bz, z)
cngrnc(W['r'][0], bz, offsetx=0)
blas.axpy(bz, x)
blas.scal(-1.0, x)
return solve
#solvers.options['show_progress'] = True
sol = solvers.coneqp(P, q, Gf, h, dims, Af, b, None, F, xdot=xdot)
primal = blas.dot(c, sol['s'])
return sol, primal
def F(W):
"""
Custom solver for the system
[ It 0 0 Xt' 0 At1' ... Atk' ][ dwt ] [ rwt ]
[ 0 0 0 -d' 0 0 ... 0 ][ db ] [ rb ]
[ 0 0 0 -I -I 0 ... 0 ][ dv ] [ rv ]
[ Xt -d -I -Wl1^-2 ][ dzl1 ] [ rl1 ]
[ 0 0 -I -Wl2^-2 ][ dzl2 ] = [ rl2 ]
[ At1 0 0 -W1^-2 ][ dz1 ] [ r1 ]
[ | | | . ][ | ] [ | ]
[ Atk 0 0 -Wk^-2 ][ dzk ] [ rk ]
where
It = [ I 0 ] Xt = [ -D*X E ] Ati = [ 0 -e_i' ]
[ 0 0 ] [ -Pi 0 ]
dwt = [ dw ] rwt = [ rw ]
[ dt ] [ rt ].
"""
# scalings and 'intermediate' vectors
# db = inv(Wl1)^2 + inv(Wl2)^2
db = W['di'][:m]**2 + W['di'][m:2 * m]**2
dbi = div(1.0, db)
# dt = I - inv(Wl1)*Dbi*inv(Wl1)
dt = 1.0 - mul(W['di'][:m]**2, dbi)
dtsqrt = sqrt(dt)
# lam = Dt*inv(Wl1)*d
lam = mul(dt, mul(W['di'][:m], d))
# lt = E'*inv(Wl1)*lam
lt = matrix(0.0, (k, 1))
base.gemv(E, mul(W['di'][:m], lam), lt, trans='T')
# Xs = sqrt(Dt)*inv(Wl1)*X
tmp = mul(dtsqrt, W['di'][:m])
Xs = spmatrix(tmp, range(m), range(m)) * X
# Es = D*sqrt(Dt)*inv(Wl1)*E
Es = spmatrix(mul(d, tmp), range(m), range(m)) * E
# form Ab = I + sum((1/bi)^2*(Pi'*Pi + 4*(v'*v + 1)*Pi'*y*y'*Pi)) + Xs'*Xs
# and Bb = -sum((1/bi)^2*(4*ui*v'*v*Pi'*y*ei')) - Xs'*Es
# and D2 = Es'*Es + sum((1/bi)^2*(1+4*ui^2*(v'*v - 1))
Ab = matrix(0.0, (n, n))
Ab[::n + 1] = 1.0
base.syrk(Xs, Ab, trans='T', beta=1.0)
Bb = matrix(0.0, (n, k))
Bb = -Xs.T * Es # inefficient!?
D2 = spmatrix(0.0, range(k), range(k))
base.syrk(Es, D2, trans='T', partial=True)
d2 = +D2.V
del D2
py = matrix(0.0, (n, 1))
for i in range(k):
binvsq = (1.0 / W['beta'][i])**2
Ab += binvsq * Pt[i]
dvv = blas.dot(W['v'][i], W['v'][i])
blas.gemv(P[i], W['v'][i][1:], py, trans='T', alpha=1.0, beta=0.0)
blas.syrk(py, Ab, alpha=4 * binvsq * (dvv + 1), beta=1.0)
Bb[:, i] -= 4 * binvsq * W['v'][i][0] * dvv * py
d2[i] += binvsq * (1 + 4 * (W['v'][i][0]**2) * (dvv - 1))
d2i = div(1.0, d2)
d2isqrt = sqrt(d2i)
# compute a = alpha - lam'*inv(Wl1)*E*inv(D2)*E'*inv(Wl1)*lam
alpha = blas.dot(lam, mul(W['di'][:m], d))
tmp = matrix(0.0, (k, 1))
base.gemv(E, mul(W['di'][:m], lam), tmp, trans='T')
tmp = mul(tmp, d2isqrt) #tmp = inv(D2)^(1/2)*E'*inv(Wl1)*lam
a = alpha - blas.dot(tmp, tmp)
# compute M12 = X'*D*inv(Wl1)*lam + Bb*inv(D2)*E'*inv(Wl1)*lam
tmp = mul(tmp, d2isqrt)
M12 = matrix(0.0, (n, 1))
blas.gemv(Bb, tmp, M12, alpha=1.0)
tmp = mul(d, mul(W['di'][:m], lam))
blas.gemv(X, tmp, M12, trans='T', alpha=1.0, beta=1.0)
# form and factor M
sBb = Bb * spmatrix(d2isqrt, range(k), range(k))
base.syrk(sBb, Ab, alpha=-1.0, beta=1.0)
M = matrix([[Ab, M12.T], [M12, a]])
lapack.potrf(M)
def f(x, y, z):
# residuals
rwt = x[:n + k]
rb = x[n + k]
rv = x[n + k + 1:n + k + 1 + m]
iw_rl1 = mul(W['di'][:m], z[:m])
iw_rl2 = mul(W['di'][m:2 * m], z[m:2 * m])
ri = [
z[2 * m + i * (n + 1):2 * m + (i + 1) * (n + 1)]
for i in range(k)
]
# compute 'derived' residuals
# rbwt = rwt + sum(Ai'*inv(Wi)^2*ri) + [-X'*D; E']*inv(Wl1)^2*rl1
rbwt = +rwt
for i in range(k):
tmp = +ri[i]
qscal(tmp, W['beta'][i], W['v'][i], inv=True)
qscal(tmp, W['beta'][i], W['v'][i], inv=True)
rbwt[n + i] -= tmp[0]
blas.gemv(P[i], tmp[1:], rbwt, trans='T', alpha=-1.0, beta=1.0)
tmp = mul(W['di'][:m], iw_rl1)
tmp2 = matrix(0.0, (k, 1))
base.gemv(E, tmp, tmp2, trans='T')
rbwt[n:] += tmp2
tmp = mul(d, tmp) # tmp = D*inv(Wl1)^2*rl1
blas.gemv(X, tmp, rbwt, trans='T', alpha=-1.0, beta=1.0)
# rbb = rb - d'*inv(Wl1)^2*rl1
rbb = rb - sum(tmp)
# rbv = rv - inv(Wl2)*rl2 - inv(Wl1)^2*rl1
rbv = rv - mul(W['di'][m:2 * m], iw_rl2) - mul(W['di'][:m], iw_rl1)
# [rtw;rtt] = rbwt + [-X'*D; E']*inv(Wl1)^2*inv(Db)*rbv
tmp = mul(W['di'][:m]**2, mul(dbi, rbv))
rtt = +rbwt[n:]
base.gemv(E, tmp, rtt, trans='T', alpha=1.0, beta=1.0)
rtw = +rbwt[:n]
tmp = mul(d, tmp)
blas.gemv(X, tmp, rtw, trans='T', alpha=-1.0, beta=1.0)
# rtb = rbb - d'*inv(Wl1)^2*inv(Db)*rbv
rtb = rbb - sum(tmp)
# solve M*[dw;db] = [rtw - Bb*inv(D2)*rtt; rtb + lt'*inv(D2)*rtt]
tmp = mul(d2i, rtt)
tmp2 = matrix(0.0, (n, 1))
blas.gemv(Bb, tmp, tmp2)
dwdb = matrix([rtw - tmp2, rtb + blas.dot(mul(d2i, lt), rtt)])
lapack.potrs(M, dwdb)
# compute dt = inv(D2)*(rtt - Bb'*dw + lt*db)
tmp2 = matrix(0.0, (k, 1))
blas.gemv(Bb, dwdb[:n], tmp2, trans='T')
dt = mul(d2i, rtt - tmp2 + lt * dwdb[-1])
# compute dv = inv(Db)*(rbv + inv(Wl1)^2*(E*dt - D*X*dw - d*db))
dv = matrix(0.0, (m, 1))
blas.gemv(X, dwdb[:n], dv, alpha=-1.0)
dv = mul(d, dv) - d * dwdb[-1]
base.gemv(E, dt, dv, beta=1.0)
tmp = +dv # tmp = E*dt - D*X*dw - d*db
dv = mul(dbi, rbv + mul(W['di'][:m]**2, dv))
# compute wdz1 = inv(Wl1)*(E*dt - D*X*dw - d*db - dv - rl1)
wdz1 = mul(W['di'][:m], tmp - dv) - iw_rl1
# compute wdz2 = - inv(Wl2)*(dv + rl2)
wdz2 = -mul(W['di'][m:2 * m], dv) - iw_rl2
# compute wdzi = inv(Wi)*([-ei'*dt; -Pi*dw] - ri)
wdzi = []
tmp = matrix(0.0, (n, 1))
for i in range(k):
blas.gemv(P[i], dwdb[:n], tmp, alpha=-1.0, beta=0.0)
tmp1 = matrix([-dt[i], tmp])
blas.axpy(ri[i], tmp1, alpha=-1.0)
qscal(tmp1, W['beta'][i], W['v'][i], inv=True)
wdzi.append(tmp1)
# solution
x[:n] = dwdb[:n]
x[n:n + k] = dt
x[n + k] = dwdb[-1]
x[n + k + 1:] = dv
z[:m] = wdz1
z[m:2 * m] = wdz2
for i in range(k):
z[2 * m + i * (n + 1):2 * m + (i + 1) * (n + 1)] = wdzi[i]
return f
def F(W):
# SVD R[j] = U[j] * diag(sig[j]) * Vt[j]
lapack.gesvd(+W['r'][0], sv, jobu='A', jobvt='A', U=U, Vt=Vt)
# Vt[j] := diag(sig[j])^-1 * Vt[j]
for k in xrange(ns):
blas.tbsv(sv, Vt, n=ns, k=0, ldA=1, offsetx=k * ns)
# Gamma[j] is an ns[j] x ns[j] symmetric matrix
#
# (sig[j] * sig[j]') ./ sqrt(1 + rho * (sig[j] * sig[j]').^2)
# S = sig[j] * sig[j]'
S = matrix(0.0, (ns, ns))
blas.syrk(sv, S)
Gamma = div(S, sqrt(1.0 + rho * S**2))
symmetrize(Gamma, ns)
# As represents the scaled mapping
#
# As(x) = A(u * (Gamma .* x) * u')
# As'(y) = Gamma .* (u' * A'(y) * u)
#
# stored in a similar format as A, except that we use packed
# storage for the columns of As[i][j].
if type(A) is spmatrix:
blas.scal(0.0, As)
try:
As[VecAIndex] = +A['s'][VecAIndex]
except:
As[VecAIndex] = +A[VecAIndex]
else:
blas.copy(A, As)
# As[i][j][:,k] = diag( diag(Gamma[j]))*As[i][j][:,k]
# As[i][j][l,:] = Gamma[j][l,l]*As[i][j][l,:]
for k in xrange(ms):
cngrnc(U, As, trans='T', offsetx=k * (ns2))
blas.tbmv(Gamma, As, n=ns2, k=0, ldA=1, offsetx=k * (ns2))
misc.pack(As, Aspkd, {'l': 0, 'q': [], 's': [ns] * ms})
# H is an m times m block matrix with i, k block
#
# Hik = sum_j As[i,j]' * As[k,j]
#
# of size ms[i] x ms[k]. Hik = 0 if As[i,j] or As[k,j]
# are zero for all j
H = matrix(0.0, (ms, ms))
blas.syrk(Aspkd, H, trans='T', beta=1.0, k=ns * (ns + 1) / 2)
lapack.potrf(H)
def solve(x, y, z):
"""
Returns solution of
rho * ux + A'(uy) - r^-T * uz * r^-1 = bx
A(ux) = by
-ux - r * uz * r' = bz.
On entry, x = bx, y = by, z = bz.
On exit, x = ux, y = uy, z = uz.
"""
# bz is a copy of z in the format of x
blas.copy(z, bz)
blas.axpy(bz, x, alpha=rho)
# x := Gamma .* (u' * x * u)
# = Gamma .* (u' * (bx + rho * bz) * u)
cngrnc(U, x, trans='T', offsetx=0)
blas.tbmv(Gamma, x, n=ns2, k=0, ldA=1, offsetx=0)
# y := y - As(x)
# := by - As( Gamma .* u' * (bx + rho * bz) * u)
#blas.copy(x,xp)
#pack_ip(xp,n = ns,m=1,nl=nl)
misc.pack(x, xp, {'l': 0, 'q': [], 's': [ns]})
blas.gemv(Aspkd, xp, y, trans = 'T',alpha = -1.0, beta = 1.0, \
m = ns*(ns+1)/2, n = ms,offsetx = 0)
# y := -y - A(bz)
# = -by - A(bz) + As(Gamma .* (u' * (bx + rho * bz) * u)
Af(bz, y, alpha=-1.0, beta=-1.0)
# y := H^-1 * y
# = H^-1 ( -by - A(bz) + As(Gamma.* u'*(bx + rho*bz)*u) )
# = uy
blas.trsv(H, y)
blas.trsv(H, y, trans='T')
# bz = Vt' * vz * Vt
# = uz where
# vz := Gamma .* ( As'(uy) - x )
# = Gamma .* ( As'(uy) - Gamma .* (u'*(bx + rho *bz)*u) )
# = Gamma.^2 .* ( u' * (A'(uy) - bx - rho * bz) * u ).
#blas.copy(x,xp)
#pack_ip(xp,n=ns,m=1,nl=nl)
misc.pack(x, xp, {'l': 0, 'q': [], 's': [ns]})
blas.scal(-1.0, xp)
blas.gemv(Aspkd,
y,
xp,
alpha=1.0,
beta=1.0,
m=ns * (ns + 1) / 2,
n=ms,
offsety=0)
# bz[j] is xp unpacked and multiplied with Gamma
misc.unpack(xp, bz, {'l': 0, 'q': [], 's': [ns]})
blas.tbmv(Gamma, bz, n=ns2, k=0, ldA=1, offsetx=0)
# bz = Vt' * bz * Vt
# = uz
cngrnc(Vt, bz, trans='T', offsetx=0)
symmetrize(bz, ns, offset=0)
# x = -bz - r * uz * r'
# z contains r.h.s. bz; copy to x
blas.copy(z, x)
blas.copy(bz, z)
cngrnc(W['r'][0], bz, offsetx=0)
blas.axpy(bz, x)
blas.scal(-1.0, x)
return solve
def utility(x, y):
return (1.1 * sqrt(x) + 0.8 * sqrt(y)) / 1.9
def F(W):
"""
Generate a solver for
A'(uz0) = bx[0]
-uz0 - uz1 = bx[1]
A(ux[0]) - ux[1] - r0*r0' * uz0 * r0*r0' = bz0
- ux[1] - r1*r1' * uz1 * r1*r1' = bz1.
uz0, uz1, bz0, bz1 are symmetric m x m-matrices.
ux[0], bx[0] are n-vectors.
ux[1], bx[1] are symmetric m x m-matrices.
We first calculate a congruence that diagonalizes r0*r0' and r1*r1':
U' * r0 * r0' * U = I, U' * r1 * r1' * U = S.
We then make a change of variables
usx[0] = ux[0],
usx[1] = U' * ux[1] * U
usz0 = U^-1 * uz0 * U^-T
usz1 = U^-1 * uz1 * U^-T
and define
As() = U' * A() * U'
bsx[1] = U^-1 * bx[1] * U^-T
bsz0 = U' * bz0 * U
bsz1 = U' * bz1 * U.
This gives
As'(usz0) = bx[0]
-usz0 - usz1 = bsx[1]
As(usx[0]) - usx[1] - usz0 = bsz0
-usx[1] - S * usz1 * S = bsz1.
1. Eliminate usz0, usz1 using equations 3 and 4,
usz0 = As(usx[0]) - usx[1] - bsz0
usz1 = -S^-1 * (usx[1] + bsz1) * S^-1.
This gives two equations in usx[0] an usx[1].
As'(As(usx[0]) - usx[1]) = bx[0] + As'(bsz0)
-As(usx[0]) + usx[1] + S^-1 * usx[1] * S^-1
= bsx[1] - bsz0 - S^-1 * bsz1 * S^-1.
2. Eliminate usx[1] using equation 2:
usx[1] + S * usx[1] * S
= S * ( As(usx[0]) + bsx[1] - bsz0 ) * S - bsz1
i.e., with Gamma[i,j] = 1.0 + S[i,i] * S[j,j],
usx[1] = ( S * As(usx[0]) * S ) ./ Gamma
+ ( S * ( bsx[1] - bsz0 ) * S - bsz1 ) ./ Gamma.
This gives an equation in usx[0].
As'( As(usx[0]) ./ Gamma )
= bx0 + As'(bsz0) +
As'( (S * ( bsx[1] - bsz0 ) * S - bsz1) ./ Gamma )
= bx0 + As'( ( bsz0 - bsz1 + S * bsx[1] * S ) ./ Gamma ).
"""
# Calculate U s.t.
#
# U' * r0*r0' * U = I, U' * r1*r1' * U = diag(s).
# Cholesky factorization r0 * r0' = L * L'
blas.syrk(W['r'][0], L)
lapack.potrf(L)
# SVD L^-1 * r1 = U * diag(s) * V'
blas.copy(W['r'][1], U)
blas.trsm(L, U)
lapack.gesvd(U, s, jobu = 'O')
# s := s**2
s[:] = s**2
# Uti := U
blas.copy(U, Uti)
# U := L^-T * U
blas.trsm(L, U, transA = 'T')
# Uti := L * Uti = U^-T
blas.trmm(L, Uti)
# Us := U * diag(s)^-1
blas.copy(U, Us)
for i in range(m):
blas.tbsv(s, Us, n = m, k = 0, ldA = 1, incx = m, offsetx = i)
# S is m x m with lower triangular entries s[i] * s[j]
# sqrtG is m x m with lower triangular entries sqrt(1.0 + s[i]*s[j])
# Upper triangular entries are undefined but nonzero.
blas.scal(0.0, S)
blas.syrk(s, S)
Gamma = 1.0 + S
sqrtG = sqrt(Gamma)
# Asc[i] = (U' * Ai * * U ) ./ sqrtG, for i = 1, ..., n
# = Asi ./ sqrt(Gamma)
blas.copy(A, Asc)
misc.scale(Asc, # only 'r' part of the dictionary is used
{'dnl': matrix(0.0, (0, 1)), 'dnli': matrix(0.0, (0, 1)),
'd': matrix(0.0, (0, 1)), 'di': matrix(0.0, (0, 1)),
'v': [], 'beta': [], 'r': [ U ], 'rti': [ U ]})
for i in range(n):
blas.tbsv(sqrtG, Asc, n = msq, k = 0, ldA = 1, offsetx = i*msq)
# Convert columns of Asc to packed storage
misc.pack2(Asc, {'l': 0, 'q': [], 's': [ m ]})
# Cholesky factorization of Asc' * Asc.
H = matrix(0.0, (n, n))
blas.syrk(Asc, H, trans = 'T', k = mpckd)
lapack.potrf(H)
def solve(x, y, z):
"""
1. Solve for usx[0]:
Asc'(Asc(usx[0]))
= bx0 + Asc'( ( bsz0 - bsz1 + S * bsx[1] * S ) ./ sqrtG)
= bx0 + Asc'( ( bsz0 + S * ( bsx[1] - bssz1) S )
./ sqrtG)
where bsx[1] = U^-1 * bx[1] * U^-T, bsz0 = U' * bz0 * U,
bsz1 = U' * bz1 * U, bssz1 = S^-1 * bsz1 * S^-1
2. Solve for usx[1]:
usx[1] + S * usx[1] * S
= S * ( As(usx[0]) + bsx[1] - bsz0 ) * S - bsz1
usx[1]
= ( S * (As(usx[0]) + bsx[1] - bsz0) * S - bsz1) ./ Gamma
= -bsz0 + (S * As(usx[0]) * S) ./ Gamma
+ (bsz0 - bsz1 + S * bsx[1] * S ) . / Gamma
= -bsz0 + (S * As(usx[0]) * S) ./ Gamma
+ (bsz0 + S * ( bsx[1] - bssz1 ) * S ) . / Gamma
Unscale ux[1] = Uti * usx[1] * Uti'
3. Compute usz0, usz1
r0' * uz0 * r0 = r0^-1 * ( A(ux[0]) - ux[1] - bz0 ) * r0^-T
r1' * uz1 * r1 = r1^-1 * ( -ux[1] - bz1 ) * r1^-T
"""
# z0 := U' * z0 * U
# = bsz0
__cngrnc(U, z, trans = 'T')
# z1 := Us' * bz1 * Us
# = S^-1 * U' * bz1 * U * S^-1
# = S^-1 * bsz1 * S^-1
__cngrnc(Us, z, trans = 'T', offsetx = msq)
# x[1] := Uti' * x[1] * Uti
# = bsx[1]
__cngrnc(Uti, x[1], trans = 'T')
# x[1] := x[1] - z[msq:]
# = bsx[1] - S^-1 * bsz1 * S^-1
blas.axpy(z, x[1], alpha = -1.0, offsetx = msq)
# x1 = (S * x[1] * S + z[:msq] ) ./ sqrtG
# = (S * ( bsx[1] - S^-1 * bsz1 * S^-1) * S + bsz0 ) ./ sqrtG
# = (S * bsx[1] * S - bsz1 + bsz0 ) ./ sqrtG
# in packed storage
blas.copy(x[1], x1)
blas.tbmv(S, x1, n = msq, k = 0, ldA = 1)
blas.axpy(z, x1, n = msq)
blas.tbsv(sqrtG, x1, n = msq, k = 0, ldA = 1)
misc.pack2(x1, {'l': 0, 'q': [], 's': [m]})
# x[0] := x[0] + Asc'*x1
# = bx0 + Asc'( ( bsz0 - bsz1 + S * bsx[1] * S ) ./ sqrtG)
# = bx0 + As'( ( bz0 - bz1 + S * bx[1] * S ) ./ Gamma )
blas.gemv(Asc, x1, x[0], m = mpckd, trans = 'T', beta = 1.0)
# x[0] := H^-1 * x[0]
# = ux[0]
lapack.potrs(H, x[0])
# x1 = Asc(x[0]) .* sqrtG (unpacked)
# = As(x[0])
blas.gemv(Asc, x[0], tmp, m = mpckd)
misc.unpack(tmp, x1, {'l': 0, 'q': [], 's': [m]})
blas.tbmv(sqrtG, x1, n = msq, k = 0, ldA = 1)
# usx[1] = (x1 + (x[1] - z[:msq])) ./ sqrtG**2
# = (As(ux[0]) + bsx[1] - bsz0 - S^-1 * bsz1 * S^-1)
# ./ Gamma
# x[1] := x[1] - z[:msq]
# = bsx[1] - bsz0 - S^-1 * bsz1 * S^-1
blas.axpy(z, x[1], -1.0, n = msq)
# x[1] := x[1] + x1
# = As(ux) + bsx[1] - bsz0 - S^-1 * bsz1 * S^-1
blas.axpy(x1, x[1])
# x[1] := x[1] / Gammma
# = (As(ux) + bsx[1] - bsz0 + S^-1 * bsz1 * S^-1 ) / Gamma
# = S^-1 * usx[1] * S^-1
blas.tbsv(Gamma, x[1], n = msq, k = 0, ldA = 1)
# z[msq:] := r1' * U * (-z[msq:] - x[1]) * U * r1
# := -r1' * U * S^-1 * (bsz1 + ux[1]) * S^-1 * U * r1
# := -r1' * uz1 * r1
blas.axpy(x[1], z, n = msq, offsety = msq)
blas.scal(-1.0, z, offset = msq)
__cngrnc(U, z, offsetx = msq)
__cngrnc(W['r'][1], z, trans = 'T', offsetx = msq)
# x[1] := S * x[1] * S
# = usx1
blas.tbmv(S, x[1], n = msq, k = 0, ldA = 1)
# z[:msq] = r0' * U' * ( x1 - x[1] - z[:msq] ) * U * r0
# = r0' * U' * ( As(ux) - usx1 - bsz0 ) * U * r0
# = r0' * U' * usz0 * U * r0
# = r0' * uz0 * r0
blas.axpy(x1, z, -1.0, n = msq)
blas.scal(-1.0, z, n = msq)
blas.axpy(x[1], z, -1.0, n = msq)
__cngrnc(U, z)
__cngrnc(W['r'][0], z, trans = 'T')
# x[1] := Uti * x[1] * Uti'
# = ux[1]
__cngrnc(Uti, x[1])
return solve
# least squares solution: minimize || A*x - b ||_2^2 xls = +b lapack.gels(+A, xls) xls = xls[:n] # Tikhonov solution: minimize || A*x - b ||_2^2 + 0.1*||x||^2_2 xtik = A.T * b S = A.T * A S[::n + 1] += 0.1 lapack.posv(S, xtik) # Worst case solution xwc = wcls(A, Ap, b) notrials = 100000 r = sqrt(uniform(1, notrials)) theta = 2.0 * pi * uniform(1, notrials) u = matrix(0.0, (2, notrials)) u[0, :] = mul(r, cos(theta)) u[1, :] = mul(r, sin(theta)) # LS solution q = A * xls - b P = matrix(0.0, (m, 2)) P[:, 0], P[:, 1] = Ap[0] * xls, Ap[1] * xls r = P * u + q[:, notrials * [0]] resls = sqrt(matrix(1.0, (1, m)) * mul(r, r)) q = A * xtik - b P[:, 0], P[:, 1] = Ap[0] * xtik, Ap[1] * xtik r = P * u + q[:, notrials * [0]]
def kkt(W):
"""
KKT solver for
X*X' * ux + uy * 1_m' + mat(uz) = bx
ux * 1_m = by
ux - d.^2 .* mat(uz) = mat(bz).
ux and bx are N x m matrices.
uy and by are N-vectors.
uz and bz are N*m-vectors. mat(uz) is the N x m matrix that
satisfies mat(uz)[:] = uz.
d = mat(W['d']) a positive N x m matrix.
If we eliminate uz from the last equation using
mat(uz) = (ux - mat(bz)) ./ d.^2
we get two equations in ux, uy:
X*X' * ux + ux ./ d.^2 + uy * 1_m' = bx + mat(bz) ./ d.^2
ux * 1_m = by.
From the 1st equation,
uxk = (X*X' + Dk^-2)^-1 * (-uy + bxk + Dk^-2 * bzk)
= Dk * (I + Xk*Xk')^-1 * Dk * (-uy + bxk + Dk^-2 * bzk)
for k = 1, ..., m, where Dk = diag(d[:,k]), Xk = Dk * X,
uxk is column k of ux, and bzk is column k of mat(bz).
We use the matrix inversion lemma
( I + Xk * Xk' )^-1 = I - Xk * (I + Xk' * Xk)^-1 * Xk'
= I - Xk * Hk^-1 * Xk'
= I - Xk * Lk^-T * Lk^-1 * Xk'
where Hk = I + Xk' * Xk = Lk * Lk' to write this as
uxk = Dk * (I - Xk * Hk^-1 * Xk') * Dk *
(-uy + bxk + Dk^-2 * bzk)
= (Dk^2 - Dk^2 * X * Hk^-1 * X' * Dk^2) *
(-uy + bxk + Dk^-2 * bzk).
Substituting this in the second equation gives an equation
for uy:
sum_k (Dk^2 - Dk^2 * X * Hk^-1 * X' * Dk^2 ) * uy
= -by + sum_k (Dk^2 - Dk^2 * X * Hk^-1 * X' * Dk^2) *
( bxk + Dk^-2 * bzk ),
i.e., with D = (sum_k Dk^2)^1/2, Yk = D^-1 * Dk^2 * X * Lk^-T,
D * ( I - sum_k Yk * Yk' ) * D * uy
= -by + sum_k (Dk^2 - Dk^2 * X * Hk^-1 * X' * Dk^2) *
( bxk + Dk^-2 *bzk ).
Another application of the matrix inversion lemma gives
uy = D^-1 * (I + Y * S^-1 * Y') * D^-1 *
( -by + sum_k ( Dk^2 - Dk^2 * X * Hk^-1 * X' * Dk^2 ) *
( bxk + Dk^-2 *bzk ) )
with S = I - Y' * Y, Y = [ Y1 ... Ym ].
Summary:
1. Compute
uy = D^-1 * (I + Y * S^-1 * Y') * D^-1 *
( -by + sum_k (Dk^2 - Dk^2 * X * Hk^-1 * X' * Dk^2)
* ( bxk + Dk^-2 *bzk ) )
2. For k = 1, ..., m:
uxk = (Dk^2 - Dk^2 * X * Hk^-1 * X' * Dk^2) *
(-uy + bxk + Dk^-2 * bzk)
3. Solve for uz
d .* uz = ( ux - mat(bz) ) ./ d.
Return ux, uy, d .* uz.
"""
###
utime0, stime0 = cputime()
###
d = matrix(W['d'], (N, m))
dsq = matrix(W['d']**2, (N, m))
# Factor the matrices
#
# H[k] = I + Xk' * Xk
# = I + X' * Dk^2 * X.
#
# Dk = diag(d[:,k]).
for k in range(m):
# H[k] = I
blas.scal(0.0, H[k])
H[k][::n + 1] = 1.0
# Xs = Dk * X
# = diag(d[:,k]]) * X
blas.copy(X, Xs)
for j in range(n):
blas.tbmv(d,
Xs,
n=N,
k=0,
ldA=1,
offsetA=k * N,
offsetx=j * N)
# H[k] := H[k] + Xs' * Xs
# = I + Xk' * Xk
blas.syrk(Xs, H[k], trans='T', beta=1.0)
# Factorization H[k] = Lk * Lk'
lapack.potrf(H[k])
###
utime, stime = cputime()
print("Factor Hk's: utime = %.2f, stime = %.2f" \
%(utime-utime0, stime-stime0))
utime0, stime0 = cputime()
###
# diag(D) = ( sum_k d[:,k]**2 ) ** 1/2
# = ( sum_k Dk^2) ** 1/2.
blas.gemv(dsq, ones, D)
D[:] = sqrt(D)
###
# utime, stime = cputime()
# print("Compute D: utime = %.2f, stime = %.2f" \
# %(utime-utime0, stime-stime0))
utime0, stime0 = cputime()
###
# S = I - Y'* Y is an m x m block matrix.
# The i,j block of Y' * Y is
#
# Yi' * Yj = Li^-1 * X' * Di^2 * D^-1 * Dj^2 * X * Lj^-T.
#
# We compute only the lower triangular blocks in Y'*Y.
blas.scal(0.0, S)
for i in range(m):
for j in range(i + 1):
# Xs = Di * Dj * D^-1 * X
blas.copy(X, Xs)
blas.copy(d, wN, n=N, offsetx=i * N)
blas.tbmv(d, wN, n=N, k=0, ldA=1, offsetA=j * N)
blas.tbsv(D, wN, n=N, k=0, ldA=1)
for k in range(n):
blas.tbmv(wN, Xs, n=N, k=0, ldA=1, offsetx=k * N)
# block i, j of S is Xs' * Xs (as nonsymmetric matrix so we
# get the correct multiple after scaling with Li, Lj)
blas.gemm(Xs,
Xs,
S,
transA='T',
ldC=m * n,
offsetC=(j * n) * m * n + i * n)
###
utime, stime = cputime()
print("Form S: utime = %.2f, stime = %.2f" \
%(utime-utime0, stime-stime0))
utime0, stime0 = cputime()
###
for i in range(m):
# multiply block row i of S on the left with Li^-1
blas.trsm(H[i],
S,
m=n,
n=(i + 1) * n,
ldB=m * n,
offsetB=i * n)
# multiply block column i of S on the right with Li^-T
blas.trsm(H[i],
S,
side='R',
transA='T',
m=(m - i) * n,
n=n,
ldB=m * n,
offsetB=i * n * (m * n + 1))
blas.scal(-1.0, S)
S[::(m * n + 1)] += 1.0
###
utime, stime = cputime()
print("Form S (2): utime = %.2f, stime = %.2f" \
%(utime-utime0, stime-stime0))
utime0, stime0 = cputime()
###
# S = L*L'
lapack.potrf(S)
###
utime, stime = cputime()
print("Factor S: utime = %.2f, stime = %.2f" \
%(utime-utime0, stime-stime0))
utime0, stime0 = cputime()
###
def f(x, y, z):
"""
1. Compute
uy = D^-1 * (I + Y * S^-1 * Y') * D^-1 *
( -by + sum_k (Dk^2 - Dk^2 * X * Hk^-1 * X' * Dk^2)
* ( bxk + Dk^-2 *bzk ) )
2. For k = 1, ..., m:
uxk = (Dk^2 - Dk^2 * X * Hk^-1 * X' * Dk^2) *
(-uy + bxk + Dk^-2 * bzk)
3. Solve for uz
d .* uz = ( ux - mat(bz) ) ./ d.
Return ux, uy, d .* uz.
"""
###
utime0, stime0 = cputime()
###
# xk := Dk^2 * xk + zk
# = Dk^2 * bxk + bzk
blas.tbmv(dsq, x, n=N * m, k=0, ldA=1)
blas.axpy(z, x)
# y := -y + sum_k ( I - Dk^2 * X * Hk^-1 * X' ) * xk
# = -y + x*ones - sum_k Dk^2 * X * Hk^-1 * X' * xk
# y := -y + x*ones
blas.gemv(x, ones, y, alpha=1.0, beta=-1.0)
# wnm = X' * x (wnm interpreted as an n x m matrix)
blas.gemm(X, x, wnm, m=n, k=N, n=m, transA='T', ldB=N, ldC=n)
# wnm[:,k] = Hk \ wnm[:,k] (for wnm as an n x m matrix)
for k in range(m):
lapack.potrs(H[k], wnm, offsetB=k * n)
for k in range(m):
# wN = X * wnm[:,k]
blas.gemv(X, wnm, wN, offsetx=n * k)
# wN = Dk^2 * wN
blas.tbmv(dsq[:, k], wN, n=N, k=0, ldA=1)
# y := y - wN
blas.axpy(wN, y, -1.0)
# y = D^-1 * (I + Y * S^-1 * Y') * D^-1 * y
#
# Y = [Y1 ... Ym ], Yk = D^-1 * Dk^2 * X * Lk^-T.
# y := D^-1 * y
blas.tbsv(D, y, n=N, k=0, ldA=1)
# wnm = Y' * y (interpreted as an Nm vector)
# = [ L1^-1 * X' * D1^2 * D^-1 * y;
# L2^-1 * X' * D2^2 * D^-1 * y;
# ...
# Lm^-1 * X' * Dm^2 * D^-1 * y ]
for k in range(m):
# wN = D^-1 * Dk^2 * y
blas.copy(y, wN)
blas.tbmv(dsq, wN, n=N, k=0, ldA=1, offsetA=k * N)
blas.tbsv(D, wN, n=N, k=0, ldA=1)
# wnm[:,k] = X' * wN
blas.gemv(X, wN, wnm, trans='T', offsety=k * n)
# wnm[:,k] = Lk^-1 * wnm[:,k]
blas.trsv(H[k], wnm, offsetx=k * n)
# wnm := S^-1 * wnm (an mn-vector)
lapack.potrs(S, wnm)
# y := y + Y * wnm
# = y + D^-1 * [ D1^2 * X * L1^-T ... D2^k * X * Lk^-T]
# * wnm
for k in range(m):
# wnm[:,k] = Lk^-T * wnm[:,k]
blas.trsv(H[k], wnm, trans='T', offsetx=k * n)
# wN = X * wnm[:,k]
blas.gemv(X, wnm, wN, offsetx=k * n)
# wN = D^-1 * Dk^2 * wN
blas.tbmv(dsq, wN, n=N, k=0, ldA=1, offsetA=k * N)
blas.tbsv(D, wN, n=N, k=0, ldA=1)
# y += wN
blas.axpy(wN, y)
# y := D^-1 * y
blas.tbsv(D, y, n=N, k=0, ldA=1)
# For k = 1, ..., m:
#
# xk = (I - Dk^2 * X * Hk^-1 * X') * (-Dk^2 * y + xk)
# x = x - [ D1^2 * y ... Dm^2 * y] (as an N x m matrix)
for k in range(m):
blas.copy(y, wN)
blas.tbmv(dsq, wN, n=N, k=0, ldA=1, offsetA=k * N)
blas.axpy(wN, x, -1.0, offsety=k * N)
# wnm = X' * x (as an n x m matrix)
blas.gemm(X, x, wnm, transA='T', m=n, n=m, k=N, ldB=N, ldC=n)
# wnm[:,k] = Hk^-1 * wnm[:,k]
for k in range(m):
lapack.potrs(H[k], wnm, offsetB=n * k)
for k in range(m):
# wN = X * wnm[:,k]
blas.gemv(X, wnm, wN, offsetx=k * n)
# wN = Dk^2 * wN
blas.tbmv(dsq, wN, n=N, k=0, ldA=1, offsetA=k * N)
# x[:,k] := x[:,k] - wN
blas.axpy(wN, x, -1.0, n=N, offsety=k * N)
# z := ( x - z ) ./ d
blas.axpy(x, z, -1.0)
blas.scal(-1.0, z)
blas.tbsv(d, z, n=N * m, k=0, ldA=1)
###
utime, stime = cputime()
print("Solve: utime = %.2f, stime = %.2f" \
%(utime-utime0, stime-stime0))
###
return f
def _build_conelp(self):
Nx = self.nbus
self.Nx = Nx
nflow_constr = len(self.branches_with_flow_constraints())
npad_constr = len(self.branches_with_pad_constraints())
ngen_var_p = len(self.generators_with_var_real_power())
ngen_qcost = len(self.generators_with_var_real_power_and_quadratic_cost())
ngen_lcost = len(self.generators_with_var_real_power_and_linear_cost())
self._ngen_var_p = ngen_var_p
ngen_var_q = len(self.generators_with_var_reactive_power())
self._ngen_var_q = ngen_var_q
dims = {}
dims['l'] = 2*self.nbus + 2*ngen_var_p + 2*ngen_var_q + ngen_qcost + 2*npad_constr
dims['q'] = 2*nflow_constr*[3]
dims['q'] += ngen_qcost*[3]
dims['s'] = [Nx]
offset = {}
offset['t'] = 0 # t = aux. vars for epigraph formulation of quad. gen. power cost
offset['wpl'] = offset['t'] + ngen_qcost # wpl = slack lower bnd: Pmin[i] + wpl[i] = Pg[i]
offset['wpu'] = offset['wpl'] + ngen_var_p # wpu = slack upper bnd: Pg[i] + wpu[i] = Pmax[i]
offset['wql'] = offset['wpu'] + ngen_var_p # wql = slack lower bnd: Qmin[i] + wql[i] = Qg[i]
offset['wqu'] = offset['wql'] + ngen_var_q # wqu = slack upper bnd: Qg[i] + wqu[i] = Qmax[i]
offset['ul'] = offset['wqu'] + ngen_var_q # ul = slack lower bnd: Vmin[i]**2 + ul[i] = abs(V[i])
offset['uu'] = offset['ul'] + self.nbus # uu = slack upper bnd: abs(V[i]) + uu[i] = Vmax[i]**2
offset['lpad'] = offset['uu'] + self.nbus
offset['upad'] = offset['lpad'] + npad_constr
offset['z'] = offset['upad'] + npad_constr # z = line flow const: z[k*3:(k+1)*3] in SOC of dim 3
offset['w'] = offset['z'] + sum(dims['q'])-3*ngen_qcost # w = epigraph of quad. gen cost: w[k*3:(k+1)*3] in SOC of dim 3
offset['X'] = offset['w'] + 3*ngen_qcost # X = SDR of X = V*V.H
N = offset['X'] + Nx**2
self.offset = offset
dual_offset = [0]
# power balance
dual_offset.append(dual_offset[-1] + 2*self.ngen)
# gen. limits (real)
dual_offset.append(dual_offset[-1] + len(self.generators_with_var_real_power()))
# gen. limits (reactive)
dual_offset.append(dual_offset[-1] + len(self.generators_with_var_reactive_power()))
# voltage constraints
dual_offset.append(dual_offset[-1] + 2*self.nbus)
# line constraints
dual_offset.append(dual_offset[-1] + 6*len(self.branches_with_flow_constraints()))
# pad pad_constraints
dual_offset.append(dual_offset[-1] + 2*npad_constr)
# quad. cost
dual_offset.append(dual_offset[-1] + 3*len(self.generators_with_var_real_power_and_quadratic_cost()))
self.dual_offset = matrix(dual_offset)
##
## Build h and initialize c
##
I,V = [],[]
for k,gen in enumerate(self. generators_with_var_real_power()):
beta = gen['Pcost']['coef'][-2]
if gen['Pcost']['ncoef'] > 2: beta += 2.0*self.baseMVA*gen['Pcost']['coef'][-3]*gen['Pmin']
I.append(offset['wpl'] + gen['pslack'])
V.append(beta)
h = spmatrix(ngen_qcost*[1.0]+V, list(range(ngen_qcost))+I, (ngen_qcost+len(I))*[0], (N,1))
c = []
##
## Initialize lists for triplet storage of columns in G
##
I,V = [],[]
##
## Power balance constraints
##
rp,ci,val = self.Ybus.T.CCS
def conj(l): return [li.conjugate() for li in l]
def smul(l): return [-li for li in l]
def jmul(l):
j = complex(0.0,1.0)
return [j*li for li in l]
def Yijv(k):
jj = list(ci[rp[k]:rp[k+1]])
ii = len(jj)*[k]
vv = list(0.5*val[rp[k]:rp[k+1]])
t = jj.index(k)
Iret = t*[k]+jj+(len(jj)-1-t)*[k]
Jret = jj[:t]+len(jj)*[k]+jj[t+1:]
Vret = vv[:t] + conj(vv[:t]) + [2.0*vv[t].real] + conj(vv[t+1:]) + vv[t+1:]
Vbret = vv[:t] + smul(conj(vv[:t])) + [complex(0.0,2.0*vv[t].imag)] + smul(conj(vv[t+1:])) + vv[t+1:]
return Iret,Jret,Vret,jmul(Vbret)
for k, bus in enumerate(self.busses):
YI,YJ,YV,YbV = Yijv(k)
if bus['id'] in self.bus_id_to_genlist:
genlist = [self.generators[i] for i in self.bus_id_to_genlist[bus['id']]]
L = [offset['X'] + ii + jj*Nx for ii,jj in zip(YI,YJ)]
I.append([offset['wpl']+gen['pslack'] for gen in genlist if gen['pslack'] is not None] + L)
V.append(len([1 for gen in genlist if gen['pslack'] is not None])*[-1.0] + YV)
c.append(bus['Pd'] - sum([gen['Pmin'] for gen in genlist]))
L = [offset['X'] + ii + jj*Nx for ii,jj in zip(YI,YJ)]
I.append([offset['wql']+gen['qslack'] for gen in genlist if gen['qslack'] is not None] + L)
V.append(len([1 for gen in genlist if gen['qslack'] is not None])*[-1.0] + YbV)
c.append(bus['Qd'] - sum([gen['Qmin'] for gen in genlist]))
else:
L = [offset['X'] + ii + jj*Nx for ii,jj in zip(YI,YJ)]
I.append(L)
V.append(YV)
c.append(bus['Pd'])
L = [offset['X'] + ii + jj*Nx for ii,jj in zip(YI,YJ)]
I.append(L)
V.append(YbV)
c.append(bus['Qd'])
##
## Power generation limits
##
for k, gen in enumerate(self.generators_with_var_real_power()):
I.append([offset['wpl']+k, offset['wpu']+k])
V.append([1.0, 1.0])
c.append(gen['Pmin'] - gen['Pmax'])
for k, gen in enumerate(self.generators_with_var_reactive_power()):
I.append([offset['wql']+k, offset['wqu']+k])
V.append([1.0, 1.0])
c.append(gen['Qmin'] - gen['Qmax'])
##
## Voltage constraints
##
for k, bus in enumerate(self.busses):
I.append([offset['ul']+k, offset['X']+k*(self.nbus+1)])
V.append([1.0, -1.0])
c.append(bus['minVm']**2)
for k, bus in enumerate(self.busses):
I.append([offset['uu']+k, offset['X']+k*(self.nbus+1)])
V.append([1.0, 1.0])
c.append(-bus['maxVm']**2)
##
## Line constraints
##
rpf,cif,valf = self.Yf.T.CCS
rpt,cit,valt = self.Yt.T.CCS
def Tijv(k, fr, to, flow = 'ft'):
if flow == 'ft':
y = self.Ybr[k][0]
if fr > to:
y.reverse()
fr,to = to,fr
flow = 'tf'
elif flow == 'tf':
y = self.Ybr[k][1]
if fr > to:
y.reverse()
fr,to = to,fr
flow = 'ft'
if flow == 'ft':
Iret = [fr,to,fr]
Jret = [fr,fr,to]
Tret = [y[0].real, 0.5*y[1].conjugate(), 0.5*y[1]]
Tbret = [-y[0].imag, -complex(0.0,0.5)*y[1].conjugate(), complex(0.0,0.5)*y[1]]
elif flow == 'tf':
Iret = [to,fr,to]
Jret = [fr,to,to]
Tret = [0.5*y[0],0.5*y[0].conjugate(),y[1].real]
Tbret = [complex(0.0,0.5)*y[0],-complex(0.0,0.5)*y[0].conjugate(),-y[1].imag]
return Iret,Jret,Tret,Tbret
for kk,kbranch in enumerate(self.branches_with_flow_constraints()):
k,branch = kbranch
fr = self.bus_id_to_index[branch['from']]
to = self.bus_id_to_index[branch['to']]
# flow at "from" end
Ir,Jr,Vr,Vbr = Tijv(k,fr,to,'ft')
I.append([offset['z']+6*kk])
V.append([1.0])
c.append(-branch['rateA']/self.baseMVA)
I.append([offset['z']+6*kk+1] + [offset['X'] + ii + jj*Nx for ii,jj in zip(Ir,Jr)])
V.append([-1.0] + Vr)
c.append(0.0)
I.append([offset['z']+6*kk+2] + [offset['X'] + ii + jj*Nx for ii,jj in zip(Ir,Jr)])
V.append([-1.0] + Vbr)
c.append(0.0)
# flow at "to" end
Ir,Jr,Vr,Vbr = Tijv(k,fr,to,'tf')
I.append([offset['z']+6*kk+3])
V.append([1.0])
c.append(-branch['rateA']/self.baseMVA)
I.append([offset['z']+6*kk+4] + [offset['X'] + ii + jj*Nx for ii,jj in zip(Ir,Jr)])
V.append([-1.0] + Vr)
c.append(0.0)
I.append([offset['z']+6*kk+5] + [offset['X'] + ii + jj*Nx for ii,jj in zip(Ir,Jr)])
V.append([-1.0] + Vbr)
c.append(0.0)
##
## Phase angle difference constraints
##
for kk,kbranch in enumerate(self.branches_with_pad_constraints()):
k,branch = kbranch
tan_amin = tan(branch['angle_min']*pi/180.0)
f = self.bus_id_to_index[branch['from']]
t = self.bus_id_to_index[branch['to']]
I.append([offset['lpad']+kk] + [offset['X']+f+t*Nx,offset['X']+t+f*Nx])
V.append([1.0] + [complex(0.5*tan_amin,0.5), complex(0.5*tan_amin,-0.5)])
c.append(0.0)
for kk,kbranch in enumerate(self.branches_with_pad_constraints()):
k,branch = kbranch
tan_amax = tan(branch['angle_max']*pi/180.0)
f = self.bus_id_to_index[branch['from']]
t = self.bus_id_to_index[branch['to']]
I.append([offset['upad']+kk] + [offset['X']+f+t*Nx,offset['X']+t+f*Nx])
V.append([1.0] + [-complex(0.5*tan_amax,0.5), -complex(0.5*tan_amax,-0.5)])
c.append(0.0)
##
## Quadratic cost -- epigraph
##
for k, gen in enumerate(self.generators_with_var_real_power_and_quadratic_cost()):
assert gen['Pcost']['ncoef'] == 3, "only quadratic cost implemented"
assert gen['Pcost']['model'] == 2, "only quadratic cost implemented"
ak = gen['Pcost']['coef'][-3]*self.baseMVA
I.append([offset['t']+k, offset['w']+3*k])
V.append([1.0, -1.0])
c.append(0.5)
I.append([offset['t']+k, offset['w']+3*k+1])
V.append([-1.0, -1.0])
c.append(0.5)
I.append([offset['wpl']+gen['pslack'], offset['w']+3*k+2])
V.append([sqrt(2.0*ak), -1.0])
c.append(0.0)
##
## Fixed cost
##
self.const_cost = 0.0
for kk,gen in enumerate(self.generators):
if gen['Pcost']['ncoef'] == 2 or (gen['Pcost']['ncoef'] == 3 and gen['Pcost']['coef'][0] == 0.0):
self.const_cost += gen['Pcost']['coef'][-1]/self.baseMVA + gen['Pcost']['coef'][-2]*gen['Pmin']
elif gen['Pcost']['ncoef'] == 3:
self.const_cost += gen['Pcost']['coef'][-1]/self.baseMVA \
+ gen['Pcost']['coef'][-2]*gen['Pmin'] \
+ self.baseMVA*gen['Pcost']['coef'][-3]*gen['Pmin']**2
self.const_cost *= self.baseMVA
##
## Build c, h, and G
##
c = matrix(c)
J = [len(Ii)*[j] for j,Ii in enumerate(I)]
G = spmatrix([v for v in chain(*V)],
[v for v in chain(*I)],
[v for v in chain(*J)],(N,len(c)))
self.problem_data = (c, G, h, dims)
return
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 F(W):
for j in xrange(N):
# SVD R[j] = U[j] * diag(sig[j]) * Vt[j]
lapack.gesvd(+W['r'][j],
sv[j],
jobu='A',
jobvt='A',
U=U[j],
Vt=Vt[j])
# Vt[j] := diag(sig[j])^-1 * Vt[j]
for k in xrange(ns[j]):
blas.tbsv(sv[j], Vt[j], n=ns[j], k=0, ldA=1, offsetx=k * ns[j])
# Gamma[j] is an ns[j] x ns[j] symmetric matrix
#
# (sig[j] * sig[j]') ./ sqrt(1 + rho * (sig[j] * sig[j]').^2)
# S = sig[j] * sig[j]'
S = matrix(0.0, (ns[j], ns[j]))
blas.syrk(sv[j], S)
Gamma[j][:] = div(S, sqrt(1.0 + rho * S**2))[:]
symmetrize(Gamma[j], ns[j])
# As represents the scaled mapping
#
# As(x) = A(u * (Gamma .* x) * u')
# As'(y) = Gamma .* (u' * A'(y) * u)
#
# stored in a similar format as A, except that we use packed
# storage for the columns of As[i][j].
for i in xrange(M):
if (type(A[i][j]) is matrix) or (type(A[i][j]) is spmatrix):
# As[i][j][:,k] = vec(
# (U[j]' * mat( A[i][j][:,k] ) * U[j]) .* Gamma[j])
copy(A[i][j], As[i][j])
As[i][j] = matrix(As[i][j])
for k in xrange(ms[i]):
cngrnc(U[j],
As[i][j],
trans='T',
offsetx=k * (ns[j]**2),
n=ns[j])
blas.tbmv(Gamma[j],
As[i][j],
n=ns[j]**2,
k=0,
ldA=1,
offsetx=k * (ns[j]**2))
# pack As[i][j] in place
#pack_ip(As[i][j], ns[j])
for k in xrange(As[i][j].size[1]):
tmp = +As[i][j][:, k]
misc.pack2(tmp, {'l': 0, 'q': [], 's': [ns[j]]})
As[i][j][:, k] = tmp
else:
As[i][j] = 0.0
# H is an m times m block matrix with i, k block
#
# Hik = sum_j As[i,j]' * As[k,j]
#
# of size ms[i] x ms[k]. Hik = 0 if As[i,j] or As[k,j]
# are zero for all j
H = spmatrix([], [], [], (sum(ms), sum(ms)))
rowid = 0
for i in xrange(M):
colid = 0
for k in xrange(i + 1):
sparse_block = True
Hik = matrix(0.0, (ms[i], ms[k]))
for j in xrange(N):
if (type(As[i][j]) is matrix) and \
(type(As[k][j]) is matrix):
sparse_block = False
# Hik += As[i,j]' * As[k,j]
if i == k:
blas.syrk(As[i][j],
Hik,
trans='T',
beta=1.0,
k=ns[j] * (ns[j] + 1) / 2,
ldA=ns[j]**2)
else:
blas.gemm(As[i][j],
As[k][j],
Hik,
transA='T',
beta=1.0,
k=ns[j] * (ns[j] + 1) / 2,
ldA=ns[j]**2,
ldB=ns[j]**2)
if not (sparse_block):
H[rowid:rowid+ms[i], colid:colid+ms[k]] \
= sparse(Hik)
colid += ms[k]
rowid += ms[i]
HF = cholmod.symbolic(H)
cholmod.numeric(H, HF)
def solve(x, y, z):
"""
Returns solution of
rho * ux + A'(uy) - r^-T * uz * r^-1 = bx
A(ux) = by
-ux - r * uz * r' = bz.
On entry, x = bx, y = by, z = bz.
On exit, x = ux, y = uy, z = uz.
"""
# bz is a copy of z in the format of x
blas.copy(z, bz)
# x := x + rho * bz
# = bx + rho * bz
blas.axpy(bz, x, alpha=rho)
# x := Gamma .* (u' * x * u)
# = Gamma .* (u' * (bx + rho * bz) * u)
offsetj = 0
for j in xrange(N):
cngrnc(U[j], x, trans='T', offsetx=offsetj, n=ns[j])
blas.tbmv(Gamma[j], x, n=ns[j]**2, k=0, ldA=1, offsetx=offsetj)
offsetj += ns[j]**2
# y := y - As(x)
# := by - As( Gamma .* u' * (bx + rho * bz) * u)
blas.copy(x, xp)
offsetj = 0
for j in xrange(N):
misc.pack2(xp, {'l': offsetj, 'q': [], 's': [ns[j]]})
offsetj += ns[j]**2
offseti = 0
for i in xrange(M):
offsetj = 0
for j in xrange(N):
if type(As[i][j]) is matrix:
blas.gemv(As[i][j],
xp,
y,
trans='T',
alpha=-1.0,
beta=1.0,
m=ns[j] * (ns[j] + 1) / 2,
n=ms[i],
ldA=ns[j]**2,
offsetx=offsetj,
offsety=offseti)
offsetj += ns[j]**2
offseti += ms[i]
# y := -y - A(bz)
# = -by - A(bz) + As(Gamma .* (u' * (bx + rho * bz) * u)
Af(bz, y, alpha=-1.0, beta=-1.0)
# y := H^-1 * y
# = H^-1 ( -by - A(bz) + As(Gamma.* u'*(bx + rho*bz)*u) )
# = uy
cholmod.solve(HF, y)
# bz = Vt' * vz * Vt
# = uz where
# vz := Gamma .* ( As'(uy) - x )
# = Gamma .* ( As'(uy) - Gamma .* (u'*(bx + rho *bz)*u) )
# = Gamma.^2 .* ( u' * (A'(uy) - bx - rho * bz) * u ).
blas.copy(x, xp)
offsetj = 0
for j in xrange(N):
# xp is -x[j] = -Gamma .* (u' * (bx + rho*bz) * u)
# in packed storage
misc.pack2(xp, {'l': offsetj, 'q': [], 's': [ns[j]]})
offsetj += ns[j]**2
blas.scal(-1.0, xp)
offsetj = 0
for j in xrange(N):
# xp += As'(uy)
offseti = 0
for i in xrange(M):
if type(As[i][j]) is matrix:
blas.gemv(As[i][j], y, xp, alpha = 1.0,
beta = 1.0, m = ns[j]*(ns[j]+1)/2, \
n = ms[i],ldA = ns[j]**2, \
offsetx = offseti, offsety = offsetj)
offseti += ms[i]
# bz[j] is xp unpacked and multiplied with Gamma
#unpack(xp, bz[j], ns[j])
misc.unpack(xp,
bz, {
'l': 0,
'q': [],
's': [ns[j]]
},
offsetx=offsetj,
offsety=offsetj)
blas.tbmv(Gamma[j],
bz,
n=ns[j]**2,
k=0,
ldA=1,
offsetx=offsetj)
# bz = Vt' * bz * Vt
# = uz
cngrnc(Vt[j], bz, trans='T', offsetx=offsetj, n=ns[j])
symmetrize(bz, ns[j], offset=offsetj)
offsetj += ns[j]**2
# x = -bz - r * uz * r'
blas.copy(z, x)
blas.copy(bz, z)
offsetj = 0
for j in xrange(N):
cngrnc(+W['r'][j], bz, offsetx=offsetj, n=ns[j])
offsetj += ns[j]**2
blas.axpy(bz, x)
blas.scal(-1.0, x)
return solve
def Fkkt(W):
"""
Custom solver:
v := alpha * 2*A'*A * u + beta * v
"""
global mmS
mmS = matrix(0.0, (iR, iR))
global vvV
vvV = matrix(0.0, (iR, 1))
# Factor
#
# S = A*D^-1*A' + I
#
# where D = 2*D1*D2*(D1+D2)^-1, D1 = d[:n]**2, D2 = d[n:]**2.
mmAsc = matrix(0.0, (iR, iC))
d1, d2 = W["di"][:iC] ** 2, W["di"][iC:] ** 2
# ds is square root of diagonal of D
ds = sqrt(2.0) * div(mul(W["di"][:iC], W["di"][iC:]), sqrt(d1 + d2))
d3 = div(d2 - d1, d1 + d2)
# Asc = A*diag(d)^-1/2
blas.copy(mmTh, mmAsc)
for k in range(iR):
blas.tbsv(ds, mmAsc, n=iC, k=0, ldA=1, incx=iR, offsetx=k)
# S = I + A * D^-1 * A'
blas.syrk(mmAsc, mmS)
mmS[:: iR + 1] += 1.0
lapack.potrf(mmS)
def g(x, y, z):
x[:iC] = 0.5 * (
x[:iC] - mul(d3, x[iC:]) + mul(d1, z[:iC] + mul(d3, z[:iC])) - mul(d2, z[iC:] - mul(d3, z[iC:]))
)
x[:iC] = div(x[:iC], ds)
# Solve
#
# S * v = 0.5 * A * D^-1 * ( bx[:n]
# - (D2-D1)*(D1+D2)^-1 * bx[n:]
# + D1 * ( I + (D2-D1)*(D1+D2)^-1 ) * bz[:n]
# - D2 * ( I - (D2-D1)*(D1+D2)^-1 ) * bz[n:] )
blas.gemv(mmAsc, x, vvV)
lapack.potrs(mmS, vvV)
# x[:n] = D^-1 * ( rhs - A'*v ).
blas.gemv(mmAsc, vvV, x, alpha=-1.0, beta=1.0, trans="T")
x[:iC] = div(x[:iC], ds)
# x[n:] = (D1+D2)^-1 * ( bx[n:] - D1*bz[:n] - D2*bz[n:] )
# - (D2-D1)*(D1+D2)^-1 * x[:n]
x[iC:] = div(x[iC:] - mul(d1, z[:iC]) - mul(d2, z[iC:]), d1 + d2) - mul(d3, x[:iC])
# z[:n] = D1^1/2 * ( x[:n] - x[n:] - bz[:n] )
# z[n:] = D2^1/2 * ( -x[:n] - x[n:] - bz[n:] ).
z[:iC] = mul(W["di"][:iC], x[:iC] - x[iC:] - z[:iC])
z[iC:] = mul(W["di"][iC:], -x[:iC] - x[iC:] - z[iC:])
return g
