def G(x, y, alpha=1.0, beta=0.0, trans='N'):
minor = 0
if not helpers.sp_minor_empty():
minor = helpers.sp_minor_top()
else:
global loopg
loopg += 1
minor = loopg
helpers.sp_create("00-Gfunc", minor)
y *= beta
if trans == 'N':
# y += alpha * G * x
y[:n] += alpha * (x[:n] - x[n:2 * n])
y[n:2 * n] += alpha * (-x[:n] - x[n:2 * n])
y[2 * n + 1:] -= alpha * A * x[:n]
else:
# y += alpha * G'*x
y[:n] += alpha * (x[:n] - x[n:2 * n] - A.T * x[-m:])
y[n:] -= alpha * (x[:n] + x[n:2 * n])
helpers.sp_create("10-Gfunc", minor)
def G(x, y, alpha = 1.0, beta = 0.0, trans = 'N'):
minor = 0
if not helpers.sp_minor_empty():
minor = helpers.sp_minor_top()
else:
global loopg
loopg += 1
minor = loopg
helpers.sp_create("00-Gfunc", minor)
y *= beta
if trans=='N':
# y += alpha * G * x
y[:n] += alpha * (x[:n] - x[n:2*n])
y[n:2*n] += alpha * (-x[:n] - x[n:2*n])
y[2*n+1:] -= alpha * A*x[:n]
else:
# y += alpha * G'*x
y[:n] += alpha * (x[:n] - x[n:2*n] - A.T * x[-m:])
y[n:] -= alpha * (x[:n] + x[n:2*n])
helpers.sp_create("10-Gfunc", minor)
def Fkkt(W):
# Returns a function f(x, y, z) that solves
#
# [ 0 G' ] [ x ] = [ bx ]
# [ G -W'*W ] [ z ] [ bz ].
# First factor
#
# S = G' * W**-1 * W**-T * G
# = [0; -A]' * W3^-2 * [0; -A] + 4 * (W1**2 + W2**2)**-1
#
# where
#
# W1 = diag(d1) with d1 = W['d'][:n] = 1 ./ W['di'][:n]
# W2 = diag(d2) with d2 = W['d'][n:] = 1 ./ W['di'][n:]
# W3 = beta * (2*v*v' - J), W3^-1 = 1/beta * (2*J*v*v'*J - J)
# with beta = W['beta'][0], v = W['v'][0], J = [1, 0; 0, -I].
# As = W3^-1 * [ 0 ; -A ] = 1/beta * ( 2*J*v * v' - I ) * [0; A]
minor = 0
if not helpers.sp_minor_empty():
minor = helpers.sp_minor_top()
beta, v = W['beta'][0], W['v'][0]
As = 2 * v * (v[1:].T * A)
As[1:, :] *= -1.0
As[1:, :] -= A
As /= beta
# S = As'*As + 4 * (W1**2 + W2**2)**-1
S = As.T * As
helpers.sp_add_var("S", S)
d1, d2 = W['d'][:n], W['d'][n:]
d = 4.0 * (d1**2 + d2**2)**-1
S[::n + 1] += d
lapack.potrf(S)
helpers.sp_create("00-Fkkt", minor)
def f(x, y, z):
minor = 0
if not helpers.sp_minor_empty():
minor = helpers.sp_minor_top()
else:
global loopf
loopf += 1
minor = loopf
helpers.sp_create("00-f", minor)
# z := - W**-T * z
z[:n] = -div(z[:n], d1)
z[n:2 * n] = -div(z[n:2 * n], d2)
z[2 * n:] -= 2.0 * v * (v[0] * z[2 * n] -
blas.dot(v[1:], z[2 * n + 1:]))
z[2 * n + 1:] *= -1.0
z[2 * n:] /= beta
# x := x - G' * W**-1 * z
x[:n] -= div(z[:n], d1) - div(z[n:2 * n], d2) + As.T * z[-(m + 1):]
x[n:] += div(z[:n], d1) + div(z[n:2 * n], d2)
helpers.sp_create("15-f", minor)
# Solve for x[:n]:
#
# S*x[:n] = x[:n] - (W1**2 - W2**2)(W1**2 + W2**2)^-1 * x[n:]
x[:n] -= mul(div(d1**2 - d2**2, d1**2 + d2**2), x[n:])
helpers.sp_create("25-f", minor)
lapack.potrs(S, x)
helpers.sp_create("30-f", minor)
# Solve for x[n:]:
#
# (d1**-2 + d2**-2) * x[n:] = x[n:] + (d1**-2 - d2**-2)*x[:n]
x[n:] += mul(d1**-2 - d2**-2, x[:n])
helpers.sp_create("35-f", minor)
x[n:] = div(x[n:], d1**-2 + d2**-2)
helpers.sp_create("40-f", minor)
# z := z + W^-T * G*x
z[:n] += div(x[:n] - x[n:2 * n], d1)
helpers.sp_create("44-f", minor)
z[n:2 * n] += div(-x[:n] - x[n:2 * n], d2)
helpers.sp_create("48-f", minor)
z[2 * n:] += As * x[:n]
helpers.sp_create("50-f", minor)
return f
def f(x, y, z):
minor = 0
if not helpers.sp_minor_empty():
minor = helpers.sp_minor_top()
else:
global loopf
loopf += 1
minor = loopf
helpers.sp_create("00-f", minor)
# z := - W**-T * z
z[:n] = -div(z[:n], d1)
z[n:2 * n] = -div(z[n:2 * n], d2)
z[2 * n:] -= 2.0 * v * (v[0] * z[2 * n] -
blas.dot(v[1:], z[2 * n + 1:]))
z[2 * n + 1:] *= -1.0
z[2 * n:] /= beta
# x := x - G' * W**-1 * z
x[:n] -= div(z[:n], d1) - div(z[n:2 * n], d2) + As.T * z[-(m + 1):]
x[n:] += div(z[:n], d1) + div(z[n:2 * n], d2)
helpers.sp_create("15-f", minor)
# Solve for x[:n]:
#
# S*x[:n] = x[:n] - (W1**2 - W2**2)(W1**2 + W2**2)^-1 * x[n:]
x[:n] -= mul(div(d1**2 - d2**2, d1**2 + d2**2), x[n:])
helpers.sp_create("25-f", minor)
lapack.potrs(S, x)
helpers.sp_create("30-f", minor)
# Solve for x[n:]:
#
# (d1**-2 + d2**-2) * x[n:] = x[n:] + (d1**-2 - d2**-2)*x[:n]
x[n:] += mul(d1**-2 - d2**-2, x[:n])
helpers.sp_create("35-f", minor)
x[n:] = div(x[n:], d1**-2 + d2**-2)
helpers.sp_create("40-f", minor)
# z := z + W^-T * G*x
z[:n] += div(x[:n] - x[n:2 * n], d1)
helpers.sp_create("44-f", minor)
z[n:2 * n] += div(-x[:n] - x[n:2 * n], d2)
helpers.sp_create("48-f", minor)
z[2 * n:] += As * x[:n]
helpers.sp_create("50-f", minor)
def Fkkt(W):
# Returns a function f(x, y, z) that solves
#
# [ 0 G' ] [ x ] = [ bx ]
# [ G -W'*W ] [ z ] [ bz ].
# First factor
#
# S = G' * W**-1 * W**-T * G
# = [0; -A]' * W3^-2 * [0; -A] + 4 * (W1**2 + W2**2)**-1
#
# where
#
# W1 = diag(d1) with d1 = W['d'][:n] = 1 ./ W['di'][:n]
# W2 = diag(d2) with d2 = W['d'][n:] = 1 ./ W['di'][n:]
# W3 = beta * (2*v*v' - J), W3^-1 = 1/beta * (2*J*v*v'*J - J)
# with beta = W['beta'][0], v = W['v'][0], J = [1, 0; 0, -I].
# As = W3^-1 * [ 0 ; -A ] = 1/beta * ( 2*J*v * v' - I ) * [0; A]
minor = 0
if not helpers.sp_minor_empty():
minor = helpers.sp_minor_top()
beta, v = W['beta'][0], W['v'][0]
As = 2 * v * (v[1:].T * A)
As[1:,:] *= -1.0
As[1:,:] -= A
As /= beta
# S = As'*As + 4 * (W1**2 + W2**2)**-1
S = As.T * As
helpers.sp_add_var("S", S)
d1, d2 = W['d'][:n], W['d'][n:]
d = 4.0 * (d1**2 + d2**2)**-1
S[::n+1] += d
lapack.potrf(S)
helpers.sp_create("00-Fkkt", minor)
def f(x, y, z):
minor = 0
if not helpers.sp_minor_empty():
minor = helpers.sp_minor_top()
else:
global loopf
loopf += 1
minor = loopf
helpers.sp_create("00-f", minor)
# z := - W**-T * z
z[:n] = -div( z[:n], d1 )
z[n:2*n] = -div( z[n:2*n], d2 )
z[2*n:] -= 2.0*v*( v[0]*z[2*n] - blas.dot(v[1:], z[2*n+1:]) )
z[2*n+1:] *= -1.0
z[2*n:] /= beta
# x := x - G' * W**-1 * z
x[:n] -= div(z[:n], d1) - div(z[n:2*n], d2) + As.T * z[-(m+1):]
x[n:] += div(z[:n], d1) + div(z[n:2*n], d2)
helpers.sp_create("15-f", minor)
# Solve for x[:n]:
#
# S*x[:n] = x[:n] - (W1**2 - W2**2)(W1**2 + W2**2)^-1 * x[n:]
x[:n] -= mul( div(d1**2 - d2**2, d1**2 + d2**2), x[n:])
helpers.sp_create("25-f", minor)
lapack.potrs(S, x)
helpers.sp_create("30-f", minor)
# Solve for x[n:]:
#
# (d1**-2 + d2**-2) * x[n:] = x[n:] + (d1**-2 - d2**-2)*x[:n]
x[n:] += mul( d1**-2 - d2**-2, x[:n])
helpers.sp_create("35-f", minor)
x[n:] = div( x[n:], d1**-2 + d2**-2)
helpers.sp_create("40-f", minor)
# z := z + W^-T * G*x
z[:n] += div( x[:n] - x[n:2*n], d1)
helpers.sp_create("44-f", minor)
z[n:2*n] += div( -x[:n] - x[n:2*n], d2)
helpers.sp_create("48-f", minor)
z[2*n:] += As*x[:n]
helpers.sp_create("50-f", minor)
return f
def f(x, y, z):
minor = 0
if not helpers.sp_minor_empty():
minor = helpers.sp_minor_top()
else:
global loopf
loopf += 1
minor = loopf
helpers.sp_create("00-f", minor)
# z := - W**-T * z
z[:n] = -div( z[:n], d1 )
z[n:2*n] = -div( z[n:2*n], d2 )
z[2*n:] -= 2.0*v*( v[0]*z[2*n] - blas.dot(v[1:], z[2*n+1:]) )
z[2*n+1:] *= -1.0
z[2*n:] /= beta
# x := x - G' * W**-1 * z
x[:n] -= div(z[:n], d1) - div(z[n:2*n], d2) + As.T * z[-(m+1):]
x[n:] += div(z[:n], d1) + div(z[n:2*n], d2)
helpers.sp_create("15-f", minor)
# Solve for x[:n]:
#
# S*x[:n] = x[:n] - (W1**2 - W2**2)(W1**2 + W2**2)^-1 * x[n:]
x[:n] -= mul( div(d1**2 - d2**2, d1**2 + d2**2), x[n:])
helpers.sp_create("25-f", minor)
lapack.potrs(S, x)
helpers.sp_create("30-f", minor)
# Solve for x[n:]:
#
# (d1**-2 + d2**-2) * x[n:] = x[n:] + (d1**-2 - d2**-2)*x[:n]
x[n:] += mul( d1**-2 - d2**-2, x[:n])
helpers.sp_create("35-f", minor)
x[n:] = div( x[n:], d1**-2 + d2**-2)
helpers.sp_create("40-f", minor)
# z := z + W^-T * G*x
z[:n] += div( x[:n] - x[n:2*n], d1)
helpers.sp_create("44-f", minor)
z[n:2*n] += div( -x[:n] - x[n:2*n], d2)
helpers.sp_create("48-f", minor)
z[2*n:] += As*x[:n]
helpers.sp_create("50-f", minor)