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:])
def f(x, y, z):
# 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)
# 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:])
lapack.potrs(S, x)
# 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])
x[n:] = div( x[n:], d1**-2 + d2**-2)
# z := z + W^-T * G*x
z[:n] += div( x[:n] - x[n:2*n], d1)
z[n:2*n] += div( -x[:n] - x[n:2*n], d2)
z[2*n:] += As*x[:n]
def f(x, y, z):
x[:n] += P.T * (mul(div(d2**2 - d1**2, d1**2 + d2**2), x[n:]) +
mul(.5 * D, z[:m] - z[m:]))
lapack.potrs(A, x)
u = P * x[:n]
x[n:] = div(
x[n:] - div(z[:m], d1**2) - div(z[m:], d2**2) +
mul(d1**-2 - d2**-2, u), d1**-2 + d2**-2)
z[:m] = div(u - x[n:] - z[:m], d1)
z[m:] = div(-u - x[n:] - z[m:], d2)
def test_basic(self):
import kvxopt
a = kvxopt.matrix([1.0, 2.0, 3.0])
assert list(a) == [1.0, 2.0, 3.0]
b = kvxopt.matrix([3.0, -2.0, -1.0])
c = kvxopt.spmatrix([1.0, -2.0, 3.0], [0, 2, 4], [1, 2, 4], (6, 5))
d = kvxopt.spmatrix([1.0, 2.0, 5.0], [0, 1, 2], [0, 0, 0], (3, 1))
e = kvxopt.mul(a, b)
self.assertEqualLists(e, [3.0, -4.0, -3.0])
self.assertAlmostEqualLists(list(kvxopt.div(a, b)),
[1.0 / 3.0, -1.0, -3.0])
self.assertAlmostEqual(kvxopt.div([1.0, 2.0, 0.25]), 2.0)
self.assertEqualLists(list(kvxopt.min(a, b)), [1.0, -2.0, -1.0])
self.assertEqualLists(list(kvxopt.max(a, b)), [3.0, 2.0, 3.0])
self.assertEqual(kvxopt.max([1.0, 2.0]), 2.0)
self.assertEqual(kvxopt.max(a), 3.0)
self.assertEqual(kvxopt.max(c), 3.0)
self.assertEqual(kvxopt.max(d), 5.0)
self.assertEqual(kvxopt.min([1.0, 2.0]), 1.0)
self.assertEqual(kvxopt.min(a), 1.0)
self.assertEqual(kvxopt.min(c), -2.0)
self.assertEqual(kvxopt.min(d), 1.0)
self.assertEqual(len(c.imag()), 0)
with self.assertRaises(OverflowError):
kvxopt.matrix(1.0, (32780 * 4, 32780))
with self.assertRaises(OverflowError):
kvxopt.spmatrix(1.0, (0, 32780 * 4), (0, 32780)) + 1
def Hf(u, v, alpha=1.0, beta=0.0):
"""
v := alpha * (A'*A*u + 2*((1+w)./(1-w)).*u + beta *v
"""
v *= beta
v += 2.0 * alpha * mul(div(1.0 + w, (1.0 - w)**2), u)
blas.gemv(A, u, r)
blas.gemv(A, r, v, alpha=alpha, beta=1.0, trans='T')
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:])
def acent(A,b):
"""
Computes analytic center of A*x <= b with A m by n of rank n.
We assume that b > 0 and the feasible set is bounded.
"""
MAXITERS = 100
ALPHA = 0.01
BETA = 0.5
TOL = 1e-8
ntdecrs = []
m, n = A.size
x = matrix(0.0, (n,1))
H = matrix(0.0, (n,n))
for iter in range(MAXITERS):
# Gradient is g = A^T * (1./(b-A*x)).
d = (b-A*x)**-1
g = A.T * d
# Hessian is H = A^T * diag(1./(b-A*x))^2 * A.
Asc = mul( d[:,n*[0]], A)
blas.syrk(Asc, H, trans='T')
# Newton step is v = H^-1 * g.
v = -g
lapack.posv(H, v)
# Directional derivative and Newton decrement.
lam = blas.dot(g, v)
ntdecrs += [ sqrt(-lam) ]
print("%2d. Newton decr. = %3.3e" %(iter,ntdecrs[-1]))
if ntdecrs[-1] < TOL: return x, ntdecrs
# Backtracking line search.
y = mul(A*v, d)
step = 1.0
while 1-step*max(y) < 0: step *= BETA
while True:
if -sum(log(1-step*y)) < ALPHA*step*lam: break
step *= BETA
x += step*v
def F(x=None, z=None):
if x is None: return 5, matrix(17 * [0.0] + 5 * [1.0])
if min(x[17:]) <= 0.0: return None
f = -x[12:17] + div(Amin, x[17:])
Df = matrix(0.0, (5, 22))
Df[:, 12:17] = spmatrix(-1.0, range(5), range(5))
Df[:, 17:] = spmatrix(-div(Amin, x[17:]**2), range(5), range(5))
if z is None: return f, Df
H = spmatrix(2.0 * mul(z, div(Amin, x[17::]**3)), range(17, 22),
range(17, 22))
return f, Df, H
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 covsel(Y):
"""
Returns the solution of
minimize -log det K + tr(KY)
subject to K_ij = 0 if (i,j) not in zip(I, J).
Y is a symmetric sparse matrix with nonzero diagonal elements.
I = Y.I, J = Y.J.
"""
cholmod.options['supernodal'] = 2
I, J = Y.I, Y.J
n, m = Y.size[0], len(I)
# non-zero positions for one-argument indexing
N = I + J*n
# position of diagonal elements
D = [ k for k in range(m) if I[k]==J[k] ]
# starting point: symmetric identity with nonzero pattern I,J
K = spmatrix(0.0, I, J)
K[::n+1] = 1.0
# Kn is used in the line search
Kn = spmatrix(0.0, I, J)
# symbolic factorization of K
F = cholmod.symbolic(K)
# Kinv will be the inverse of K
Kinv = matrix(0.0, (n,n))
for iters in range(100):
# numeric factorization of K
cholmod.numeric(K, F)
d = cholmod.diag(F)
# compute Kinv by solving K*X = I
Kinv[:] = 0.0
Kinv[::n+1] = 1.0
cholmod.solve(F, Kinv)
# solve Newton system
grad = 2 * (Y.V - Kinv[N])
hess = 2 * ( mul(Kinv[I,J], Kinv[J,I]) +
mul(Kinv[I,I], Kinv[J,J]) )
v = -grad
lapack.posv(hess,v)
# stopping criterion
sqntdecr = -blas.dot(grad,v)
print("Newton decrement squared:%- 7.5e" %sqntdecr)
if (sqntdecr < 1e-12):
print("number of iterations: %d" %(iters+1))
break
# line search
dx = +v
dx[D] *= 2
f = -2.0*sum(log(d)) # f = -log det K
s = 1
for lsiter in range(50):
Kn.V = K.V + s*dx
try:
cholmod.numeric(Kn, F)
except ArithmeticError:
s *= 0.5
else:
d = cholmod.diag(F)
fn = -2.0 * sum(log(d)) + 2*s*blas.dot(v,Y.V)
if (fn < f - 0.01*s*sqntdecr): break
else: s *= 0.5
K.V = Kn.V
return K
def g(x, y, z):
x[:] = mul(x, ds) / a
blas.gemv(Asc, x, v)
lapack.potrs(S, v)
blas.gemv(Asc, v, x, alpha=-1.0, beta=1.0, trans='T')
x[:] = mul(x, ds)