def sparseeigs(m, ndims):
"""Does a sparse eigenvalue problem, with the given number of dimensions"""
import cvxopt
import cvxopt.lapack as lp
log(' Starting sparse eigendecomposition on matrix of shape %s now using %d dims'
% (m.shape, ndims))
t1 = time.time()
N = len(m)
if ndims < 0 or ndims > N:
ndims = N
m = cvxopt.base.matrix(m)
evals = cvxopt.base.matrix([0.0] * N)
evecs = cvxopt.base.matrix(0.0, (N, ndims))
ret = lp.syevx(m,
evals,
jobz='V',
range='I',
il=N - ndims + 1,
iu=N,
Z=evecs)
log(' Finished sparse eigendecomposition in %0.3f secs' %
(time.time() - t1))
evals = array(evals)[:ndims, 0]
evecs = array(evecs)
return sortvals(evals, evecs)
def sparseeigs(m, ndims):
"""Does a sparse eigenvalue problem, with the given number of dimensions"""
import cvxopt
import cvxopt.lapack as lp
log(' Starting sparse eigendecomposition on matrix of shape %s now using %d dims' % (m.shape, ndims))
t1 = time.time()
N = len(m)
if ndims < 0 or ndims > N:
ndims = N
m = cvxopt.base.matrix(m)
evals = cvxopt.base.matrix([0.0]*N)
evecs = cvxopt.base.matrix(0.0, (N, ndims))
ret = lp.syevx(m, evals, jobz='V', range='I', il=N-ndims+1, iu=N, Z=evecs)
log(' Finished sparse eigendecomposition in %0.3f secs' % (time.time()-t1))
evals = array(evals)[:ndims, 0]
evecs = array(evecs)
return sortvals(evals, evecs)
def mcsdp(w):
"""
Returns solution x, z to
(primal) minimize sum(x)
subject to w + diag(x) >= 0
(dual) maximize -tr(w*z)
subject to diag(z) = 1
z >= 0.
"""
n = w.size[0]
def Fs(x, y, alpha = 1.0, beta = 0.0, trans = 'N'):
"""
y := alpha*(-diag(x)) + beta*y.
"""
if trans=='N':
# x is a vector; y is a matrix.
blas.scal(beta, y)
blas.axpy(x, y, alpha = -alpha, incy = n+1)
else:
# x is a matrix; y is a vector.
blas.scal(beta, y)
blas.axpy(x, y, alpha = -alpha, incx = n+1)
def cngrnc(r, x, alpha = 1.0):
"""
Congruence transformation
x := alpha * r'*x*r.
r and x are square matrices.
"""
# Scale diagonal of x by 1/2.
x[::n+1] *= 0.5
# a := tril(x)*r
a = +r
tx = matrix(x, (n,n))
blas.trmm(tx, a, side='L')
# x := alpha*(a*r' + r*a')
blas.syr2k(r, a, tx, trans = 'T', alpha = alpha)
x[:] = tx[:]
def Fkkt(W):
rti = W['rti'][0]
# t = rti*rti' as a nonsymmetric matrix.
t = matrix(0.0, (n,n))
blas.gemm(rti, rti, t, transB = 'T')
# Cholesky factorization of tsq = t.*t.
tsq = t**2
lapack.potrf(tsq)
def f(x, y, z):
"""
Solve
-diag(z) = bx
-diag(x) - inv(rti*rti') * z * inv(rti*rti') = bs
On entry, x and z contain bx and bs.
On exit, they contain the solution, with z scaled
(inv(rti)'*z*inv(rti) is returned instead of z).
We first solve
((rti*rti') .* (rti*rti')) * x = bx - diag(t*bs*t)
and take z = -rti' * (diag(x) + bs) * rti.
"""
# tbst := t * zs * t = t * bs * t
tbst = matrix(z, (n,n))
cngrnc(t, tbst)
# x := x - diag(tbst) = bx - diag(rti*rti' * bs * rti*rti')
x -= tbst[::n+1]
# x := (t.*t)^{-1} * x = (t.*t)^{-1} * (bx - diag(t*bs*t))
lapack.potrs(tsq, x)
# z := z + diag(x) = bs + diag(x)
z[::n+1] += x
# z := -rti' * z * rti = -rti' * (diag(x) + bs) * rti
cngrnc(rti, z, alpha = -1.0)
return f
c = matrix(1.0, (n,1))
# Initial feasible x: x = 1.0 - min(lambda(w)).
lmbda = matrix(0.0, (n,1))
lapack.syevx(+w, lmbda, range='I', il=1, iu=1)
x0 = matrix(-lmbda[0]+1.0, (n,1))
s0 = +w
s0[::n+1] += x0
# Initial feasible z is identity.
z0 = matrix(0.0, (n,n))
z0[::n+1] = 1.0
dims = {'l': 0, 'q': [], 's': [n]}
sol = solvers.conelp(c, Fs, w[:], dims, kktsolver = Fkkt,
primalstart = {'x': x0, 's': s0[:]}, dualstart = {'z': z0[:]})
return sol['x'], matrix(sol['z'], (n,n))
def mcsdp(w):
"""
Returns solution x, z to
(primal) minimize sum(x)
subject to w + diag(x) >= 0
(dual) maximize -tr(w*z)
subject to diag(z) = 1
z >= 0.
"""
n = w.size[0]
def Fs(x, y, alpha=1.0, beta=0.0, trans='N'):
"""
y := alpha*(-diag(x)) + beta*y.
"""
if trans == 'N':
# x is a vector; y is a matrix.
blas.scal(beta, y)
blas.axpy(x, y, alpha=-alpha, incy=n + 1)
else:
# x is a matrix; y is a vector.
blas.scal(beta, y)
blas.axpy(x, y, alpha=-alpha, incx=n + 1)
def cngrnc(r, x, alpha=1.0):
"""
Congruence transformation
x := alpha * r'*x*r.
r and x are square matrices.
"""
# Scale diagonal of x by 1/2.
x[::n + 1] *= 0.5
# a := tril(x)*r
a = +r
tx = matrix(x, (n, n))
blas.trmm(tx, a, side='L')
# x := alpha*(a*r' + r*a')
blas.syr2k(r, a, tx, trans='T', alpha=alpha)
x[:] = tx[:]
def Fkkt(W):
rti = W['rti'][0]
# t = rti*rti' as a nonsymmetric matrix.
t = matrix(0.0, (n, n))
blas.gemm(rti, rti, t, transB='T')
# Cholesky factorization of tsq = t.*t.
tsq = t**2
lapack.potrf(tsq)
def f(x, y, z):
"""
Solve
-diag(z) = bx
-diag(x) - inv(rti*rti') * z * inv(rti*rti') = bs
On entry, x and z contain bx and bs.
On exit, they contain the solution, with z scaled
(inv(rti)'*z*inv(rti) is returned instead of z).
We first solve
((rti*rti') .* (rti*rti')) * x = bx - diag(t*bs*t)
and take z = -rti' * (diag(x) + bs) * rti.
"""
# tbst := t * zs * t = t * bs * t
tbst = matrix(z, (n, n))
cngrnc(t, tbst)
# x := x - diag(tbst) = bx - diag(rti*rti' * bs * rti*rti')
x -= tbst[::n + 1]
# x := (t.*t)^{-1} * x = (t.*t)^{-1} * (bx - diag(t*bs*t))
lapack.potrs(tsq, x)
# z := z + diag(x) = bs + diag(x)
z[::n + 1] += x
# z := -rti' * z * rti = -rti' * (diag(x) + bs) * rti
cngrnc(rti, z, alpha=-1.0)
return f
c = matrix(1.0, (n, 1))
# Initial feasible x: x = 1.0 - min(lambda(w)).
lmbda = matrix(0.0, (n, 1))
lapack.syevx(+w, lmbda, range='I', il=1, iu=1)
x0 = matrix(-lmbda[0] + 1.0, (n, 1))
s0 = +w
s0[::n + 1] += x0
# Initial feasible z is identity.
z0 = matrix(0.0, (n, n))
z0[::n + 1] = 1.0
dims = {'l': 0, 'q': [], 's': [n]}
sol = solvers.conelp(c,
Fs,
w[:],
dims,
kktsolver=Fkkt,
primalstart={
'x': x0,
's': s0[:]
},
dualstart={'z': z0[:]})
return sol['x'], matrix(sol['z'], (n, n))
