def res(ux, uy, uz, us, vx, vy, vz, vs, W, lmbda):
# Evaluates residual in Newton equations:
#
# [ vx ] [ vx ] [ 0 ] [ P A' G' ] [ ux ]
# [ vy ] := [ vy ] - [ 0 ] - [ A 0 0 ] * [ uy ]
# [ vz ] [ vz ] [ W'*us ] [ G 0 0 ] [ W^{-1}*uz ]
#
# vs := vs - lmbda o (uz + us).
# vx := vx - P*ux - A'*uy - G'*W^{-1}*uz
fP(ux, vx, alpha=-1.0, beta=1.0)
fA(uy, vx, alpha=-1.0, beta=1.0, trans='T')
blas.copy(uz, wz3)
misc.scale(wz3, W, inverse='I')
fG(wz3, vx, alpha=-1.0, beta=1.0, trans='T')
# vy := vy - A*ux
fA(ux, vy, alpha=-1.0, beta=1.0)
# vz := vz - G*ux - W'*us
fG(ux, vz, alpha=-1.0, beta=1.0)
blas.copy(us, ws3)
misc.scale(ws3, W, trans='T')
blas.axpy(ws3, vz, alpha=-1.0)
# vs := vs - lmbda o (uz + us)
blas.copy(us, ws3)
blas.axpy(uz, ws3)
misc.sprod(ws3, lmbda, dims, diag='D')
blas.axpy(ws3, vs, alpha=-1.0)
def f4_no_ir(x, y, z, s):
# Solve
#
# [ P A' G' ] [ ux ] [ bx ]
# [ A 0 0 ] [ uy ] = [ by ]
# [ G 0 -W'*W ] [ W^{-1}*uz ] [ bz - W'*(lmbda o\ bs) ]
#
# us = lmbda o\ bs - uz.
#
# On entry, x, y, z, s contains bx, by, bz, bs.
# On exit they contain x, y, z, s.
# s := lmbda o\ s
# = lmbda o\ bs
misc.sinv(s, lmbda, dims)
# z := z - W'*s
# = bz - W'*(lambda o\ bs)
ws3 = matrix(np.copy(s))
misc.scale(ws3, W, trans='T')
blas.axpy(ws3, z, alpha=-1.0)
# Solve for ux, uy, uz
f3(x, y, z)
# s := s - z
# = lambda o\ bs - uz.
blas.axpy(z, s, alpha=-1.0)
def solve(x, y, z):
# Solve
#
# [ P GG'*W^{-1} ] [ ux ] [ bx ]
# [ W^{-T}*GG -I ] [ W*uz ] [ W^{-T}*bz ]
#
# and return ux, uy, W*uz.
#
# On entry, x, y, z contain bx, by, bz. On exit, they contain
# the solution ux, uy, W*uz.
#
# z=W^-T z
scale(z, W, trans = 'T', inverse = 'I')
# x=x+Gs^T z
# blas.gemv(Gs, z, x, beta = 1.0, trans = 'T', m = n)
WGGW_C.Gs_mv(Gs, z, x, 1,1, n)
#solve Kx
lapack.potrs(K, x, n = n, offsetA = 0, offsetB = 0)
# z=z-Gs^T x
#blas.gemv(Gs, x, z, alpha = 1.0, beta = -1.0, m = n)
WGGW_C.Gs_mv(Gs, x, z, -1,0, n)
def factor(W, H = None, Df = None):
blas.scal(0.0, K)
if H is not None: K[:n, :n] = H
K[n:n+p, :n] = A
for k in range(n):
if mnl: g[:mnl] = Df[:,k]
g[mnl:] = G[:,k]
scale(g, W, trans = 'T', inverse = 'I')
pack(g, K, dims, mnl, offsety = k*ldK + n + p)
K[(ldK+1)*(p+n) :: ldK+1] = -1.0
# Add positive regularization in 1x1 block and negative in 2x2 block.
blas.copy(K, Ktilde)
Ktilde[0 : (ldK+1)*n : ldK+1] += EPSILON
Ktilde[(ldK+1)*n :: ldK+1] += -EPSILON
lapack.sytrf(Ktilde, ipiv)
def solve(x, y, z):
# Solve
#
# [ H A' GG'*W^{-1} ] [ ux ] [ bx ]
# [ A 0 0 ] * [ uy [ = [ by ]
# [ W^{-T}*GG 0 -I ] [ W*uz ] [ W^{-T}*bz ]
#
# and return ux, uy, W*uz.
#
# On entry, x, y, z contain bx, by, bz. On exit, they contain
# the solution ux, uy, W*uz.
blas.scal(0.0, sltn)
blas.copy(x, u)
blas.copy(y, u, offsety = n)
scale(z, W, trans = 'T', inverse = 'I')
pack(z, u, dims, mnl, offsety = n + p)
blas.copy(u, r)
# Iterative refinement algorithm:
# Init: sltn = 0, r_0 = [bx; by; W^{-T}*bz]
# 1. u_k = Ktilde^-1 * r_k
# 2. sltn += u_k
# 3. r_k+1 = r - K*sltn
# Repeat until exceed MAX_ITER iterations or ||r|| <= ERROR_BOUND
iteration = 0
resid_norm = 1
while iteration <= MAX_ITER and resid_norm > ERROR_BOUND:
lapack.sytrs(Ktilde, ipiv, u)
blas.axpy(u, sltn, alpha = 1.0)
blas.copy(r, u)
blas.symv(K, sltn, u, alpha = -1.0, beta = 1.0)
resid_norm = math.sqrt(blas.dot(u, u))
iteration += 1
blas.copy(sltn, x, n = n)
blas.copy(sltn, y, offsetx = n, n = p)
unpack(sltn, z, dims, mnl, offsetx = n + p)
return solve
def solve(x, y, z):
# Solve
#
# [ H A' GG'*W^{-1} ] [ ux ] [ bx ]
# [ A 0 0 ] * [ uy [ = [ by ]
# [ W^{-T}*GG 0 -I ] [ W*uz ] [ W^{-T}*bz ]
#
# and return ux, uy, W*uz.
#
# On entry, x, y, z contain bx, by, bz. On exit, they contain
# the solution ux, uy, W*uz.
blas.copy(x, u)
blas.copy(y, u, offsety=n)
scale(z, W, trans='T', inverse='I')
pack(z, u, dims, mnl, offsety=n + p)
lapack.sytrs(K, ipiv, u)
blas.copy(u, x, n=n)
blas.copy(u, y, offsetx=n, n=p)
unpack(u, z, dims, mnl, offsetx=n + p)
def solve(x, y, z):
# Solve
#
# [ H A' GG'*W^{-1} ] [ ux ] [ bx ]
# [ A 0 0 ] * [ uy [ = [ by ]
# [ W^{-T}*GG 0 -I ] [ W*uz ] [ W^{-T}*bz ]
#
# and return ux, uy, W*uz.
#
# On entry, x, y, z contain bx, by, bz. On exit, they contain
# the solution ux, uy, W*uz.
blas.copy(x, u)
blas.copy(y, u, offsety=n)
scale(z, W, trans='T', inverse='I')
pack(z, u, dims, mnl, offsety=n + p)
lapack.sytrs(K, ipiv, u)
blas.copy(u, x, n=n)
blas.copy(u, y, offsetx=n, n=p)
unpack(u, z, dims, mnl, offsetx=n + p)
def factor(W, H=None, Df=None):
blas.scal(0.0, K)
if H is not None:
K[:n, :n] = H
K[n:n+p, :n] = A
for k in range(n):
if mnl:
g[:mnl] = Df[:, k]
g[mnl:] = G[:, k]
scale(g, W, trans='T', inverse='I')
pack(g, K, dims, mnl, offsety=k*ldK + n + p)
K[(ldK+1)*(p+n):: ldK+1] = -1.0
# Add positive regularization in 1x1 block and negative in 2x2 block.
K[0: (ldK+1)*n: ldK+1] += REG_EPS
K[(ldK+1)*n:: ldK+1] += -REG_EPS
lapack.sytrf(K, ipiv)
def solve(x, y, z):
# Solve
#
# [ H A' GG'*W^{-1} ] [ ux ] [ bx ]
# [ A 0 0 ] * [ uy [ = [ by ]
# [ W^{-T}*GG 0 -I ] [ W*uz ] [ W^{-T}*bz ]
#
# and return ux, uy, W*uz.
#
# On entry, x, y, z contain bx, by, bz. On exit, they contain
# the solution ux, uy, W*uz.
blas.copy(x, u)
blas.copy(y, u, offsety=n)
scale(z, W, trans='T', inverse='I')
pack(z, u, dims, mnl, offsety=n + p)
lapack.sytrs(K, ipiv, u)
blas.copy(u, x, n=n)
blas.copy(u, y, offsetx=n, n=p)
unpack(u, z, dims, mnl, offsetx=n + p)
return solve
def factor(W, H=None, Df=None):
blas.scal(0.0, K)
if H is not None:
K[:n, :n] = H
K[n:n+p, :n] = A
for k in range(n):
if mnl:
g[:mnl] = Df[:, k]
g[mnl:] = G[:, k]
scale(g, W, trans='T', inverse='I')
pack(g, K, dims, mnl, offsety=k*ldK + n + p)
K[(ldK+1)*(p+n):: ldK+1] = -1.0
# Add positive regularization in 1x1 block and negative in 2x2 block.
K[0: (ldK+1)*n: ldK+1] += REG_EPS
K[(ldK+1)*n:: ldK+1] += -REG_EPS
lapack.sytrf(K, ipiv)
def solve(x, y, z):
# Solve
#
# [ H A' GG'*W^{-1} ] [ ux ] [ bx ]
# [ A 0 0 ] * [ uy [ = [ by ]
# [ W^{-T}*GG 0 -I ] [ W*uz ] [ W^{-T}*bz ]
#
# and return ux, uy, W*uz.
#
# On entry, x, y, z contain bx, by, bz. On exit, they contain
# the solution ux, uy, W*uz.
blas.copy(x, u)
blas.copy(y, u, offsety=n)
scale(z, W, trans='T', inverse='I')
pack(z, u, dims, mnl, offsety=n + p)
lapack.sytrs(K, ipiv, u)
blas.copy(u, x, n=n)
blas.copy(u, y, offsetx=n, n=p)
unpack(u, z, dims, mnl, offsetx=n + p)
return solve
def solve(x, y, z):
# Solve
#
# [ H A' GG'*W^{-1} ] [ ux ] [ bx ]
# [ A 0 0 ] * [ uy [ = [ by ]
# [ W^{-T}*GG 0 -I ] [ W*uz ] [ W^{-T}*bz ]
#
# and return ux, uy, W*uz.
#
# On entry, x, y, z contain bx, by, bz. On exit, they contain
# the solution ux, uy, W*uz.
blas.scal(0.0, sltn)
blas.copy(x, u)
blas.copy(y, u, offsety = n)
scale(z, W, trans = 'T', inverse = 'I')
pack(z, u, dims, mnl, offsety = n + p)
blas.copy(u, r)
# Iterative refinement algorithm:
# Init: sltn = 0, r_0 = [bx; by; W^{-T}*bz]
# 1. u_k = Ktilde^-1 * r_k
# 2. sltn += u_k
# 3. r_k+1 = r - K*sltn
# Repeat until exceed MAX_ITER iterations or ||r|| <= ERROR_BOUND
iteration = 0
resid_norm = 1
while iteration <= MAX_ITER and resid_norm > ERROR_BOUND:
lapack.sytrs(Ktilde, ipiv, u)
blas.axpy(u, sltn, alpha = 1.0)
blas.copy(r, u)
blas.symv(K, sltn, u, alpha = -1.0, beta = 1.0)
resid_norm = math.sqrt(blas.dot(u, u))
iteration += 1
blas.copy(sltn, x, n = n)
blas.copy(sltn, y, offsetx = n, n = p)
unpack(sltn, z, dims, mnl, offsetx = n + p)
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 kkt_solver(x, y, z):
if (TEST_KKT):
x0 = matrix(0., x.size)
y0 = matrix(0., y.size)
z0 = matrix(0., z.size)
x0[:] = x[:]
y0[:] = y[:]
z0[:] = z[:]
# Get default solver solutions.
xp = matrix(0., x.size)
yp = matrix(0., y.size)
zp = matrix(0., z.size)
xp[:] = x[:]
yp[:] = y[:]
zp[:] = z[:]
default_solver(xp, yp, zp)
offset = K * (K + 1) / 2
for i in xrange(K):
symmetrize_matrix(zp, K + 1, offset)
offset += (K + 1) * (K + 1)
# pab = x[:K*(K+1)/2] # p_{ab} 1<=a<=b<=K
# pis = x[K*(K+1)/2:] # \pi_i 1<=i<=K
# z_{ab} := d_{ab}^{-1} z_{ab}
# \mat{z}_i = r_i^{-1} \mat{z}_i r_i^{-t}
misc.scale(z, W, trans='T', inverse='I')
l = z[:]
# l_{ab} := d_{ab}^{-2} z_{ab}
# \mat{z}_i := r_i^{-t}r_i^{-1} \mat{z}_i r_i^{-t} r_i^{-1}
misc.scale(l, W, trans='N', inverse='I')
# The RHS of equations
#
# d_{ab}^{-2}n_{ab} + vec(V_{ab})^t . vec( \sum_i R_i* F R_i*)
# + \sum_i vec(V_{ab})^t . vec( g_i g_i^t) u_i + y
# = -d_{ab}^{-2} l_{ab} + ( p_{ab} - vec(V_{ab})^t . vec(\sum_i L_i*)
#
###
# Lsum := \sum_i L_i
moffset = K * (K + 1) / 2
Lsum = np.sum(np.array(l[moffset:]).reshape((K, (K + 1) * (K + 1))),
axis=0)
Lsum = matrix(Lsum, (K + 1, K + 1))
Ls = Lsum[:K, :K]
x[:K * (K + 1) / 2] -= l[:K * (K + 1) / 2]
dL = matrix(0., (K * (K + 1) / 2, 1))
ab = 0
for a in xrange(K):
dL[ab] = Ls[a, a]
ab += 1
for b in xrange(a + 1, K):
dL[ab] = Ls[a, a] + Ls[b, b] - 2 * Ls[b, a]
ab += 1
x[:K * (K + 1) / 2] -= cvxopt.mul(si2ab, dL)
# The RHS of equations
# g_i^t F g_i + R_{i,K+1,K+1}^2 u_i = pi - L_{i,K+1,K+1}
x[K * (K + 1) / 2:] -= l[K * (K + 1) / 2 + (K + 1) * (K + 1) -
1::(K + 1) * (K + 1)]
# x := B^{-1} Cv
lapack.sytrs(Bm, ipiv, x)
# lapack.potrs( Bm, x)
# y := (oz'.B^{-1}.Cv[:-1] - y)/(oz'.B^{-1}.oz)
y[0] = (blas.dotu(oz, x) - y[0]) / blas.dotu(oz, iB1)
# x := B^{-1} Cv - B^{-1}.oz y
blas.axpy(iB1, x, -y[0])
# Solve for -n_{ab} - d_{ab}^2 z_{ab} = l_{ab}
# We need to return scaled d*z.
# z := d_{ab} d_{ab}^{-2}(n_{ab} + l_{ab})
# = d_{ab}^{-1}n_{ab} + d_{ab}^{-1}l_{ab}
z[:K * (K + 1) / 2] += cvxopt.mul(dis, x[:K * (K + 1) / 2])
z[:K * (K + 1) / 2] *= -1.
# Solve for \mat{z}_i = -R_i (\mat{l}_i + diag(F, u_i)) R_i
# = -L_i - R_i diag(F, u_i) R_i
# We return
# r_i^t \mat{z}_i r_i = -r_i^{-1} (\mat{l}_i + diag(F, u_i)) r_i^{-t}
ui = x[-K:]
nab = tri2symm(x, K)
F = Fisher_matrix(si2, nab)
offset = K * (K + 1) / 2
for i in xrange(K):
start, end = i * (K + 1) * (K + 1), (i + 1) * (K + 1) * (K + 1)
Fu = matrix(0.0, (K + 1, K + 1))
Fu[:K, :K] = F
Fu[K, K] = ui[i]
Fu = matrix(Fu, ((K + 1) * (K + 1), 1))
# Fu := -r_i^{-1} diag( F, u_i) r_i^{-t}
cngrnc(rtis[i], Fu, K + 1, alpha=-1.)
# Fu := -r_i^{-1} (\mat{l}_i + diag( F, u_i )) r_i^{-t}
blas.axpy(z[offset + start:offset + end], Fu, alpha=-1.)
z[offset + start:offset + end] = Fu
if (TEST_KKT):
offset = K * (K + 1) / 2
for i in xrange(K):
symmetrize_matrix(z, K + 1, offset)
offset += (K + 1) * (K + 1)
dz = np.max(np.abs(z - zp))
dx = np.max(np.abs(x - xp))
dy = np.max(np.abs(y - yp))
tol = 1e-5
if dx > tol:
print 'dx='
print dx
print x
print xp
if dy > tol:
print 'dy='
print dy
print y
print yp
if dz > tol:
print 'dz='
print dz
print z
print zp
if dx > tol or dy > tol or dz > tol:
for i, (r, rti) in enumerate(zip(ris, rtis)):
print 'r[%d]=' % i
print r
print 'rti[%d]=' % i
print rti
print 'rti.T*r='
print rti.T * r
for i, d in enumerate(ds):
print 'd[%d]=%g' % (i, d)
print 'x0, y0, z0='
print x0
print y0
print z0
print Bm0
def qp(P, q, G=None, h=None, A=None, b=None):
"""
Solves a pair of primal and dual convex quadratic cone programs
minimize (1/2)*x'*P*x + q'*x
subject to G*x + s = h
A*x = b
s >= 0
maximize -(1/2)*(q + G'*z + A'*y)' * pinv(P) * (q + G'*z + A'*y)
- h'*z - b'*y
subject to q + G'*z + A'*y in range(P)
z >= 0.
The inequalities are with respect to a cone C defined as the Cartesian
product of N + M + 1 cones:
C = C_0 x C_1 x .... x C_N x C_{N+1} x ... x C_{N+M}.
The first cone C_0 is the nonnegative orthant of dimension ml.
The next N cones are 2nd order cones of dimension mq[0], ..., mq[N-1].
The second order cone of dimension m is defined as
{ (u0, u1) in R x R^{m-1} | u0 >= ||u1||_2 }.
The next M cones are positive semidefinite cones of order ms[0], ...,
ms[M-1] >= 0.
Input arguments (basic usage).
P is a dense or sparse 'd' matrix of size (n,n) with the lower
triangular part of the Hessian of the objective stored in the
lower triangle. Must be positive semidefinite.
q is a dense 'd' matrix of size (n,1).
dims is a dictionary with the dimensions of the components of C.
It has three fields.
- dims['l'] = ml, the dimension of the nonnegative orthant C_0.
(ml >= 0.)
- dims['q'] = mq = [ mq[0], mq[1], ..., mq[N-1] ], a list of N
integers with the dimensions of the second order cones
C_1, ..., C_N. (N >= 0 and mq[k] >= 1.)
- dims['s'] = ms = [ ms[0], ms[1], ..., ms[M-1] ], a list of M
integers with the orders of the semidefinite cones
C_{N+1}, ..., C_{N+M}. (M >= 0 and ms[k] >= 0.)
The default value of dims = {'l': G.size[0], 'q': [], 's': []}.
G is a dense or sparse 'd' matrix of size (K,n), where
K = ml + mq[0] + ... + mq[N-1] + ms[0]**2 + ... + ms[M-1]**2.
Each column of G describes a vector
v = ( v_0, v_1, ..., v_N, vec(v_{N+1}), ..., vec(v_{N+M}) )
in V = R^ml x R^mq[0] x ... x R^mq[N-1] x S^ms[0] x ... x S^ms[M-1]
stored as a column vector
[ v_0; v_1; ...; v_N; vec(v_{N+1}); ...; vec(v_{N+M}) ].
Here, if u is a symmetric matrix of order m, then vec(u) is the
matrix u stored in column major order as a vector of length m**2.
We use BLAS unpacked 'L' storage, i.e., the entries in vec(u)
corresponding to the strictly upper triangular entries of u are
not referenced.
h is a dense 'd' matrix of size (K,1), representing a vector in V,
in the same format as the columns of G.
A is a dense or sparse 'd' matrix of size (p,n). The default
value is a sparse 'd' matrix of size (0,n).
b is a dense 'd' matrix of size (p,1). The default value is a
dense 'd' matrix of size (0,1).
It is assumed that rank(A) = p and rank([P; A; G]) = n.
The other arguments are normally not needed. They make it possible
to exploit certain types of structure, as described below.
Output arguments.
Returns a dictionary with keys 'status', 'x', 's', 'z', 'y',
'primal objective', 'dual objective', 'gap', 'relative gap',
'primal infeasibility', 'dual infeasibility', 'primal slack',
'dual slack', 'iterations'.
The 'status' field has values 'optimal' or 'unknown'. 'iterations'
is the number of iterations taken.
If the status is 'optimal', 'x', 's', 'y', 'z' are an approximate
solution of the primal and dual optimality conditions
G*x + s = h, A*x = b
P*x + G'*z + A'*y + q = 0
s >= 0, z >= 0
s'*z = 0.
If the status is 'unknown', 'x', 'y', 's', 'z' are the last
iterates before termination. These satisfy s > 0 and z > 0,
but are not necessarily feasible.
The values of the other fields are defined as follows.
- 'primal objective': the primal objective (1/2)*x'*P*x + q'*x.
- 'dual objective': the dual objective
L(x,y,z) = (1/2)*x'*P*x + q'*x + z'*(G*x - h) + y'*(A*x-b).
- 'gap': the duality gap s'*z.
- 'relative gap': the relative gap, defined as
gap / -primal objective
if the primal objective is negative,
gap / dual objective
if the dual objective is positive, and None otherwise.
- 'primal infeasibility': the residual in the primal constraints,
defined as the maximum of the residual in the inequalities
|| G*x + s + h || / max(1, ||h||)
and the residual in the equalities
|| A*x - b || / max(1, ||b||).
- 'dual infeasibility': the residual in the dual constraints,
defined as
|| P*x + G'*z + A'*y + q || / max(1, ||q||).
- 'primal slack': the smallest primal slack, sup {t | s >= t*e },
where
e = ( e_0, e_1, ..., e_N, e_{N+1}, ..., e_{M+N} )
is the identity vector in C. e_0 is an ml-vector of ones,
e_k, k = 1,..., N, is the unit vector (1,0,...,0) of length
mq[k], and e_k = vec(I) where I is the identity matrix of order
ms[k].
- 'dual slack': the smallest dual slack, sup {t | z >= t*e }.
If the exit status is 'optimal', then the primal and dual
infeasibilities are guaranteed to be less than
Termination with status 'unknown' indicates that the algorithm
failed to find a solution that satisfies the specified tolerances.
In some cases, the returned solution may be fairly accurate. If
the primal and dual infeasibilities, the gap, and the relative gap
are small, then x, y, s, z are close to optimal.
Advanced usage.
Three mechanisms are provided to express problem structure.
1. The user can provide a customized routine for solving linear
equations (`KKT systems')
[ P A' G' ] [ ux ] [ bx ]
[ A 0 0 ] [ uy ] = [ by ].
[ G 0 -W'*W ] [ uz ] [ bz ]
W is a scaling matrix, a block diagonal mapping
W*u = ( W0*u_0, ..., W_{N+M}*u_{N+M} )
defined as follows.
- For the 'l' block (W_0):
W_0 = diag(d),
with d a positive vector of length ml.
- For the 'q' blocks (W_{k+1}, k = 0, ..., N-1):
W_{k+1} = beta_k * ( 2 * v_k * v_k' - J )
where beta_k is a positive scalar, v_k is a vector in R^mq[k]
with v_k[0] > 0 and v_k'*J*v_k = 1, and J = [1, 0; 0, -I].
- For the 's' blocks (W_{k+N}, k = 0, ..., M-1):
W_k * u = vec(r_k' * mat(u) * r_k)
where r_k is a nonsingular matrix of order ms[k], and mat(x) is
the inverse of the vec operation.
The optional argument kktsolver is a Python function that will be
called as g = kktsolver(W). W is a dictionary that contains
the parameters of the scaling:
- W['d'] is a positive 'd' matrix of size (ml,1).
- W['di'] is a positive 'd' matrix with the elementwise inverse of
W['d'].
- W['beta'] is a list [ beta_0, ..., beta_{N-1} ]
- W['v'] is a list [ v_0, ..., v_{N-1} ]
- W['r'] is a list [ r_0, ..., r_{M-1} ]
- W['rti'] is a list [ rti_0, ..., rti_{M-1} ], with rti_k the
inverse of the transpose of r_k.
The call g = kktsolver(W) should return a function g that solves
the KKT system by g(x, y, z). On entry, x, y, z contain the
righthand side bx, by, bz. On exit, they contain the solution,
with uz scaled, the argument z contains W*uz. In other words,
on exit x, y, z are the solution of
[ P A' G'*W^{-1} ] [ ux ] [ bx ]
[ A 0 0 ] [ uy ] = [ by ].
[ G 0 -W' ] [ uz ] [ bz ]
2. The linear operators P*u, G*u and A*u can be specified
by providing Python functions instead of matrices. This can only
be done in combination with 1. above, i.e., it requires the
kktsolver argument.
If P is a function, the call P(u, v, alpha, beta) should evaluate
the matrix-vectors product
v := alpha * P * u + beta * v.
The arguments u and v are required. The other arguments have
default values alpha = 1.0, beta = 0.0.
If G is a function, the call G(u, v, alpha, beta, trans) should
evaluate the matrix-vector products
v := alpha * G * u + beta * v if trans is 'N'
v := alpha * G' * u + beta * v if trans is 'T'.
The arguments u and v are required. The other arguments have
default values alpha = 1.0, beta = 0.0, trans = 'N'.
If A is a function, the call A(u, v, alpha, beta, trans) should
evaluate the matrix-vectors products
v := alpha * A * u + beta * v if trans is 'N'
v := alpha * A' * u + beta * v if trans is 'T'.
The arguments u and v are required. The other arguments
have default values alpha = 1.0, beta = 0.0, trans = 'N'.
If X is the vector space of primal variables x, then:
If this option is used, the argument q must be in the same format
as x, the argument P must be a Python function, the arguments A
and G must be Python functions or None, and the argument
kktsolver is required.
If Y is the vector space of primal variables y:
If this option is used, the argument b must be in the same format
as y, the argument A must be a Python function or None, and the
argument kktsolver is required.
"""
from cvxopt import base, blas, misc
from cvxopt.base import matrix, spmatrix
dims = None
kktsolver = 'chol2'
# Argument error checking depends on level of customization.
customkkt = not isinstance(kktsolver, str)
matrixP = isinstance(P, (matrix, spmatrix))
matrixG = isinstance(G, (matrix, spmatrix))
matrixA = isinstance(A, (matrix, spmatrix))
if (not matrixP or (not matrixG and G is not None) or
(not matrixA and A is not None)) and not customkkt:
raise ValueError("use of function valued P, G, A requires a "
"user-provided kktsolver")
if False and (matrixA or not customkkt):
raise ValueError("use of non vector type for y requires "
"function valued A and user-provided kktsolver")
if (not isinstance(q, matrix) or q.typecode != 'd' or q.size[1] != 1):
raise TypeError("'q' must be a 'd' matrix with one column")
if matrixP:
if P.typecode != 'd' or P.size != (q.size[0], q.size[0]):
raise TypeError("'P' must be a 'd' matrix of size (%d, %d)"
% (q.size[0], q.size[0]))
def fP(x, y, alpha=1.0, beta=0.0):
base.symv(P, x, y, alpha=alpha, beta=beta)
else:
fP = P
if h is None:
h = matrix(0.0, (0, 1))
if not isinstance(h, matrix) or h.typecode != 'd' or h.size[1] != 1:
raise TypeError("'h' must be a 'd' matrix with one column")
if not dims:
dims = {'l': h.size[0], 'q': [], 's': []}
if not isinstance(dims['l'], (int, long)) or dims['l'] < 0:
raise TypeError("'dims['l']' must be a nonnegative integer")
if [k for k in dims['q'] if not isinstance(k, (int, long)) or k < 1]:
raise TypeError("'dims['q']' must be a list of positive integers")
if [k for k in dims['s'] if not isinstance(k, (int, long)) or k < 0]:
raise TypeError("'dims['s']' must be a list of nonnegative "
"integers")
if dims['q'] or dims['s']:
refinement = 1
else:
refinement = 0
cdim = dims['l'] + sum(dims['q']) + sum([k ** 2 for k in dims['s']])
if h.size[0] != cdim:
raise TypeError("'h' must be a 'd' matrix of size (%d,1)" % cdim)
# Data for kth 'q' constraint are found in rows indq[k]:indq[k+1] of G.
indq = [dims['l']]
for k in dims['q']:
indq = indq + [indq[-1] + k]
# Data for kth 's' constraint are found in rows inds[k]:inds[k+1] of G.
inds = [indq[-1]]
for k in dims['s']:
inds = inds + [inds[-1] + k ** 2]
if G is None:
G = spmatrix([], [], [], (0, q.size[0]))
matrixG = True
if matrixG:
if G.typecode != 'd' or G.size != (cdim, q.size[0]):
raise TypeError("'G' must be a 'd' matrix of size (%d, %d)"
% (cdim, q.size[0]))
def fG(x, y, trans='N', alpha=1.0, beta=0.0):
misc.sgemv(G, x, y, dims, trans=trans, alpha=alpha,
beta=beta)
else:
fG = G
if A is None:
A = spmatrix([], [], [], (0, q.size[0]))
matrixA = True
if matrixA:
if A.typecode != 'd' or A.size[1] != q.size[0]:
raise TypeError("'A' must be a 'd' matrix with %d columns"
% q.size[0])
def fA(x, y, trans='N', alpha=1.0, beta=0.0):
base.gemv(A, x, y, trans=trans, alpha=alpha, beta=beta)
else:
fA = A
if b is None:
b = matrix(0.0, (0, 1))
if not isinstance(b, matrix) or b.typecode != 'd' or b.size[1] != 1:
raise TypeError("'b' must be a 'd' matrix with one column")
if matrixA and b.size[0] != A.size[0]:
raise TypeError("'b' must have length %d" % A.size[0])
ws3, wz3 = matrix(0.0, (cdim, 1)), matrix(0.0, (cdim, 1))
def res(ux, uy, uz, us, vx, vy, vz, vs, W, lmbda):
# Evaluates residual in Newton equations:
#
# [ vx ] [ vx ] [ 0 ] [ P A' G' ] [ ux ]
# [ vy ] := [ vy ] - [ 0 ] - [ A 0 0 ] * [ uy ]
# [ vz ] [ vz ] [ W'*us ] [ G 0 0 ] [ W^{-1}*uz ]
#
# vs := vs - lmbda o (uz + us).
# vx := vx - P*ux - A'*uy - G'*W^{-1}*uz
fP(ux, vx, alpha=-1.0, beta=1.0)
fA(uy, vx, alpha=-1.0, beta=1.0, trans='T')
blas.copy(uz, wz3)
misc.scale(wz3, W, inverse='I')
fG(wz3, vx, alpha=-1.0, beta=1.0, trans='T')
# vy := vy - A*ux
fA(ux, vy, alpha=-1.0, beta=1.0)
# vz := vz - G*ux - W'*us
fG(ux, vz, alpha=-1.0, beta=1.0)
blas.copy(us, ws3)
misc.scale(ws3, W, trans='T')
blas.axpy(ws3, vz, alpha=-1.0)
# vs := vs - lmbda o (uz + us)
blas.copy(us, ws3)
blas.axpy(uz, ws3)
misc.sprod(ws3, lmbda, dims, diag='D')
blas.axpy(ws3, vs, alpha=-1.0)
# kktsolver(W) returns a routine for solving
#
# [ P A' G'*W^{-1} ] [ ux ] [ bx ]
# [ A 0 0 ] [ uy ] = [ by ].
# [ G 0 -W' ] [ uz ] [ bz ]
factor = kkt_chol2(G, dims, A)
def kktsolver(W):
return factor(W, P)
resx0 = max(1.0, math.sqrt(np.dot(q.T, q)))
resy0 = max(1.0, math.sqrt(np.dot(b.T, b)))
resz0 = max(1.0, misc.snrm2(h, dims))
if cdim == 0:
return solve_only_equalities_qp(kktsolver, fP, fA, resx0, resy0, dims)
x, y = matrix(q), matrix(b)
s, z = matrix(0.0, (cdim, 1)), matrix(0.0, (cdim, 1))
# Factor
#
# [ P A' G' ]
# [ A 0 0 ].
# [ G 0 -I ]
W = {}
W['d'] = matrix(1.0, (dims['l'], 1))
W['di'] = matrix(1.0, (dims['l'], 1))
W['v'] = [matrix(0.0, (m, 1)) for m in dims['q']]
W['beta'] = len(dims['q']) * [1.0]
for v in W['v']:
v[0] = 1.0
W['r'] = [matrix(0.0, (m, m)) for m in dims['s']]
W['rti'] = [matrix(0.0, (m, m)) for m in dims['s']]
for r in W['r']:
r[::r.size[0] + 1] = 1.0
for rti in W['rti']:
rti[::rti.size[0] + 1] = 1.0
try:
f = kktsolver(W)
except ArithmeticError:
raise ValueError("Rank(A) < p or Rank([P; A; G]) < n")
# Solve
#
# [ P A' G' ] [ x ] [ -q ]
# [ A 0 0 ] * [ y ] = [ b ].
# [ G 0 -I ] [ z ] [ h ]
x = matrix(np.copy(q))
x *= -1.0
y = matrix(np.copy(b))
z = matrix(np.copy(h))
try:
f(x, y, z)
except ArithmeticError:
raise ValueError("Rank(A) < p or Rank([P; G; A]) < n")
s = matrix(np.copy(z))
blas.scal(-1.0, s)
nrms = misc.snrm2(s, dims)
ts = misc.max_step(s, dims)
if ts >= -1e-8 * max(nrms, 1.0):
a = 1.0 + ts
s[:dims['l']] += a
s[indq[:-1]] += a
ind = dims['l'] + sum(dims['q'])
for m in dims['s']:
s[ind: ind + m * m: m + 1] += a
ind += m ** 2
nrmz = misc.snrm2(z, dims)
tz = misc.max_step(z, dims)
if tz >= -1e-8 * max(nrmz, 1.0):
a = 1.0 + tz
z[:dims['l']] += a
z[indq[:-1]] += a
ind = dims['l'] + sum(dims['q'])
for m in dims['s']:
z[ind: ind + m * m: m + 1] += a
ind += m ** 2
rx, ry, rz = matrix(q), matrix(b), matrix(0.0, (cdim, 1))
dx, dy = matrix(x), matrix(y)
dz, ds = matrix(0.0, (cdim, 1)), matrix(0.0, (cdim, 1))
lmbda = matrix(0.0, (dims['l'] + sum(dims['q']) + sum(dims['s']), 1))
lmbdasq = matrix(0.0, (dims['l'] + sum(dims['q']) + sum(dims['s']), 1))
sigs = matrix(0.0, (sum(dims['s']), 1))
sigz = matrix(0.0, (sum(dims['s']), 1))
if show_progress:
print("% 10s% 12s% 10s% 8s% 7s" % ("pcost", "dcost", "gap", "pres",
"dres"))
gap = misc.sdot(s, z, dims)
for iters in range(MAXITERS + 1):
# f0 = (1/2)*x'*P*x + q'*x + r and rx = P*x + q + A'*y + G'*z.
rx = matrix(np.copy(q))
fP(x, rx, beta=1.0)
f0 = 0.5 * (np.dot(x.T, rx) + np.dot(x.T, q))
fA(y, rx, beta=1.0, trans='T')
fG(z, rx, beta=1.0, trans='T')
resx = math.sqrt(np.dot(rx.T, rx))
# ry = A*x - b
ry = matrix(np.copy(b))
fA(x, ry, alpha=1.0, beta=-1.0)
resy = math.sqrt(np.dot(ry.T, ry))
# rz = s + G*x - h
rz = matrix(np.copy(s))
blas.axpy(h, rz, alpha=-1.0)
fG(x, rz, beta=1.0)
resz = misc.snrm2(rz, dims)
# Statistics for stopping criteria.
# pcost = (1/2)*x'*P*x + q'*x
# dcost = (1/2)*x'*P*x + q'*x + y'*(A*x-b) + z'*(G*x-h)
# = (1/2)*x'*P*x + q'*x + y'*(A*x-b) + z'*(G*x-h+s) - z'*s
# = (1/2)*x'*P*x + q'*x + y'*ry + z'*rz - gap
pcost = f0
dcost = f0 + np.dot(y.T, ry) + misc.sdot(z, rz, dims) - gap
if pcost < 0.0:
relgap = gap / -pcost
elif dcost > 0.0:
relgap = gap / dcost
else:
relgap = None
pres = max(resy / resy0, resz / resz0)
dres = resx / resx0
if show_progress:
print("%2d: % 8.4e % 8.4e % 4.0e% 7.0e% 7.0e"
% (iters, pcost, dcost, gap, pres, dres))
if (pres <= FEASTOL and dres <= FEASTOL and (gap <= ABSTOL or
(relgap is not None and relgap <= RELTOL))) or \
iters == MAXITERS:
ind = dims['l'] + sum(dims['q'])
for m in dims['s']:
misc.symm(s, m, ind)
misc.symm(z, m, ind)
ind += m ** 2
ts = misc.max_step(s, dims)
tz = misc.max_step(z, dims)
if iters == MAXITERS:
if show_progress:
print("Terminated (maximum number of iterations "
"reached).")
status = 'unknown'
else:
if show_progress:
print("Optimal solution found.")
status = 'optimal'
return {'x': x, 'y': y, 's': s, 'z': z, 'status': status,
'gap': gap, 'relative gap': relgap,
'primal objective': pcost, 'dual objective': dcost,
'primal infeasibility': pres,
'dual infeasibility': dres, 'primal slack': -ts,
'dual slack': -tz, 'iterations': iters}
# Compute initial scaling W and scaled iterates:
#
# W * z = W^{-T} * s = lambda.
#
# lmbdasq = lambda o lambda.
if iters == 0:
W = misc.compute_scaling(s, z, lmbda, dims)
misc.ssqr(lmbdasq, lmbda, dims)
# f3(x, y, z) solves
#
# [ P A' G' ] [ ux ] [ bx ]
# [ A 0 0 ] [ uy ] = [ by ].
# [ G 0 -W'*W ] [ W^{-1}*uz ] [ bz ]
#
# On entry, x, y, z containg bx, by, bz.
# On exit, they contain ux, uy, uz.
try:
f3 = kktsolver(W)
except ArithmeticError:
if iters == 0:
raise ValueError("Rank(A) < p or Rank([P; A; G]) < n")
else:
ind = dims['l'] + sum(dims['q'])
for m in dims['s']:
misc.symm(s, m, ind)
misc.symm(z, m, ind)
ind += m ** 2
ts = misc.max_step(s, dims)
tz = misc.max_step(z, dims)
if show_progress:
print("Terminated (singular KKT matrix).")
return {'x': x, 'y': y, 's': s, 'z': z,
'status': 'unknown', 'gap': gap,
'relative gap': relgap, 'primal objective': pcost,
'dual objective': dcost, 'primal infeasibility': pres,
'dual infeasibility': dres, 'primal slack': -ts,
'dual slack': -tz, 'iterations': iters}
# f4_no_ir(x, y, z, s) solves
#
# [ 0 ] [ P A' G' ] [ ux ] [ bx ]
# [ 0 ] + [ A 0 0 ] * [ uy ] = [ by ]
# [ W'*us ] [ G 0 0 ] [ W^{-1}*uz ] [ bz ]
#
# lmbda o (uz + us) = bs.
#
# On entry, x, y, z, s contain bx, by, bz, bs.
# On exit, they contain ux, uy, uz, us.
def f4_no_ir(x, y, z, s):
# Solve
#
# [ P A' G' ] [ ux ] [ bx ]
# [ A 0 0 ] [ uy ] = [ by ]
# [ G 0 -W'*W ] [ W^{-1}*uz ] [ bz - W'*(lmbda o\ bs) ]
#
# us = lmbda o\ bs - uz.
#
# On entry, x, y, z, s contains bx, by, bz, bs.
# On exit they contain x, y, z, s.
# s := lmbda o\ s
# = lmbda o\ bs
misc.sinv(s, lmbda, dims)
# z := z - W'*s
# = bz - W'*(lambda o\ bs)
ws3 = matrix(np.copy(s))
misc.scale(ws3, W, trans='T')
blas.axpy(ws3, z, alpha=-1.0)
# Solve for ux, uy, uz
f3(x, y, z)
# s := s - z
# = lambda o\ bs - uz.
blas.axpy(z, s, alpha=-1.0)
# f4(x, y, z, s) solves the same system as f4_no_ir, but applies
# iterative refinement.
if iters == 0:
if refinement:
wx, wy = matrix(q), matrix(b)
wz, ws = matrix(0.0, (cdim, 1)), matrix(0.0, (cdim, 1))
if refinement:
wx2, wy2 = matrix(q), matrix(b)
wz2, ws2 = matrix(0.0, (cdim, 1)), matrix(0.0, (cdim, 1))
def f4(x, y, z, s):
if refinement:
wx = matrix(np.copy(x))
wy = matrix(np.copy(y))
wz = matrix(np.copy(z))
ws = matrix(np.copy(s))
f4_no_ir(x, y, z, s)
for i in range(refinement):
wx2 = matrix(np.copy(wx))
wy2 = matrix(np.copy(wy))
wz2 = matrix(np.copy(wz))
ws2 = matrix(np.copy(ws))
res(x, y, z, s, wx2, wy2, wz2, ws2, W, lmbda)
f4_no_ir(wx2, wy2, wz2, ws2)
y += wx2
y += wy2
blas.axpy(wz2, z)
blas.axpy(ws2, s)
mu = gap / (dims['l'] + len(dims['q']) + sum(dims['s']))
sigma, eta = 0.0, 0.0
for i in [0, 1]:
# Solve
#
# [ 0 ] [ P A' G' ] [ dx ]
# [ 0 ] + [ A 0 0 ] * [ dy ] = -(1 - eta) * r
# [ W'*ds ] [ G 0 0 ] [ W^{-1}*dz ]
#
# lmbda o (dz + ds) = -lmbda o lmbda + sigma*mu*e (i=0)
# lmbda o (dz + ds) = -lmbda o lmbda - dsa o dza
# + sigma*mu*e (i=1) where dsa, dza
# are the solution for i=0.
# ds = -lmbdasq + sigma * mu * e (if i is 0)
# = -lmbdasq - dsa o dza + sigma * mu * e (if i is 1),
# where ds, dz are solution for i is 0.
blas.scal(0.0, ds)
if i == 1:
blas.axpy(ws3, ds, alpha=-1.0)
blas.axpy(lmbdasq, ds, n=dims['l'] + sum(dims['q']),
alpha=-1.0)
ds[:dims['l']] += sigma * mu
ind = dims['l']
for m in dims['q']:
ds[ind] += sigma * mu
ind += m
ind2 = ind
for m in dims['s']:
blas.axpy(lmbdasq, ds, n=m, offsetx=ind2, offsety=ind, incy=m + 1, alpha=-1.0)
ds[ind: ind + m * m: m + 1] += sigma * mu
ind += m * m
ind2 += m
# (dx, dy, dz) := -(1 - eta) * (rx, ry, rz)
dx *= 0.0
dx += (-1.0 + eta) * rx
dy *= 0.0
dy += (-1.0 + eta) * ry
blas.scal(0.0, dz)
blas.axpy(rz, dz, alpha=-1.0 + eta)
try:
f4(dx, dy, dz, ds)
except ArithmeticError:
if iters == 0:
raise ValueError("Rank(A) < p or Rank([P; A; G]) < n")
else:
ind = dims['l'] + sum(dims['q'])
for m in dims['s']:
misc.symm(s, m, ind)
misc.symm(z, m, ind)
ind += m ** 2
ts = misc.max_step(s, dims)
tz = misc.max_step(z, dims)
if show_progress:
print("Terminated (singular KKT matrix).")
return {'x': x, 'y': y, 's': s, 'z': z,
'status': 'unknown', 'gap': gap,
'relative gap': relgap, 'primal objective': pcost,
'dual objective': dcost,
'primal infeasibility': pres,
'dual infeasibility': dres, 'primal slack': -ts,
'dual slack': -tz, 'iterations': iters}
dsdz = misc.sdot(ds, dz, dims)
# Save ds o dz for Mehrotra correction
if i == 0:
ws3 = matrix(np.copy(ds))
misc.sprod(ws3, dz, dims)
# Maximum steps to boundary.
#
# If i is 1, also compute eigenvalue decomposition of the
# 's' blocks in ds,dz. The eigenvectors Qs, Qz are stored in
# dsk, dzk. The eigenvalues are stored in sigs, sigz.
misc.scale2(lmbda, ds, dims)
misc.scale2(lmbda, dz, dims)
if i == 0:
ts = misc.max_step(ds, dims)
tz = misc.max_step(dz, dims)
else:
ts = misc.max_step(ds, dims, sigma=sigs)
tz = misc.max_step(dz, dims, sigma=sigz)
t = max([0.0, ts, tz])
if t == 0:
step = 1.0
else:
if i == 0:
step = min(1.0, 1.0 / t)
else:
step = min(1.0, STEP / t)
if i == 0:
sigma = min(1.0, max(0.0,
1.0 - step + dsdz / gap * step ** 2)) ** EXPON
eta = 0.0
x += step * dx
y += step * dy
# We will now replace the 'l' and 'q' blocks of ds and dz with
# the updated iterates in the current scaling.
# We also replace the 's' blocks of ds and dz with the factors
# Ls, Lz in a factorization Ls*Ls', Lz*Lz' of the updated variables
# in the current scaling.
# ds := e + step*ds for nonlinear, 'l' and 'q' blocks.
# dz := e + step*dz for nonlinear, 'l' and 'q' blocks.
blas.scal(step, ds, n=dims['l'] + sum(dims['q']))
blas.scal(step, dz, n=dims['l'] + sum(dims['q']))
ind = dims['l']
ds[:ind] += 1.0
dz[:ind] += 1.0
for m in dims['q']:
ds[ind] += 1.0
dz[ind] += 1.0
ind += m
# ds := H(lambda)^{-1/2} * ds and dz := H(lambda)^{-1/2} * dz.
#
# This replaced the 'l' and 'q' components of ds and dz with the
# updated iterates in the current scaling.
# The 's' components of ds and dz are replaced with
#
# diag(lmbda_k)^{1/2} * Qs * diag(lmbda_k)^{1/2}
# diag(lmbda_k)^{1/2} * Qz * diag(lmbda_k)^{1/2}
#
misc.scale2(lmbda, ds, dims, inverse='I')
misc.scale2(lmbda, dz, dims, inverse='I')
# sigs := ( e + step*sigs ) ./ lambda for 's' blocks.
# sigz := ( e + step*sigz ) ./ lmabda for 's' blocks.
blas.scal(step, sigs)
blas.scal(step, sigz)
sigs += 1.0
sigz += 1.0
blas.tbsv(lmbda, sigs, n=sum(dims['s']), k=0, ldA=1, offsetA=dims['l'] + sum(dims['q']))
blas.tbsv(lmbda, sigz, n=sum(dims['s']), k=0, ldA=1, offsetA=dims['l'] + sum(dims['q']))
# dsk := Ls = dsk * sqrt(sigs).
# dzk := Lz = dzk * sqrt(sigz).
ind2, ind3 = dims['l'] + sum(dims['q']), 0
for k in range(len(dims['s'])):
m = dims['s'][k]
for i in range(m):
blas.scal(math.sqrt(sigs[ind3 + i]), ds, offset=ind2 + m * i,
n=m)
blas.scal(math.sqrt(sigz[ind3 + i]), dz, offset=ind2 + m * i,
n=m)
ind2 += m * m
ind3 += m
# Update lambda and scaling.
misc.update_scaling(W, lmbda, ds, dz)
# Unscale s, z (unscaled variables are used only to compute
# feasibility residuals).
blas.copy(lmbda, s, n=dims['l'] + sum(dims['q']))
ind = dims['l'] + sum(dims['q'])
ind2 = ind
for m in dims['s']:
blas.scal(0.0, s, offset=ind2)
blas.copy(lmbda, s, offsetx=ind, offsety=ind2, n=m,
incy=m + 1)
ind += m
ind2 += m * m
misc.scale(s, W, trans='T')
blas.copy(lmbda, z, n=dims['l'] + sum(dims['q']))
ind = dims['l'] + sum(dims['q'])
ind2 = ind
for m in dims['s']:
blas.scal(0.0, z, offset=ind2)
blas.copy(lmbda, z, offsetx=ind, offsety=ind2, n=m,
incy=m + 1)
ind += m
ind2 += m * m
misc.scale(z, W, inverse='I')
gap = blas.dot(lmbda, lmbda)
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