def l1(P, q):
"""
Returns the solution u of the ell-1 approximation problem
(primal) minimize ||P*u - q||_1
(dual) maximize q'*w
subject to P'*w = 0
||w||_infty <= 1.
"""
m, n = P.size
# Solve equivalent LP
#
# minimize [0; 1]' * [u; v]
# subject to [P, -I; -P, -I] * [u; v] <= [q; -q]
#
# maximize -[q; -q]' * z
# subject to [P', -P']*z = 0
# [-I, -I]*z + 1 = 0
# z >= 0
c = matrix(n * [0.0] + m * [1.0])
h = matrix([q, -q])
def Fi(x, y, alpha=1.0, beta=0.0, trans='N'):
if trans == 'N':
# y := alpha * [P, -I; -P, -I] * x + beta*y
u = P * x[:n]
y[:m] = alpha * (u - x[n:]) + beta * y[:m]
y[m:] = alpha * (-u - x[n:]) + beta * y[m:]
else:
# y := alpha * [P', -P'; -I, -I] * x + beta*y
y[:n] = alpha * P.T * (x[:m] - x[m:]) + beta * y[:n]
y[n:] = -alpha * (x[:m] + x[m:]) + beta * y[n:]
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 -W1^2 0 ] [ z[:m] ] = [ bz[:m] ]
# [-P -I 0 -W2 ] [ z[m:] ] [ bz[m:] ]
#
# On entry bx, bz are stored in x, z.
# On exit x, z contain the solution, with z scaled (W['di'] .* z is
# returned instead of z).
d1, d2 = W['d'][:m], W['d'][m:]
D = 4 * (d1**2 + d2**2)**-1
A = P.T * spdiag(D) * P
lapack.potrf(A)
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)
return f
# Initial primal and dual points from least-squares solution.
# uls minimizes ||P*u-q||_2; rls is the LS residual.
uls = +q
lapack.gels(+P, uls)
rls = P * uls[:n] - q
# x0 = [ uls; 1.1*abs(rls) ]; s0 = [q;-q] - [P,-I; -P,-I] * x0
x0 = matrix([uls[:n], 1.1 * abs(rls)])
s0 = +h
Fi(x0, s0, alpha=-1, beta=1)
# z0 = [ (1+w)/2; (1-w)/2 ] where w = (.9/||rls||_inf) * rls
# if rls is nonzero and w = 0 otherwise.
if max(abs(rls)) > 1e-10:
w = .9 / max(abs(rls)) * rls
else:
w = matrix(0.0, (m, 1))
z0 = matrix([.5 * (1 + w), .5 * (1 - w)])
dims = {'l': 2 * m, 'q': [], 's': []}
sol = solvers.conelp(c,
Fi,
h,
dims,
kktsolver=Fkkt,
primalstart={
'x': x0,
's': s0
},
dualstart={'z': z0})
return sol['x'][:n], sol['status']
def l1blas(P, q):
"""
Returns the solution u of the ell-1 approximation problem
(primal) minimize ||P*u - q||_1
(dual) maximize q'*w
subject to P'*w = 0
||w||_infty <= 1.
"""
m, n = P.size
# Solve equivalent LP
#
# minimize [0; 1]' * [u; v]
# subject to [P, -I; -P, -I] * [u; v] <= [q; -q]
#
# maximize -[q; -q]' * z
# subject to [P', -P']*z = 0
# [-I, -I]*z + 1 = 0
# z >= 0
c = matrix(n * [0.0] + m * [1.0])
h = matrix([q, -q])
u = matrix(0.0, (m, 1))
Ps = matrix(0.0, (m, n))
A = matrix(0.0, (n, n))
def Fi(x, y, alpha=1.0, beta=0.0, trans='N'):
if trans == 'N':
# y := alpha * [P, -I; -P, -I] * x + beta*y
blas.gemv(P, x, u)
y[:m] = alpha * (u - x[n:]) + beta * y[:m]
y[m:] = alpha * (-u - x[n:]) + beta * y[m:]
else:
# y := alpha * [P', -P'; -I, -I] * x + beta*y
blas.copy(x[:m] - x[m:], u)
blas.gemv(P, u, y, alpha=alpha, beta=beta, trans='T')
y[n:] = -alpha * (x[:m] + x[m:]) + beta * y[n:]
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
# Initial primal and dual points from least-squares solution.
# uls minimizes ||P*u-q||_2; rls is the LS residual.
uls = +q
lapack.gels(+P, uls)
rls = P * uls[:n] - q
# x0 = [ uls; 1.1*abs(rls) ]; s0 = [q;-q] - [P,-I; -P,-I] * x0
x0 = matrix([uls[:n], 1.1 * abs(rls)])
s0 = +h
Fi(x0, s0, alpha=-1, beta=1)
# z0 = [ (1+w)/2; (1-w)/2 ] where w = (.9/||rls||_inf) * rls
# if rls is nonzero and w = 0 otherwise.
if max(abs(rls)) > 1e-10:
w = .9 / max(abs(rls)) * rls
else:
w = matrix(0.0, (m, 1))
z0 = matrix([.5 * (1 + w), .5 * (1 - w)])
dims = {'l': 2 * m, 'q': [], 's': []}
sol = solvers.conelp(c,
Fi,
h,
dims,
kktsolver=Fkkt,
primalstart={
'x': x0,
's': s0
},
dualstart={'z': z0})
return sol['x'][:n], sol['status']
def qcl1(A, b):
"""
Returns the solution u, z of
(primal) minimize || u ||_1
subject to || A * u - b ||_2 <= 1
(dual) maximize b^T z - ||z||_2
subject to || A'*z ||_inf <= 1.
Exploits structure, assuming A is m by n with m >= n.
"""
m, n = A.size
# Solve equivalent cone LP with variables x = [u; v]:
#
# minimize [0; 1]' * x
# subject to [ I -I ] * x <= [ 0 ] (componentwise)
# [-I -I ] * x <= [ 0 ] (componentwise)
# [ 0 0 ] * x <= [ 1 ] (SOC)
# [-A 0 ] [ -b ].
#
# maximize -t + b' * w
# subject to z1 - z2 = A'*w
# z1 + z2 = 1
# z1 >= 0, z2 >=0, ||w||_2 <= t.
c = matrix(n*[0.0] + n*[1.0])
h = matrix( 0.0, (2*n + m + 1, 1))
h[2*n] = 1.0
h[2*n+1:] = -b
def G(x, y, alpha = 1.0, beta = 0.0, trans = 'N'):
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])
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]
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
d1, d2 = W['d'][:n], W['d'][n:]
d = 4.0 * (d1**2 + d2**2)**-1
S[::n+1] += d
lapack.potrf(S)
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]
return f
dims = {'l': 2*n, 'q': [m+1], 's': []}
sol = solvers.conelp(c, G, h, dims, kktsolver = Fkkt)
if sol['status'] == 'optimal':
return sol['x'][:n], sol['z'][-m:]
else:
return None, None
# The small linear cone program of section 8.1 (Linear cone programs).
from kvxopt import matrix, solvers
c = matrix([-6., -4., -5.])
G = matrix([[ 16., 7., 24., -8., 8., -1., 0., -1., 0., 0., 7.,
-5., 1., -5., 1., -7., 1., -7., -4.],
[-14., 2., 7., -13., -18., 3., 0., 0., -1., 0., 3.,
13., -6., 13., 12., -10., -6., -10., -28.],
[ 5., 0., -15., 12., -6., 17., 0., 0., 0., -1., 9.,
6., -6., 6., -7., -7., -6., -7., -11.]])
h = matrix( [ -3., 5., 12., -2., -14., -13., 10., 0., 0., 0., 68.,
-30., -19., -30., 99., 23., -19., 23., 10.] )
dims = {'l': 2, 'q': [4, 4], 's': [3]}
sol = solvers.conelp(c, G, h, dims)
print("\nStatus: " + sol['status'])
print("\nx = \n")
print(sol['x'])
print("\nz = \n")
print(sol['z'])
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))