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# The quadratically constrained 1-norm minimization example of section 8.7
# (Exploiting structure).
from cvxopt import blas, lapack, solvers, matrix, mul, div, setseed, normal
from math import sqrt
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
setseed()
m, n = 100, 100
A, b = normal(m,n), normal(m,1)
x, z = qcl1(A, b)
if x is None: print("infeasible")
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