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l1.py
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from cvxopt import blas, lapack, solvers
from cvxopt import matrix, spdiag, mul, div, sparse
from cvxopt import spmatrix, sqrt, base
try:
import mosek
import sys
__MOSEK = True
except: __MOSEK = False
if __MOSEK:
def l1mosek(P, q):
"""
minimize e'*v
subject to P*u - v <= q
-P*u - v <= -q
"""
from mosek.array import zeros
m, n = P.size
task = env.Task(0,0)
task.set_Stream(mosek.streamtype.log, lambda x: sys.stdout.write(x))
task.append(mosek.accmode.var, n + m) # number of variables
task.append(mosek.accmode.con, 2*m) # number of constraints
task.putclist(range(n+m), n*[0.0] + m*[1.0]) # setup objective
# input A matrix row by row
for i in range(m):
task.putavec(mosek.accmode.con, i,
range(n) + [n+i] , list(P[i,:]) + [-1.0])
task.putavec(mosek.accmode.con, i+m,
range(n) + [n+i] , list(-P[i,:]) + [-1.0])
# setup bounds on constraints
task.putboundslice(mosek.accmode.con,
0, 2*m, 2*m*[mosek.boundkey.up], 2*m*[0.0], list(q)+list(-q))
# setup variable bounds
task.putboundslice(mosek.accmode.var,
0, n+m, (n+m)*[mosek.boundkey.fr], (n+m)*[0.0], (n+m)*[0.0])
# optimize the task
task.putobjsense(mosek.objsense.minimize)
task.putintparam(mosek.iparam.optimizer, mosek.optimizertype.intpnt)
task.putintparam(mosek.iparam.intpnt_basis, mosek.basindtype.never)
task.optimize()
task.solutionsummary(mosek.streamtype.log)
x = zeros(n, float)
task.getsolutionslice(mosek.soltype.itr, mosek.solitem.xx, 0, n, x)
return matrix(x)
def l1mosek2(P, q):
"""
minimize e'*s + e'*t
subject to P*u - q = s - t
s, t >= 0
"""
from mosek.array import zeros
m, n = P.size
# env = mosek.Env()
task = env.Task(0,0)
task.set_Stream(mosek.streamtype.log, lambda x: sys.stdout.write(x))
task.append(mosek.accmode.var, n + 2*m) # number of variables
task.append(mosek.accmode.con, m) # number of constraints
task.putclist(range(n+2*m), n*[0.0] + 2*m*[1.0]) # setup objective
# input A matrix row by row
for i in range(m):
task.putavec(mosek.accmode.con, i,
range(n) + [n+i, n+m+i] , list(P[i,:]) + [-1.0, 1.0])
# setup bounds on constraints
task.putboundslice(mosek.accmode.con,
0, m, m*[mosek.boundkey.fx], list(q), list(q))
# setup variable bounds
task.putboundslice(mosek.accmode.var,
0, n, n*[mosek.boundkey.fr], n*[0.0], n*[0.0])
task.putboundslice(mosek.accmode.var,
n, n+2*m, 2*m*[mosek.boundkey.lo], 2*m*[0.0], 2*m*[0.0])
# optimize the task
task.putobjsense(mosek.objsense.minimize)
task.putintparam(mosek.iparam.optimizer, mosek.optimizertype.intpnt)
task.putintparam(mosek.iparam.intpnt_basis, mosek.basindtype.never)
task.optimize()
task.solutionsummary(mosek.streamtype.log)
x = zeros(n, float)
task.getsolutionslice(mosek.soltype.itr, mosek.solitem.xx, 0, n, x)
return matrix(x)
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]
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]