def ADMM_ConstrBP(A,s,epsilon):
"""
Alternating Direction Method of Multipliers for Constrained Basis Pursuit
Solve the problem:
min ||x||_1 s.t. ||Ax-s||_2 <= epsilon
i.e., find the minimum-norm vector s.t. the reconstruction error is under epsilon.
The problem is transformed into:
min ||y||_1 + 1/2||z-s||_2 s.t x = y, Ax = z
"""
(M,N) = A.shape
x,y,z,u,v = zeros((N,1)), zeros((N,1)), zeros((M,1)), zeros((N,1)), zeros((M,1))
max_it = 200
gamma = epsilon
sigma = epsilon
k = 0
xks,yks,zks,uks,vks = [x],[y],[z],[u],[v]
prim_resid = 100
dual_resid = 100
prim_epsi = 0
dual_epsi = 0
epsi_rel = 0.001
epsi_abs = 0.0001
while k < max_it: #TODO stopping criterion
# Update x
rhs = gamma*(yks[k] - uks[k]) + sigma*dot(A.T,(zks[k]-vks[k]))
x_star = solve.lemma_solve(A,rhs,gamma,PL,U)
xks.append(x_star)
# Update y
t = xks[k+1] + uks[k]
y_star = sparse.shrinkage(t, 1/gamma)
yks.append(y_star)
# Update z
b = dot(A,xks[k+1]) + vks[k]
if linalg.norm(b-s) > epsilon:
z_star = s + epsilon*(b - s)/linalg.norm(b-s)
else:
z_star = b
zks.append(z_star)
# Update u
u_star = uks[k] + xks[k+1] - yks[k+1]
uks.append(u_star)
# Update v
v_star = vks[k] + dot(A,xks[k+1]) - zks[k+1]
vks.append(v_star)
prim_resid = linalg.norm(xks[k+1] - yks[k+1])
dual_resid = linalg.norm(rho*(yks[k] - yks[k+1]))
print k, linalg.norm(x_star,1)
print linalg.norm(dot(A,x_star)-s)
k += 1
prim_epsi = sqrt(N)*epsi_abs + epsi_rel*max(linalg.norm(xks[k]),linalg.norm(yks[k]))
dual_epsi = sqrt(N)*epsi_abs + epsi_rel*linalg.norm(uks[k])
print prim_resid, dual_resid
print prim_epsi, dual_epsi
return xks[k]
def ADMM_BPDN(A,s,l, max_it=1000):
"""
Alternating Direction Method of Multipliers for Basis Pursuit Denoising
Solve the problem:
min 1/2 ||Ax-s||^2 + lambda*||x||_1
transformed into:
min 1/2 ||Ax-s||^2 + lambda*||y||_1 s.t. x = y
"""
(M,N) = A.shape
x,y,u = zeros((N,1)), zeros((N,1)), zeros((N,1))
rho = l
k = 0
xks,yks,uks = [x],[y],[u]
assert(s.shape == (M,1))
# Cached factorizations when M < N
if M < N:
C = dot(A,A.T) + rho*eye(M)
PL,U = scipy.linalg.lu(C, permute_l=True)
As = dot(A.T,s)
prim_resid = 100
dual_resid = 100
prim_epsi = 0
dual_epsi = 0
epsi_rel = 0.001
epsi_abs = 0.0001
while k < max_it and (prim_resid >= prim_epsi or dual_resid >= dual_epsi):
# Update x
rhs = As + rho*(yks[k] - uks[k])
if M < N:
x_star = solve.lemma_solve(A,rhs,rho,PL,U)
else:
x_star = solve.direct_solve(A,rhs,rho)
xks.append(x_star)
# Update y
z = xks[k+1] + uks[k]
y_star = sparse.shrinkage(z, l/rho)
yks.append(y_star)
# Update u
u_star = uks[k] + xks[k+1] - yks[k+1]
uks.append(u_star)
prim_resid = linalg.norm(xks[k+1] - yks[k+1])
dual_resid = linalg.norm(rho*(yks[k] - yks[k+1]))
#TODO change rho to balance primal/dual residual
print k, 1/2*linalg.norm(dot(A,x_star)-s) + l*linalg.norm(x_star,1)
k += 1
prim_epsi = sqrt(N)*epsi_abs + epsi_rel*max(linalg.norm(xks[k]),linalg.norm(yks[k]))
dual_epsi = sqrt(N)*epsi_abs + epsi_rel*linalg.norm(uks[k])
print prim_resid, dual_resid
print prim_epsi, dual_epsi
return xks[k]
