def solve_full(self, bi, lmbd, show_progress=True):
solvers.options['show_progress'] = show_progress
# load problem data
self.bi = bi
self.lmbd = lmbd
self.c[1 + self.Nm:1 + self.Nm + self.Nb] = lmbd
self.h[1:1 + self.Nm] = -bi
t1 = time()
self.sol = solvers.conelp(
self.c, self.G, self.h, self.dims, self.A, self.b)
t2 = time()
t3 = time()
if self.sol['status'] in ('optimal'):
S = self.sol['s'][1 + self.Nm:]
self.sA[:] = S[self.I11] + 1j*S[self.I21]
lapack.heevr(
self.sA, self.sW, jobz='V', range='I', uplo='L',
vl=0.0, vu=0.0, il=self.Nb, iu=self.Nb, Z=self.sZ)
self.beta_hat[:] = math.sqrt(self.sW[0])*self.sZ[:, 0]
else:
raise RuntimeError('numerical problems')
t4 = time()
self.solver_time = t2 - t1
self.eig_time = t4 - t3
def solve(opts):
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.
])
A = matrix(0.0, (0, c.size[0]))
b = matrix(0.0, (0, 1))
dims = {'l': 2, 'q': [4, 4], 's': [3]}
#localcones.options.update(opts)
#sol = localcones.conelp(c, G, h, dims, kktsolver='qr')
solvers.options.update(opts)
sol = solvers.conelp(c, G, h, dims)
print("\nStatus: " + sol['status'])
if sol['status'] == 'optimal':
print "x=\n", helpers.strSpe(sol['x'], "%.5f")
print "s=\n", helpers.strSpe(sol['s'], "%.5f")
print "z=\n", helpers.strSpe(sol['z'], "%.5f")
rungotest(sol)
def solve_full(self, bi, lmbd, show_progress=True):
solvers.options['show_progress'] = show_progress
# load problem data
self.bi = bi
self.lmbd = lmbd
self.c[1 + self.Nm:1 + self.Nm + self.Nb] = lmbd
self.h[1:1 + self.Nm] = -bi
t1 = time()
self.sol = solvers.conelp(self.c, self.G, self.h, self.dims, self.A,
self.b)
t2 = time()
t3 = time()
if self.sol['status'] in ('optimal'):
S = self.sol['s'][1 + self.Nm:]
self.sA[:] = S[self.I11] + 1j * S[self.I21]
lapack.heevr(self.sA,
self.sW,
jobz='V',
range='I',
uplo='L',
vl=0.0,
vu=0.0,
il=self.Nb,
iu=self.Nb,
Z=self.sZ)
self.beta_hat[:] = math.sqrt(self.sW[0]) * self.sZ[:, 0]
else:
raise RuntimeError('numerical problems')
t4 = time()
self.solver_time = t2 - t1
self.eig_time = t4 - t3
def call_solver(p,quiet):
"""
Calls solver.
:param p: Convex cvxpy_program.
Assumed to be expanded.
:param quiet: Boolean.
"""
# Set printing format for cvxopt sparse matrices
opt.spmatrix_str = opt.printing.spmatrix_str_triplet
# Expand objects defined via partial minimization
constr_list = cvxpy_list(pm_expand(p.constraints))
# Get variables
variables = constr_list.variables
variables.sort()
# Count variables
n = len(variables)
# Create variable-index map
var_to_index = {}
for i in range(0,n,1):
var_to_index[variables[i]] = i
# Construct objective vector
c = construct_c(p.objective,var_to_index,n,p.action)
# Construct Ax == b
A,b = construct_Ab(constr_list._get_eq(),var_to_index,n)
# Construct Gx <= h
G,h,dim_l,dim_q,dim_s = construct_Gh(constr_list._get_ineq_in(),
var_to_index,n)
# Construct F
F = construct_F(constr_list._get_ineq_in(),var_to_index,n)
# Call cvxopt
solvers.options['maxiters'] = p.options['maxiters']
solvers.options['abstol'] = p.options['abstol']
solvers.options['reltol'] = p.options['reltol']
solvers.options['feastol'] = p.options['feastol']
solvers.options['show_progress'] = not quiet
dims = {'l':dim_l, 'q':dim_q, 's':dim_s}
if F is None:
r = solvers.conelp(c,G,h,dims,A,b)
else:
r = solvers.cpl(c,F,G,h,dims,A,b)
# Store numerical values
if r['status'] != PRIMAL_INFEASIBLE:
for v in variables:
v.value = r['x'][var_to_index[v]]
# Return result
return r
def call_solver(p, quiet):
"""
Calls solver.
:param p: Convex cvxpy_program.
Assumed to be expanded.
:param quiet: Boolean.
"""
# Set printing format for cvxopt sparse matrices
opt.spmatrix_str = opt.printing.spmatrix_str_triplet
# Expand objects defined via partial minimization
constr_list = cvxpy_list(pm_expand(p.constraints))
# Get variables
variables = constr_list.variables
variables.sort()
# Count variables
n = len(variables)
# Create variable-index map
var_to_index = {}
for i in range(0, n, 1):
var_to_index[variables[i]] = i
# Construct objective vector
c = construct_c(p.objective, var_to_index, n, p.action)
# Construct Ax == b
A, b = construct_Ab(constr_list._get_eq(), var_to_index, n)
# Construct Gx <= h
G, h, dim_l, dim_q, dim_s = construct_Gh(constr_list._get_ineq_in(),
var_to_index, n)
# Construct F
F = construct_F(constr_list._get_ineq_in(), var_to_index, n)
# Call cvxopt
solvers.options['maxiters'] = p.options['maxiters']
solvers.options['abstol'] = p.options['abstol']
solvers.options['reltol'] = p.options['reltol']
solvers.options['feastol'] = p.options['feastol']
solvers.options['show_progress'] = not quiet
dims = {'l': dim_l, 'q': dim_q, 's': dim_s}
if F is None:
r = solvers.conelp(c, G, h, dims, A, b)
else:
r = solvers.cpl(c, F, G, h, dims, A, b)
# Store numerical values
if r['status'] != PRIMAL_INFEASIBLE:
for v in variables:
v.value = r['x'][var_to_index[v]]
# Return result
return r
def test_conelp(self):
from cvxopt import matrix, msk, 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]}
self.assertAlmostEqualLists(list(solvers.conelp(c, G, h, dims)['x']),list(msk.conelp(c, G, h, dims)[1]))
def run(self, lam=10.0, mu=0.0, eps=0.0, s0_val=0.001):
G_tmp, h_tmp = get_G_h(self.var_No)
h_eps = matrix(0.0, (self.G_No, 1))
G_x = []
G_i = []
G_j = []
for G_name in self.SFG:
G = self.SFG.node[G_name]
if G['type'] == 'G':
g_cnt = G['cnt']
h_eps[g_cnt] = G['intensity'] + eps
for I_name in self.SFG[G_name]:
i_cnt = self.SFG.node[I_name]['cnt']
G_x.append(1.0)
G_i.append(g_cnt)
G_j.append(i_cnt)
G_x.append(-1.0)
G_i.append(g_cnt)
G_j.append(self.GI_No + self.M_No + g_cnt)
G_tmp2 = spmatrix(G_x, G_i, G_j, size=(self.G_No, self.var_No))
G = sparse([G_tmp, G_tmp2])
h = matrix([h_tmp, h_eps])
A_tmp, b = get_A_b(self.SFG, self.M_No, self.I_No, self.GI_No)
A_eps = spmatrix([], [], [], (self.I_No, self.G_No))
A = sparse([[A_tmp], [A_eps]])
x0 = get_initvals(self.var_No)
x0['s'] = matrix(s0_val, size=h.size)
c = matrix([
matrix(-1.0, size=(self.GI_No, 1)),
matrix(mu, size=(self.M_No, 1)),
matrix((1.0 + lam), size=(self.G_No, 1))
])
self.sol = solvers.conelp(c=c, G=G, h=h, A=A, b=b, primalstart=x0)
Xopt = self.sol['x']
alphas = [] # reporting results
for N_name in self.SFG:
N = self.SFG.node[N_name]
if N['type'] == 'M':
N['estimate'] = Xopt[self.GI_No + N['cnt']]
alphas.append(N.copy())
if N['type'] == 'G':
for I_name in self.SFG[N_name]:
NI = self.SFG.edge[N_name][I_name]
NI['estimate'] = Xopt[NI['cnt']]
g_cnt = self.GI_No + self.M_No + N['cnt']
N['relaxation'] = Xopt[g_cnt]
# fit error: evaluation of the cost function at the minimizer
error = self.get_mean_square_error()
return alphas, error, self.sol['status']
def find_naive_one_time_best_team(Q, A, B, G, H):
# Tracer()()
Q = matrix(Q)
best_team = conelp(Q, G, H, A=A, b=B)
team = [int(round(s)) for s in best_team['x']]
players_np = np.array(players)
scores_np = np.array(scores)
team_players = players_np[np.nonzero(team)].tolist()
team_player_scores = scores_np[np.nonzero(team)].tolist()
print team_player_scores
print team_players
def find_naive_one_time_best_team(Q,A,B,G,H): # Tracer()() Q = matrix(Q) best_team = conelp(Q,G,H,A=A,b=B) team = [int(round(s)) for s in best_team['x']] players_np = np.array(players) scores_np = np.array(scores) team_players = players_np[np.nonzero(team)].tolist() team_player_scores = scores_np[np.nonzero(team)].tolist() print team_player_scores print team_players
def balanced_cut(A, delta=1.0, cut='min', solver="conelp_custom"):
"""
Solves semidefinite relaxation of the following
balance-constrained min/max cut problem
"""
N = A.shape[0]
prob = balanced_cut_conelp(A, N, delta, cut)
if solver == "mosek":
sol = msk.conelp(*prob)
Z = matrix(sol[2][1:], (N, N), tc='d')
elif solver == "conelp":
sol = solvers.conelp(*prob)
Z = matrix(sol['z'][1:], (N, N), tc='d')
elif solver == "conelp_custom":
sol = solvers.conelp(*prob,
kktsolver=custom_kkt,
options={'refinement': 3})
Z = matrix(sol['z'][1:], (N, N), tc='d')
else:
raise ValueError("Unknown solver")
return Z
def Optimise(self, obj, n, order=None):
c, G, h, A, b, dims = self.ConstructSDP(obj, n, order, self.ineqcons,
self.eqcons)
solvers.options['show_progress'] = False
sol = solvers.conelp(c, G, h, dims, A, b)
solution = {'status': None, 'obj': None, 'iterations': None, 'x': None}
xopt = np.array(sol['x'][:n])
solution['iterations'] = sol['iterations']
if sol['status'] == 'optimal':
solution['status'] = sol['status']
solution['x'] = np.array(xopt)
solution['obj'] = sol['primal objective'] + obj[0][0, 0]
return solution
def solve(self):
if len(self.aux.G) == 0:
raise ArithmeticError('No constraints')
A = solvers.conelp(c=matrix(self.__c),
G=matrix(numpy.array(self.aux.G)),
h=matrix(numpy.array(self.aux.h)),
dims=self.aux.dims)
if A["status"] == "optimal":
return numpy.array(A['x']).transpose()[0], A
else:
raise ArithmeticError("Solution not optimal: " + A["status"])
def l1(P, q):
m, n = P.size
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':
u = P*x[:n]
y[:m] = alpha * ( u - x[n:]) + beta*y[:m]
y[m:] = alpha * (-u - x[n:]) + beta*y[m:]
else:
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):
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
uls = +q
lapack.gels(+P, uls)
rls = P*uls[:n] - q
x0 = matrix( [uls[:n], 1.1*abs(rls)] )
s0 = +h
Fi(x0, s0, alpha=-1, beta=1)
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 optimise(self,obj,n,order=None,ineqcons=[],eqcons=[]):
c,G,h,A,b,dims = self.constructsdp(obj,n,ineqcons,eqcons,order)
sol = solvers.conelp(c, G, h, dims, A, b)
solution = {'status': None, 'objective': None, 'iterations': None, 'xopt': None}
solution['status'] = sol['status']
solution['iterations'] = sol['iterations']
if solution['status'] is 'optimal':
solution['objective'] = sol['primal objective'] + obj[0][0,0]
solution['xopt'] = np.array(sol['x'][:n])
return solution
else:
return solution
def check_point(self, p, A1, B_s, u_s):
dims = {
'l': self.size_tb() + 2*self.size_x(), # Pure inequality constraints
# No com cone
'q': [4]*len(self.contacts)*len(self.gravity_envelope),
's': [] # No sd cones
}
size_cones = self.size_x()*4 // 3
#Min x ~ who cares, we just want to know if there is a solution
c = matrix(np.ones((self.size_x(), 1)))
A1_diag = block_diag(*([A1]*len(self.gravity_envelope)))
A2 = np.vstack([self.computeA2(self.gravity+e)
for e in self.gravity_envelope])
T = np.vstack([self.computeT(self.gravity+e, height=self.height)
for e in self.gravity_envelope]) - A2.dot(p)
A = matrix(A1_diag)
g_s = []
h_s = []
if self.L_s:
g_s.append(np.vstack(self.L_s))
h_s.append(np.vstack(self.tb_s))
g_force = np.vstack([np.eye(self.size_x()), -np.eye(self.size_x())])
g_s.append(g_force)
h_s.append(self.force_lim*self.mass*9.81*np.ones((2*self.size_x(), 1)))
#B = diag{[u_i b_i.T].T}
blocks = [-np.vstack([u.T, B]) for u, B in zip(u_s, B_s)]*len(self.gravity_envelope)
block = block_diag(*blocks)
g_s.append(block)
h_cones = np.zeros((size_cones, 1))
h_s.append(h_cones)
g = np.vstack(g_s)
h = np.vstack(h_s)
sol = solvers.conelp(c, G=matrix(g), h=matrix(h),
A=A, b=matrix(T), dims=dims)
return sol
def QP(self, expert, features, epsilon=None):
if epsilon is None:
epsilon = self.M.epsilon
assert expert.shape[-1] == np.array(features).shape[-1]
c = matrix(np.eye(len(expert) + 1)[-1] * -1)
G_i = []
h_i = []
for k in range(len(expert)):
G_i.append([0])
G_i.append([-1])
h_i.append(0)
for j in range(len(features)):
for k in range(len(expert)):
G_i[k].append(-expert[k] + features[j][k])
G_i[len(expert)].append(1)
h_i.append(0)
for k in range(len(expert)):
G_i[k] = G_i[k] + [0.0] * (k + 1) + [
-1.0
] + [0.0] * (len(expert) + 1 - k - 1)
G_i[len(expert)] = G_i[len(expert)] + [0.0] * (1 + len(expert)) + [0.0]
h_i = h_i + [1] + (1 + len(expert)) * [0.0]
G = matrix(G_i)
h = matrix(h_i)
dims = {'l': 1 + len(features), 'q': [len(expert) + 1, 1], 's': []}
start = time.time()
sol = solvers.conelp(c, G, h, dims)
end = time.time()
print("QP operation time = " + str(end - start))
solution = np.array(sol['x'])
if solution is not None:
solution = solution.reshape([len(expert) + 1]).tolist()
w = solution[0:-1]
t = solution[-1]
else:
w = None
t = None
return w, t
def l2_bound_layer1(self, weight, bias, x):
""" Given input weights and biases will compute if each neuron can be
on or off
We do this using cvxopt conelp, but it's a messy formulation
the G matrix looks like
[ -I | I | 0row | -I]^T
and the h matrix looks like
[-lo_bounds, u_bounds, radius, -x]
"""
G_list = [
-1 * np.eye(self.dimension),
np.eye(self.dimension),
np.zeros((1, self.dimension)), -1 * np.eye(self.dimension)
]
G = matrix(np.vstack(G_list).astype(np.float))
h_list = [
-1 * self.box_low, self.box_high,
np.array([self.l2_radius]), -x
]
h = matrix(np.hstack(h_list).astype(np.float))
dims = {'l': self.dimension * 2, 'q': [self.dimension + 1], 's': [0]}
m = weight.shape[0]
# Do lowers first
new_lows, new_highs = [], []
for scale, working_list in [(1, new_lows), (-1, new_highs)]:
for i in range(m):
if (i % 10) == 0:
print(scale, i)
c = matrix(weight[i].astype(np.float) * scale)
import time
start = time.time()
solve_out = solvers.conelp(c, G, h, dims, solver='mosek')
print(i, time.time() - start)
working_list.append(scale * solve_out['primal objective'] +
bias[i])
return new_lows, new_highs
def s_SDP(self, Sigma):
p = Sigma.shape[0]
c = -np.ones(p)
c = matrix(c)
G = np.zeros((2 * p + p**2, p))
G[:p, :] = np.eye(p)
G[p:2 * p, :] = -np.eye(p)
for i in range(p):
G[2 * p + p * i, i] = 1
G = matrix(G)
h = np.ones(2 * p + p**2)
h[p:2 * p] *= 0
h[2 * p:] *= 2 * (Sigma).reshape(-1)
h = matrix(h)
dims = {'l': 2 * p, 'q': [], 's': [p]}
solvers.options['show_progress'] = False
sol = conelp(c, G, h, dims)
s = np.array(sol['x']).reshape(-1)
return s
def test_conelp(self):
from cvxopt import matrix, msk, 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]}
self.assertAlmostEqualLists(list(solvers.conelp(c, G, h, dims)['x']),
list(msk.conelp(c, G, h, dims)[1]))
def block_socp(self, a, A1, A2, t, B_s, u_s):
dims = {
'l': 0, # No pure inequality constraints
'q': [4]*len(self.contacts), # Size of the 2nd order cones : x,y,z+1
's': [] # No sd cones
}
#Max a^T z ~ min -a^T z
c = matrix(np.vstack([np.zeros((self.size_x(), 1)), -a]))
A = matrix(np.hstack([A1, A2]))
#B = diag{[u_i b_i.T].T}
blocks = [-np.vstack([u.T, B]) for u, B in zip(u_s, B_s)]
block = block_diag(*blocks)
g = np.hstack([block, np.zeros((self.size_x()*4 // 3, self.size_z()))])
h = np.zeros((self.size_x()*4 // 3, 1))
sol = solvers.conelp(c, G=matrix(g), h=matrix(h),
A=A, b=matrix(t), dims=dims)
return sol
def _intersection_point(T, radii, alpha, values):
lines = list()
for i in range(0, len(T)):
lines.append([(1 - values[i]) * T[i][alpha[i] - 1][k] +
values[i] * T[i][alpha[i]][k]
for k in range(0, T[i].dimensions)])
c = matrix([0.] * T[0].dimensions)
G = []
for i in range(0, T[0].dimensions):
m = len(T) * ([0.] + i * [0.] + [-1] +
(T[0].dimensions - i - 1) * [0.])
G.append(m)
G = matrix(G)
h = []
for i in range(0, len(T)):
h += [radii[i]] + [-lines[i][j] for j in range(0, T[i].dimensions)]
h = matrix(h)
dims = {'l': 0, 'q': [1 + T[0].dimensions] * len(T), 's': [0]}
sol = solvers.conelp(c, G, h, dims)
return sol['status'] == 'optimal', np.array(
[s for s in sol['x']]) if sol['x'] is not None else None
def solve(opts):
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.] )
A = matrix(0.0, (0, c.size[0]))
b = matrix(0.0, (0, 1))
dims = {'l': 2, 'q': [4, 4], 's': [3]}
solvers.options.update(opts)
sol = solvers.conelp(c, G, h, dims, kktsolver='ldl')
print("\nStatus: " + sol['status'])
if sol['status'] == 'optimal':
print "x=\n", helpers.str2(sol['x'])
print "s=\n", helpers.str2(sol['s'])
print "z=\n", helpers.str2(sol['z'])
helpers.run_go_test("../testconelp", {'x': sol['x'], 's': sol['s'], 'z': sol['z']})
def solve_convex(p,mode):
"""
Description
-----------
Solves the convex program p. It assumes p is the convex part
of an expanded and transformed program.
Arguments
---------
p: convex cvxpy_program.
mode: 'rel' (relaxation) or 'scp' (sequential convex programming).
"""
# Select options
if(mode == 'scp'):
options = p.options['SCP_SOL']
else:
options = p.options['REL_SOL']
quiet = p.options['quiet']
# Printing format for cvxopt sparse matrices
opt.spmatrix_str = opt.printing.spmatrix_str_triplet
# Partial minimization expansion
constr_list = pm_expand(p.constr)
# Get variables
variables = constr_list.get_vars()
# Count variables
n = len(variables)
# Create a map (var - pos)
if(options['show steps'] and not quiet):
print '\nCreating variable - index map'
var_to_index = {}
for i in range(0,n,1):
if(options['show steps'] and not quiet):
print variables[i],'<-->',i
var_to_index[variables[i]] = i
# Construct objective vector
c = construct_c(p.obj,var_to_index,n,p.action)
if(options['show steps'] and not quiet):
print '\nConstructing c vector'
print 'c = '
print c
# Construct Ax == b
A,b = construct_Ab(constr_list._get_eq(),var_to_index,n,options)
if(options['show steps'] and not quiet):
print '\nConstructing Ax == b'
print 'A ='
print A
print 'b ='
print b
# Construct Gx <= h
G,h,dim_l,dim_q,dim_s = construct_Gh(constr_list._get_ineq_in(),
var_to_index,n)
if(options['show steps'] and not quiet):
print '\nConstructing Gx <= h'
print 'G ='
print G
print 'h ='
print h
# Construct F
F = construct_F(constr_list._get_ineq_in(),var_to_index,n)
if(options['show steps'] and not quiet):
print '\nConstructing F'
# Call cvxopt
solvers.options['show_progress'] = options['solver progress'] and not quiet
solvers.options['maxiters'] = options['maxiters']
solvers.options['abstol'] = options['abstol']
solvers.options['reltol'] = options['reltol']
solvers.options['feastol'] = options['feastol']
dims = {'l':dim_l, 'q':dim_q, 's':dim_s}
if(F is None):
if(options['show steps'] and not quiet):
print '\nCalling cvxopt conelp solver'
r = solvers.conelp(c,G,h,dims,A,b)
else:
if(options['show steps'] and not quiet):
print '\nCalling cvxopt cpl solver'
r = solvers.cpl(c,F,G,h,dims,A,b)
# Store numerical values
if(r['status'] != 'primal infeasible'):
if(options['show steps'] and not quiet):
print '\nStoring numerical values:'
for v in variables:
value = r['x'][var_to_index[v]]
v.data = value
if(options['show steps']and not quiet):
print v,' <- ', v.data
# Return result
return r
def ubsdp(c, A, B, pstart = None, dstart = None):
"""
minimize c'*x + tr(X)
s.t. sum_{i=1}^n xi * Ai - X <= B
X >= 0
maximize -tr(B * Z0)
s.t. tr(Ai * Z0) + ci = 0, i = 1, ..., n
-Z0 - Z1 + I = 0
Z0 >= 0, Z1 >= 0.
c is an n-vector.
A is an m^2 x n-matrix.
B is an m x m-matrix.
"""
msq, n = A.size
m = int(math.sqrt(msq))
mpckd = int(m * (m+1) / 2)
dims = {'l': 0, 'q': [], 's': [m, m]}
# The primal variable is stored as a tuple (x, X).
cc = (c, matrix(0.0, (m, m)))
cc[1][::m+1] = 1.0
def xnewcopy(u):
return (+u[0], +u[1])
def xdot(u, v):
return blas.dot(u[0], v[0]) + misc.sdot2(u[1], v[1])
def xscal(alpha, u):
blas.scal(alpha, u[0])
blas.scal(alpha, u[1])
def xaxpy(u, v, alpha = 1.0):
blas.axpy(u[0], v[0], alpha)
blas.axpy(u[1], v[1], alpha)
def G(u, v, alpha = 1.0, beta = 0.0, trans = 'N'):
"""
If trans is 'N':
v[:msq] := alpha * (A*u[0] - u[1][:]) + beta * v[:msq]
v[msq:] := -alpha * u[1][:] + beta * v[msq:].
If trans is 'T':
v[0] := alpha * A' * u[:msq] + beta * v[0]
v[1][:] := alpha * (-u[:msq] - u[msq:]) + beta * v[1][:].
"""
if trans == 'N':
blas.gemv(A, u[0], v, alpha = alpha, beta = beta)
blas.axpy(u[1], v, alpha = -alpha)
blas.scal(beta, v, offset = msq)
blas.axpy(u[1], v, alpha = -alpha, offsety = msq)
else:
misc.sgemv(A, u, v[0], dims = {'l': 0, 'q': [], 's': [m]},
alpha = alpha, beta = beta, trans = 'T')
blas.scal(beta, v[1])
blas.axpy(u, v[1], alpha = -alpha, n = msq)
blas.axpy(u, v[1], alpha = -alpha, n = msq, offsetx = msq)
h = matrix(0.0, (2*msq, 1))
blas.copy(B, h)
L = matrix(0.0, (m, m))
U = matrix(0.0, (m, m))
Us = matrix(0.0, (m, m))
Uti = matrix(0.0, (m, m))
s = matrix(0.0, (m, 1))
Asc = matrix(0.0, (msq, n))
S = matrix(0.0, (m, m))
tmp = matrix(0.0, (m, m))
x1 = matrix(0.0, (m**2, 1))
H = matrix(0.0, (n, n))
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
solvers.options['show_progress'] = 0
sol = solvers.conelp(cc, G, h, dims = {'l': 0, 's': [m, m], 'q': []},
kktsolver = F, xnewcopy = xnewcopy, xdot = xdot, xaxpy = xaxpy,
xscal = xscal, primalstart = pstart, dualstart = dstart)
return matrix(sol['x'][1], (n,n))
c = matrix([-2., 1., 5.])
G = [ matrix( [[12., 13., 12.], [6., -3., -12.], [-5., -5., 6.]] ) ]
G += [ matrix( [[3., 3., -1., 1.], [-6., -6., -9., 19.], [10., -2., -2., -3.]] ) ]
h = [ matrix( [-12., -3., -2.] ), matrix( [27., 0., 3., -42.] ) ]
sol = solvers.socp(c, Gq = G, hq = h)
sol['status']
c = matrix([-2., 1., 5.])
G = matrix( [[12., 13., 12., 3., 3., -1., 1.],
[6., -3., -12., -6., -6., -9., 19.],
[-5., -5., 6., 10., -2., -2., -3.]])
h = matrix( [-12., -3., -2., 27., 0., 3., -42.] )
dims = {'l': 0, 'q': [3,4], 's': []}
sol2 = solvers.conelp(c, G, h, dims)
## least sqaures problem, as a QP
A = matrix([ [ .3, -.4, -.2, -.4, 1.3 ],
[ .6, 1.2, -1.7, .3, -.3 ],
[-.3, .0, .6, -1.2, -2.0 ] ])
b = matrix([ 1.5, .0, -1.2, -.7, .0])
m, n = A.size
I = matrix(0.0, (n,n))
I[::n+1] = 1.0
G = matrix([-I, matrix(0.0, (1,n)), I])
h = matrix(n*[0.0] + [1.0] + n*[0.0])
dimsQP = {'l': n, 'q': [n+1], 's': []}
def _updateTrustRegionSOCP(x, fx, oldFx, oldDeltaX, p, radius, g, oldGrad, H, func, grad, z, G, h, y, A, b):
if A is not None:
bTemp = b - A.dot(x)
else:
bTemp = None
GTemp = numpy.append(numpy.zeros((1,p)), numpy.eye(p), axis=0)
hTemp = numpy.zeros(p+1)
hTemp[0] += radius
if G is not None:
GTemp = numpy.append(G, GTemp, axis=0)
hTemp = numpy.append(h - G.dot(x), hTemp)
dims1 = {'l': G.shape[0], 'q': [p+1,p+1], 's': []}
else:
dims1 = {'l': 0, 'q': [p+1,p+1], 's': []}
# now we have finished the setup process, we reformulate
# the problem as a SOCP
m,n = GTemp.shape
c = matrix([1.0] + [0.0]*n)
hTemp1 = matrix([0.0]+(-g.flatten()).tolist())
GTemp1 = matrix(numpy.array(scipy.sparse.bmat([
[[-1.0],None],
[None,H]
]).todense()))
GTemp1 = matrix(numpy.append(numpy.append(numpy.array([0]*m).reshape(m,1),numpy.array(GTemp),axis=1),
numpy.array(GTemp1),
axis=0))
hTemp1 = matrix(numpy.append(hTemp,hTemp1))
if A is not None:
out = solvers.conelp(c, GTemp1, hTemp1, dims1, matrix(A), matrix(bTemp))
else:
out = solvers.conelp(c, GTemp1, hTemp1, dims1)
# exact the descent diretion and do a line search
deltaX = numpy.array(out['x'][1::])
M = _diffM(g.flatten(), H)
newFx = func(x + deltaX)
predRatio = (fx - newFx) / M(deltaX)
if predRatio>=0.75:
radius = min(2.0*radius, maxRadius)
elif predRatio<=0.25:
radius *= 0.25
if predRatio>=0.25:
oldGrad = g.copy()
x += deltaX
oldFx = fx
fx = newFx
update = True
else:
update = False
if G is not None:
# only want the information for the inequalities
# and not the two cones - trust region, objective functionn
z[:] = numpy.array(out['z'])[G.shape[0]]
if A is not None:
y[:] = numpy.array(out['y'])
# print numpy.append(numpy.array(out['s']),numpy.array(s),axis=1)
return x, update, radius, deltaX, z, y, fx, oldFx, oldGrad, out['iterations']
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]
def solve(opts):
c = matrix([-6.0, -4.0, -5.0])
G = matrix(
[
[16.0, 7.0, 24.0, -8.0, 8.0, -1.0, 0.0, -1.0, 0.0, 0.0, 7.0, -5.0, 1.0, -5.0, 1.0, -7.0, 1.0, -7.0, -4.0],
[
-14.0,
2.0,
7.0,
-13.0,
-18.0,
3.0,
0.0,
0.0,
-1.0,
0.0,
3.0,
13.0,
-6.0,
13.0,
12.0,
-10.0,
-6.0,
-10.0,
-28.0,
],
[
5.0,
0.0,
-15.0,
12.0,
-6.0,
17.0,
0.0,
0.0,
0.0,
-1.0,
9.0,
6.0,
-6.0,
6.0,
-7.0,
-7.0,
-6.0,
-7.0,
-11.0,
],
]
)
h = matrix(
[
-3.0,
5.0,
12.0,
-2.0,
-14.0,
-13.0,
10.0,
0.0,
0.0,
0.0,
68.0,
-30.0,
-19.0,
-30.0,
99.0,
23.0,
-19.0,
23.0,
10.0,
]
)
A = matrix(0.0, (0, c.size[0]))
b = matrix(0.0, (0, 1))
dims = {"l": 2, "q": [4, 4], "s": [3]}
# localcones.options.update(opts)
# sol = localcones.conelp(c, G, h, dims, kktsolver='qr')
solvers.options.update(opts)
sol = solvers.conelp(c, G, h, dims)
print ("\nStatus: " + sol["status"])
if sol["status"] == "optimal":
print "x=\n", helpers.strSpe(sol["x"], "%.5f")
print "s=\n", helpers.strSpe(sol["s"], "%.5f")
print "z=\n", helpers.strSpe(sol["z"], "%.5f")
rungotest(sol)
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 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))
def solve_convex(p, mode):
"""
Description
-----------
Solves the convex program p. It assumes p is the convex part
of an expanded and transformed program.
Arguments
---------
p: convex cvxpy_program.
mode: 'rel' (relaxation) or 'scp' (sequential convex programming).
"""
# Select options
if (mode == 'scp'):
options = p.options['SCP_SOL']
else:
options = p.options['REL_SOL']
quiet = p.options['quiet']
# Printing format for cvxopt sparse matrices
opt.spmatrix_str = opt.printing.spmatrix_str_triplet
# Partial minimization expansion
constr_list = pm_expand(p.constr)
# Get variables
variables = constr_list.get_vars()
# Count variables
n = len(variables)
# Create a map (var - pos)
if (options['show steps'] and not quiet):
print '\nCreating variable - index map'
var_to_index = {}
for i in range(0, n, 1):
if (options['show steps'] and not quiet):
print variables[i], '<-->', i
var_to_index[variables[i]] = i
# Construct objective vector
c = construct_c(p.obj, var_to_index, n, p.action)
if (options['show steps'] and not quiet):
print '\nConstructing c vector'
print 'c = '
print c
# Construct Ax == b
A, b = construct_Ab(constr_list._get_eq(), var_to_index, n, options)
if (options['show steps'] and not quiet):
print '\nConstructing Ax == b'
print 'A ='
print A
print 'b ='
print b
# Construct Gx <= h
G, h, dim_l, dim_q, dim_s = construct_Gh(constr_list._get_ineq_in(),
var_to_index, n)
if (options['show steps'] and not quiet):
print '\nConstructing Gx <= h'
print 'G ='
print G
print 'h ='
print h
# Construct F
F = construct_F(constr_list._get_ineq_in(), var_to_index, n)
if (options['show steps'] and not quiet):
print '\nConstructing F'
# Call cvxopt
solvers.options['show_progress'] = options['solver progress'] and not quiet
solvers.options['maxiters'] = options['maxiters']
solvers.options['abstol'] = options['abstol']
solvers.options['reltol'] = options['reltol']
solvers.options['feastol'] = options['feastol']
dims = {'l': dim_l, 'q': dim_q, 's': dim_s}
if (F is None):
if (options['show steps'] and not quiet):
print '\nCalling cvxopt conelp solver'
r = solvers.conelp(c, G, h, dims, A, b)
else:
if (options['show steps'] and not quiet):
print '\nCalling cvxopt cpl solver'
r = solvers.cpl(c, F, G, h, dims, A, b)
# Store numerical values
if (r['status'] != 'primal infeasible'):
if (options['show steps'] and not quiet):
print '\nStoring numerical values:'
for v in variables:
value = r['x'][var_to_index[v]]
v.data = value
if (options['show steps'] and not quiet):
print v, ' <- ', v.data
# Return result
return r
A = sp_rand(n,n,0.015) + spmatrix(1.0,range(n),range(n))
I = cp.tril(A)[:].I
N = len(I)/50 # each data matrix has 1/50 of total nonzeros in pattern
Ig = []; Jg = []
for j in range(m):
Ig += sorted(random.sample(I,N))
Jg += N*[j]
G = spmatrix(normal(len(Ig),1),Ig,Jg,(n**2,m))
h = G*normal(m,1) + spmatrix(1.0,range(n),range(n))[:]
c = normal(m,1)
dims = {'l':0, 'q':[], 's': [n]};
# solve SDP with CVXOPT
print("Solving SDP with CVXOPT..")
prob = (c, G, matrix(h), dims)
sol = solvers.conelp(*prob)
Z1 = matrix(sol['z'], (n,n))
# convert SDP and solve
prob2, blocks_to_sparse, symbs = cp.convert_conelp(*prob)
print("Solving converted SDP (no merging)..")
sol2 = solvers.conelp(*prob2)
# convert block-diagonal solution to spmatrix
blki,I,J,bn = blocks_to_sparse[0]
Z2 = spmatrix(sol2['z'][blki],I,J)
# compute completion
symb = cp.symbolic(Z2, p=cp.maxcardsearch)
Z2c = cp.psdcompletion(cp.cspmatrix(symb)+Z2, reordered=False)
def MOQP(self, expert, features, K, epsilon = None):
if epsilon is None:
epsilon = self.M.epsilon
cexs = features['cexs']
cands = features['cands']
safes = features['safes']
if True:
#G_i_j=[[], [], [], []]
G_i_j_k = []
for e in range(len(expert) + 2):
G_i_j_k.append([])
h_i_j_k = []
c = matrix(- ((K) * np.eye(len(expert) + 2)[-2] + (1 - K) * np.eye(len(expert) + 2)[-1]))
for m in range(len(cands)):
for e in range(len(expert)):
G_i_j_k[e].append(K * (- expert[e] + cands[m][e]))
G_i_j_k[len(expert)].append(1)
G_i_j_k[len(expert) + 1].append(0)
h_i_j_k.append(0)
#G_i_j[0].append(- (expert[0] - cands[m][0]))
#G_i_j[1].append(- (expert[1] - cands[m][1]))
#G_i_j[2].append(- (expert[2] - cands[m][2]))
#G_i_j[3].append(- (expert[3] - cands[m][3]))
for j in range(len(cexs)):
for k in range(len(safes)):
for e in range(len(expert)):
G_i_j_k[e].append((1.0 - K) * (- safes[k][e] + cexs[j][e]))
G_i_j_k[len(expert)].append(0)
G_i_j_k[len(expert) + 1].append(1)
h_i_j_k.append(0)
#G_i_j[0].append(cands[k][0] - mu_Bs[j][0] - (cands[l][0] - mu_Bs[n][0]))
#G_i_j[1].append(cands[k][1] - mu_Bs[j][1] - (cands[l][1] - mu_Bs[n][1]))
#G_i_j[2].append(cands[k][2] - mu_Bs[j][2] - (cands[l][2] - mu_Bs[n][2]))
#G_i_j[3].append(cands[k][3] - mu_Bs[j][3] - (cands[l][3] - mu_Bs[n][3]))
#h_i_j.append(0)
for e in range(len(expert)):
G_i_j_k[e] = G_i_j_k[e] + [0.0] * (e + 1) + [-1.0] + [0.0] * (len(expert) + 2 - e - 1)
G_i_j_k[len(expert)] = G_i_j_k[len(expert)] + [0.0] * (len(expert) + 3)
G_i_j_k[len(expert) + 1] = G_i_j_k[len(expert) + 1] + [0.0] * (len(expert) + 3)
h_i_j_k = h_i_j_k + [10] + (2 + len(expert)) * [0.0]
#G_i_j[0]= G_i_j[0] + [0., -1., 0., 0., 0.]
#G_i_j[1]= G_i_j[1] + [0., 0., -1., 0., 0.]
#G_i_j[2]= G_i_j[2] + [0., 0., 0., -1., 0.]
#G_i_j[3]= G_i_j[3] + [0., 0., 0., 0., -1.]
#h_i_j = h_i_j + [1., 0., 0., 0., 0.]
G = matrix(G_i_j_k)
h = matrix(h_i_j_k)
#G = matrix(G_i_j)
# h = matrix([-1 * penalty, 1., 0., 0., 0.])
#h = matrix(h_i_j)
dims = {'l': len(cands) + len(cexs) * len(safes), 'q': [2 + len(expert), 1], 's': []}
sol = solvers.conelp(c, G, h, dims)
sol['status']
solution = np.array(sol['x'])
if solution is not None:
solution=solution.reshape(len(expert) + 2)
t=(1 - K) * solution[-1] + K * solution[-2]
w=solution[:-2]
else:
solution = None
t = None
w = None
return w, t
def solve_linmap(self, bi, lmbd, linmap, show_progress=True):
solvers.options['show_progress'] = show_progress
# save all problem data
self.bi, self.lmbd = bi, lmbd
# count alive in map
Nm2 = int(sum(linmap))
self.Nm2 = Nm2
self.linmap = linmap
n_x1 = self.Nb*(self.Nb + 1)//2
n_x2 = self.Nb*(self.Nb - 1)//2
n_x = 1 + Nm2 + n_x1 + n_x2
c = matrix(0.0, (1 + Nm2 + n_x1 + n_x2, 1))
c[0] = 1.0
c[1 + Nm2:1 + Nm2 + self.Nb] = lmbd
G = matrix(0.0, (1 + Nm2 + (2*self.Nb)**2, n_x))
G[:1 + Nm2, :1 + Nm2] = -eye(1 + Nm2)
G[1 + Nm2:, 1 + Nm2:] = self.G[1 + self.Nm:, 1 + self.Nm:]
h = matrix(0.0, (1 + Nm2 + (2*self.Nb)**2, 1))
A = matrix(0.0, (Nm2, n_x))
lbi = matrix(0.0, (Nm2, 1))
A[:, 1:1 + Nm2] = eye(Nm2)
pos = 0
for i in range(self.Nm):
if linmap[i] != 0.0:
A[pos, 1 + Nm2:] = self.A[i, 1 + self.Nm:]
lbi[pos] = bi[i]
h[1 + pos] = -bi[i]
pos += 1
assert(pos == Nm2)
b = matrix(0.0, (Nm2, 1))
dims = {'l': 0, 'q': [1 + Nm2], 's': [2*self.Nb]}
t1 = time()
self.sol = solvers.conelp(c, G, h, dims, A, b)
t2 = time()
self.lc = c
self.lG = G
self.lh = h
self.ldims = dims
self.lA = A
self.lb = b
self.lNm = Nm2
self.lbi = lbi
t3 = time()
if self.sol['status'] in ('optimal', 'unknown'):
S = self.sol['s'][1 + Nm2:]
self.sA[:] = S[self.I11] + 1j*S[self.I21]
lapack.heevr(
self.sA, self.sW, jobz='V', range='I', uplo='L',
vl=0.0, vu=0.0, il=self.Nb, iu=self.Nb, Z=self.sZ)
self.beta_hat[:] = math.sqrt(self.sW[0])*self.sZ[:, 0]
else:
raise RuntimeError('numerical problems')
t4 = time()
self.solver_time = t2 - t1
self.eig_time = t4 - t3
def solve_linmap(self, bi, lmbd, linmap, show_progress=True):
solvers.options['show_progress'] = show_progress
# save all problem data
self.bi, self.lmbd = bi, lmbd
# count alive in map
Nm2 = int(sum(linmap))
self.Nm2 = Nm2
self.linmap = linmap
n_x1 = self.Nb * (self.Nb + 1) // 2
n_x2 = self.Nb * (self.Nb - 1) // 2
n_x = 1 + Nm2 + n_x1 + n_x2
c = matrix(0.0, (1 + Nm2 + n_x1 + n_x2, 1))
c[0] = 1.0
c[1 + Nm2:1 + Nm2 + self.Nb] = lmbd
G = matrix(0.0, (1 + Nm2 + (2 * self.Nb)**2, n_x))
G[:1 + Nm2, :1 + Nm2] = -eye(1 + Nm2)
G[1 + Nm2:, 1 + Nm2:] = self.G[1 + self.Nm:, 1 + self.Nm:]
h = matrix(0.0, (1 + Nm2 + (2 * self.Nb)**2, 1))
A = matrix(0.0, (Nm2, n_x))
lbi = matrix(0.0, (Nm2, 1))
A[:, 1:1 + Nm2] = eye(Nm2)
pos = 0
for i in range(self.Nm):
if linmap[i] != 0.0:
A[pos, 1 + Nm2:] = self.A[i, 1 + self.Nm:]
lbi[pos] = bi[i]
h[1 + pos] = -bi[i]
pos += 1
assert (pos == Nm2)
b = matrix(0.0, (Nm2, 1))
dims = {'l': 0, 'q': [1 + Nm2], 's': [2 * self.Nb]}
t1 = time()
self.sol = solvers.conelp(c, G, h, dims, A, b)
t2 = time()
self.lc = c
self.lG = G
self.lh = h
self.ldims = dims
self.lA = A
self.lb = b
self.lNm = Nm2
self.lbi = lbi
t3 = time()
if self.sol['status'] in ('optimal', 'unknown'):
S = self.sol['s'][1 + Nm2:]
self.sA[:] = S[self.I11] + 1j * S[self.I21]
lapack.heevr(self.sA,
self.sW,
jobz='V',
range='I',
uplo='L',
vl=0.0,
vu=0.0,
il=self.Nb,
iu=self.Nb,
Z=self.sZ)
self.beta_hat[:] = math.sqrt(self.sW[0]) * self.sZ[:, 0]
else:
raise RuntimeError('numerical problems')
t4 = time()
self.solver_time = t2 - t1
self.eig_time = t4 - t3
def SDP():
return solvers.conelp(c, G, h, dims)
figure(2)
clf()
spy(matrix(Cun))
title('Sparsity of C')
figure(3)
clf()
spy(matrix(Aun[:, 10], Cun.size))
title('Sparsity of A_10')
print "(Close figure plots to continue)\n"
show()
#Solve unconverted problem using CVXOPT
print "Solve unconverted problem using CVXOPT"
solvers.options['show_progress'] = True
usol = solvers.conelp(-matrix(bun), Aun, matrix(Cun[:]), dims=dims_un)
print "Primal objective: %f" % -usol['primal objective']
print '(Press any key to continue)\n'
raw_input()
#Converted problem
print "Converted problem"
Acon, bcon, Ccon, Lcon, dims_con = convert(Aun,
bun,
Cun,
Spun,
dims_un,
Atype=matrix)
#Plot sparsity pattern for converted pattern.
print "Plot sparsity pattern for converted pattern."
def nrmapp(A, B, C = None, d = None, G = None, h = None):
"""
Solves the regularized nuclear norm approximation problem
minimize || A(x) + B ||_* + 1/2 x'*C*x + d'*x
subject to G*x <= h
and its dual
maximize -h'*z + tr(B'*Z) - 1/2 v'*C*v
subject to d + G'*z + A'(Z) = C*v
z >= 0
|| Z || <= 1.
A(x) is a linear mapping that maps n-vectors x to (p x q)-matrices A(x).
||.||_* is the nuclear norm (sum of singular values).
A'(Z) is the adjoint mapping of A(x).
||.|| is the maximum singular value norm.
INPUT
A real dense or sparse matrix of size (p*q, n). Its columns are
the coefficients A_i of the mapping
A: reals^n --> reals^pxq, A(x) = sum_i=1^n x_i * A_i,
stored in column-major order, as p*q-vectors.
B real dense or sparse matrix of size (p, q), with p >= q.
C real symmetric positive semidefinite dense or sparse matrix of
order n. Only the lower triangular part of C is accessed.
The default value is a zero matrix.
d real dense matrix of size (n, 1). The default value is a zero
vector.
G real dense or sparse matrix of size (m, n), with m >= 0.
The default value is a matrix of size (0, n).
h real dense matrix of size (m, 1). The default value is a
matrix of size (0, 1).
OUTPUT
status 'optimal', 'primal infeasible', or 'unknown'.
x 'd' matrix of size (n, 1) if status is 'optimal';
None otherwise.
z 'd' matrix of size (m, 1) if status is 'optimal' or 'primal
infeasible'; None otherwise.
Z 'd' matrix of size (p, q) if status is 'optimal' or 'primal
infeasible'; None otherwise.
If status is 'optimal', then x, z, Z are approximate solutions of the
optimality conditions
C * x + G' * z + A'(Z) + d = 0
G * x <= h
z >= 0, || Z || < = 1
z' * (h - G*x) = 0
tr (Z' * (A(x) + B)) = || A(x) + B ||_*.
The last (complementary slackness) condition can be replaced by the
following. If the singular value decomposition of A(x) + B is
A(x) + B = [ U1 U2 ] * diag(s, 0) * [ V1 V2 ]',
with s > 0, then
Z = U1 * V1' + U2 * W * V2', || W || <= 1.
If status is 'primal infeasible', then Z = 0 and z is a certificate of
infeasibility for the inequalities G * x <= h, i.e., a vector that
satisfies
h' * z = 1, G' * z = 0, z >= 0.
"""
if type(B) not in (matrix, spmatrix) or B.typecode is not 'd':
raise TypeError, "B must be a real dense or sparse matrix"
p, q = B.size
if p < q:
raise ValueError, "row dimension of B must be greater than or "\
"equal to column dimension"
if type(A) not in (matrix, spmatrix) or A.typecode is not 'd' or \
A.size[0] != p*q:
raise TypeError, "A must be a real dense or sparse matrix with "\
"p*q rows if B has size (p, q)"
n = A.size[1]
if G is None: G = spmatrix([], [], [], (0, n))
if h is None: h = matrix(0.0, (0, 1))
if type(h) is not matrix or h.typecode is not 'd' or h.size[1] != 1:
raise TypeError, "h must be a real dense matrix with one column"
m = h.size[0]
if type(G) not in (matrix, spmatrix) or G.typecode is not 'd' or \
G.size != (m, n):
raise TypeError, "G must be a real dense matrix or sparse matrix "\
"of size (m, n) if h has length m and A has n columns"
if C is None: C = spmatrix(0.0, [], [], (n,n))
if d is None: d = matrix(0.0, (n, 1))
if type(C) not in (matrix, spmatrix) or C.typecode is not 'd' or \
C.size != (n,n):
raise TypeError, "C must be real dense or sparse matrix of size "\
"(n, n) if A has n columns"
if type(d) is not matrix or d.typecode is not 'd' or d.size != (n,1):
raise TypeError, "d must be a real matrix of size (n, 1) if A has "\
"n columns"
# The problem is solved as a cone program
#
# minimize (1/2) * x'*C*x + d'*x + (1/2) * (tr X1 + tr X2)
# subject to G*x <= h
# [ X1 (A(x) + B)' ]
# [ A(x) + B X2 ] >= 0.
#
# The primal variable is stored as a list [ x, X1, X2 ].
def xnewcopy(u):
return [ matrix(u[0]), matrix(u[1]), matrix(u[2]) ]
def xdot(u,v):
return blas.dot(u[0], v[0]) + misc.sdot2(u[1], v[1]) + \
misc.sdot2(u[2], v[2])
def xscal(alpha, u):
blas.scal(alpha, u[0])
blas.scal(alpha, u[1])
blas.scal(alpha, u[2])
def xaxpy(u, v, alpha = 1.0):
blas.axpy(u[0], v[0], alpha)
blas.axpy(u[1], v[1], alpha)
blas.axpy(u[2], v[2], alpha)
def Pf(u, v, alpha = 1.0, beta = 0.0):
base.symv(C, u[0], v[0], alpha = alpha, beta = beta)
blas.scal(beta, v[1])
blas.scal(beta, v[2])
c = [ d, matrix(0.0, (q,q)), matrix(0.0, (p,p)) ]
c[1][::q+1] = 0.5
c[2][::p+1] = 0.5
# If V is a p+q x p+q matrix
#
# [ V11 V12 ]
# V = [ ]
# [ V21 V22 ]
#
# with V11 q x q, V21 p x q, V12 q x p, and V22 p x p, then I11, I21,
# I22 are the index sets defined by
#
# V[I11] = V11[:], V[I21] = V21[:], V[I22] = V22[:].
#
I11 = matrix([ i + j*(p+q) for j in xrange(q) for i in xrange(q) ])
I21 = matrix([ q + i + j*(p+q) for j in xrange(q) for i in xrange(p) ])
I22 = matrix([ (p+q)*q + q + i + j*(p+q) for j in xrange(p) for
i in xrange(p) ])
dims = {'l': m, 'q': [], 's': [p+q]}
hh = matrix(0.0, (m + (p+q)**2, 1))
hh[:m] = h
hh[m + I21] = B[:]
def Gf(u, v, alpha = 1.0, beta = 0.0, trans = 'N'):
if trans == 'N':
# v[:m] := alpha * G * u[0] + beta * v[:m]
base.gemv(G, u[0], v, alpha = alpha, beta = beta)
# v[m:] := alpha * [-u[1], -A(u[0])'; -A(u[0]), -u[2]]
# + beta * v[m:]
blas.scal(beta, v, offset = m)
v[m + I11] -= alpha * u[1][:]
v[m + I21] -= alpha * A * u[0]
v[m + I22] -= alpha * u[2][:]
else:
# v[0] := alpha * ( G.T * u[:m] - 2.0 * A.T * u[m + I21] )
# + beta v[1]
base.gemv(G, u, v[0], trans = 'T', alpha = alpha, beta = beta)
base.gemv(A, u[m + I21], v[0], trans = 'T', alpha = -2.0*alpha,
beta = 1.0)
# v[1] := -alpha * u[m + I11] + beta * v[1]
blas.scal(beta, v[1])
blas.axpy(u[m + I11], v[1], alpha = -alpha)
# v[2] := -alpha * u[m + I22] + beta * v[2]
blas.scal(beta, v[2])
blas.axpy(u[m + I22], v[2], alpha = -alpha)
def Af(u, v, alpha = 1.0, beta = 0.0, trans = 'N'):
if trans == 'N':
pass
else:
blas.scal(beta, v[0])
blas.scal(beta, v[1])
blas.scal(beta, v[2])
L1 = matrix(0.0, (q, q))
L2 = matrix(0.0, (p, p))
T21 = matrix(0.0, (p, q))
s = matrix(0.0, (q, 1))
SS = matrix(0.0, (q, q))
V1 = matrix(0.0, (q, q))
V2 = matrix(0.0, (p, p))
As = matrix(0.0, (p*q, n))
As2 = matrix(0.0, (p*q, n))
tmp = matrix(0.0, (p, q))
a = matrix(0.0, (p+q, p+q))
H = matrix(0.0, (n,n))
Gs = matrix(0.0, (m, n))
Q1 = matrix(0.0, (q, p+q))
Q2 = matrix(0.0, (p, p+q))
tau1 = matrix(0.0, (q,1))
tau2 = matrix(0.0, (p,1))
bz11 = matrix(0.0, (q,q))
bz22 = matrix(0.0, (p,p))
bz21 = matrix(0.0, (p,q))
# Suppose V = [V1; V2] is p x q with V1 q x q. If v = V[:] then
# v[Itriu] are the strict upper triangular entries of V1 stored
# columnwise.
Itriu = [ i + j*p for j in xrange(1,q) for i in xrange(j) ]
# v[Itril] are the strict lower triangular entries of V1 stored rowwise.
Itril = [ j + i*p for j in xrange(1,q) for i in xrange(j) ]
# v[Idiag] are the diagonal entries of V1.
Idiag = [ i*(p+1) for i in xrange(q) ]
# v[Itriu2] are the upper triangular entries of V1, with the diagonal
# entries stored first, followed by the strict upper triangular entries
# stored columnwise.
Itriu2 = Idiag + Itriu
# If V is a q x q matrix and v = V[:], then v[Itril2] are the strict
# lower triangular entries of V stored columnwise and v[Itril3] are
# the strict lower triangular entries stored rowwise.
Itril2 = [ i + j*q for j in xrange(q) for i in xrange(j+1,q) ]
Itril3 = [ i + j*q for i in xrange(q) for j in xrange(i) ]
P = spmatrix(0.0, Itriu, Itril, (p*q, p*q))
D = spmatrix(1.0, range(p*q), range(p*q))
DV = matrix(1.0, (p*q, 1))
def F(W):
"""
Create a solver for the linear equations
C * ux + G' * uzl - 2*A'(uzs21) = bx
-uzs11 = bX1
-uzs22 = bX2
G * ux - Dl^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ uzs11 uzs21' ]
[ ] - r*r' * [ ] * r*r' = bzs
[ -A(ux) -uX2 ] [ uzs21 uzs22 ]
where Dl = diag(W['l']), r = W['r'][0].
On entry, x = (bx, bX1, bX2) and z = [ bzl; bzs[:] ].
On exit, x = (ux, uX1, uX2) and z = [ Dl*uzl; (r'*uzs*r)[:] ].
1. Compute matrices V1, V2 such that (with T = r*r')
[ V1 0 ] [ T11 T21' ] [ V1' 0 ] [ I S' ]
[ ] [ ] [ ] = [ ]
[ 0 V2' ] [ T21 T22 ] [ 0 V2 ] [ S I ]
and S = [ diag(s); 0 ], s a positive q-vector.
2. Factor the mapping X -> X + S * X' * S:
X + S * X' * S = L( L'( X )).
3. Compute scaled mappings: a matrix As with as its columns the
coefficients of the scaled mapping
L^-1( V2' * A() * V1' )
and the matrix Gs = Dl^-1 * G.
4. Cholesky factorization of H = C + Gs'*Gs + 2*As'*As.
"""
# 1. Compute V1, V2, s.
r = W['r'][0]
# LQ factorization R[:q, :] = L1 * Q1.
lapack.lacpy(r, Q1, m = q)
lapack.gelqf(Q1, tau1)
lapack.lacpy(Q1, L1, n = q, uplo = 'L')
lapack.orglq(Q1, tau1)
# LQ factorization R[q:, :] = L2 * Q2.
lapack.lacpy(r, Q2, m = p, offsetA = q)
lapack.gelqf(Q2, tau2)
lapack.lacpy(Q2, L2, n = p, uplo = 'L')
lapack.orglq(Q2, tau2)
# V2, V1, s are computed from an SVD: if
#
# Q2 * Q1' = U * diag(s) * V',
#
# then V1 = V' * L1^-1 and V2 = L2^-T * U.
# T21 = Q2 * Q1.T
blas.gemm(Q2, Q1, T21, transB = 'T')
# SVD T21 = U * diag(s) * V'. Store U in V2 and V' in V1.
lapack.gesvd(T21, s, jobu = 'A', jobvt = 'A', U = V2, Vt = V1)
# # Q2 := Q2 * Q1' without extracting Q1; store T21 in Q2
# this will requires lapack.ormlq or lapack.unmlq
# V2 = L2^-T * U
blas.trsm(L2, V2, transA = 'T')
# V1 = V' * L1^-1
blas.trsm(L1, V1, side = 'R')
# 2. Factorization X + S * X' * S = L( L'( X )).
#
# The factor L is stored as a diagonal matrix D and a sparse lower
# triangular matrix P, such that
#
# L(X)[:] = D**-1 * (I + P) * X[:]
# L^-1(X)[:] = D * (I - P) * X[:].
# SS is q x q with SS[i,j] = si*sj.
blas.scal(0.0, SS)
blas.syr(s, SS)
# For a p x q matrix X, P*X[:] is Y[:] where
#
# Yij = si * sj * Xji if i < j
# = 0 otherwise.
#
P.V = SS[Itril2]
# For a p x q matrix X, D*X[:] is Y[:] where
#
# Yij = Xij / sqrt( 1 - si^2 * sj^2 ) if i < j
# = Xii / sqrt( 1 + si^2 ) if i = j
# = Xij otherwise.
#
DV[Idiag] = sqrt(1.0 + SS[::q+1])
DV[Itriu] = sqrt(1.0 - SS[Itril3]**2)
D.V = DV**-1
# 3. Scaled linear mappings
# Ask := V2' * Ask * V1'
blas.scal(0.0, As)
base.axpy(A, As)
for i in xrange(n):
# tmp := V2' * As[i, :]
blas.gemm(V2, As, tmp, transA = 'T', m = p, n = q, k = p,
ldB = p, offsetB = i*p*q)
# As[:,i] := tmp * V1'
blas.gemm(tmp, V1, As, transB = 'T', m = p, n = q, k = q,
ldC = p, offsetC = i*p*q)
# As := D * (I - P) * As
# = L^-1 * As.
blas.copy(As, As2)
base.gemm(P, As, As2, alpha = -1.0, beta = 1.0)
base.gemm(D, As2, As)
# Gs := Dl^-1 * G
blas.scal(0.0, Gs)
base.axpy(G, Gs)
for k in xrange(n):
blas.tbmv(W['di'], Gs, n = m, k = 0, ldA = 1, offsetx = k*m)
# 4. Cholesky factorization of H = C + Gs' * Gs + 2 * As' * As.
blas.syrk(As, H, trans = 'T', alpha = 2.0)
blas.syrk(Gs, H, trans = 'T', beta = 1.0)
base.axpy(C, H)
lapack.potrf(H)
def f(x, y, z):
"""
Solve
C * ux + G' * uzl - 2*A'(uzs21) = bx
-uzs11 = bX1
-uzs22 = bX2
G * ux - D^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ uzs11 uzs21' ]
[ ] - T * [ ] * T = bzs.
[ -A(ux) -uX2 ] [ uzs21 uzs22 ]
On entry, x = (bx, bX1, bX2) and z = [ bzl; bzs[:] ].
On exit, x = (ux, uX1, uX2) and z = [ D*uzl; (r'*uzs*r)[:] ].
Define X = uzs21, Z = T * uzs * T:
C * ux + G' * uzl - 2*A'(X) = bx
[ 0 X' ] [ bX1 0 ]
T * [ ] * T - Z = T * [ ] * T
[ X 0 ] [ 0 bX2 ]
G * ux - D^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ Z11 Z21' ]
[ ] - [ ] = bzs
[ -A(ux) -uX2 ] [ Z21 Z22 ]
Return x = (ux, uX1, uX2), z = [ D*uzl; (rti'*Z*rti)[:] ].
We use the congruence transformation
[ V1 0 ] [ T11 T21' ] [ V1' 0 ] [ I S' ]
[ ] [ ] [ ] = [ ]
[ 0 V2' ] [ T21 T22 ] [ 0 V2 ] [ S I ]
and the factorization
X + S * X' * S = L( L'(X) )
to write this as
C * ux + G' * uzl - 2*A'(X) = bx
L'(V2^-1 * X * V1^-1) - L^-1(V2' * Z21 * V1') = bX
G * ux - D^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ Z11 Z21' ]
[ ] - [ ] = bzs,
[ -A(ux) -uX2 ] [ Z21 Z22 ]
or
C * ux + Gs' * uuzl - 2*As'(XX) = bx
XX - ZZ21 = bX
Gs * ux - uuzl = D^-1 * bzl
-As(ux) - ZZ21 = bbzs_21
-uX1 - Z11 = bzs_11
-uX2 - Z22 = bzs_22
if we introduce scaled variables
uuzl = D * uzl
XX = L'(V2^-1 * X * V1^-1)
= L'(V2^-1 * uzs21 * V1^-1)
ZZ21 = L^-1(V2' * Z21 * V1')
and define
bbzs_21 = L^-1(V2' * bzs_21 * V1')
[ bX1 0 ]
bX = L^-1( V2' * (T * [ ] * T)_21 * V1').
[ 0 bX2 ]
Eliminating Z21 gives
C * ux + Gs' * uuzl - 2*As'(XX) = bx
Gs * ux - uuzl = D^-1 * bzl
-As(ux) - XX = bbzs_21 - bX
-uX1 - Z11 = bzs_11
-uX2 - Z22 = bzs_22
and eliminating uuzl and XX gives
H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bX - bbzs_21)
Gs * ux - uuzl = D^-1 * bzl
-As(ux) - XX = bbzs_21 - bX
-uX1 - Z11 = bzs_11
-uX2 - Z22 = bzs_22.
In summary, we can use the following algorithm:
1. bXX := bX - bbzs21
[ bX1 0 ]
= L^-1( V2' * ((T * [ ] * T)_21 - bzs_21) * V1')
[ 0 bX2 ]
2. Solve H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bXX).
3. From ux, compute
uuzl = Gs*ux - D^-1 * bzl and
X = V2 * L^-T(-As(ux) + bXX) * V1.
4. Return ux, uuzl,
rti' * Z * rti = r' * [ -bX1, X'; X, -bX2 ] * r
and uX1 = -Z11 - bzs_11, uX2 = -Z22 - bzs_22.
"""
# Save bzs_11, bzs_22, bzs_21.
lapack.lacpy(z, bz11, uplo = 'L', m = q, n = q, ldA = p+q,
offsetA = m)
lapack.lacpy(z, bz21, m = p, n = q, ldA = p+q, offsetA = m+q)
lapack.lacpy(z, bz22, uplo = 'L', m = p, n = p, ldA = p+q,
offsetA = m + (p+q+1)*q)
# zl := D^-1 * zl
# = D^-1 * bzl
blas.tbmv(W['di'], z, n = m, k = 0, ldA = 1)
# zs := r' * [ bX1, 0; 0, bX2 ] * r.
# zs := [ bX1, 0; 0, bX2 ]
blas.scal(0.0, z, offset = m)
lapack.lacpy(x[1], z, uplo = 'L', m = q, n = q, ldB = p+q,
offsetB = m)
lapack.lacpy(x[2], z, uplo = 'L', m = p, n = p, ldB = p+q,
offsetB = m + (p+q+1)*q)
# scale diagonal of zs by 1/2
blas.scal(0.5, z, inc = p+q+1, offset = m)
# a := tril(zs)*r
blas.copy(r, a)
blas.trmm(z, a, side = 'L', m = p+q, n = p+q, ldA = p+q, ldB =
p+q, offsetA = m)
# zs := a'*r + r'*a
blas.syr2k(r, a, z, trans = 'T', n = p+q, k = p+q, ldB = p+q,
ldC = p+q, offsetC = m)
# bz21 := L^-1( V2' * ((r * zs * r')_21 - bz21) * V1')
#
# [ bX1 0 ]
# = L^-1( V2' * ((T * [ ] * T)_21 - bz21) * V1').
# [ 0 bX2 ]
# a = [ r21 r22 ] * z
# = [ r21 r22 ] * r' * [ bX1, 0; 0, bX2 ] * r
# = [ T21 T22 ] * [ bX1, 0; 0, bX2 ] * r
blas.symm(z, r, a, side = 'R', m = p, n = p+q, ldA = p+q,
ldC = p+q, offsetB = q)
# bz21 := -bz21 + a * [ r11, r12 ]'
# = -bz21 + (T * [ bX1, 0; 0, bX2 ] * T)_21
blas.gemm(a, r, bz21, transB = 'T', m = p, n = q, k = p+q,
beta = -1.0, ldA = p+q, ldC = p)
# bz21 := V2' * bz21 * V1'
# = V2' * (-bz21 + (T*[bX1, 0; 0, bX2]*T)_21) * V1'
blas.gemm(V2, bz21, tmp, transA = 'T', m = p, n = q, k = p,
ldB = p)
blas.gemm(tmp, V1, bz21, transB = 'T', m = p, n = q, k = q,
ldC = p)
# bz21[:] := D * (I-P) * bz21[:]
# = L^-1 * bz21[:]
# = bXX[:]
blas.copy(bz21, tmp)
base.gemv(P, bz21, tmp, alpha = -1.0, beta = 1.0)
base.gemv(D, tmp, bz21)
# Solve H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bXX).
# x[0] := x[0] + Gs'*zl + 2*As'(bz21)
# = bx + G' * D^-1 * bzl + 2 * As'(bXX)
blas.gemv(Gs, z, x[0], trans = 'T', alpha = 1.0, beta = 1.0)
blas.gemv(As, bz21, x[0], trans = 'T', alpha = 2.0, beta = 1.0)
# x[0] := H \ x[0]
# = ux
lapack.potrs(H, x[0])
# uuzl = Gs*ux - D^-1 * bzl
blas.gemv(Gs, x[0], z, alpha = 1.0, beta = -1.0)
# bz21 := V2 * L^-T(-As(ux) + bz21) * V1
# = X
blas.gemv(As, x[0], bz21, alpha = -1.0, beta = 1.0)
blas.tbsv(DV, bz21, n = p*q, k = 0, ldA = 1)
blas.copy(bz21, tmp)
base.gemv(P, tmp, bz21, alpha = -1.0, beta = 1.0, trans = 'T')
blas.gemm(V2, bz21, tmp)
blas.gemm(tmp, V1, bz21)
# zs := -zs + r' * [ 0, X'; X, 0 ] * r
# = r' * [ -bX1, X'; X, -bX2 ] * r.
# a := bz21 * [ r11, r12 ]
# = X * [ r11, r12 ]
blas.gemm(bz21, r, a, m = p, n = p+q, k = q, ldA = p, ldC = p+q)
# z := -z + [ r21, r22 ]' * a + a' * [ r21, r22 ]
# = rti' * uzs * rti
blas.syr2k(r, a, z, trans = 'T', beta = -1.0, n = p+q, k = p,
offsetA = q, offsetC = m, ldB = p+q, ldC = p+q)
# uX1 = -Z11 - bzs_11
# = -(r*zs*r')_11 - bzs_11
# uX2 = -Z22 - bzs_22
# = -(r*zs*r')_22 - bzs_22
blas.copy(bz11, x[1])
blas.copy(bz22, x[2])
# scale diagonal of zs by 1/2
blas.scal(0.5, z, inc = p+q+1, offset = m)
# a := r*tril(zs)
blas.copy(r, a)
blas.trmm(z, a, side = 'R', m = p+q, n = p+q, ldA = p+q, ldB =
p+q, offsetA = m)
# x[1] := -x[1] - a[:q,:] * r[:q, :]' - r[:q,:] * a[:q,:]'
# = -bzs_11 - (r*zs*r')_11
blas.syr2k(a, r, x[1], n = q, alpha = -1.0, beta = -1.0)
# x[2] := -x[2] - a[q:,:] * r[q:, :]' - r[q:,:] * a[q:,:]'
# = -bzs_22 - (r*zs*r')_22
blas.syr2k(a, r, x[2], n = p, alpha = -1.0, beta = -1.0,
offsetA = q, offsetB = q)
# scale diagonal of zs by 1/2
blas.scal(2.0, z, inc = p+q+1, offset = m)
return f
if C:
sol = solvers.coneqp(Pf, c, Gf, hh, dims, Af, kktsolver = F,
xnewcopy = xnewcopy, xdot = xdot, xaxpy = xaxpy, xscal = xscal)
else:
sol = solvers.conelp(c, Gf, hh, dims, Af, kktsolver = F,
xnewcopy = xnewcopy, xdot = xdot, xaxpy = xaxpy, xscal = xscal)
if sol['status'] is 'optimal':
x = sol['x'][0]
z = sol['z'][:m]
Z = sol['z'][m:]
Z.size = (p + q, p + q)
Z = -2.0 * Z[-p:, :q]
elif sol['status'] is 'primal infeasible':
x = None
z = sol['z'][:m]
Z = sol['z'][m:]
Z.size = (p + q, p + q)
Z = -2.0 * Z[-p:, :q]
else:
x, z, Z = None, None, None
return {'status': sol['status'], 'x': x, 'z': z, 'Z': Z }
def A_optimize_fast(sij,
N=1.,
nsofar=None,
delta=None,
only_include_measurements=None,
maxiters=100,
feastol=1e-6):
'''
Find the A-optimal of the difference network that minimizes the trace of
the covariance matrix. This corresponds to minimizing the average error.
In an iterative optimization of the difference network, the
optimal allocation is updated with the estimate of s_{ij}, and we
need to allocate the next iteration of sampling based on what has
already been sampled for each pair.
This implementation uses a customized KKT solver. The time complexity is
O(K^5), memory complexity is O(K^4).
Args:
sij: KxK symmetric matrix, where the measurement variance of the
difference between i and j is proportional to s[i][j]^2 =
s[j][i]^2, and the measurement variance of i is proportional to
s[i][i]^2.
nadd: float, Nadd gives the additional number of samples to be collected in
the next iteration.
nsofar: KxK symmetric matrix, where nsofar[i,j] is the number of samples
that has already been collected for (i,j) pair.
delta: a length K vector. delta[i] is the measurement uncertainty on the
quantity x[i] from an independent experiment; if no independent experiment
provides a value for x[i], delta[i] is set to numpy.infty.
only_include_measurements: set of pairs, if not None, indicate which
pairs should be considered in the optimal network. Any pair (i,j) not in
the set will be excluded in the allocation (i.e. dn[i,j] = 0). The pair
(i,j) in the set must be ordered so that i<=j.
Return:
KxK symmetric matrix of float, the (i,j) element of which gives the
number of samples to be allocated to the measurement of (i,j) difference
in the next iteration.
'''
si2 = cvxopt.div(1., sij**2)
K = si2.size[0]
if delta is not None:
di2 = np.array([
1. / delta[i]**2 if delta[i] is not None else 0. for i in xrange(K)
])
else:
di2 = None
if only_include_measurements is not None:
for i in xrange(K):
for j in xrange(i, K):
if not (i, j) in only_include_measurements:
# Set the s[i,j] to infinity, thus excluding the pair.
si2[i, j] = si2[j, i] = 0.
Gm, hv, Am = Aopt_GhA(si2, nsofar, di2=di2, G_as_function=True)
dims = dict(l=K * (K + 1) / 2, q=[], s=[K + 1] * K)
cv = matrix(np.concatenate([np.zeros(K * (K + 1) / 2),
np.ones(K)]), (K * (K + 1) / 2 + K, 1))
bv = matrix(float(N), (1, 1))
def default_kkt_solver(W):
return misc.kkt_ldl(Gm, dims, Am)(W)
sol = solvers.conelp(cv,
Gm,
hv,
dims,
Am,
bv,
options=dict(maxiters=maxiters, feastol=feastol),
kktsolver=lambda W: Aopt_KKT_solver(si2, W))
return conelp_solution_to_nij(sol['x'], K)
c = matrix([1.0] + [0.0]*n)
hTemp1 = matrix([0.0]+(-g).tolist())
GTemp1 = matrix(numpy.array(scipy.sparse.bmat([
[[-1.0],None],
[None,H]
]).todense()))
GTemp1 = matrix(numpy.append(numpy.append(numpy.array([0]*m).reshape(m,1),numpy.array(GTemp),axis=1),
numpy.array(GTemp1),
axis=0))
hTemp1 = matrix(numpy.append(hTemp,hTemp1))
dims1 = {'l': G.shape[0], 'q': [n+1,n+1], 's': []}
solSOCP = solvers.conelp(c, GTemp1, hTemp1, dims1)
print solSOCP['x'][1::]
## testing the change in objective function
print F(matrix(theta))[0]
print F(matrix(theta) + solSOCP['x'][1::])[0]
print F(matrix(theta) + sol1['x'])[0]
blas.nrm2(solSOCP['x'][1::])
blas.nrm2(sol1['x'])
blas.nrm2(qpOut['x'])
blas.nrm2(hTemp1[-n::] - matrix(GTemp)[-n::,:] * solSOCP['x'][1::] )
# The small linear cone program of section 8.1 (Linear cone programs).
from cvxopt 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 ubsdp(c, A, B, pstart=None, dstart=None):
"""
minimize c'*x + tr(X)
s.t. sum_{i=1}^n xi * Ai - X <= B
X >= 0
maximize -tr(B * Z0)
s.t. tr(Ai * Z0) + ci = 0, i = 1, ..., n
-Z0 - Z1 + I = 0
Z0 >= 0, Z1 >= 0.
c is an n-vector.
A is an m^2 x n-matrix.
B is an m x m-matrix.
"""
msq, n = A.size
m = int(math.sqrt(msq))
mpckd = int(m * (m + 1) / 2)
dims = {'l': 0, 'q': [], 's': [m, m]}
# The primal variable is stored as a tuple (x, X).
cc = (c, matrix(0.0, (m, m)))
cc[1][::m + 1] = 1.0
def xnewcopy(u):
return (+u[0], +u[1])
def xdot(u, v):
return blas.dot(u[0], v[0]) + misc.sdot2(u[1], v[1])
def xscal(alpha, u):
blas.scal(alpha, u[0])
blas.scal(alpha, u[1])
def xaxpy(u, v, alpha=1.0):
blas.axpy(u[0], v[0], alpha)
blas.axpy(u[1], v[1], alpha)
def G(u, v, alpha=1.0, beta=0.0, trans='N'):
"""
If trans is 'N':
v[:msq] := alpha * (A*u[0] - u[1][:]) + beta * v[:msq]
v[msq:] := -alpha * u[1][:] + beta * v[msq:].
If trans is 'T':
v[0] := alpha * A' * u[:msq] + beta * v[0]
v[1][:] := alpha * (-u[:msq] - u[msq:]) + beta * v[1][:].
"""
if trans == 'N':
blas.gemv(A, u[0], v, alpha=alpha, beta=beta)
blas.axpy(u[1], v, alpha=-alpha)
blas.scal(beta, v, offset=msq)
blas.axpy(u[1], v, alpha=-alpha, offsety=msq)
else:
misc.sgemv(A,
u,
v[0],
dims={
'l': 0,
'q': [],
's': [m]
},
alpha=alpha,
beta=beta,
trans='T')
blas.scal(beta, v[1])
blas.axpy(u, v[1], alpha=-alpha, n=msq)
blas.axpy(u, v[1], alpha=-alpha, n=msq, offsetx=msq)
h = matrix(0.0, (2 * msq, 1))
blas.copy(B, h)
L = matrix(0.0, (m, m))
U = matrix(0.0, (m, m))
Us = matrix(0.0, (m, m))
Uti = matrix(0.0, (m, m))
s = matrix(0.0, (m, 1))
Asc = matrix(0.0, (msq, n))
S = matrix(0.0, (m, m))
tmp = matrix(0.0, (m, m))
x1 = matrix(0.0, (m**2, 1))
H = matrix(0.0, (n, n))
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
sol = solvers.conelp(cc,
G,
h,
dims={
'l': 0,
's': [m, m],
'q': []
},
kktsolver=F,
xnewcopy=xnewcopy,
xdot=xdot,
xaxpy=xaxpy,
xscal=xscal,
primalstart=pstart,
dualstart=dstart)
return matrix(sol['x'][1], (n, n))
def nrmapp(A, B, C=None, d=None, G=None, h=None):
"""
Solves the regularized nuclear norm approximation problem
minimize || A(x) + B ||_* + 1/2 x'*C*x + d'*x
subject to G*x <= h
and its dual
maximize -h'*z + tr(B'*Z) - 1/2 v'*C*v
subject to d + G'*z + A'(Z) = C*v
z >= 0
|| Z || <= 1.
A(x) is a linear mapping that maps n-vectors x to (p x q)-matrices A(x).
||.||_* is the nuclear norm (sum of singular values).
A'(Z) is the adjoint mapping of A(x).
||.|| is the maximum singular value norm.
INPUT
A real dense or sparse matrix of size (p*q, n). Its columns are
the coefficients A_i of the mapping
A: reals^n --> reals^pxq, A(x) = sum_i=1^n x_i * A_i,
stored in column-major order, as p*q-vectors.
B real dense or sparse matrix of size (p, q), with p >= q.
C real symmetric positive semidefinite dense or sparse matrix of
order n. Only the lower triangular part of C is accessed.
The default value is a zero matrix.
d real dense matrix of size (n, 1). The default value is a zero
vector.
G real dense or sparse matrix of size (m, n), with m >= 0.
The default value is a matrix of size (0, n).
h real dense matrix of size (m, 1). The default value is a
matrix of size (0, 1).
OUTPUT
status 'optimal', 'primal infeasible', or 'unknown'.
x 'd' matrix of size (n, 1) if status is 'optimal';
None otherwise.
z 'd' matrix of size (m, 1) if status is 'optimal' or 'primal
infeasible'; None otherwise.
Z 'd' matrix of size (p, q) if status is 'optimal' or 'primal
infeasible'; None otherwise.
If status is 'optimal', then x, z, Z are approximate solutions of the
optimality conditions
C * x + G' * z + A'(Z) + d = 0
G * x <= h
z >= 0, || Z || < = 1
z' * (h - G*x) = 0
tr (Z' * (A(x) + B)) = || A(x) + B ||_*.
The last (complementary slackness) condition can be replaced by the
following. If the singular value decomposition of A(x) + B is
A(x) + B = [ U1 U2 ] * diag(s, 0) * [ V1 V2 ]',
with s > 0, then
Z = U1 * V1' + U2 * W * V2', || W || <= 1.
If status is 'primal infeasible', then Z = 0 and z is a certificate of
infeasibility for the inequalities G * x <= h, i.e., a vector that
satisfies
h' * z = 1, G' * z = 0, z >= 0.
"""
if type(B) not in (matrix, spmatrix) or B.typecode is not 'd':
raise TypeError, "B must be a real dense or sparse matrix"
p, q = B.size
if p < q:
raise ValueError, "row dimension of B must be greater than or "\
"equal to column dimension"
if type(A) not in (matrix, spmatrix) or A.typecode is not 'd' or \
A.size[0] != p*q:
raise TypeError, "A must be a real dense or sparse matrix with "\
"p*q rows if B has size (p, q)"
n = A.size[1]
if G is None: G = spmatrix([], [], [], (0, n))
if h is None: h = matrix(0.0, (0, 1))
if type(h) is not matrix or h.typecode is not 'd' or h.size[1] != 1:
raise TypeError, "h must be a real dense matrix with one column"
m = h.size[0]
if type(G) not in (matrix, spmatrix) or G.typecode is not 'd' or \
G.size != (m, n):
raise TypeError, "G must be a real dense matrix or sparse matrix "\
"of size (m, n) if h has length m and A has n columns"
if C is None: C = spmatrix(0.0, [], [], (n, n))
if d is None: d = matrix(0.0, (n, 1))
if type(C) not in (matrix, spmatrix) or C.typecode is not 'd' or \
C.size != (n,n):
raise TypeError, "C must be real dense or sparse matrix of size "\
"(n, n) if A has n columns"
if type(d) is not matrix or d.typecode is not 'd' or d.size != (n, 1):
raise TypeError, "d must be a real matrix of size (n, 1) if A has "\
"n columns"
# The problem is solved as a cone program
#
# minimize (1/2) * x'*C*x + d'*x + (1/2) * (tr X1 + tr X2)
# subject to G*x <= h
# [ X1 (A(x) + B)' ]
# [ A(x) + B X2 ] >= 0.
#
# The primal variable is stored as a list [ x, X1, X2 ].
def xnewcopy(u):
return [matrix(u[0]), matrix(u[1]), matrix(u[2])]
def xdot(u, v):
return blas.dot(u[0], v[0]) + misc.sdot2(u[1], v[1]) + \
misc.sdot2(u[2], v[2])
def xscal(alpha, u):
blas.scal(alpha, u[0])
blas.scal(alpha, u[1])
blas.scal(alpha, u[2])
def xaxpy(u, v, alpha=1.0):
blas.axpy(u[0], v[0], alpha)
blas.axpy(u[1], v[1], alpha)
blas.axpy(u[2], v[2], alpha)
def Pf(u, v, alpha=1.0, beta=0.0):
base.symv(C, u[0], v[0], alpha=alpha, beta=beta)
blas.scal(beta, v[1])
blas.scal(beta, v[2])
c = [d, matrix(0.0, (q, q)), matrix(0.0, (p, p))]
c[1][::q + 1] = 0.5
c[2][::p + 1] = 0.5
# If V is a p+q x p+q matrix
#
# [ V11 V12 ]
# V = [ ]
# [ V21 V22 ]
#
# with V11 q x q, V21 p x q, V12 q x p, and V22 p x p, then I11, I21,
# I22 are the index sets defined by
#
# V[I11] = V11[:], V[I21] = V21[:], V[I22] = V22[:].
#
I11 = matrix([i + j * (p + q) for j in xrange(q) for i in xrange(q)])
I21 = matrix([q + i + j * (p + q) for j in xrange(q) for i in xrange(p)])
I22 = matrix([(p + q) * q + q + i + j * (p + q) for j in xrange(p)
for i in xrange(p)])
dims = {'l': m, 'q': [], 's': [p + q]}
hh = matrix(0.0, (m + (p + q)**2, 1))
hh[:m] = h
hh[m + I21] = B[:]
def Gf(u, v, alpha=1.0, beta=0.0, trans='N'):
if trans == 'N':
# v[:m] := alpha * G * u[0] + beta * v[:m]
base.gemv(G, u[0], v, alpha=alpha, beta=beta)
# v[m:] := alpha * [-u[1], -A(u[0])'; -A(u[0]), -u[2]]
# + beta * v[m:]
blas.scal(beta, v, offset=m)
v[m + I11] -= alpha * u[1][:]
v[m + I21] -= alpha * A * u[0]
v[m + I22] -= alpha * u[2][:]
else:
# v[0] := alpha * ( G.T * u[:m] - 2.0 * A.T * u[m + I21] )
# + beta v[1]
base.gemv(G, u, v[0], trans='T', alpha=alpha, beta=beta)
base.gemv(A,
u[m + I21],
v[0],
trans='T',
alpha=-2.0 * alpha,
beta=1.0)
# v[1] := -alpha * u[m + I11] + beta * v[1]
blas.scal(beta, v[1])
blas.axpy(u[m + I11], v[1], alpha=-alpha)
# v[2] := -alpha * u[m + I22] + beta * v[2]
blas.scal(beta, v[2])
blas.axpy(u[m + I22], v[2], alpha=-alpha)
def Af(u, v, alpha=1.0, beta=0.0, trans='N'):
if trans == 'N':
pass
else:
blas.scal(beta, v[0])
blas.scal(beta, v[1])
blas.scal(beta, v[2])
L1 = matrix(0.0, (q, q))
L2 = matrix(0.0, (p, p))
T21 = matrix(0.0, (p, q))
s = matrix(0.0, (q, 1))
SS = matrix(0.0, (q, q))
V1 = matrix(0.0, (q, q))
V2 = matrix(0.0, (p, p))
As = matrix(0.0, (p * q, n))
As2 = matrix(0.0, (p * q, n))
tmp = matrix(0.0, (p, q))
a = matrix(0.0, (p + q, p + q))
H = matrix(0.0, (n, n))
Gs = matrix(0.0, (m, n))
Q1 = matrix(0.0, (q, p + q))
Q2 = matrix(0.0, (p, p + q))
tau1 = matrix(0.0, (q, 1))
tau2 = matrix(0.0, (p, 1))
bz11 = matrix(0.0, (q, q))
bz22 = matrix(0.0, (p, p))
bz21 = matrix(0.0, (p, q))
# Suppose V = [V1; V2] is p x q with V1 q x q. If v = V[:] then
# v[Itriu] are the strict upper triangular entries of V1 stored
# columnwise.
Itriu = [i + j * p for j in xrange(1, q) for i in xrange(j)]
# v[Itril] are the strict lower triangular entries of V1 stored rowwise.
Itril = [j + i * p for j in xrange(1, q) for i in xrange(j)]
# v[Idiag] are the diagonal entries of V1.
Idiag = [i * (p + 1) for i in xrange(q)]
# v[Itriu2] are the upper triangular entries of V1, with the diagonal
# entries stored first, followed by the strict upper triangular entries
# stored columnwise.
Itriu2 = Idiag + Itriu
# If V is a q x q matrix and v = V[:], then v[Itril2] are the strict
# lower triangular entries of V stored columnwise and v[Itril3] are
# the strict lower triangular entries stored rowwise.
Itril2 = [i + j * q for j in xrange(q) for i in xrange(j + 1, q)]
Itril3 = [i + j * q for i in xrange(q) for j in xrange(i)]
P = spmatrix(0.0, Itriu, Itril, (p * q, p * q))
D = spmatrix(1.0, range(p * q), range(p * q))
DV = matrix(1.0, (p * q, 1))
def F(W):
"""
Create a solver for the linear equations
C * ux + G' * uzl - 2*A'(uzs21) = bx
-uzs11 = bX1
-uzs22 = bX2
G * ux - Dl^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ uzs11 uzs21' ]
[ ] - r*r' * [ ] * r*r' = bzs
[ -A(ux) -uX2 ] [ uzs21 uzs22 ]
where Dl = diag(W['l']), r = W['r'][0].
On entry, x = (bx, bX1, bX2) and z = [ bzl; bzs[:] ].
On exit, x = (ux, uX1, uX2) and z = [ Dl*uzl; (r'*uzs*r)[:] ].
1. Compute matrices V1, V2 such that (with T = r*r')
[ V1 0 ] [ T11 T21' ] [ V1' 0 ] [ I S' ]
[ ] [ ] [ ] = [ ]
[ 0 V2' ] [ T21 T22 ] [ 0 V2 ] [ S I ]
and S = [ diag(s); 0 ], s a positive q-vector.
2. Factor the mapping X -> X + S * X' * S:
X + S * X' * S = L( L'( X )).
3. Compute scaled mappings: a matrix As with as its columns the
coefficients of the scaled mapping
L^-1( V2' * A() * V1' )
and the matrix Gs = Dl^-1 * G.
4. Cholesky factorization of H = C + Gs'*Gs + 2*As'*As.
"""
# 1. Compute V1, V2, s.
r = W['r'][0]
# LQ factorization R[:q, :] = L1 * Q1.
lapack.lacpy(r, Q1, m=q)
lapack.gelqf(Q1, tau1)
lapack.lacpy(Q1, L1, n=q, uplo='L')
lapack.orglq(Q1, tau1)
# LQ factorization R[q:, :] = L2 * Q2.
lapack.lacpy(r, Q2, m=p, offsetA=q)
lapack.gelqf(Q2, tau2)
lapack.lacpy(Q2, L2, n=p, uplo='L')
lapack.orglq(Q2, tau2)
# V2, V1, s are computed from an SVD: if
#
# Q2 * Q1' = U * diag(s) * V',
#
# then V1 = V' * L1^-1 and V2 = L2^-T * U.
# T21 = Q2 * Q1.T
blas.gemm(Q2, Q1, T21, transB='T')
# SVD T21 = U * diag(s) * V'. Store U in V2 and V' in V1.
lapack.gesvd(T21, s, jobu='A', jobvt='A', U=V2, Vt=V1)
# # Q2 := Q2 * Q1' without extracting Q1; store T21 in Q2
# this will requires lapack.ormlq or lapack.unmlq
# V2 = L2^-T * U
blas.trsm(L2, V2, transA='T')
# V1 = V' * L1^-1
blas.trsm(L1, V1, side='R')
# 2. Factorization X + S * X' * S = L( L'( X )).
#
# The factor L is stored as a diagonal matrix D and a sparse lower
# triangular matrix P, such that
#
# L(X)[:] = D**-1 * (I + P) * X[:]
# L^-1(X)[:] = D * (I - P) * X[:].
# SS is q x q with SS[i,j] = si*sj.
blas.scal(0.0, SS)
blas.syr(s, SS)
# For a p x q matrix X, P*X[:] is Y[:] where
#
# Yij = si * sj * Xji if i < j
# = 0 otherwise.
#
P.V = SS[Itril2]
# For a p x q matrix X, D*X[:] is Y[:] where
#
# Yij = Xij / sqrt( 1 - si^2 * sj^2 ) if i < j
# = Xii / sqrt( 1 + si^2 ) if i = j
# = Xij otherwise.
#
DV[Idiag] = sqrt(1.0 + SS[::q + 1])
DV[Itriu] = sqrt(1.0 - SS[Itril3]**2)
D.V = DV**-1
# 3. Scaled linear mappings
# Ask := V2' * Ask * V1'
blas.scal(0.0, As)
base.axpy(A, As)
for i in xrange(n):
# tmp := V2' * As[i, :]
blas.gemm(V2,
As,
tmp,
transA='T',
m=p,
n=q,
k=p,
ldB=p,
offsetB=i * p * q)
# As[:,i] := tmp * V1'
blas.gemm(tmp,
V1,
As,
transB='T',
m=p,
n=q,
k=q,
ldC=p,
offsetC=i * p * q)
# As := D * (I - P) * As
# = L^-1 * As.
blas.copy(As, As2)
base.gemm(P, As, As2, alpha=-1.0, beta=1.0)
base.gemm(D, As2, As)
# Gs := Dl^-1 * G
blas.scal(0.0, Gs)
base.axpy(G, Gs)
for k in xrange(n):
blas.tbmv(W['di'], Gs, n=m, k=0, ldA=1, offsetx=k * m)
# 4. Cholesky factorization of H = C + Gs' * Gs + 2 * As' * As.
blas.syrk(As, H, trans='T', alpha=2.0)
blas.syrk(Gs, H, trans='T', beta=1.0)
base.axpy(C, H)
lapack.potrf(H)
def f(x, y, z):
"""
Solve
C * ux + G' * uzl - 2*A'(uzs21) = bx
-uzs11 = bX1
-uzs22 = bX2
G * ux - D^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ uzs11 uzs21' ]
[ ] - T * [ ] * T = bzs.
[ -A(ux) -uX2 ] [ uzs21 uzs22 ]
On entry, x = (bx, bX1, bX2) and z = [ bzl; bzs[:] ].
On exit, x = (ux, uX1, uX2) and z = [ D*uzl; (r'*uzs*r)[:] ].
Define X = uzs21, Z = T * uzs * T:
C * ux + G' * uzl - 2*A'(X) = bx
[ 0 X' ] [ bX1 0 ]
T * [ ] * T - Z = T * [ ] * T
[ X 0 ] [ 0 bX2 ]
G * ux - D^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ Z11 Z21' ]
[ ] - [ ] = bzs
[ -A(ux) -uX2 ] [ Z21 Z22 ]
Return x = (ux, uX1, uX2), z = [ D*uzl; (rti'*Z*rti)[:] ].
We use the congruence transformation
[ V1 0 ] [ T11 T21' ] [ V1' 0 ] [ I S' ]
[ ] [ ] [ ] = [ ]
[ 0 V2' ] [ T21 T22 ] [ 0 V2 ] [ S I ]
and the factorization
X + S * X' * S = L( L'(X) )
to write this as
C * ux + G' * uzl - 2*A'(X) = bx
L'(V2^-1 * X * V1^-1) - L^-1(V2' * Z21 * V1') = bX
G * ux - D^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ Z11 Z21' ]
[ ] - [ ] = bzs,
[ -A(ux) -uX2 ] [ Z21 Z22 ]
or
C * ux + Gs' * uuzl - 2*As'(XX) = bx
XX - ZZ21 = bX
Gs * ux - uuzl = D^-1 * bzl
-As(ux) - ZZ21 = bbzs_21
-uX1 - Z11 = bzs_11
-uX2 - Z22 = bzs_22
if we introduce scaled variables
uuzl = D * uzl
XX = L'(V2^-1 * X * V1^-1)
= L'(V2^-1 * uzs21 * V1^-1)
ZZ21 = L^-1(V2' * Z21 * V1')
and define
bbzs_21 = L^-1(V2' * bzs_21 * V1')
[ bX1 0 ]
bX = L^-1( V2' * (T * [ ] * T)_21 * V1').
[ 0 bX2 ]
Eliminating Z21 gives
C * ux + Gs' * uuzl - 2*As'(XX) = bx
Gs * ux - uuzl = D^-1 * bzl
-As(ux) - XX = bbzs_21 - bX
-uX1 - Z11 = bzs_11
-uX2 - Z22 = bzs_22
and eliminating uuzl and XX gives
H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bX - bbzs_21)
Gs * ux - uuzl = D^-1 * bzl
-As(ux) - XX = bbzs_21 - bX
-uX1 - Z11 = bzs_11
-uX2 - Z22 = bzs_22.
In summary, we can use the following algorithm:
1. bXX := bX - bbzs21
[ bX1 0 ]
= L^-1( V2' * ((T * [ ] * T)_21 - bzs_21) * V1')
[ 0 bX2 ]
2. Solve H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bXX).
3. From ux, compute
uuzl = Gs*ux - D^-1 * bzl and
X = V2 * L^-T(-As(ux) + bXX) * V1.
4. Return ux, uuzl,
rti' * Z * rti = r' * [ -bX1, X'; X, -bX2 ] * r
and uX1 = -Z11 - bzs_11, uX2 = -Z22 - bzs_22.
"""
# Save bzs_11, bzs_22, bzs_21.
lapack.lacpy(z, bz11, uplo='L', m=q, n=q, ldA=p + q, offsetA=m)
lapack.lacpy(z, bz21, m=p, n=q, ldA=p + q, offsetA=m + q)
lapack.lacpy(z,
bz22,
uplo='L',
m=p,
n=p,
ldA=p + q,
offsetA=m + (p + q + 1) * q)
# zl := D^-1 * zl
# = D^-1 * bzl
blas.tbmv(W['di'], z, n=m, k=0, ldA=1)
# zs := r' * [ bX1, 0; 0, bX2 ] * r.
# zs := [ bX1, 0; 0, bX2 ]
blas.scal(0.0, z, offset=m)
lapack.lacpy(x[1], z, uplo='L', m=q, n=q, ldB=p + q, offsetB=m)
lapack.lacpy(x[2],
z,
uplo='L',
m=p,
n=p,
ldB=p + q,
offsetB=m + (p + q + 1) * q)
# scale diagonal of zs by 1/2
blas.scal(0.5, z, inc=p + q + 1, offset=m)
# a := tril(zs)*r
blas.copy(r, a)
blas.trmm(z,
a,
side='L',
m=p + q,
n=p + q,
ldA=p + q,
ldB=p + q,
offsetA=m)
# zs := a'*r + r'*a
blas.syr2k(r,
a,
z,
trans='T',
n=p + q,
k=p + q,
ldB=p + q,
ldC=p + q,
offsetC=m)
# bz21 := L^-1( V2' * ((r * zs * r')_21 - bz21) * V1')
#
# [ bX1 0 ]
# = L^-1( V2' * ((T * [ ] * T)_21 - bz21) * V1').
# [ 0 bX2 ]
# a = [ r21 r22 ] * z
# = [ r21 r22 ] * r' * [ bX1, 0; 0, bX2 ] * r
# = [ T21 T22 ] * [ bX1, 0; 0, bX2 ] * r
blas.symm(z,
r,
a,
side='R',
m=p,
n=p + q,
ldA=p + q,
ldC=p + q,
offsetB=q)
# bz21 := -bz21 + a * [ r11, r12 ]'
# = -bz21 + (T * [ bX1, 0; 0, bX2 ] * T)_21
blas.gemm(a,
r,
bz21,
transB='T',
m=p,
n=q,
k=p + q,
beta=-1.0,
ldA=p + q,
ldC=p)
# bz21 := V2' * bz21 * V1'
# = V2' * (-bz21 + (T*[bX1, 0; 0, bX2]*T)_21) * V1'
blas.gemm(V2, bz21, tmp, transA='T', m=p, n=q, k=p, ldB=p)
blas.gemm(tmp, V1, bz21, transB='T', m=p, n=q, k=q, ldC=p)
# bz21[:] := D * (I-P) * bz21[:]
# = L^-1 * bz21[:]
# = bXX[:]
blas.copy(bz21, tmp)
base.gemv(P, bz21, tmp, alpha=-1.0, beta=1.0)
base.gemv(D, tmp, bz21)
# Solve H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bXX).
# x[0] := x[0] + Gs'*zl + 2*As'(bz21)
# = bx + G' * D^-1 * bzl + 2 * As'(bXX)
blas.gemv(Gs, z, x[0], trans='T', alpha=1.0, beta=1.0)
blas.gemv(As, bz21, x[0], trans='T', alpha=2.0, beta=1.0)
# x[0] := H \ x[0]
# = ux
lapack.potrs(H, x[0])
# uuzl = Gs*ux - D^-1 * bzl
blas.gemv(Gs, x[0], z, alpha=1.0, beta=-1.0)
# bz21 := V2 * L^-T(-As(ux) + bz21) * V1
# = X
blas.gemv(As, x[0], bz21, alpha=-1.0, beta=1.0)
blas.tbsv(DV, bz21, n=p * q, k=0, ldA=1)
blas.copy(bz21, tmp)
base.gemv(P, tmp, bz21, alpha=-1.0, beta=1.0, trans='T')
blas.gemm(V2, bz21, tmp)
blas.gemm(tmp, V1, bz21)
# zs := -zs + r' * [ 0, X'; X, 0 ] * r
# = r' * [ -bX1, X'; X, -bX2 ] * r.
# a := bz21 * [ r11, r12 ]
# = X * [ r11, r12 ]
blas.gemm(bz21, r, a, m=p, n=p + q, k=q, ldA=p, ldC=p + q)
# z := -z + [ r21, r22 ]' * a + a' * [ r21, r22 ]
# = rti' * uzs * rti
blas.syr2k(r,
a,
z,
trans='T',
beta=-1.0,
n=p + q,
k=p,
offsetA=q,
offsetC=m,
ldB=p + q,
ldC=p + q)
# uX1 = -Z11 - bzs_11
# = -(r*zs*r')_11 - bzs_11
# uX2 = -Z22 - bzs_22
# = -(r*zs*r')_22 - bzs_22
blas.copy(bz11, x[1])
blas.copy(bz22, x[2])
# scale diagonal of zs by 1/2
blas.scal(0.5, z, inc=p + q + 1, offset=m)
# a := r*tril(zs)
blas.copy(r, a)
blas.trmm(z,
a,
side='R',
m=p + q,
n=p + q,
ldA=p + q,
ldB=p + q,
offsetA=m)
# x[1] := -x[1] - a[:q,:] * r[:q, :]' - r[:q,:] * a[:q,:]'
# = -bzs_11 - (r*zs*r')_11
blas.syr2k(a, r, x[1], n=q, alpha=-1.0, beta=-1.0)
# x[2] := -x[2] - a[q:,:] * r[q:, :]' - r[q:,:] * a[q:,:]'
# = -bzs_22 - (r*zs*r')_22
blas.syr2k(a,
r,
x[2],
n=p,
alpha=-1.0,
beta=-1.0,
offsetA=q,
offsetB=q)
# scale diagonal of zs by 1/2
blas.scal(2.0, z, inc=p + q + 1, offset=m)
return f
if C:
sol = solvers.coneqp(Pf,
c,
Gf,
hh,
dims,
Af,
kktsolver=F,
xnewcopy=xnewcopy,
xdot=xdot,
xaxpy=xaxpy,
xscal=xscal)
else:
sol = solvers.conelp(c,
Gf,
hh,
dims,
Af,
kktsolver=F,
xnewcopy=xnewcopy,
xdot=xdot,
xaxpy=xaxpy,
xscal=xscal)
if sol['status'] is 'optimal':
x = sol['x'][0]
z = sol['z'][:m]
Z = sol['z'][m:]
Z.size = (p + q, p + q)
Z = -2.0 * Z[-p:, :q]
elif sol['status'] is 'primal infeasible':
x = None
z = sol['z'][:m]
Z = sol['z'][m:]
Z.size = (p + q, p + q)
Z = -2.0 * Z[-p:, :q]
else:
x, z, Z = None, None, None
return {'status': sol['status'], 'x': x, 'z': z, 'Z': Z}
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 block_socp(self, a, A1, A2, t, B_s, u_s):
dims = {
'l': self.size_tb() + 2*self.size_x() + sum([cc.size for cc in self.cube_constraints]), # Pure inequality constraints
# Com cone is now 3d, Size of the cones: x,y,z+1
'q': [dc.size for dc in self.dist_constraints]+[4]*len(self.contacts)*len(self.gravity_envelope)+[self.size_z()+1]*self.size_s()+[fc.size for fc in self.force_constraints],
's': [] # No sd cones
}
size_cones = self.size_x()*4 // 3
#Optimization vector [ f1 ... fn CoM s1 ... sn]
#Max a^T z ~ min -a^T z
#Final cost : min -a^T z + s
c = np.vstack([np.zeros((self.size_x(), 1)), -a, np.zeros((self.size_s(), 1))])
A1_diag = block_diag(*([A1]*len(self.gravity_envelope)))
A2 = np.vstack([self.computeA2(self.gravity+e)
for e in self.gravity_envelope])
T = np.vstack([self.computeT(self.gravity+e, height=self.height)
for e in self.gravity_envelope])
A = np.hstack([A1_diag, A2, np.zeros((A2.shape[0], self.size_s()))])
g_s = []
h_s = []
#Torque constraint
if self.L_s:
g_s.extend(self.L_s)
h_s.extend(self.tb_s)
#Linear force constraint
# <x> <z> <1>
# <x> [[ I 0] <= <x> lim*mg 2*<x> [ 1]
# <x> [[ -I 0]
g_force = np.vstack([np.eye(self.size_x()), -np.eye(self.size_x())])
g_s.append(np.hstack([g_force, np.zeros((2*self.size_x(), self.size_z()+self.size_s()))]))
h_s.append(self.force_lim*self.mass*9.81*np.ones((2*self.size_x(), 1)))
#Cube constraints
if self.C_s:
g_s.append(np.vstack(self.C_s))
h_s.append(np.vstack(self.d_s))
#These are additional dist constraints
g_s.extend(self.S_s)
h_s.extend(self.r_s)
#B = diag{[u_i b_i.T].T}
blocks = [-np.vstack([u.T, B]) for u, B in zip(u_s, B_s)]*len(self.gravity_envelope)
block = block_diag(*blocks)
g_contacts = np.hstack([block, np.zeros((size_cones, self.size_z()+self.size_s()))])
h_cones = np.zeros((size_cones, 1))
g_slacks, h_slacks = None, None
if self.robust_sphere > 0:
g_slacks = np.zeros((4*self.size_s(), self.nrVars()))
h_slacks = np.zeros((4*self.size_s(), 1))
for i, contact in enumerate(self.contacts):
robusticity = -self.robust_sphere*self.mass*(self.theta_bar+self.sigma_bar*contact.mu)
g_contacts[4*i, self.size_x()+self.size_z()+i] = -robusticity
h_cones[4*i, 0] = robusticity
#Add slack variables cones
# || z || < s
g_slacks[4*i, self.size_x()+self.size_z()+i] = -1
g_slacks[4*i+1:4*i+1+self.size_z(), self.size_x():self.size_x()+self.size_z()] = -np.eye(self.size_z())
g_s.append(g_contacts)
h_s.append(h_cones)
if g_slacks is not None:
g_s.append(g_slacks)
h_s.append(h_slacks)
#Force Constraints
g_s.extend(self.F_s)
h_s.extend(self.f_s)
#Concat everything
g = np.vstack(g_s)
h = np.vstack(h_s)
sol = solvers.conelp(matrix(c), G=matrix(g), h=matrix(h),
A=matrix(A), b=matrix(T), dims=dims)
return sol
def SDP():
return solvers.conelp(c, G, h, dims)
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
def l1(P, q):
"""
Returns the solution u, w 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 -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)
A = P.T * spdiag(4 * D) * P
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:]) ).
x[:n] += P.T * (mul(div(d1 - d2, d1 + d2), x[n:]) +
mul(2 * D, z[:m] - z[m:]))
lapack.potrs(A, x)
# x[n:] := (D1+D2)^{-1} * (bx[n:] - D1*bz[:m] - D2*bz[m:]
# + (D1-D2)*P*x[:n])
u = P * x[:n]
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['z'][m:] - sol['z'][:m]
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
