def test_dsdp(self):
# Setup problem
c = matrix([1., -1., 1.])
G = [
matrix([[-7., -11., -11., 3.], [7., -18., -18., 8.],
[-2., -8., -8., 1.]])
]
G += [
matrix([[-21., -11., 0., -11., 10., 8., 0., 8., 5.],
[0., 10., 16., 10., -10., -10., 16., -10., 3.],
[-5., 2., -17., 2., -6., 8., -17., 8., 6.]])
]
h = [matrix([[33., -9.], [-9., 26.]])]
h += [matrix([[14., 9., 40.], [9., 91., 10.], [40., 10., 15.]])]
# Solve with default solver
sol = solvers.sdp(c, Gs=G, hs=h)
# Solve with DSDP
sol_dsdp = solvers.sdp(c, Gs=G, hs=h, solver='dsdp')
# Compare solutions
self.assertMatrixAlmostEqual(sol['x'], sol_dsdp['x'])
self.assertMatrixAlmostEqual(sol['zs'][0], sol_dsdp['zs'][0])
self.assertMatrixAlmostEqual(sol['zs'][1], sol_dsdp['zs'][1])
def __solve(self,c,F0,F,real=False,altsolver=None): c = cvxopt.matrix(c,tc='d') h = self.__F0toh(F0,real) G = self.__FtoG(F,real) if not altsolver: return solvers.sdp(c,Gs=G,hs=h) else: return solvers.sdp(c,Gs=G,hs=h,solver='dsdp')
def test_sdp(self):
from cvxopt import solvers, dsdp
c, Gs, hs = self._prob_data
sol_ref1 = solvers.sdp(c, None, None, Gs, hs)
self.assertTrue(sol_ref1['status'] == 'optimal')
sol_ref2 = solvers.sdp(c, Gs=Gs, hs=hs)
self.assertTrue(sol_ref2['status'] == 'optimal')
sol1 = solvers.sdp(c, None, None, Gs, hs, solver='dsdp')
self.assertTrue(sol1['status'] == 'optimal')
sol2 = solvers.sdp(c, Gs=Gs, hs=hs, solver='dsdp')
self.assertTrue(sol2['status'] == 'optimal')
sol3 = dsdp.sdp(c, None, None, Gs, hs)
self.assertTrue(sol3[0] == 'DSDP_PDFEASIBLE')
sol4 = dsdp.sdp(c, Gs=Gs, hs=hs)
self.assertTrue(sol4[0] == 'DSDP_PDFEASIBLE')
def test_sdp(self):
from cvxopt import solvers, dsdp
c,Gs,hs = self._prob_data
sol_ref1 = solvers.sdp(c,None,None,Gs,hs)
self.assertTrue(sol_ref1['status']=='optimal')
sol_ref2 = solvers.sdp(c,Gs=Gs,hs=hs)
self.assertTrue(sol_ref2['status']=='optimal')
sol1 = solvers.sdp(c,None,None,Gs,hs,solver='dsdp')
self.assertTrue(sol1['status']=='optimal')
sol2 = solvers.sdp(c,Gs=Gs,hs=hs,solver='dsdp')
self.assertTrue(sol2['status']=='optimal')
sol3 = dsdp.sdp(c,None,None,Gs,hs)
self.assertTrue(sol3[0] == 'DSDP_PDFEASIBLE')
sol4 = dsdp.sdp(c,Gs=Gs,hs=hs)
self.assertTrue(sol4[0] == 'DSDP_PDFEASIBLE')
def solve_sdp(c, Gl=None, hl=None, Gsl=[], hsl=[], A=None, b=None):
"""
Solve Semi-Definite Prgramming
:param c:
:param Gl:
:param hl:
:param Gsl:
:param hsl:
:param A:
:param b:
:return:
"""
if Gl is None:
Gl = np.zeros((1, len(c)))
hl = np.zeros((1, 1))
if not Gsl:
Gsl = [np.zeros((1, len(c)))]
hsl = [np.zeros((1, 1))]
args = [
matrix(c),
matrix(Gl),
matrix(hl), [matrix(Gs) for Gs in Gsl], [matrix(hs) for hs in hsl]
]
if A is not None:
args.extend([matrix(A), matrix(b)])
sol = solvers.sdp(*args)
if 'optimal' not in sol['status']:
print("CVXOPT FAIL:", sol['status'])
return None
return np.array(sol['x']).reshape((len(c), ))
def SDP_query_distribution(A, lambda_, X_pool, k):
"""Solving SDP problem in FIR-based active learning
to obtain the query distribution
"""
n = len(A)
tau = A[0].shape[0]
"""Preparing the variables"""
# matrix inequality constraints
G, h = inequality_cvx_matrix(A)
# equality constraint (for having probabilities)
if lambda_ > 0:
d = X_pool.shape[0]
pool_norms = np.sum(X_pool**2, axis=0)
# vector c (in the objective)
cvec = matrix(np.concatenate((-lambda_ * pool_norms, np.ones(tau))))
# A matrix (LHS of equality)
left_A = np.concatenate((X_pool, np.ones((1, n))), axis=0)
A_eq = matrix(np.concatenate((left_A, np.zeros((d + 1, tau))), axis=1))
# b vector (RHS of equality)
b_eq = np.zeros(d + 1)
b_eq[-1] = 1.
b_eq = matrix(b_eq)
else:
cvec = matrix(np.concatenate((np.zeros(n), np.ones(tau))))
A_eq = matrix(np.concatenate((np.ones(n), np.zeros(tau)))).trans()
b_eq = matrix(1.)
"""Solving SDP"""
soln = solvers.sdp(cvec, Gs=G, hs=h, A=A_eq, b=b_eq)
return soln
def test_mmsolvers():
# simple test to make sure things run
print 'testing simple unimixture with a skipped observation, just to test that things run'
x = sp.symbols('x')
M = MomentMatrix(3, [x], morder='grevlex')
constrs = [x-1.5, x**2-2.5, x**4-8.5]
#constrs = [x-1.5, x**2-2.5, x**3-4.5, x**4-8.5]
cin = get_cvxopt_inputs(M, constrs, slack = 1e-5)
#import MomentMatrixSolver
#print 'joint_alternating_solver...'
#y,L = MomentMatrixSolver.sgd_solver(M, constrs, 2, maxiter=101, eta = 0.001)
#y,X = MomentMatrixSolver.convex_projection_solver(M, constrs, 2, maxiter=2000)
#print y
#print X
gs = [2-x, 2+x]
locmatrices = [LocalizingMatrix(M, g) for g in gs]
Ghs = [get_cvxopt_Gh(lm) for lm in locmatrices]
Gs=cin['G'] + [Gh['G'] for Gh in Ghs]
hs=cin['h'] + [Gh['h'] for Gh in Ghs]
sol = cvxsolvers.sdp(cin['c'], Gs=Gs, \
hs=hs, A=cin['A'], b=cin['b'])
print sol['x']
print abs(sol['x'][3]-4.5)
assert(abs(sol['x'][3]-4.5) <= 1e-3)
import extractors
print extractors.extract_solutions_lasserre(M, sol['x'], Kmax = 2)
print 'true values are 1 and 2'
def test_unimixture():
print 'testing simple unimixture with a skipped observation'
x = sp.symbols('x')
constrs = [x + x**2 - 2, x + 2 * x**4 - 3,
2 * x**2 + 3 * x**4 - 5] # always solvable
constrs = [
x + x**2 - 1, x**2 + 2 * x**3 - 1, 2 * x**3 + 3 * x**4 - 3,
2 * x + x**3 - 0
] # always solvable
constrs = [x - 1.5, x**2 - 2.5,
x**5 - 16.5] # solvable with deg=3, but not deg=2
constrs = [x - 1.5, x**2 - 2.5,
x**4 - 8.5] # solvable with deg=3, but not deg=2
M = mp.MomentMatrix(3, [x], morder='grevlex')
cin = mp.solvers.get_cvxopt_inputs(M, constrs)
sol = solvers.sdp(matrix(sc.rand(*cin['c'].size)), Gs=cin['G'], \
hs=cin['h'], A=cin['A'], b=cin['b'])
M.pretty_print(sol)
print mp.extractors.extract_solutions_lasserre(M, sol['x'], Kmax=2)
return M, sol
def CVXOPT_SDP_Solver(p, solverName):
if solverName == 'native_CVXOPT_SDP_Solver': solverName = None
cvxopt_solvers.options['maxiters'] = p.maxIter
cvxopt_solvers.options['feastol'] = p.contol
cvxopt_solvers.options['abstol'] = p.ftol
if p.iprint <= 0:
cvxopt_solvers.options['show_progress'] = False
cvxopt_solvers.options['LPX_K_MSGLEV'] = 0
#cvxopt_solvers.options['MSK_IPAR_LOG'] = 0
xBounds2Matrix(p)
#FIXME: if problem is search for MAXIMUM, not MINIMUM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
f = copy(p.f).reshape(-1, 1)
# CVXOPT have some problems with x0 so currently I decided to avoid using the one
sol = cvxopt_solvers.sdp(Matrix(p.f), Matrix(p.A), Matrix(p.b), \
converter_to_CVXOPT_SDP_Matrices_from_OO_SDP_Class(p.S, p.n), \
DictToList(p.d), Matrix(p.Aeq), Matrix(p.beq), solverName)
p.msg = sol['status']
if p.msg == 'optimal':
p.istop = SOLVED_WITH_UNIMPLEMENTED_OR_UNKNOWN_REASON
else:
p.istop = -100
if sol['x'] is not None:
p.xf = asarray(sol['x']).flatten()
p.ff = sum(p.dotmult(p.f, p.xf))
#p.duals = concatenate((asarray(sol['y']).flatten(), asarray(sol['z']).flatten()))
else:
p.ff = nan
p.xf = nan * ones(p.n)
def test_cvx_sdp():
"""
Test that the CVX SDP optimization works.
"""
c = matrix([1., -1., 1.])
G = [
matrix([[-7., -11., -11., 3.], [7., -18., -18., 8.],
[-2., -8., -8., 1.]])
]
G += [
matrix([[-21., -11., 0., -11., 10., 8., 0., 8., 5.],
[0., 10., 16., 10., -10., -10., 16., -10., 3.],
[-5., 2., -17., 2., -6., 8., -17., 8., 6.]])
]
h = [matrix([[33., -9.], [-9., 26.]])]
h += [matrix([[14., 9., 40.], [9., 91., 10.], [40., 10., 15.]])]
sol = solvers.sdp(c, Gs=G, hs=h)
x = np.matrix(sol['x'])
x_ = np.matrix([
[-3.68e-01],
[1.90e+00],
[-8.88e-01],
])
print x, x_
assert np.allclose(x, x_, atol=1e-2)
def inv_solver(xs, ps, xmax):
"""Optimization w.r.t. (Q,r)
Parameters
----------
xs: list of (optimal) demand vectors
ps: list of price vectors
xmax: maximum demand vector (for each product)
"""
n, N = len(xs[0]), len(xs)
# constraint Q <= 0
G, h = conic_constraint(n)
# constraint Q*xmax + r >= 0
A = -linear_constraint(xmax)
b = matrix(0., (n,1))
# constraints Q*x^j + r <= p^j
for j in range(N):
A = matrix([A, linear_constraint(xs[j])])
b = matrix([b, matrix(ps[j])])
# constraint r >= 0
tmp = matrix([[matrix(0., (n, (n*(n+1))/2))], [-spdiag([1.]*n)]])
A = matrix([A, tmp])
b = matrix([b, matrix(0., (n,1))])
# linear objective
c = linear_objective(xs)
x = solvers.sdp(c, A, b, G, h)['x']
Q, r = matrix(0., (n,n)), matrix(0., (n,1))
for i in range(n):
Q[i,i] = x[(n+1)*i-(i*(i+1))/2]
r[i] = x[(n*(n+1))/2+i]
for j in range(i+1,n):
Q[i,j] = x[n*i+j-(i*(i+1))/2]; Q[j,i] = Q[i,j]
return Q, r
def CVXOPT_SDP_Solver(p, solverName):
if solverName == 'native_CVXOPT_SDP_Solver': solverName = None
cvxopt_solvers.options['maxiters'] = p.maxIter
cvxopt_solvers.options['feastol'] = p.contol
cvxopt_solvers.options['abstol'] = p.ftol
if p.iprint <= 0:
cvxopt_solvers.options['show_progress'] = False
cvxopt_solvers.options['LPX_K_MSGLEV'] = 0
#cvxopt_solvers.options['MSK_IPAR_LOG'] = 0
xBounds2Matrix(p)
#FIXME: if problem is search for MAXIMUM, not MINIMUM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
f = copy(p.f).reshape(-1,1)
# CVXOPT have some problems with x0 so currently I decided to avoid using the one
sol = cvxopt_solvers.sdp(Matrix(p.f), Matrix(p.A), Matrix(p.b), \
converter_to_CVXOPT_SDP_Matrices_from_OO_SDP_Class(p.S, p.n), \
DictToList(p.d), Matrix(p.Aeq), Matrix(p.beq), solverName)
p.msg = sol['status']
if p.msg == 'optimal' : p.istop = SOLVED_WITH_UNIMPLEMENTED_OR_UNKNOWN_REASON
else: p.istop = -100
if sol['x'] is not None:
p.xf = asarray(sol['x']).flatten()
p.ff = sum(p.dotmult(p.f, p.xf))
#p.duals = concatenate((asarray(sol['y']).flatten(), asarray(sol['z']).flatten()))
else:
p.ff = nan
p.xf = nan*ones(p.n)
def solve_sdp_si_cvx(G,
data,
p0,
p1,
a_b_ratio,
rho=0.1,
max_iter=100,
tol=1e-5):
B = construct_B(G, 1.0)
B_tilde = construct_B_tilde(B, data, p0, p1, a_b_ratio)
n = B.shape[0]
h = [matrix(-B_tilde)]
c = matrix(np.hstack((np.ones(n + 1), np.zeros(n + 1))))
g_matrix = matrix(0.0, ((n + 1)**2, 2 * n + 2))
for i in range(n + 1):
g_matrix[(n + 1) * i + i, i] = -1
for _i in range(n + 1, 2 * n + 2):
i = _i - n - 1
for j in range(1, n + 1):
g_matrix[(n + 1) * i + j, _i] = -1
g_matrix[(n + 1) * j + i, _i] = -1
solvers.options['abstol'] = tol
solvers.options['maxiters'] = max_iter
sol = solvers.sdp(c, Gs=[g_matrix], hs=h)
X = np.array(sol['zs'][0])
labels = X[0, 1:] > 0
return labels.astype(np.int)
def solve_ith_GMP(MM, objective, gs, hs, slack=0):
"""
Generalized moment problem solver
@param - objective: a sympy expression that is the objective
@param - gs: list of contraints defining the semialgebraic set K,
each g corresponds to localizing matrices.
@param - hs: constraints on the moments,
not needed for polynomial optimization.
@param - slack: add equalities as pairs of inequalities with slack separation.
"""
cin = get_cvxopt_inputs(MM, hs, slack=slack)
Bf = MM.get_Bflat()
objcoeff, __ = MM.get_Ab([objective], cvxoptmode=False)
locmatrices = [LocalizingMatrix(MM, g) for g in gs]
Ghs = [get_cvxopt_Gh(lm) for lm in locmatrices]
# list addition
Gs = cin['G'] + [Gh['G'] for Gh in Ghs]
hs = cin['h'] + [Gh['h'] for Gh in Ghs]
# print Ghs
solsdp = cvxsolvers.sdp(matrix(objcoeff[0, :]),
Gs=Gs,
hs=hs,
A=cin['A'],
b=cin['b'])
#ipdb.set_trace()
return solsdp
def solve_Q(x_dim, A, B, D):
# x = [s vec(Z) vec(Q)]
MAX_ITERS=30
c_dim = 1 + 2 * x_dim*(x_dim+1)/2
c = zeros(c_dim)
c[0] = x_dim
prev = 1
for i in range(x_dim):
vec_pos = prev + i * (i+1)/2 + i
c[vec_pos] = 1
cm = matrix(c)
G = construct_coeff_matrix(x_dim, B)
G = -G # set negative since s = h - Gx in cvxopt's sdp solver
#print "G-shape"
#print shape(G)
Gs = [matrix(G)]
h,_ = construct_const_matrix(x_dim, A, B, D)
#print "h-shape"
#print shape(h)
hs = [matrix(h)]
solvers.options['maxiters'] = MAX_ITERS
sol = solvers.sdp(cm, Gs = Gs, hs=hs)
#print sol['x']
return sol, c, G, h
def sdp(Sigma):
p = Sigma.shape[0]
identity_p = np.identity(p)
zero_one = np.repeat(0., p**3)
zero_one2 = zero_one.copy()
indexes = np.arange(p) * (
1 + p + p**2) # np.arange(p)*p+ np.arange(p)*(p**2) + np.arange(p)
zero_one[indexes] = 1.
block_identity = zero_one.reshape(p * p, p)
zero_one2[indexes] = -1.
mblock_identity = zero_one2.reshape(p * p, p)
c = matrix(np.repeat(-1., p))
G = [matrix(block_identity)] + [matrix(mblock_identity)
] + [matrix(block_identity)]
h = [matrix(2. * Sigma)] + [matrix(np.zeros(
(p, p)))] + [matrix(identity_p)]
solvers.options['show_progress'] = False
sol = solvers.sdp(c, Gs=G, hs=h)['x']
sol = np.array(sol).flatten()
sol[sol > 1.] = 1.
sol[sol < 0.] = 0.
# print(os.getpid(), "\n")
return (sol)
def test_mmsolvers():
# simple test to make sure things run
print 'testing simple unimixture with a skipped observation, just to test that things run'
x = sp.symbols('x')
M = MomentMatrix(3, [x], morder='grevlex')
constrs = [x - 1.5, x**2 - 2.5, x**4 - 8.5]
#constrs = [x-1.5, x**2-2.5, x**3-4.5, x**4-8.5]
cin = get_cvxopt_inputs(M, constrs, slack=1e-5)
#import MomentMatrixSolver
#print 'joint_alternating_solver...'
#y,L = MomentMatrixSolver.sgd_solver(M, constrs, 2, maxiter=101, eta = 0.001)
#y,X = MomentMatrixSolver.convex_projection_solver(M, constrs, 2, maxiter=2000)
#print y
#print X
gs = [2 - x, 2 + x]
locmatrices = [LocalizingMatrix(M, g) for g in gs]
Ghs = [get_cvxopt_Gh(lm) for lm in locmatrices]
Gs = cin['G'] + [Gh['G'] for Gh in Ghs]
hs = cin['h'] + [Gh['h'] for Gh in Ghs]
sol = cvxsolvers.sdp(cin['c'], Gs=Gs, \
hs=hs, A=cin['A'], b=cin['b'])
print sol['x']
print abs(sol['x'][3] - 4.5)
assert (abs(sol['x'][3] - 4.5) <= 1e-3)
import extractors
print extractors.extract_solutions_lasserre(M, sol['x'], Kmax=2)
print 'true values are 1 and 2'
def solve_moments_with_convexiterations(MM, constraints, maxrank = 3, slack = 1e-3, maxiter = 200):
"""
Solve using the moment matrix iteratively using the rank constrained convex iterations
Use @symbols with basis bounded by degree @deg.
Also use the constraints.
"""
cin = get_cvxopt_inputs(MM, constraints)
Bf = MM.get_Bflat()
R = np.random.rand(len(MM), len(MM))
#W = R.dot(R.T)
W = np.eye(len(MM))
tau = []
for i in xrange(maxiter):
w = Bf.dot(W.flatten())
#solsdp = cvxsolvers.sdp(matrix(w), Gs=cin['G'], hs=cin['h'], A=cin['A'], b=cin['b'])
solsdp = cvxsolvers.sdp(matrix(w), Gs=cin['G'], hs=cin['h'], Gl=cin['Gl'], hl=cin['hl'])
Xstar = MM.numeric_instance(solsdp['x'])
W,solW = solve_W(Xstar, maxrank)
W = np.array(W)
ctau = np.sum(W * Xstar.flatten())
if ctau < 1e-3:
break
tau.append(ctau)
#ipdb.set_trace()
print tau
return solsdp
def testsdp(opts):
c = matrix([1., -1., 1.])
G = [
matrix([[-7., -11., -11., 3.], [7., -18., -18., 8.],
[-2., -8., -8., 1.]])
]
G += [
matrix([[-21., -11., 0., -11., 10., 8., 0., 8., 5.],
[0., 10., 16., 10., -10., -10., 16., -10., 3.],
[-5., 2., -17., 2., -6., 8., -17., -7., 6.]])
]
h = [matrix([[33., -9.], [-9., 26.]])]
h += [matrix([[14., 9., 40.], [9., 91., 10.], [40., 10., 15.]])]
sol = solvers.sdp(c, Gs=G, hs=h)
print "x = \n", str2(sol['x'], "%.9f")
print "zs[0] = \n", helpers.str2(sol['zs'][0], "%.9f")
print "zs[1] = \n", helpers.str2(sol['zs'][1], "%.9f")
helpers.run_go_test(
"../testsdp", {
'x': sol['x'],
'ss0': sol['ss'][0],
'ss1': sol['ss'][1],
'zs0': sol['zs'][0],
'zs0': sol['zs'][1]
})
def solve_moments_with_convexiterations(MM,
constraints,
maxrank=3,
slack=1e-3,
maxiter=200):
"""
Solve using the moment matrix iteratively using the rank constrained convex iterations
Use @symbols with basis bounded by degree @deg.
Also use the constraints.
"""
cin = get_cvxopt_inputs(MM, constraints)
Bf = MM.get_Bflat()
R = np.random.rand(len(MM), len(MM))
#W = R.dot(R.T)
W = np.eye(len(MM))
tau = []
for i in xrange(maxiter):
w = Bf.dot(W.flatten())
#solsdp = cvxsolvers.sdp(matrix(w), Gs=cin['G'], hs=cin['h'], A=cin['A'], b=cin['b'])
solsdp = cvxsolvers.sdp(matrix(w),
Gs=cin['G'],
hs=cin['h'],
Gl=cin['Gl'],
hl=cin['hl'])
Xstar = MM.numeric_instance(solsdp['x'])
W, solW = solve_W(Xstar, maxrank)
W = np.array(W)
ctau = np.sum(W * Xstar.flatten())
if ctau < 1e-3:
break
tau.append(ctau)
#ipdb.set_trace()
print tau
return solsdp
def optimize_polynomial(pol):
r"""
Optimization polynomial by representing it as an SDP in the
appropriate basis
TODO: Support constraints
\min_y c^\top y
subject to M(y) \succeq 0
y_1 = 1
@param poly - polynomial to optimize
@param basis - basis for the polynomial (can be 'power',
'symmetric', 'symmetric_irreducible')
@returns (p*, y*) - the optimal value of the polynomial and
y* = E_\mu[b(x)].
"""
# basis in sorted order from 1 to largest
# Get the largest monomial term
max_degree = get_max_degree(pol)
half_degree = map(lambda deg: int(np.ceil(deg / 2.)), max_degree)
# Generate basis
basis = list(it.product(*[range(deg + 1) for deg in max_degree]))
half_basis = list(it.product(*[range(deg + 1) for deg in half_degree]))
N, M = len(basis), len(half_basis)
# The coefficient term
c = np.matrix([float(pol.nth(*elem)) for elem in basis]).T
# The inequality term, enforcing sos-ness
G = np.zeros((M**2, N))
for i, j in it.product(xrange(M), xrange(M)):
e1, e2 = half_basis[i], half_basis[j]
monom = tuple_add(e1, e2)
k = basis.index(monom)
if k != -1:
G[M * i + j, k] = 1
h = np.zeros((M, M))
# Finally, y_1 = 1
A = np.zeros((1, N))
A[0, 0] = 1
b = np.zeros(1)
b[0] = 1
sol = solvers.sdp(
c=matrix(c),
Gs=[matrix(-G)],
hs=[matrix(h)],
A=matrix(A),
b=matrix(b),
)
assert sol['status'] == 'optimal'
return dict([(get_monom(pol, coeff), val)
for coeff, val in zip(basis, list(sol['x']))])
def cheb(A, b, Sigma):
# Calculates Chebyshev lower bound on Prob(A*x <= b) where
# x in R^2 has mean zero and covariance Sigma.
#
# maximize 1 - tr(Sigma*P) - r
# subject to [ P, q - (tauk/2)*ak ]
# [ (q - (tauk/2)*ak)', r - 1 + tauk*bk ] >= 0,
# k = 0,...,m-1
# [ P, q ]
# [ q', r ] >= 0
# tauk >= 0, k=0,...,m-1.
#
# variables P[0,0], P[1,0], P[1,1], q[0], q[1], r, tau[0], ...,
# tau[m-1].
m = A.size[0]
novars = 3 + 2 + 1 + m
# Cost function.
c = matrix(0.0, (novars, 1))
c[0], c[1], c[2] = Sigma[0, 0], 2 * Sigma[1, 0], Sigma[1, 1]
c[5] = 1.0
Gs = [spmatrix([], [], [], (9, novars)) for k in range(m + 1)]
# Coefficients of P, q, r in LMI constraints.
for k in range(m + 1):
Gs[k][0, 0] = -1.0 # P[0,0]
Gs[k][1, 1] = -1.0 # P[1,0]
Gs[k][4, 2] = -1.0 # P[1,1]
Gs[k][2, 3] = -1.0 # q[0]
Gs[k][5, 4] = -1.0 # q[1]
Gs[k][8, 5] = -1.0 # r
# Coefficients of tau.
for k in range(m):
Gs[k][2, 6 + k] = 0.5 * A[k, 0]
Gs[k][5, 6 + k] = 0.5 * A[k, 1]
Gs[k][8, 6 + k] = -b[k]
hs = [ matrix(8*[0.0] + [-1.0], (3,3)) for k in range(m) ] + \
[ matrix(0.0, (3,3)) ]
# Constraints tau >= 0.
Gl, hl = spmatrix(-1.0, range(m), range(6, 6 + m)), matrix(0.0, (m, 1))
sol = solvers.sdp(c, Gl, hl, Gs, hs)
P = matrix(sol['x'][[0, 1, 1, 2]], (2, 2))
q = matrix(sol['x'][[3, 4]], (2, 1))
r = sol['x'][5]
bound = 1.0 - Sigma[0] * P[0] - 2 * Sigma[1] * P[1] - Sigma[3] * P[3] - r
# Worst-case distribution from dual solution.
X = [Z[2, :2].T / Z[2, 2] for Z in sol['zs'] if Z[2, 2] > 1e-5]
return bound, P, q, r, X
def cheb(A, b, Sigma):
# Calculates Chebyshev lower bound on Prob(A*x <= b) where
# x in R^2 has mean zero and covariance Sigma.
#
# maximize 1 - tr(Sigma*P) - r
# subject to [ P, q - (tauk/2)*ak ]
# [ (q - (tauk/2)*ak)', r - 1 + tauk*bk ] >= 0,
# k = 0,...,m-1
# [ P, q ]
# [ q', r ] >= 0
# tauk >= 0, k=0,...,m-1.
#
# variables P[0,0], P[1,0], P[1,1], q[0], q[1], r, tau[0], ...,
# tau[m-1].
m = A.size[0]
novars = 3 + 2 + 1 + m
# Cost function.
c = matrix(0.0, (novars,1))
c[0], c[1], c[2] = Sigma[0,0], 2*Sigma[1,0], Sigma[1,1]
c[5] = 1.0
Gs = [ spmatrix([],[],[], (9,novars)) for k in range(m+1) ]
# Coefficients of P, q, r in LMI constraints.
for k in range(m+1):
Gs[k][0,0] = -1.0 # P[0,0]
Gs[k][1,1] = -1.0 # P[1,0]
Gs[k][4,2] = -1.0 # P[1,1]
Gs[k][2,3] = -1.0 # q[0]
Gs[k][5,4] = -1.0 # q[1]
Gs[k][8,5] = -1.0 # r
# Coefficients of tau.
for k in range(m):
Gs[k][2, 6+k] = 0.5 * A[k,0]
Gs[k][5, 6+k] = 0.5 * A[k,1]
Gs[k][8, 6+k] = -b[k]
hs = [ matrix(8*[0.0] + [-1.0], (3,3)) for k in range(m) ] + \
[ matrix(0.0, (3,3)) ]
# Constraints tau >= 0.
Gl, hl = spmatrix(-1.0, range(m), range(6,6+m)), matrix(0.0, (m,1))
sol = solvers.sdp(c, Gl, hl, Gs, hs)
P = matrix(sol['x'][[0,1,1,2]], (2,2))
q = matrix(sol['x'][[3,4]], (2,1))
r = sol['x'][5]
bound = 1.0 - Sigma[0]*P[0] - 2*Sigma[1]*P[1] - Sigma[3]*P[3] - r
# Worst-case distribution from dual solution.
X = [ Z[2,:2].T / Z[2,2] for Z in sol['zs'] if Z[2,2] > 1e-5 ]
return bound, P, q, r, X
def solve_sdp_cvx(G):
B = construct_B(G)
n = B.shape[0]
h = [matrix(-B)]
c = matrix(1.0, (n, 1))
g_matrix = spmatrix(-1, [n * i + i for i in range(n)],
range(n),
size=(n * n, n))
sol = solvers.sdp(c, Gs=[g_matrix], hs=h)
return get_labels_sdp2(np.array(sol['zs'][0]))
def SDP_diagonal(cov_matrix, multiKN_par):
cor_matrix = np.diag(1/np.sqrt(np.diag(cov_matrix))).dot(cov_matrix.dot(np.diag(1/np.sqrt(np.diag(cov_matrix)))))
dim = cov_matrix.shape[0]
c = matrix(-np.ones(dim))
Gl = matrix(np.vstack([-np.identity(dim), np.identity(dim)]))
hl = matrix(np.hstack([np.zeros(dim), np.ones(dim)]))
Gs = [matrix(np.vstack([flatten_Ei(l,dim) for l in np.arange(dim)]).T)]
hs = [matrix(multiKN_par*cor_matrix)]
result = solvers.sdp(c,Gl=Gl,hl=hl,Gs=Gs,hs=hs)
return np.diag((np.array(result['x'])[:,0]))*0.9999*np.diag(cov_matrix)
def optimize_polynomial(pol):
r"""
Optimization polynomial by representing it as an SDP in the
appropriate basis
TODO: Support constraints
\min_y c^\top y
subject to M(y) \succeq 0
y_1 = 1
@param poly - polynomial to optimize
@param basis - basis for the polynomial (can be 'power',
'symmetric', 'symmetric_irreducible')
@returns (p*, y*) - the optimal value of the polynomial and
y* = E_\mu[b(x)].
"""
# basis in sorted order from 1 to largest
# Get the largest monomial term
max_degree = get_max_degree(pol)
half_degree = map( lambda deg: int(np.ceil(deg/2.)), max_degree)
# Generate basis
basis = list(it.product( *[range(deg + 1) for deg in max_degree] ) )
half_basis = list(it.product( *[range(deg + 1) for deg in half_degree] ) )
N, M = len(basis), len(half_basis)
# The coefficient term
c = np.matrix([float(pol.nth(*elem)) for elem in basis]).T
# The inequality term, enforcing sos-ness
G = np.zeros( ( M**2, N ) )
for i, j in it.product(xrange(M), xrange(M)):
e1, e2 = half_basis[i], half_basis[j]
monom = tuple_add(e1,e2)
k = basis.index(monom)
if k != -1:
G[ M * i + j, k ] = 1
h = np.zeros((M, M))
# Finally, y_1 = 1
A = np.zeros((1, N))
A[0,0] = 1
b = np.zeros(1)
b[0] = 1
sol = solvers.sdp(
c = matrix(c),
Gs = [matrix(-G)],
hs = [matrix(h)],
A = matrix(A),
b = matrix(b),
)
assert sol['status'] == 'optimal'
return dict([ (get_monom(pol,coeff), val) for coeff, val in zip(basis, list(sol['x'])) ] )
def test_1dmog(mus=[-2., 2.], sigs=[1., np.sqrt(3.)], pis=[0.5, 0.5], deg = 3, obsdeg = 6):
print 'testing 1d mixture of Gaussians'
# constraints on parameters here
K = len(mus)
mu,sig = sp.symbols('mu,sigma')
M = mm.MomentMatrix(deg, [mu, sig], morder='grevlex')
num_constrs = obsdeg + 1; # so observe num_constrs-1 moments
H = abs(hermite_coeffs(num_constrs))
constrs = [0]*num_constrs
for order in range(num_constrs):
for i in range(order+1):
constrs[order] = constrs[order] + H[order,i]* mu**(i) * sig**(order-i)
constrsval = 0
for k in range(K):
constrsval += pis[k]*constrs[order].evalf(subs={mu:mus[k], sig:sigs[k]})
constrs[order] -= constrsval
print constrs
cin = M.get_cvxopt_inputs(constrs[1:])
gs = [sig-0.2, 5-sig, sig+5, 10-mu, mu+10]
#gs = [sig-0.2]
locmatrices = [mm.LocalizingMatrix(M, g) for g in gs]
Ghs = [lm.get_cvxopt_Gh() for lm in locmatrices]
Gs=cin['G']# + [Gh['G'] for Gh in Ghs]
hs=cin['h']# + [Gh['h'] for Gh in Ghs]
sol = solvers.sdp(cin['c'],Gs=Gs, \
hs=hs, A=cin['A'], b=cin['b'])
## Q = cin['A'].T*cin['A']
## weight = 1;
## dims = {}
## dims['l'] = 0; dims['q'] = []; dims['s'] = [len(M.row_monos)]
## ipdb.set_trace()
## sol = solvers.coneqp(Q*weight, cin['A'].T*cin['b']*weight + cin['c'],G=Gs[0], \
## h=hs[0][:], dims = dims, A=cin['A'], b=cin['b'])
for i,mono in enumerate(M.matrix_monos):
trueval = 0;
if i>0:
for k in range(K):
trueval += pis[k]*mono.evalf(subs={mu:mus[k], sig:sigs[k]})
else:
trueval = 1
print '%s:\t%f\t%f' % (str(mono), sol['x'][i], trueval)
print M.extract_solutions_lasserre(sol['x'], Kmax=len(mus))
#import MomentMatrixSolver as mmsolver
#print mmsolver.alternating_solver(M, constrs, 2, maxiter=30000)
return M,sol
def test_dsdp(self):
# Setup problem
c = matrix([1., -1., 1.])
G = [matrix([[-7., -11., -11., 3.],
[7., -18., -18., 8.],
[-2., -8., -8., 1.]])]
G += [matrix([[-21., -11., 0., -11., 10., 8., 0., 8., 5.],
[0., 10., 16., 10., -10., -10., 16., -10., 3.],
[-5., 2., -17., 2., -6., 8., -17., 8., 6.]])]
h = [matrix([[33., -9.], [-9., 26.]])]
h += [matrix([[14., 9., 40.], [9., 91., 10.], [40., 10., 15.]])]
# Solve with default solver
sol = solvers.sdp(c, Gs=G, hs=h)
# Solve with DSDP
sol_dsdp = solvers.sdp(c, Gs=G, hs=h, solver='dsdp')
# Compare solutions
self.assertMatrixAlmostEqual(sol['x'], sol_dsdp['x'])
self.assertMatrixAlmostEqual(sol['zs'][0], sol_dsdp['zs'][0])
self.assertMatrixAlmostEqual(sol['zs'][1], sol_dsdp['zs'][1])
def _calculate_lovasz_embeddings_(A):
"""Calculate the lovasz embeddings for the given graph.
Parameters
----------
A : np.array, ndim=2
The adjacency matrix.
Returns
-------
lovasz_theta : float
Returns the lovasz theta number.
optimization_solution : np.array
Returns the slack variable solution
of the primal optimization program.
"""
if A.shape[0] == 1:
return 1.0
else:
tf_adjm = (np.abs(A) <= min_weight)
np.fill_diagonal(tf_adjm, False)
i_list, j_list = np.nonzero(np.triu(tf_adjm.astype(int), k=1))
nv = A.shape[0]
ne = len(i_list)
x_list = list()
e_list = list()
for (e, (i, j)) in enumerate(zip(i_list, j_list)):
e_list.append(int(e)), x_list.append(int(i * nv + j))
if tf_adjm[i, j]:
e_list.append(int(e)), x_list.append(int(i * nv + j))
# Add on the last row, diagonal elements
e_list = e_list + (nv * [ne])
x_list = x_list + [int(i * nv + i) for i in range(nv)]
# initialise g sparse (to values -1, based on two list that
# define index and one that defines shape
g_sparse = spmatrix(-1, x_list, e_list, (nv * nv, ne + 1))
# Initialise optimization parameters
h = matrix(-1.0, (nv, nv))
c = matrix([0.0] * ne + [1.0])
# Solve the convex optimization problem
sol = sdp(c, Gs=[g_sparse], hs=[h])
return np.array(sol['ss'][0]), sol['x'][ne]
def test_unimixture():
print 'testing simple unimixture with a skipped observation'
x = sp.symbols('x')
M = mm.MomentMatrix(3, [x], morder='grevlex')
constrs = [x-1.5, x**2-2.5, x**4-8.5]
cin = M.get_cvxopt_inputs(constrs)
sol = solvers.sdp(cin['c'], Gs=cin['G'], \
hs=cin['h'], A=cin['A'], b=cin['b'])
print sol['x']
print abs(sol['x'][3]-4.5)
assert(abs(sol['x'][3]-4.5) <= 1e-5)
print M.extract_solutions_lasserre(sol['x'], Kmax = 2)
return M,sol
def test_unimixture():
print 'testing simple unimixture with a skipped observation'
x = sp.symbols('x')
M = mm.MomentMatrix(3, [x], morder='grevlex')
constrs = [x - 1.5, x**2 - 2.5, x**4 - 8.5]
cin = M.get_cvxopt_inputs(constrs)
sol = solvers.sdp(cin['c'], Gs=cin['G'], \
hs=cin['h'], A=cin['A'], b=cin['b'])
print sol['x']
print abs(sol['x'][3] - 4.5)
assert (abs(sol['x'][3] - 4.5) <= 1e-5)
print M.extract_solutions_lasserre(sol['x'], Kmax=2)
return M, sol
def solve_basic_constraints(MM, constraints, slack = 1e-2):
"""
Solve using the moment matrix.
Use @symbols with basis bounded by degree @deg.
Also use the constraints.
"""
cin = get_cvxopt_inputs(MM, constraints, slack=slack)
Bf = MM.get_Bflat()
R = np.random.rand(len(MM), len(MM))
W = R.dot(R.T)
#W = np.eye(len(MM))
w = Bf.dot(W.flatten())
solsdp = cvxsolvers.sdp(c=cin['c'], Gs=cin['G'], hs=cin['h'], Gl=cin['Gl'], hl=cin['hl'])
#ipdb.set_trace()
return solsdp
def test_bimixture():
print 'testing simple unimixture with a skipped observation'
x,y = sp.symbols('x,y')
constrs = [y+0.5, x-0.5, x+y-0, x**2+y-0, x+y**2-1, x+x**2-1, x*y + y**2, x-4*y - 2.5, y**2*x -0.5] # solvable with deg=3, but not deg=2
M = mp.MomentMatrix(2, (x,y), morder='grevlex')
cin = mp.solvers.get_cvxopt_inputs(M, constrs)
sol = solvers.sdp(matrix(sc.rand(*cin['c'].size)), Gs=cin['G'], \
hs=cin['h'], A=cin['A'], b=cin['b'])
M.pretty_print(sol)
print mp.extractors.extract_solutions_lasserre(M, sol['x'], Kmax = 2)
return M,sol
def sdp_solver(cormatrix):
p = np.shape(cormatrix)[0]
c = -np.ones(p)
G1 = np.zeros((p, p * p))
for i in range(p):
G1[i, i * p + i] = 1.0
c = matrix(c)
Gs = [matrix(G1.T)] + [matrix(G1.T)]
hs = [matrix(2 * cormatrix)] + [matrix(np.identity(p))]
G0 = matrix(-(np.identity(p)))
h0 = matrix(np.zeros(p))
sol = solvers.sdp(c, G0, h0, Gs, hs)
return np.array(sol['x'])
def optimizePSD(distMat, numdata):
G0 = matrix(
np.concatenate([-np.eye(numdata * numdata),
np.eye(numdata * numdata)]))
h0 = matrix(
np.concatenate([
np.zeros([numdata * numdata, 1]),
np.ones([numdata * numdata, 1])
]))
wopt = []
for nvar in range(len(distMat)):
c = np.reshape(distMat[nvar], [numdata * numdata, 1])
stat = np.sort(np.reshape(c, [numdata * numdata]))
c = matrix(c) - np.median(c)
sol = solvers.sdp(c, Gl=G0, hl=h0)
#print (np.round((np.reshape(sol['x'], [numdata, numdata])), 3))
wopt.append(np.round(np.reshape(sol['x'], [numdata, numdata]), 3))
return wopt
def test_bimixture():
print 'testing simple unimixture with a skipped observation'
x, y = sp.symbols('x,y')
constrs = [
y + 0.5, x - 0.5, x + y - 0, x**2 + y - 0, x + y**2 - 1, x + x**2 - 1,
x * y + y**2, x - 4 * y - 2.5, y**2 * x - 0.5
] # solvable with deg=3, but not deg=2
M = mp.MomentMatrix(2, (x, y), morder='grevlex')
cin = mp.solvers.get_cvxopt_inputs(M, constrs)
sol = solvers.sdp(matrix(sc.rand(*cin['c'].size)), Gs=cin['G'], \
hs=cin['h'], A=cin['A'], b=cin['b'])
M.pretty_print(sol)
print mp.extractors.extract_solutions_lasserre(M, sol['x'], Kmax=2)
return M, sol
def _design_knockoff_sdp(exog):
"""
Use semidefinite programming to construct a knockoff design
matrix.
Requires cvxopt to be installed.
"""
try:
from cvxopt import solvers, matrix
except ImportError:
raise ValueError("SDP knockoff designs require installation of cvxopt")
nobs, nvar = exog.shape
# Standardize exog
xnm = np.sum(exog**2, 0)
xnm = np.sqrt(xnm)
exog /= xnm
Sigma = np.dot(exog.T, exog)
c = matrix(-np.ones(nvar))
h0 = np.concatenate((np.zeros(nvar), np.ones(nvar)))
h0 = matrix(h0)
G0 = np.concatenate((-np.eye(nvar), np.eye(nvar)), axis=0)
G0 = matrix(G0)
h1 = 2 * Sigma
h1 = matrix(h1)
i, j = np.diag_indices(nvar)
G1 = np.zeros((nvar*nvar, nvar))
G1[i*nvar + j, i] = 1
G1 = matrix(G1)
solvers.options['show_progress'] = False
sol = solvers.sdp(c, G0, h0, [G1], [h1])
sl = np.asarray(sol['x']).ravel()
xcov = np.dot(exog.T, exog)
exogn = _get_knmat(exog, xcov, sl)
return exog, exogn, sl
def _design_knockoff_sdp(exog):
"""
Use semidefinite programming to construct a knockoff design
matrix.
Requires cvxopt to be installed.
"""
try:
from cvxopt import solvers, matrix
except ImportError:
raise ValueError("SDP knockoff designs require installation of cvxopt")
nobs, nvar = exog.shape
# Standardize exog
xnm = np.sum(exog**2, 0)
xnm = np.sqrt(xnm)
exog /= xnm
Sigma = np.dot(exog.T, exog)
c = matrix(-np.ones(nvar))
h0 = np.concatenate((np.zeros(nvar), np.ones(nvar)))
h0 = matrix(h0)
G0 = np.concatenate((-np.eye(nvar), np.eye(nvar)), axis=0)
G0 = matrix(G0)
h1 = 2 * Sigma
h1 = matrix(h1)
i, j = np.diag_indices(nvar)
G1 = np.zeros((nvar * nvar, nvar))
G1[i * nvar + j, i] = 1
G1 = matrix(G1)
solvers.options['show_progress'] = False
sol = solvers.sdp(c, G0, h0, [G1], [h1])
sl = np.asarray(sol['x']).ravel()
xcov = np.dot(exog.T, exog)
exogn = _get_knmat(exog, xcov, sl)
return exog, exogn, sl
def test_unimixture():
print 'testing simple unimixture with a skipped observation'
x = sp.symbols('x')
constrs = [x+x**2-2, x+2*x**4-3, 2*x**2+3*x**4-5] # always solvable
constrs = [x+x**2-1, x**2+2*x**3-1, 2*x**3+3*x**4-3, 2*x+x**3 - 0] # always solvable
constrs = [x-1.5, x**2-2.5, x**5-16.5] # solvable with deg=3, but not deg=2
constrs = [x-1.5, x**2-2.5, x**4-8.5] # solvable with deg=3, but not deg=2
M = mp.MomentMatrix(3, [x], morder='grevlex')
cin = mp.solvers.get_cvxopt_inputs(M, constrs)
sol = solvers.sdp(matrix(sc.rand(*cin['c'].size)), Gs=cin['G'], \
hs=cin['h'], A=cin['A'], b=cin['b'])
M.pretty_print(sol)
print mp.extractors.extract_solutions_lasserre(M, sol['x'], Kmax = 2)
return M,sol
def solve_basic_constraints(MM, constraints, slack=1e-2):
"""
Solve using the moment matrix.
Use @symbols with basis bounded by degree @deg.
Also use the constraints.
"""
cin = get_cvxopt_inputs(MM, constraints, slack=slack)
Bf = MM.get_Bflat()
R = np.random.rand(len(MM), len(MM))
W = R.dot(R.T)
#W = np.eye(len(MM))
w = Bf.dot(W.flatten())
solsdp = cvxsolvers.sdp(c=cin['c'],
Gs=cin['G'],
hs=cin['h'],
Gl=cin['Gl'],
hl=cin['hl'])
#ipdb.set_trace()
return solsdp
def test_K_by_D(K=3, D=3, pis=[0.25, 0.25, 0.5], deg=1, degobs=3):
print 'testing the K by D mixture'
np.random.seed(0)
params = np.random.randint(0, 10, size=(K, D))
print 'the true parameters (K by D): ' + str(params) + '\n' + str(pis)
xs = sp.symbols('x1:' + str(D + 1))
print xs
M = mm.MomentMatrix(deg, xs, morder='grevlex')
constrs = [0] * len(M.matrix_monos)
for i, mono in enumerate(M.matrix_monos):
constrs[i] = mono
constrsval = 0
for k in range(K):
subsk = {xs[i]: params[k, i]
for i in range(D)}
constrsval += pis[k] * mono.evalf(subs=subsk)
constrs[i] -= constrsval
#print constrs
filtered_constrs = [
constr for constr in constrs[1:]
if constr.as_poly().total_degree() <= degobs
]
print filtered_constrs
cin = M.get_cvxopt_inputs(filtered_constrs)
sol = solvers.sdp(cin['c'], Gs=cin['G'], \
hs=cin['h'], A=cin['A'], b=cin['b'])
print 'Mono:\tEstimmated\tTrue'
for i, mono in enumerate(M.matrix_monos):
trueval = 0
if i > 0:
for k in range(K):
subsk = {xs[i]: params[k, i]
for i in range(D)}
trueval += pis[k] * mono.evalf(subs=subsk)
else:
trueval = 1
print '%s:\t%f\t%f' % (str(mono), sol['x'][i], trueval)
print M.extract_solutions_lasserre(sol['x'], Kmax=K)
return M, sol
def SDP(rho, q, n):
c = np.eye(2**n).flatten().tolist()
c = matrix(c)
A, b = [matrix(Amatrix(n).tolist())
], [matrix(np.zeros((1, 2**(n - 1) * (2**n - 1))).tolist())]
A, b = cvxopt.matrix(A), cvxopt.matrix(b)
G = [matrix((-1 * np.eye(2**n * 2**n)).tolist())]
h = [
matrix((-q[len(rho) - 1] * kronecker_prod(rho[len(rho) - 1])).tolist())
]
for _ in range(len(rho) - 1):
G += [matrix((-1 * np.eye(2**n * 2**n)).tolist())]
h += [matrix((-q[_] * kronecker_prod(rho[_])).tolist())]
solvers.options['show_progress'] = False
sol = solvers.sdp(c, Gs=G, hs=h, A=A, b=b)
v = []
for _ in range(2**n * 2**n):
v.append(sol['x'][_])
value = np.array(v) @ np.eye(2**n).flatten()
return value
def solve_Q(x_dim, A, B, D):
# x = [s vec(Z) vec(Q)]
c_dim = 1 + 2 * x_dim * (x_dim + 1) / 2
c = zeros(c_dim)
c[0] = x_dim
prev = 1
for i in range(x_dim):
vec_pos = prev + i * (i + 1) / 2 + i
c[vec_pos] = 1
cm = matrix(c)
G = construct_coeff_matrix(x_dim, B)
G = -G # set negative since s = h - Gx in cvxopt's sdp solver
Gs = [matrix(G)]
h, _ = construct_const_matrix(x_dim, A, B, D)
hs = [matrix(h)]
sol = solvers.sdp(cm, Gs=Gs, hs=hs)
return sol, c, G, h
def normal_SDP_solver(Z1, Z2, input_dim1, input_dim2, Z3=None, Z4=None):
C = np.matmul(np.transpose(Z2), Z1)
if (FLAGS.model_size == 'small' and Z3 is not None and Z4 is not None):
C -= np.matmul(np.transpose(Z4), Z3)
elif (Z3 is not None):
C -= np.matmul(np.transpose(Z3), Z1)
C = (C + np.transpose(C)) / 2
A = np.ones((input_dim2, input_dim1))
b = 1
c, G, h = convert_dual_SDP(C, A, b)
solvers.options['show_progress'] = False
solution = solvers.sdp(c, Gs=G, hs=h)
#print('solution zs ---- ', solution['zs'][0])
dual_theta = np.array(solution['zs'][0])
#print('theta ---- ', dual_theta)
dual_theta = np.reshape(dual_theta, (input_dim1, input_dim2))
return dual_theta
def testsdp(opts):
c = matrix([1.,-1.,1.])
G = [ matrix([[-7., -11., -11., 3.],
[ 7., -18., -18., 8.],
[-2., -8., -8., 1.]]) ]
G += [ matrix([[-21., -11., 0., -11., 10., 8., 0., 8., 5.],
[ 0., 10., 16., 10., -10., -10., 16., -10., 3.],
[ -5., 2., -17., 2., -6., 8., -17., -7., 6.]]) ]
h = [ matrix([[33., -9.], [-9., 26.]]) ]
h += [ matrix([[14., 9., 40.], [9., 91., 10.], [40., 10., 15.]]) ]
sol = solvers.sdp(c, Gs=G, hs=h)
print "x = \n", str2(sol['x'], "%.9f")
print "zs[0] = \n", helpers.str2(sol['zs'][0], "%.9f")
print "zs[1] = \n", helpers.str2(sol['zs'][1], "%.9f")
helpers.run_go_test("../testsdp", {'x': sol['x'],
'ss0': sol['ss'][0],
'ss1': sol['ss'][1],
'zs0': sol['zs'][0],
'zs0': sol['zs'][1]})
def wcls(A, Ap, b):
"""
Solves the robust least squares problem
minimize sup_{||u||<= 1} || (A + sum_k u[k]*Ap[k])*x - b ||_2^2.
A is mxn. Ap is a list of mxn matrices. b is mx1.
"""
(m, n), p = A.size, len(Ap)
# minimize t + v
# subject to [ I * * ]
# [ P(x)' v*I -* ] >= 0.
# [ (A*x-b)' 0 t ]
#
# where P(x) = [Ap[0]*x, Ap[1]*x, ..., Ap[p-1]*x].
#
# Variables x (n), v (1), t(1).
novars = n + 2
M = m + p + 1
c = matrix( n*[0.0] + 2*[1.0])
Fs = [spmatrix([],[],[], (M**2, novars))]
for k in range(n):
# coefficient of x[k]
S = spmatrix([], [], [], (M, M))
for j in range(p):
S[m+j, :m] = -Ap[j][:,k].T
S[-1, :m ] = -A[:,k].T
Fs[0][:,k] = S[:]
# coefficient of v
Fs[0][(M+1)*m : (M+1)*(m+p) : M+1, -2] = -1.0
# coefficient of t
Fs[0][-1, -1] = -1.0
hs = [matrix(0.0, (M, M))]
hs[0][:(M+1)*m:M+1] = 1.0
hs[0][-1, :m] = -b.T
return solvers.sdp(c, None, None, Fs, hs)['x'][:n]
def wcls(A, Ap, b):
"""
Solves the robust least squares problem
minimize sup_{||u||<= 1} || (A + sum_k u[k]*Ap[k])*x - b ||_2^2.
A is mxn. Ap is a list of mxn matrices. b is mx1.
"""
(m, n), p = A.size, len(Ap)
# minimize t + v
# subject to [ I * * ]
# [ P(x)' v*I -* ] >= 0.
# [ (A*x-b)' 0 t ]
#
# where P(x) = [Ap[0]*x, Ap[1]*x, ..., Ap[p-1]*x].
#
# Variables x (n), v (1), t(1).
novars = n + 2
M = m + p + 1
c = matrix(n * [0.0] + 2 * [1.0])
Fs = [spmatrix([], [], [], (M**2, novars))]
for k in range(n):
# coefficient of x[k]
S = spmatrix([], [], [], (M, M))
for j in range(p):
S[m + j, :m] = -Ap[j][:, k].T
S[-1, :m] = -A[:, k].T
Fs[0][:, k] = S[:]
# coefficient of v
Fs[0][(M + 1) * m:(M + 1) * (m + p):M + 1, -2] = -1.0
# coefficient of t
Fs[0][-1, -1] = -1.0
hs = [matrix(0.0, (M, M))]
hs[0][:(M + 1) * m:M + 1] = 1.0
hs[0][-1, :m] = -b.T
return solvers.sdp(c, None, None, Fs, hs)['x'][:n]
def solve_A(x_dim, B, C, E, D, Q, max_iters, show_display):
# x = [s vec(Z) vec(A)]
c_dim = int(1 + x_dim * (x_dim + 1) / 2 + x_dim ** 2)
c = zeros((c_dim, 1))
c[0] = x_dim
prev = 1
for i in range(x_dim):
vec_pos = int(prev + i * (i + 1) / 2 + i)
c[vec_pos] = 1.
cm = matrix(c)
# Scale objective down by T for numerical stability
eigsQinv = max([abs(1. / q) for q in eig(Q)[0]])
eigsE = max([abs(e) for e in eig(E)[0]])
eigsCB = max([abs(cb) for cb in eig(C - B)[0]])
S = max(eigsQinv, eigsE, eigsCB)
Qdown = Q / S
Edown = E / S
Cdown = C / S
Bdown = B / S
# Ensure that D doesn't have negative eigenvals
# due to numerical issues
min_D_eig = min(eig(D)[0])
if min_D_eig < 0:
# assume abs(min_D_eig) << 1
D = D + 2 * abs(min_D_eig) * eye(x_dim)
Gs, _, _, _ = construct_coeff_matrix(x_dim, Qdown, Cdown, Bdown, Edown)
for i in range(len(Gs)):
Gs[i] = -Gs[i] + 1e-6
hs = construct_const_matrix(x_dim, D)
solvers.options['maxiters'] = max_iters
solvers.options['show_progress'] = show_display
sol = solvers.sdp(cm, Gs=Gs, hs=hs)
# check norm of A:
avec = np.array(sol['x'])
avec = avec[int(1 + x_dim * (x_dim + 1) / 2):]
A = np.reshape(avec, (x_dim, x_dim), order='F')
return sol, c, Gs, hs
def sdp_expl_solve(sym_mat_list, smallest_eig=0.001):
"""
:param sym_mat_list: list of symmetric matrices G_0, G_1, ..., G_n of same size
:param smallest_eig: parameter (default 0) may be set to small positive quantity to force non-degeneracy
:return: solver_status, a string, either 'optimal', 'infeasible', or 'unknown', and sol_vec, a vector approximately
optimizing the SDP problem if solver_status is 'optimal', and nan instead
"""
obj_vec = -matrix(get_explicit_rep_objective(sym_mat_list))
_hs = sym_mat_list[0] - smallest_eig * np.eye(sym_mat_list[0].shape[0])
sym_grams = matrix(flatten(sym_mat_list[1:])).T
sol = solvers.sdp(c=obj_vec,
Gs=[-sym_grams],
hs=[matrix(_hs)],
solver='dsdp',
options=DSDP_OPTIONS)
_status = sol['status']
if _status == 'optimal':
return 'Optimal solution found', sol['x']
elif 'infeasible' in _status:
return 'infeasible', nan
else:
return 'unknown', nan
def solve_A(x_dim, B, C, E, D, Q):
# x = [s vec(Z) vec(A)]
MAX_ITERS=30
c_dim = 1 + x_dim*(x_dim+1)/2 + x_dim**2
c = zeros(c_dim)
c[0] = x_dim
prev = 1
for i in range(x_dim):
vec_pos = prev + i * (i+1)/2 + i
c[vec_pos] = 1.
cm = matrix(c)
G,_,_,_ = construct_coeff_matrix(x_dim, Q, C, B, E)
G = -G # set negative since s = h - Gx in cvxopt's sdp solver
Gs = [matrix(G)]
h = construct_const_matrix(x_dim, Q, D)
hs = [matrix(h)]
solvers.options['maxiters'] = MAX_ITERS
sol = solvers.sdp(cm, Gs = Gs, hs=hs)
return sol, c, G, h
def solve_W(Xstar, rank):
"""
minimize trace(Xstar*W)
s.t. 0 \preceq W \preceq I
trace(W) = n - rank
"""
Balpha = []
numrow = Xstar.shape[0]
lowerdiaginds = [(i,j) for (i,j) in \
itertools.product(xrange(numrow), xrange(numrow)) if i>j]
diaginds = [i+i*numrow for i in xrange(numrow)]
for i in diaginds:
indices = [i]
values = [-1]
Balpha += [spmatrix(values, [0]*len(indices), \
indices, size=(1,numrow*numrow), tc='d')]
for i,j in lowerdiaginds:
indices = [i+j*numrow, j+i*numrow]
values = [-1, -1]
Balpha += [spmatrix(values, [0]*len(indices), \
indices, size=(1,numrow*numrow), tc='d')]
Gs = [sparse(Balpha, tc='d').trans()] + [-sparse(Balpha, tc='d').trans()]
hs = [matrix(np.zeros((numrow, numrow)))] + [matrix(np.eye(numrow))]
A = sparse(spmatrix([1]*numrow, [0]*numrow, \
range(numrow), size=(1,numrow*(numrow+1)/2), tc='d'))
b = matrix([numrow - rank], size=(1,1), tc='d')
x = [np.sum(Xstar.flatten()*matrix(Balphai)) for Balphai in Balpha]
sol = cvxsolvers.sdp(-matrix(x), Gs=Gs, hs=hs, A=A, b=b)
w = sol['x']
Wstar = 0
for i,val in enumerate(w):
Wstar += -val*np.array(Balpha[i])
#ipdb.set_trace()
return Wstar,sol
def maxcutrel(A, solver=None):
from cvxopt.base import matrix as m
from cvxopt.base import spmatrix as spm
from cvxopt import solvers
if solver=='dsdp':
from cvxopt import dsdp
solvers.options['show_progress']=True
# solvers.options['show_progress']=False
# data type and dimensions
try:
A = A.weighted_adjacency_matrix()
return maxcutrel(A, solver=solver)
except (AttributeError,TypeError):
try:
A = A.adjacency_matrix()
return maxcutrel(A, solver=solver)
except (AttributeError,TypeError):
pass
pass
try:
d = A.nrows()
# SDP constraints
c = m([-1.]*d)
G = [spm(1., [i*(d+1) for i in range(d)], range(d), (d*d,d))]
sol = solvers.sdp(c, Gs=G, hs=[m(1.*A.numpy())], solver=solver)
return 0.25*(sum(sum(A))+sol['primal objective']),sol
#return sol['primal objective'], sol['status']
except AttributeError:
print "Wrong matrix format"
raise
else:
print "Unknown error"
raise
def test_K_by_D(K=3, D=3, pis=[0.25, 0.25, 0.5], deg=1, degobs=3):
print 'testing the K by D mixture'
np.random.seed(0); params = np.random.randint(0,10,size=(K,D))
print 'the true parameters (K by D): '+str(params) + '\n' + str(pis)
xs = sp.symbols('x1:'+str(D+1))
print xs
M = mm.MomentMatrix(deg, xs, morder='grevlex')
constrs = [0] * len(M.matrix_monos)
for i,mono in enumerate(M.matrix_monos):
constrs[i] = mono
constrsval = 0;
for k in range(K):
subsk = {xs[i]:params[k,i] for i in range(D)};
constrsval += pis[k]*mono.evalf(subs=subsk)
constrs[i] -= constrsval
#print constrs
filtered_constrs = [constr for constr in constrs[1:] if constr.as_poly().total_degree()<=degobs]
print filtered_constrs
cin = M.get_cvxopt_inputs(filtered_constrs)
sol = solvers.sdp(cin['c'], Gs=cin['G'], \
hs=cin['h'], A=cin['A'], b=cin['b'])
print 'Mono:\tEstimmated\tTrue'
for i,mono in enumerate(M.matrix_monos):
trueval = 0;
if i>0:
for k in range(K):
subsk = {xs[i]:params[k,i] for i in range(D)};
trueval += pis[k]*mono.evalf(subs=subsk)
else:
trueval = 1
print '%s:\t%f\t%f' % (str(mono), sol['x'][i], trueval)
print M.extract_solutions_lasserre(sol['x'], Kmax=K)
return M,sol
def solve_ith_GMP(MM, objective, gs, hs, slack = 0):
"""
Generalized moment problem solver
@param - objective: a sympy expression that is the objective
@param - gs: list of contraints defining the semialgebraic set K,
each g corresponds to localizing matrices.
@param - hs: constraints on the moments,
not needed for polynomial optimization.
@param - slack: add equalities as pairs of inequalities with slack separation.
"""
cin = get_cvxopt_inputs(MM, hs, slack=slack)
Bf = MM.get_Bflat()
objcoeff,__ = MM.get_Ab([objective], cvxoptmode = False)
locmatrices = [LocalizingMatrix(MM, g) for g in gs]
Ghs = [get_cvxopt_Gh(lm) for lm in locmatrices]
# list addition
Gs=cin['G'] + [Gh['G'] for Gh in Ghs]
hs=cin['h'] + [Gh['h'] for Gh in Ghs]
# print Ghs
solsdp = cvxsolvers.sdp(matrix(objcoeff[0,:]), Gs=Gs, hs=hs, A=cin['A'], b=cin['b'])
#ipdb.set_trace()
return solsdp
def testsdp(opts):
c = matrix([1.0, -1.0, 1.0])
G = [matrix([[-7.0, -11.0, -11.0, 3.0], [7.0, -18.0, -18.0, 8.0], [-2.0, -8.0, -8.0, 1.0]])]
G += [
matrix(
[
[-21.0, -11.0, 0.0, -11.0, 10.0, 8.0, 0.0, 8.0, 5.0],
[0.0, 10.0, 16.0, 10.0, -10.0, -10.0, 16.0, -10.0, 3.0],
[-5.0, 2.0, -17.0, 2.0, -6.0, 8.0, -17.0, -7.0, 6.0],
]
)
]
h = [matrix([[33.0, -9.0], [-9.0, 26.0]])]
h += [matrix([[14.0, 9.0, 40.0], [9.0, 91.0, 10.0], [40.0, 10.0, 15.0]])]
solvers.options.update(opts)
sol = solvers.sdp(c, Gs=G, hs=h)
# localcones.options.update(opts)
# sol = localcones.sdp(c, Gs=G, hs=h)
print "x = \n", helpers.str2(sol["x"], "%.9f")
print "zs[0] = \n", helpers.str2(sol["zs"][0], "%.9f")
print "zs[1] = \n", helpers.str2(sol["zs"][1], "%.9f")
print "\n *** running GO test ***"
rungo(sol)
'''
Created on Apr 16, 2013
@author: Nguyen Huu Hiep
'''
from cvxopt import matrix, solvers
c = matrix([1.,-1.,1.])
G = [ matrix([[-7., -11., -11., 3.],
[ 7., -18., -18., 8.],
[-2., -8., -8., 1.]]) ]
G += [ matrix([[-21., -11., 0., -11., 10., 8., 0., 8., 5.],
[ 0., 10., 16., 10., -10., -10., 16., -10., 3.],
[ -5., 2., -17., 2., -6., 8., -17., -7., 6.]]) ]
h = [ matrix([[33., -9.], [-9., 26.]]) ]
h += [ matrix([[14., 9., 40.], [9., 91., 10.], [40., 10., 15.]]) ]
sol = solvers.sdp(c, Gs=G, hs=h)
print(sol['x'])
#[-3.68e-01]
#[ 1.90e+00]
#[-8.88e-01]
print(sol['zs'][0])
#[ 3.96e-03 -4.34e-03]
#[-4.34e-03 4.75e-03]
print(sol['zs'][1])
#[ 5.58e-02 -2.41e-03 2.42e-02]
#[-2.41e-03 1.04e-04 -1.05e-03]
#[ 2.42e-02 -1.05e-03 1.05e-02]
def find_ellipsoid(self):
# return None if not enough data is available
if(self.pa.size[1] < 3 or self.pb.size[1] < 3):
return None
# homogenize coordinates
pah = self.homogenize(self.pa)
pbh = self.homogenize(self.pb)
dim = pah.size[0]
num_pah = pah.size[1]
num_pbh = pbh.size[1]
# get the c vector
num_vars = 1+(dim-1)*(dim-1)+2*num_pah+dim*dim
c = matrix([0.0]*num_vars, (num_vars,1))
c[0,0] = -self.c1
for i in xrange(dim-1):
for j in xrange(dim-1):
if(i == j):
c[1+i*(dim-1)+j,0] = self.c2
for i in xrange(num_pah):
c[1+(dim-1)*(dim-1)+i,0] = self.c3
for i in xrange(num_pah):
c[1+(dim-1)*(dim-1)+num_pah+i,0] = 0.0
for i in xrange(dim*dim):
c[1+(dim-1)*(dim-1)+num_pah+num_pah+i,0] = 0.0
# get Gl and hl
# separation constraint
Gl_t = []
for i in xrange(num_pah):
row = [0.0]*(1+(dim-1)*(dim-1)+num_pah)+[0.0]*num_pah
row[(1+(dim-1)*(dim-1)+num_pah)+i] = -1.0
for j in xrange(dim):
for k in xrange(dim):
row.append(pah[j,i]*pah[k,i])
Gl_t.append(row)
for i in xrange(num_pbh):
row = [1.0]+[0.0]*((dim-1)*(dim-1)+2*num_pah)
for j in xrange(dim):
for k in xrange(dim):
row.append(-1.0*pbh[j,i]*pbh[k,i])
Gl_t.append(row)
# l1 norm constraint
for i in xrange(num_pah):
row = [0.0]*num_vars
row[1+(dim-1)*(dim-1)+i] = -1.0
row[1+(dim-1)*(dim-1)+num_pah+i] = -1.0
Gl_t.append(row)
for i in xrange(num_pah):
row = [0.0]*num_vars
row[1+(dim-1)*(dim-1)+i] = -1.0
row[1+(dim-1)*(dim-1)+num_pah+i] = 1.0
Gl_t.append(row)
# construct Gl and hl
Gl = matrix(Gl_t).trans()
col = [0.0]*(num_pah+num_pbh)+[-1.0]*num_pah+[1.0]*num_pah
hl = matrix(col,(3*num_pah+num_pbh,1))
# get Gs and hs
# positive semidefinite constraint
# E must be positive semidefinite
Gs = []
for i in xrange(1+(dim-1)*(dim-1)+2*num_pah):
Gs.append([0.0]*(dim*dim))
for i in xrange(dim*dim):
v = [0.0]*(dim*dim)
rpos = int(math.floor(float(i)/float(dim)))
cpos = i % dim
if(rpos == cpos):
v[i] = -1.0
else:
v[i] = -0.5
v[cpos*dim+rpos] = -0.5
Gs.append(v)
Gs = [matrix(Gs)]
hs = [matrix([0.0]*(dim*dim),(dim,dim))]
# block matrix must be positive semidefinite
Gs2 = []
Gs2.append([0.0]*(((dim-1)*2)*((dim-1)*2)))
for i in xrange((dim-1)*(dim-1)):
v = [0.0]*(((dim-1)*2)*((dim-1)*2))
rpos = int(math.floor(float(i)/float(dim-1)))
cpos = i % (dim-1)
if(rpos == cpos):
v[(rpos+dim-1)*(dim-1)*2+(dim-1)+cpos] = -1.0
else:
v[(rpos+dim-1)*(dim-1)*2+(dim-1)+cpos] = -0.5
v[(cpos+dim-1)*(dim-1)*2+(dim-1)+rpos] = -0.5
Gs2.append(v)
for i in xrange(2*num_pah):
Gs2.append([0.0]*(((dim-1)*2)*((dim-1)*2)))
for i in xrange(dim*dim):
v = [0.0]*(((dim-1)*2)*((dim-1)*2))
rpos = int(math.floor(float(i)/float(dim)))
cpos = i % dim
if(rpos == 0 or cpos == 0):
Gs2.append(v)
else:
rpos = rpos-1
cpos = cpos-1
if(rpos == cpos):
v[rpos*(dim-1)*2+cpos] = -1.0
else:
v[rpos*(dim-1)*2+cpos] = -0.5
v[cpos*(dim-1)*2+rpos] = -0.5
Gs2.append(v)
Gs.append(matrix(Gs2))
zero_block = matrix([0.0]*((dim-1)*(dim-1)),((dim-1), (dim-1)))
eye_block = spmatrix(1.0, range(dim-1), range(dim-1))
hs2 = matrix([[zero_block, eye_block], [eye_block, zero_block]])
hs.append(hs2)
# get A and b
# symmetry constraint
A_t = []
for i in xrange(dim):
for j in xrange(i, dim):
if(i != j):
v = [0.0]*(num_vars)
v[1+(dim-1)*(dim-1)+2*num_pah+i*dim+j] = 1.0
v[1+(dim-1)*(dim-1)+2*num_pah+j*dim+i] = -1.0
A_t.append(v)
for i in xrange(dim-1):
for j in xrange(i, dim-1):
if(i != j):
v = [0.0]*(num_vars)
v[1+i*(dim-1)+j] = 1.0
v[1+j*(dim-1)+i] = -1.0
A_t.append(v)
A = matrix(A_t).trans()
b = matrix([0.0]*A.size[0],(A.size[0],1))
# solve it
passed = False
ntime = 10
while(passed == False and ntime > 0):
try:
sol = solvers.sdp(c, Gl, hl, Gs, hs, A, b)
passed = True
except ZeroDivisionError:
time.sleep(0.001)
ntime = ntime-1
except Exception:
time.sleep(0.001)
ntime = ntime-1
if(passed == False):
tmp_nrow = self.pa.size[0]
tmp_c = mean_ps(self.pa).trans()
tmp_E = matrix(spmatrix(1.0, range(tmp_nrow), range(tmp_nrow)))
tmp_rho = 1.0
return {'c':tmp_c, 'E':tmp_E, 'rho':tmp_rho}
# parse out solution
x = sol['x']
k = x[0]
E_hat = matrix(x[1+(dim-1)*(dim-1)+2*num_pah:], (dim,dim))
F = E_hat[1:,1:]
v = E_hat[1:,0]
s = E_hat[0,0]
ipiv = matrix(0, (dim-1,1))
gesv(-F, v, ipiv)
c = v
btm = 1-(s-c.trans()*F*c)+0.00000001
for i in xrange(F.size[0]):
for j in xrange(F.size[1]):
F[i,j] = F[i,j]/btm
E = F
rho = k
# function return
return {'c':c, 'E':E, 'rho':rho}
# Two variables (t, u).
G = matrix(0.0, ((n+1)**2, 2))
G[-1, 0] = -1.0 # coefficient of t
G[: (n+1)**2-1 : n+2, 1] = -1.0 # coefficient of u
h = matrix( [ [ A.T * A, b.T * A ], [ A.T * b, b.T * b ] ] )
c = matrix(1.0, (2,1))
nopts = 40
alpha1 = [2.0/(nopts//2-1) * alpha for alpha in range(nopts//2) ] + \
[ 2.0 + (15.0 - 2.0)/(nopts//2) * alpha for alpha in
range(1,nopts//2+1) ]
lbnds = [ blas.nrm2(b)**2 ]
for alpha in alpha1[1:]:
c[1:] = alpha
lbnds += [ -blas.dot(c, solvers.sdp(c, Gs=[G], hs=[h])['x']) ]
nopts = 10
alpha2 = [ 1.0/(nopts-1) * alpha for alpha in range(nopts) ]
ubnds = [ blas.nrm2(b)**2 ]
for alpha in alpha2[1:]:
c[1:] = alpha
ubnds += [ blas.dot(c, solvers.sdp(c, Gs=[G], hs=[-h])['x']) ]
try: import pylab
except ImportError: pass
else:
pylab.figure(1, facecolor='w')
pylab.plot(lbnds, alpha1, 'b-', ubnds, alpha2, 'b-')
kmax = max([ k for k in range(len(alpha1)) if alpha1[k] <
blas.nrm2(xls)**2 ])
# maximize w
# subject to w*I <= V*diag(x)*V'
# x >= 0
# sum(x) = 1
novars = n+1
c = matrix(0.0, (novars,1))
c[-1] = -1.0
Gs = [matrix(0.0, (4,novars))]
for k in range(n): Gs[0][:,k] = -(V[:,k]*V[:,k].T)[:]
Gs[0][[0,3],-1] = 1.0
hs = [matrix(0.0, (2,2))]
Ge = matrix(0.0, (n, novars))
Ge[:,:n] = G
Ae = matrix(n*[1.0] + [0.0], (1,novars))
sol = solvers.sdp(c, Ge, h, Gs, hs, Ae, b)
xe = sol['x'][:n]
Z = sol['zs'][0]
mu = sol['y'][0]
if pylab_installed:
pylab.figure(2, facecolor='w', figsize=(6,6))
pylab.plot(V[0,:], V[1,:],'ow', mec = 'k')
pylab.plot([0], [0], 'k+')
I = [ k for k in range(n) if xe[k] > 1e-5 ]
pylab.plot(V[0,I], V[1,I],'or')
# Enclosing ellipse follows from the solution of the dual problem:
#
# minimize mu
# subject to diag(V'*Z*V) <= mu*1
def qcqprel(P, G, r=0.0 ):
"""
min x' Q0 x + a0'x
st
x Qi x + ai x + bi <=0
Input Q size nxn
a size nx1
b is scalar
output c size (n+1)*(n+2)/2 x 1
x Q x + a'x + b <= 0
is converted to
<q,X> <= 0
where q is returned by this function
"""
def augQ(Q, a=None, b=None):
n =Q.size[0]
if Q is None:
Q=matrix(0., (n,n))
if a is None:
a=matrix(0., (n,1))
if b is None:
b=0.
Q0 =matrix([[b, a*0.5], [a.T*0.5, Q]])
jdx=0
q=matrix(0., ((n+1)*(n+2)/2, 1) )
for j in range(n+1):
idx=matrix(range(j,n+1))
tmp=Q0[idx, j]
tmp[1:]*=2
q[jdx+idx]=tmp
jdx+=len(idx)-1
return q
"""
n is integer, size of matrix we want nnd
returns the sparse 'sdp' matrix Q that
sets up the constraint
Force the matrix
[ Y ] >= H
where H is some nxn matrix, (not specified here)
"""
def sdpmat(n=None):
m=n*(n+1)/2
Q =matrix(0., (n**2,m))
J=0
for j in range(n):
for k in range(j,n):
Q[k+j*n, J]=-1.
J+=1
Q=sparse(Q)
return Q
"""
Get the objective
"""
n=P['P0'].size[0]
I=matrix(0., (n,n))
I[::n+1]=r
Q0 =augQ(P['P0']+I, P['b0'], P['c0'])
"""
Equality: force the [0,0] entry of solution to have
value 1.
"""
A=matrix(0., (1,Q0.size[0]))
A[0]=1.
b=matrix(1., (1,1))
"""
Build the inequalities
"""
Gl=matrix(0., (0, Q0.size[0]))
for j in range( len( G['P'] ) ):
Qk=augQ(G['P'][j], G['b'][j], G['c'][j])
Gl=matrix([Gl, Qk.T])
hl=matrix(0., (Gl.size[0], 1))
"""
Build the equalities
"""
for j in range( len( G['Peq'] ) ):
Qk=augQ(G['Peq'][j], G['beq'][j], G['ceq'][j])
A=matrix([A, Qk.T])
b=matrix([b, 0.])
"""
Get the 'semidef' matrix
"""
G0=sdpmat(n+1)
h0=matrix(0., (n+1,n+1))
sol=solvers.sdp(Q0, Gl, hl, [G0],[h0], A, b)
x=sol['x']
"""
Build the matrices 'x' and 'X'
"""
if x is not None:
X=x[1:n+1]
Y=x[n+1:]
YY=matrix(0., (n,n))
J=0
for j in range(n):
for k in range(j,n):
YY[j,k]=Y[J]
YY[k,j]=Y[J]
J+=1
sol['RQCQPx']=+sol['x'][1:n+1]
sol['RQCQPX']=+YY
else:
sol['RQCQPx']=None
sol['RQCQPX']=None
return sol
