def __init__(self, dim, n_labels, mode=1):
self.mode = mode
self.dim = dim
self.n_labels = n_labels
if mode == 1:
self.P = spmatrix(1, range(dim * n_labels), range(dim * n_labels))
glast = matrix(np.ones((1, dim * n_labels)))
self.G = sparse([-self.P, self.P, glast])
h1 = np.zeros(dim * n_labels)
h2 = np.ones(dim * n_labels)
self.h = matrix(np.concatenate([h1, h2, [dim]]))
elif mode == 2:
self.P = spmatrix(1, range(n_labels), range(n_labels))
glast = matrix(np.ones((1, n_labels)))
self.G = sparse([-self.P, self.P, glast])
h1 = np.zeros(n_labels)
h2 = np.ones(n_labels)
self.h = matrix(np.concatenate([h1, h2, [1]]))
elif mode == 3:
self.P = spmatrix(1, range(n_labels), range(n_labels))
self.A = matrix(np.ones((1, n_labels)))
self.G = sparse([-self.P, self.P])
h1 = np.zeros(n_labels)
h2 = np.ones(n_labels)
self.h = matrix(np.concatenate([h1, h2]))
self.b = matrix(np.ones(1))
def get_cvxopt_inputs(MM, constraints = None, slack = 0, sparsemat = True, filter = 'even'):
"""
if provided, constraints should be a list of sympy polynomials that should be 0.
@params - constraints: a list of sympy expressions representing the constraints in the same
"""
# Many optionals for what c might be, not yet determined really
if filter is None:
c = matrix(np.ones((MM.num_matrix_monos, 1)))
else:
c = matrix([monomial_filter(yi, filter='even') for yi in MM.matrix_monos], tc='d')
Anp,bnp = MM.get_Ab(constraints)
#_, residual, _, _ = scipy.linalg.lstsq(Anp, bnp)
b = matrix(bnp)
indicatorlist = MM.get_LMI_coefficients()
Glnp,hlnp = MM.get_Ab_slack(constraints, abs_slack = slack, rel_slack = slack)
hl = matrix(hlnp)
if sparsemat:
G = [sparse(indicatorlist).trans()]
A = sparse(matrix(Anp))
Gl = sparse(matrix(Glnp))
else:
G = [matrix(indicatorlist).trans()]
A = matrix(Anp)
Gl = matrix(Glnp)
num_row_monos = len(MM.row_monos)
h = [matrix(np.zeros((num_row_monos,num_row_monos)))]
return {'c':c, 'G':G, 'h':h, 'A':A, 'b':b, 'Gl':Gl, 'hl':hl}
def F(x=None, z=None):
# Case 1
if (x is None and z is None):
x0 = opt.matrix(np.ones((n, 1))) * 1.0
return (len(fs), x0)
# Case 2
elif (x is not None and z is None):
in_domain = map(lambda y: y(x), inds)
if (reduce(lambda v, w: v and w, in_domain)):
f = opt.matrix(0.0, (len(fs), 1))
for i in range(0, len(fs), 1):
f[i] = fs[i](x)
Df = opt.spmatrix(0.0, [], [], (0, n))
for i in range(0, len(grads), 1):
Df = opt.sparse([Df, grads[i](x).T])
return (f, Df)
else:
return (None, None)
# Case 3
else:
f = opt.matrix(0.0, (len(fs), 1))
for i in range(0, len(fs), 1):
f[i] = fs[i](x)
Df = opt.spmatrix(0.0, [], [], (0, n))
for i in range(0, len(grads), 1):
Df = opt.sparse([Df, grads[i](x).T])
H = opt.spmatrix(0.0, [], [], (n, n))
for i in range(0, len(hess), 1):
H = H + z[i] * hess[i](x)
return (f, Df, H)
def test_pcg():
'Test function for projected CG.'
n = 10
m = 4
H = sprandsym(n, n)
A = sp_rand(m, n, 0.9)
x0 = matrix(1, (n, 1))
b = A * x0
c = matrix(1.0, (n, 1))
x_pcg = pcg(H, c, A, b, x0)
Lhs1 = sparse([H, A])
Lhs2 = sparse([A.T, spmatrix([], [], [], (m, m))])
Lhs = sparse([[Lhs1], [Lhs2]])
rhs = -matrix([c, spmatrix([], [], [], (m, 1))])
rhs2 = copy(rhs)
linsolve(Lhs, rhs)
#print rhs[:10]
sol = solvers.qp(H, c, A=A, b=b)
print ' cvxopt qp| pCG'
print matrix([[sol['x']], [x_pcg]])
print 'Dual variables:'
print sol['y']
print 'KKT equation residuals:'
print H * sol['x'] + c + A.T * sol['y']
def gen_equalities(dm_info, k_additive=0):
"""
Input: dm_info(criteria functions, Shapley values structure, II values structure, Necessity criteria, Sufficiency criteria), k-additivity
Generate matrix rows for different information classes
Output: Matrix and a column, Aeq*v=beq
"""
dim = len(dm_info['criteria_functions'])
Alist = [equalities(2**dim)]
blist = [0., 1]
if 'Sh_values' in dm_info:
Alist.extend([shapley(2**dim, p[0]) for p in dm_info['Sh_values']])
blist.extend([p[1] for p in dm_info['Sh_values']])
Aeq = cvx.sparse(Alist)
beq = cvx.matrix(blist)
if 'necessity' in dm_info:
Aeq = cvx.sparse([Aeq, necessity(2**dim, dm_info['necessity'])])
beq = cvx.matrix(
[beq, cvx.matrix(0., (size(Aeq)[0] - size(beq)[0], 1))])
if 'sufficiency' in dm_info:
Aeq = cvx.sparse([Aeq, sufficiency(2**dim, dm_info['sufficiency'])])
beq = cvx.matrix(
[beq, cvx.matrix(1., (size(Aeq)[0] - size(beq)[0], 1))])
if k_additive:
Aeq = cvx.sparse([Aeq, k_additivity(2**dim, k_additive)])
beq = cvx.matrix(
[beq, cvx.matrix(0., (size(Aeq)[0] - size(beq)[0], 1))])
return Aeq, beq
def _get_lineq(self):
'Extract linear equation coefficients from QP and a partition'
qp = self.QP
# Concatenate active constraints with eq constraints
Aeq = sparse([qp.Aeq, -qp.A[self.cAL, :], qp.A[self.cAU, :]])
beq = matrix([qp.beq, -qp.bl[self.cAL], qp.bu[self.cAU]])
beq -= Aeq[:, self.AL] * qp.l[self.AL] + Aeq[:,
self.AU] * qp.u[self.AU]
Aeq = Aeq[:, self.I]
# [H(I,I);Aeq(:,I)]
Lhs = sparse([qp.H[self.I, self.I], Aeq])
row_Aeq, col_Aeq = Aeq.size
# [Aeq(:,I)';zeros]
if row_Aeq != 0:
AeqT0 = sparse(
[Aeq.T, spmatrix([0], [row_Aeq - 1], [row_Aeq - 1])])
else:
AeqT0 = Aeq.T
# Concatenate by collumn
Lhs = sparse([[Lhs], [AeqT0]])
# Concatenat to yield rhs
rhs = sparse(
matrix([
-qp.c[self.I] - qp.H[self.I, self.AL] * qp.l[self.AL] -
qp.H[self.I, self.AU] * qp.u[self.AU], beq
]))
x0 = matrix(
[self.x[self.I], self.y, self.czl[self.cAL], self.czu[self.cAU]])
return (Lhs, rhs, x0)
def F(x=None,z=None):
# Case 1
if(x is None and z is None):
x0 = opt.matrix(np.ones((n,1)))*1.0
return (len(fs),x0)
# Case 2
elif(x is not None and z is None):
in_domain = map(lambda y: y(x),inds)
if(reduce(lambda v,w: v and w,in_domain)):
f = opt.matrix(0.0,(len(fs),1))
for i in range(0,len(fs),1):
f[i] = fs[i](x)
Df = opt.spmatrix(0.0,[],[],(0,n))
for i in range(0,len(grads),1):
Df = opt.sparse([Df,grads[i](x).T])
return (f,Df)
else:
return (None,None)
# Case 3
else:
f = opt.matrix(0.0,(len(fs),1))
for i in range(0,len(fs),1):
f[i] = fs[i](x)
Df = opt.spmatrix(0.0,[],[],(0,n))
for i in range(0,len(grads),1):
Df = opt.sparse([Df,grads[i](x).T])
H = opt.spmatrix(0.0,[],[],(n,n))
for i in range(0,len(hess),1):
H = H + z[i]*hess[i](x)
return (f,Df,H)
def get_cvxopt_inputs(MM, constraints = None, slack = 0, sparsemat = True, filter = 'even'):
"""
if provided, constraints should be a list of sympy polynomials that should be 0.
@params - constraints: a list of sympy expressions representing the constraints in the same
"""
# Many optionals for what c might be, not yet determined really
if filter is None:
c = matrix(np.ones((MM.num_matrix_monos, 1)))
else:
c = matrix([monomial_filter(yi, filter='even') for yi in MM.matrix_monos], tc='d')
Anp,bnp = MM.get_Ab(constraints)
#_, residual, _, _ = scipy.linalg.lstsq(Anp, bnp)
b = matrix(bnp)
indicatorlist = MM.get_LMI_coefficients()
Glnp,hlnp = MM.get_Ab_slack(constraints, abs_slack = slack, rel_slack = slack)
hl = matrix(hlnp)
if sparsemat:
G = [sparse(indicatorlist).trans()]
A = sparse(matrix(Anp))
Gl = sparse(matrix(Glnp))
else:
G = [matrix(indicatorlist).trans()]
A = matrix(Anp)
Gl = matrix(Glnp)
num_row_monos = len(MM.row_monos)
h = [matrix(np.zeros((num_row_monos,num_row_monos)))]
return {'c':c, 'G':G, 'h':h, 'A':A, 'b':b, 'Gl':Gl, 'hl':hl}
def F(x=None,z=None):
# Case 1
if x is None and z is None:
x0 = opt.matrix(1., (n,1))
return len(fs),x0
# Case 2
elif x is not None and z is None:
if all(list(map(lambda y: y(x),inds))):
f = opt.matrix(0.0,(len(fs),1))
for i in range(0,len(fs),1):
f[i] = fs[i](x)
Df = opt.spmatrix(0.0,[],[],(0,n))
for i in range(0,len(grads),1):
Df = opt.sparse([Df,grads[i](x).T])
return f,Df
else:
return None,None
# Case 3
else:
f = opt.matrix(0.0,(len(fs),1))
for i in range(0,len(fs),1):
f[i] = fs[i](x)
Df = opt.spmatrix(0.0,[],[],(0,n))
for i in range(0,len(grads),1):
Df = opt.sparse([Df,grads[i](x).T])
H = opt.spmatrix(0.0,[],[],(n,n))
for i in range(0,len(hess),1):
H = H + z[i]*hess[i](x)
return f,Df,H
def _get_lineq_c(self):
'Extract linear equation coefficients from QP and a partition'
qp = self.QP
# Inactive but not free
self.realI = setdiff(self.I,self.F)
# Put in the order of [realI,F]
pI = self.realI + self.F
# Concatenate active constraints with eq constraints
Aeq = sparse([qp.Aeq,-qp.A[self.cAL,:],qp.A[self.cAU,:]])
beq = matrix([qp.beq,-qp.bl[self.cAL],qp.bu[self.cAU]])
beq -= Aeq[:,self.AL]*qp.l[self.AL] + Aeq[:,self.AU]*qp.u[self.AU]
Aeq = Aeq[:,pI]
# [H(I,I);Aeq(:,I)]
Lhs = sparse([qp.H[pI,pI], Aeq ])
row_Aeq, col_Aeq = Aeq.size
# [Aeq(:,I)';zeros]
if row_Aeq != 0:
AeqT0 = sparse([Aeq.T,spmatrix([0],[row_Aeq-1],[row_Aeq-1])])
else:
AeqT0 = Aeq.T
# Concatenate by collumn
Lhs = sparse([[Lhs],[AeqT0]])
# Concatenat to yield rhs
rhs = sparse(matrix([-qp.c[pI] -qp.H[pI,self.AL]*qp.l[self.AL] -qp.H[pI,self.AU]*qp.u[self.AU] ,beq]))
x0 = matrix([self.x[pI],self.y,self.czl[self.cAL],self.czu[self.cAU]])
return (Lhs,rhs,x0)
def F(x=None, z=None):
# Case 1
if x is None and z is None:
x0 = opt.matrix(1., (n, 1))
return len(fs), x0
# Case 2
elif x is not None and z is None:
if all(list([y(x) for y in inds])):
f = opt.matrix(0.0, (len(fs), 1))
for i in range(0, len(fs), 1):
f[i] = fs[i](x)
Df = opt.spmatrix(0.0, [], [], (0, n))
for i in range(0, len(grads), 1):
Df = opt.sparse([Df, grads[i](x).T])
return f, Df
else:
return None, None
# Case 3
else:
f = opt.matrix(0.0, (len(fs), 1))
for i in range(0, len(fs), 1):
f[i] = fs[i](x)
Df = opt.spmatrix(0.0, [], [], (0, n))
for i in range(0, len(grads), 1):
Df = opt.sparse([Df, grads[i](x).T])
H = opt.spmatrix(0.0, [], [], (n, n))
for i in range(0, len(hess), 1):
H = H + z[i] * hess[i](x)
return f, Df, H
def solve(self):
'''
"Solves" minimization problem using epigraph linear program
formulation.
'''
A, y, x_dim, y_dim = self.A, self.y, self.x_dim, self.y_dim
A = cvxopt.sparse(cvxopt.matrix(A))
I_t = cvxopt.spdiag(cvxopt.matrix(ones(y_dim)))
I_x = cvxopt.spdiag(cvxopt.matrix(ones(x_dim)))
Z = cvxopt.spmatrix([], [], [], size=(x_dim, y_dim))
A = cvxopt.sparse([
[A, -A, -I_x],
[-I_t, -I_t, Z]
]) # Sparse block matrix in COLUMN major order...for some reason
y = cvxopt.matrix(y)
z = cvxopt.matrix(zeros((x_dim, 1)))
one = cvxopt.matrix(ones((y_dim, 1)))
y = cvxopt.matrix([y, -y, z])
c = cvxopt.matrix([z, one])
result = cvxopt.solvers.lp(c, A, y, solver='glpk')
self.result = result
return result['status']
def test_pcg():
'Test function for projected CG.'
n = 10
m = 4
H = sprandsym(n,n)
A = sp_rand(m,n,0.9)
x0 = matrix(1,(n,1))
b = A*x0
c = matrix(1.0,(n,1))
x_pcg = pcg(H,c,A,b,x0)
Lhs1 = sparse([H,A])
Lhs2 = sparse([A.T,spmatrix([],[],[],(m,m))])
Lhs = sparse([[Lhs1],[Lhs2]])
rhs = -matrix([c,spmatrix([],[],[],(m,1))])
rhs2 = copy(rhs)
linsolve(Lhs,rhs)
#print rhs[:10]
sol = solvers.qp(H,c,A=A,b=b)
print ' cvxopt qp| pCG'
print matrix([[sol['x']],[x_pcg]])
print 'Dual variables:'
print sol['y']
print 'KKT equation residuals:'
print H*sol['x'] + c + A.T*sol['y']
def geteq(n,dic):
'generate the equality constraints.'
# Equality from the Laplace operator
AeqL = -Laplace(n,dic)
h = 1.0/(n-1)
# Equality from the boundary
AeqB = spmatrix([],[],[],(0,n**2+4*n))
for col in range(n):
y = spmatrix([],[],[],(2,n**2+4*n))
y[0,dic[-1,col]] = 1
y[0,dic[1,col]] = -1
y[1,dic[n-2,col]] = -1
y[1,dic[n,col]] = 1
AeqB = sparse([AeqB,y])
for row in range(n):
y = spmatrix([],[],[],(2,n**2+4*n))
y[0,dic[row,-1]] = 1
y[0,dic[row,1]] = -1
y[1,dic[row,n-2]] = -1
y[1,dic[row,n]] = 1
AeqB = sparse([AeqB,y])
AeqL = sparse([[AeqL],[spmatrix([],[],[],(n**2,n*4))]])
AeqB = sparse([[AeqB],[-identity(4*n,h*2)]])
Aeq = sparse([AeqL,AeqB])
beq = spmatrix([],[],[],(n**2+4*n,1))
return (Aeq, beq)
def build_derivative_coef_matrix_and_column_vector(self):
block_heights = Utils.calculate_block_heights(self.no_points_per_axis)
adjacent_offsets = Utils.calculate_adjacent_sub_block_offsets(
self.no_vars, self.no_points_per_axis)
matrices_list = []
for index, block_height in enumerate(block_heights):
distance = Utils.calculate_distance_between_non_zero_entries(
index, self.no_points_per_axis)
no_sub_blocks = Utils.calculate_number_of_sub_blocks(
index, self.no_points_per_axis)
matrices_list.append(
Utils.build_matrix_for_partial_derivative(
adjacent_offsets[index], block_height, no_sub_blocks,
distance))
upper_half_matrix = sparse(matrices_list)
coef_matrix = sparse([upper_half_matrix, -upper_half_matrix])
(l_b_constraints,
u_b_constraints) = self.build_constraints_from_derivative_info()
flat_l_b_constraints = matrix(l_b_constraints)
flat_u_b_constraints = matrix(u_b_constraints)
# l <= x constraint will be rewritten as -x <= -l.
column_vector = matrix([flat_u_b_constraints, -flat_l_b_constraints])
return (coef_matrix, column_vector)
def setup_input_matrix(self):
# constructs B = Bin - Bout
# loop over all sectors
self.Bout = sparse(matrix(0.0, (self.ns, self.nu)))
self.B = sparse(matrix(0.0, (self.ns, self.nu)))
for i in range(self.ns):
Bout_one_row = sparse(matrix(0.0, (1, self.nu)))
Bin_one_row = sparse(matrix(0.0, (1, self.nu)))
# current sector number
s = i + 1
# start index of u_{s_i} in the current row of Bout
Bout_start_i = (s) * self.ns - self.ns - 1
# get Neighbors
N = self.getNeighborSectors(s)
# active indices for the row in Bout corresponding to current sector
Bout_active_i = [Bout_start_i + j for j in N]
# active indices for the row in Bin corresponding to current sector
Bin_active_i = [j * self.ns - self.ns - 1 + s for j in N]
# generate row in Bout
Bout_one_row[Bout_active_i] = 1.0
Bin_one_row[Bin_active_i] = 1.0
# update Bout matrix
self.Bout[i, :] = Bout_one_row
# update B matrix, row-by-row
self.B[i, :] = Bin_one_row - Bout_one_row
#print Bout_active_i
#print Bin_active_i
self.B
return
def _prepare_and_solve_primal(self, X, y):
# Solution vector [{w},w_0,e_1,...e_i]
# See:
# http://cvxopt.org/userguide/matrices.html#sparse-matrices
# http://cvxopt.org/userguide/coneprog.html#cvxopt.solvers.qp
objects, features = X.shape
opt_space_dim = features + 1 + objects
P = spmatrix(1., range(features), range(features),
(opt_space_dim, opt_space_dim))
q = matrix(
spmatrix(self.C / objects, range(features + 1, opt_space_dim),
[0] * objects, (opt_space_dim, 1)))
y_column = -y.reshape(-1, 1)
# G is also a block matrix
Gx = sparse([[matrix(X * y_column)], [matrix(y_column)],
[spmatrix(-1., range(objects), range(objects))]])
Ge = spmatrix(-1., range(objects), range(features + 1, opt_space_dim),
(objects, opt_space_dim))
G = sparse([Gx, Ge])
hx = spmatrix(-1., range(objects), [0] * objects)
he = spmatrix(0, range(objects), [0] * objects)
h = matrix(sparse([hx, he]))
return qp(P, q, G, h)
def test_basic_complex(self):
import cvxopt
a = cvxopt.matrix([1, -2, 3])
b = cvxopt.matrix([1.0, -2.0, 3.0])
c = cvxopt.matrix([1.0 + 2j, 1 - 2j, 0 + 1j])
d = cvxopt.spmatrix(
[complex(1.0, 0.0),
complex(0.0, 1.0),
complex(2.0, -1.0)], [0, 1, 3], [0, 2, 3], (4, 4))
e = cvxopt.spmatrix(
[complex(1.0, 0.0),
complex(0.0, 1.0),
complex(2.0, -1.0)], [2, 3, 3], [1, 2, 3], (4, 4))
self.assertAlmostEqualLists(list(cvxopt.div(b, c)),
[0.2 - 0.4j, -0.4 - 0.8j, -3j])
self.assertAlmostEqualLists(list(cvxopt.div(b, 2.0j)),
[-0.5j, 1j, -1.5j])
self.assertAlmostEqualLists(list(cvxopt.div(a, c)),
[0.2 - 0.4j, -0.4 - 0.8j, -3j])
self.assertAlmostEqualLists(list(cvxopt.div(c, a)),
[(1 + 2j),
(-0.5 + 1j), 0.3333333333333333j])
self.assertAlmostEqualLists(list(cvxopt.div(c, c)), [1.0, 1.0, 1.0])
self.assertAlmostEqualLists(list(cvxopt.div(a, 2.0j)),
[-0.5j, 1j, -1.5j])
self.assertAlmostEqualLists(list(cvxopt.div(c, 1.0j)),
[2 - 1j, -2 - 1j, 1 + 0j])
self.assertAlmostEqualLists(list(cvxopt.div(1j, c)),
[0.4 + 0.2j, -0.4 + 0.2j, 1 + 0j])
self.assertTrue(len(d) + len(e) == len(cvxopt.sparse([d, e])))
self.assertTrue(len(d) + len(e) == len(cvxopt.sparse([[d], [e]])))
def setup_problem(self):
''' Sets up optimizatoin matrices in compact form
min_X C.T*X
s.t.
A*X <= b
See the implementation detatils, in the implementation notes documents
'''
start_t = time.time()
self.ns = self.Nrows * self.Ncols
self.nu = self.ns**2
self.ns = self.Nrows * self.Ncols
self.nu = self.ns**2
self.setup_grid_matrix()
self.setup_input_matrix()
self.Ad = self.setup_dynamics_constraints()
Af = self.setup_flow_constraints()
Ab, bb = self.setup_boundary_constraints()
self.setup_initial_condition_vector()
self.get_Xref()
self.setup_optimization_vector()
# construct compact form optimization matrices
self.A = sparse([Af, Ab])
self.b = sparse([self.X0, bb])
print "Setup is done in: ", time.time() - start_t, "second(s)"
return self.A, self.b
def _convert(H, f, A, b, Aeq, beq, lb, ub):
"""
Convert everything to
cvxopt-style matrices
"""
P = cvxmat(H)
q = cvxmat(f)
if Aeq is None:
A_ = None
else:
A_ = cvxmat(Aeq)
if beq is None:
b_ = None
else:
b_ = cvxmat(beq)
if lb is None and ub is None:
if A is None:
G = None
h = None
else:
G = cvxmat(A)
h = cvxmat(b)
else:
n = len(lb)
if A is None:
G = sparse([-speye(n), speye(n)])
h = cvxmat(np.vstack([-lb, ub]))
else:
G = sparse([cvxmat(A), -speye(n), speye(n)])
h = cvxmat(np.vstack([b, -lb, ub]))
return P, q, G, h, A_, b_
def build_gy(self, dae):
"""Build line Jacobian matrix"""
if not self.n:
idx = range(dae.m)
dae.set_jac(Gy, 1e-6, idx, idx)
return
Vn = polar(1.0, dae.y[self.a])
Vc = mul(dae.y[self.v], Vn)
Ic = self.Y * Vc
diagVn = spdiag(Vn)
diagVc = spdiag(Vc)
diagIc = spdiag(Ic)
dS = self.Y * diagVn
dS = diagVc * conj(dS)
dS += conj(diagIc) * diagVn
dR = diagIc
dR -= self.Y * diagVc
dR = diagVc.H.T * dR
self.gy_store = sparse([[dR.imag(), dR.real()], [dS.real(),
dS.imag()]])
# rebuild = False
return sparse(self.gy_store)
def const_to_matrix(self, value, convert_scalars=False):
"""Convert an arbitrary value into a matrix of type self.target_matrix.
Args:
value: The constant to be converted.
convert_scalars: Should scalars be converted?
Returns:
A matrix of type self.target_matrix or a scalar.
"""
if isinstance(value, (numpy.ndarray, numpy.matrix)):
return cvxopt.sparse(cvxopt.matrix(value.astype('float64')),
tc='d')
elif isinstance(value, numbers.Number):
return cvxopt.sparse(cvxopt.matrix(value), tc='d')
# Convert scipy sparse matrices to coo form first.
elif sp.issparse(value):
value = value.tocoo()
return cvxopt.spmatrix(value.data.tolist(),
value.row.tolist(),
value.col.tolist(),
size=value.shape,
tc='d')
else: # Lists.
return cvxopt.sparse(value, tc='d')
def geteq(n, dic):
'generate the equality constraints.'
# Equality from the Laplace operator
AeqL = -Laplace(n, dic)
h = 1.0 / (n - 1)
# Equality from the boundary
AeqB = spmatrix([], [], [], (0, n**2 + 4 * n))
for col in range(n):
y = spmatrix([], [], [], (2, n**2 + 4 * n))
y[0, dic[-1, col]] = 1
y[0, dic[1, col]] = -1
y[1, dic[n - 2, col]] = -1
y[1, dic[n, col]] = 1
AeqB = sparse([AeqB, y])
for row in range(n):
y = spmatrix([], [], [], (2, n**2 + 4 * n))
y[0, dic[row, -1]] = 1
y[0, dic[row, 1]] = -1
y[1, dic[row, n - 2]] = -1
y[1, dic[row, n]] = 1
AeqB = sparse([AeqB, y])
AeqL = sparse([[AeqL], [spmatrix([], [], [], (n**2, n * 4))]])
AeqB = sparse([[AeqB], [-identity(4 * n, h * 2)]])
Aeq = sparse([AeqL, AeqB])
beq = spmatrix([], [], [], (n**2 + 4 * n, 1))
return (Aeq, beq)
def l1_fit(index, y, beta_d2=1.0, beta_d1=1.0, beta_seasonal=1.0,
beta_step=5.0, period=12, growth=0.0, step_permissives=None):
assert isinstance(y, np.ndarray)
assert isinstance(index, np.ndarray)
#x must be integer type for seasonality to make sense
assert index.dtype.kind == 'i'
n = len(y)
m = n-2
p = period
ys, y_min, y_max = mu.scale_numpy(y)
D1 = mu.get_first_derivative_matrix_nes(index)
D2 = mu.get_second_derivative_matrix_nes(index)
H = mu.get_step_function_matrix(n)
T = mu.get_T_matrix(p)
B = mu.get_B_matrix_nes(index, p)
Q = B*T
#define F_matrix from blocks like in paper
zero = mu.zero_spmatrix
ident = mu.identity_spmatrix
gvec = spmatrix(growth, range(m), [0]*m)
zero_m = spmatrix(0.0, range(m), [0]*m)
zero_p = spmatrix(0.0, range(p), [0]*p)
zero_n = spmatrix(0.0, range(n), [0]*n)
step_reg = mu.get_step_function_reg(n, beta_step, permissives=step_permissives)
F_matrix = sparse([
[ident(n), -beta_d1*D1, -beta_d2*D2, zero(p, n), zero(n)],
[Q, zero(m, p-1), zero(m, p-1), -beta_seasonal*T, zero(n, p-1)],
[H, zero(m, n), zero(m, n), zero(p, n), step_reg]
])
w_vector = sparse([
mu.np2spmatrix(ys), gvec, zero_m, zero_p, zero_n
])
solution_vector = np.asarray(l1.l1(matrix(F_matrix), matrix(w_vector))).squeeze()
#separate
xbase = solution_vector[0:n]
s = solution_vector[n:n+p-1]
h = solution_vector[n+p-1:]
#scale back to original
if y_max > y_min:
scaling = y_max - y_min
else:
scaling = 1.0
xbase = xbase*scaling + y_min
s = s*scaling
h = h*scaling
seas = np.asarray(Q*matrix(s)).squeeze()
steps = np.asarray(H*matrix(h)).squeeze()
x = xbase + seas + steps
solution = {'xbase': xbase, 'seas': seas, 'steps': steps, 'x': x, 'h': h, 's': s}
return solution
def optimize(obj):
# constraints
# The equality constraint ensures the required amount of energy is delivered
A1 = matrix(0.0, (n, t * n))
A2 = matrix(0.0, (n, t * n))
b = matrix(0.0, (2 * n, 1))
for j in range(0, n):
b[j] = float(results[j][1])
for i in range(0, t):
A1[n * (t * j + i) + j] = float(timeInterval) / 60 # kWh -> kW
if i < results[j][0] or i > results[j][2]:
A2[n * (t * j + i) + j] = 1.0
A = sparse([A1, A2])
A3 = spdiag([-1] * (t * n))
A4 = spdiag([1] * (t * n))
# The inequality constraint ensures powers are positive and below a maximum
G = sparse([A3, A4])
h = []
for i in range(0, 2 * t * n):
if i < t * n:
h.append(0.0)
else:
h.append(pMax)
h = matrix(h)
# objective
if obj == 1:
q = []
for i in range(0, n):
for j in range(0, len(baseLoad)):
q.append(baseLoad[j])
q = matrix(q)
elif obj == 2:
q = matrix([0.0] * (t * n))
if obj == 3:
q = []
for i in range(0, n):
for j in range(0, len(baseLoad)):
q.append(baseLoad[j] - pv[j])
q = matrix(q)
if obj == 1 or obj == 2 or obj == 3:
I = spdiag([1] * t)
P = sparse([[I] * n] * n)
sol = solvers.qp(P, q, G, h, A, b)
X = sol['x']
return X
def const_to_matrix(self, value):
if isinstance(value, numpy.ndarray):
# ECHU: temporary workaround when travis fails
try:
retval = cvxopt.sparse(cvxopt.matrix(value), tc='d')
except TypeError:
retval = cvxopt.sparse(cvxopt.matrix(value.T.tolist()), tc='d')
return retval
return cvxopt.sparse(value, tc='d')
def cvxopt_matrix_2_cvxopt_sparse(Q=None, A_ub=None, A_eq=None):
if Q is not None:
Q = cvxopt.sparse(Q)
if A_ub is not None:
A_ub = cvxopt.sparse(A_ub)
if A_eq is not None:
A_eq = cvxopt.sparse(A_eq)
return (Q, A_ub, A_eq)
def setup_boundary_constraints(self):
# implements boundary constraints (9) in implementation notes
# identity matrix
n = (self.nu + self.ns) * self.Tp
I = spmatrix(1.0, range(n), range(n))
A_boundary = sparse([I, -1.0 * I])
b = sparse([matrix(1.0, (n, 1)), matrix(0.0, (n, 1))])
return A_boundary, b
def __init__(
self,
dim,
const_matrix_ineq=None,
const_vector_ineq=None,
const_matrix_eq=None,
const_vector_eq=None,
solver_type="cvxopt",
scipy_solver="revised simplex",
sparse_solver=False,
):
self.dim = dim
self.solver_type = solver_type
if not (solver_type == "cvxopt" or solver_type == "scipy"):
raise TypeError("Wrong solver type")
if solver_type == "cvxopt":
solvers.options["show_progress"] = False
else:
self.scipy_solver = scipy_solver
if sparse_solver and solver_type == "scipy":
raise TypeError("scipy solver cannot handle sparse matrices.")
if const_matrix_ineq is not None and const_vector_ineq is not None:
num_ineq_constraints, dim_ineq_constraints = const_matrix_ineq.shape
if not dim_ineq_constraints == self.dim:
raise ValueError(
"Dimension of the inequality constraints does not match the dimensionality of the problem."
)
self.G = const_matrix_ineq
self.h = const_vector_ineq
if solver_type == "cvxopt":
self.G = matrix(self.G,
(num_ineq_constraints, dim_ineq_constraints))
if sparse_solver:
self.G = sparse(self.G)
self.h = matrix(self.h, (num_ineq_constraints, 1))
else:
self.G = None
self.h = None
if const_matrix_eq is not None and const_vector_eq is not None:
num_eq_constraints, dim_eq_constraints = const_matrix_eq.shape
if not (dim_eq_constraints == self.dim):
raise ValueError(
"Dimension of the equality constraints does not match the dimensionality of the problem."
)
self.A = const_matrix_eq
self.b = const_vector_eq
if solver_type == "cvxopt":
self.A = matrix(self.A,
(num_eq_constraints, dim_eq_constraints))
self.b = matrix(self.b, (num_eq_constraints, 1), "d")
if sparse_solver:
self.A = sparse(self.A)
else:
self.A = None
self.b = None
def clique_hist(P, file=None):
if not P.ischordal: raise TypeError("Nonchordal")
if file == None: plt.ion()
else: plt.ioff()
V = +P.V
p = chompack.maxcardsearch(V)
#Vc,n = chompack.embed(V,p)
symb = chompack.symbolic(V, p)
#D = chompack.info(Vc); N = len(D)
N = symb.Nsn
#Ns = [len(v['S']) for v in D]; Nu = [len(v['U']) for v in D]
#Nw = [len(v['U']) + len(v['S']) for v in D]
Ns = [len(v) for v in symb.supernodes()]
Nu = [len(v) for v in symb.separators()]
Nw = [len(v) for v in symb.cliques()]
f = plt.figure()
f.clf()
f.text(0.58, 0.40, "Number of cliques: %i" % (len(Nw)))
f.text(0.61, 0.40 - 1 * 0.07, "$\sum | W_i | = %i$" % (sum(Nw)))
f.text(0.61, 0.40 - 2 * 0.07, "$\sum | V_i | = %i$" % (sum(Ns)))
f.text(0.61, 0.40 - 3 * 0.07, "$\sum | U_i | = %i$" % (sum(Nu)))
f.text(0.61, 0.40 - 4 * 0.07, "$\max_i\,| W_i | = %i$" % (max(Nw)))
plt.subplot(221)
Nmax = max(Nw)
y = matrix(0, (Nmax, 1))
for n in Nw:
y[n - 1] += 1
y = sparse(y)
plt.stem(y.I + 1, y.V, 'k')
plt.xlim(0, Nmax + 1)
plt.title("Cliques")
Nmax = max(Nu)
y = matrix(0, (Nmax, 1))
if Nmax > 0:
plt.subplot(222)
for n in Nu:
y[n - 1] += 1
y = sparse(y)
plt.stem(y.I + 1, y.V, 'k')
plt.xlim(0, Nmax + 1)
plt.title("Separators")
plt.subplot(223)
Nmax = max(Ns)
y = matrix(0, (Nmax, 1))
for n in Ns:
y[n - 1] += 1
y = sparse(y)
plt.stem(y.I + 1, y.V, 'k')
plt.xlim(0, Nmax + 1)
plt.title("Residuals")
if file: plt.savefig(file)
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 getYklm(yk):
# generates YY_k according to Molzahn dissertation (2.2a) and (2.2b)
Yk = cv.sparse(0.5 * cv.matrix([[
(yk + yk.trans()).real(), (yk - yk.trans()).imag()
], [(yk.trans() - yk).imag(), (yk + yk.trans()).real()]]))
Yk_bar = cv.sparse(-0.5 * cv.matrix([[(yk + yk.trans()).imag(),
(yk.trans() - yk).real()],
[(yk - yk.trans()).real(),
(yk + yk.trans()).imag()]]))
return Yk, Yk_bar
def project(A, r, G=None, fA=None):
'Project r to null(A) by solving the normal equation AA.t v = Ar.'
m, n = A.size
if G is None:
G = spmatrix([1] * n, range(n), range(n))
Lhs1 = sparse([G, A])
Lhs2 = sparse([A.T, spmatrix([], [], [], (m, m))])
Lhs = sparse([[Lhs1], [Lhs2]])
rhs = matrix([r, spmatrix([], [], [], (m, 1))])
linsolve(Lhs, rhs)
return rhs[:n]
def project(A,r,G = None, fA = None):
'Project r to null(A) by solving the normal equation AA.t v = Ar.'
m,n = A.size
if G is None:
G = spmatrix([1]*n,range(n),range(n))
Lhs1 = sparse([G,A])
Lhs2 = sparse([A.T, spmatrix([],[],[],(m,m))])
Lhs = sparse([[Lhs1],[Lhs2]])
rhs = matrix([r,spmatrix([],[],[],(m,1))])
linsolve(Lhs,rhs)
return rhs[:n]
def test_minres_scipy():
H = sprandsym(10)
v = sp_rand(10,3,0.8)
A = sparse([[H],[v]])
vrow = sparse([[v.T],[spmatrix([],[],[],(3,3))]])
A = sparse([A,vrow])
b = sp_rand(13,1,0.8)
As = cvxopt_to_numpy_matrix(A)
bs = cvxopt_to_numpy_matrix(matrix(b))
result = minres(As,bs,)
x = numpy_to_cvxopt_matrix(result[0])
print nrm2(A*x-b)
def clique_hist(P,file=None):
if not P.ischordal: raise TypeError, "Nonchordal"
if file==None: pylab.ion()
else: pylab.ioff()
V = +P.V
p = chompack.maxcardsearch(V)
#Vc,n = chompack.embed(V,p)
symb = chompack.symbolic(V,p)
#D = chompack.info(Vc); N = len(D)
N = symb.Nsn
#Ns = [len(v['S']) for v in D]; Nu = [len(v['U']) for v in D]
#Nw = [len(v['U']) + len(v['S']) for v in D]
Ns = [len(v) for v in symb.supernodes()]
Nu = [len(v) for v in symb.separators()]
Nw = [len(v) for v in symb.cliques()]
f = pylab.figure(); f.clf()
f.text(0.58,0.40,"Number of cliques: %i" % (len(Nw)))
f.text(0.61,0.40-1*0.07,"$\sum | W_i | = %i$" % (sum(Nw)))
f.text(0.61,0.40-2*0.07,"$\sum | V_i | = %i$" % (sum(Ns)))
f.text(0.61,0.40-3*0.07,"$\sum | U_i | = %i$" % (sum(Nu)))
f.text(0.61,0.40-4*0.07,"$\max_i\,| W_i | = %i$" % (max(Nw)))
pylab.subplot(221)
Nmax = max(Nw)
y = matrix(0,(Nmax,1))
for n in Nw :
y[n-1] += 1
y = sparse(y)
pylab.stem(y.I+1,y.V,'k'); pylab.xlim(0, Nmax+1)
pylab.title("Cliques")
Nmax = max(Nu)
y = matrix(0,(Nmax,1))
if Nmax > 0:
pylab.subplot(222)
for n in Nu :
y[n-1] += 1
y = sparse(y)
pylab.stem(y.I+1,y.V,'k'); pylab.xlim(0, Nmax+1)
pylab.title("Separators")
pylab.subplot(223)
Nmax = max(Ns)
y = matrix(0,(Nmax,1))
for n in Ns :
y[n-1] += 1
y = sparse(y)
pylab.stem(y.I+1,y.V,'k'); pylab.xlim(0, Nmax+1)
pylab.title("Residuals")
if file: pylab.savefig(file)
def solve_generalized_mom_conelp(MM,
constraints,
W=None,
absslack=1e-4,
totalslack=1e-2,
maxiter=1):
"""
solve using iterative GMM using the cone linear program
W is a specific weight matrix
we give generous bound for each constraint, and then harsh bound for
g'Wg
@params
constraints - E[g(x,X)] = f(x) - phi(X) that are supposed to be 0
Eggt - the function handle takes current f(x) and estimates
E[g(x,X)g(x,X)'] \in \Re^{n \times n}, the information matrix
maxiter - times to run the iterative GMM
"""
N = len(constraints)
D = len(MM.matrix_monos)
sr = len(MM.row_monos)
A, b = MM.get_Ab(constraints, cvxoptmode=False)
# augumented constraint matrix introduces slack variables g
A_aug = sparse(matrix(sc.hstack((A, 1 * sc.eye(N + 1)[:, :-1]))))
P = spdiag([matrix(0 * np.eye(D)), matrix(np.eye(N))])
b = matrix(b)
indicatorlist = MM.get_LMI_coefficients()
G = sparse(indicatorlist).trans()
V, I, J = G.V, G.I, G.J,
Gaug = sparse(spmatrix(V, I, J, size=(sr * sr, N + D)))
h = matrix(np.zeros((sr * sr, 1)))
dims = {}
dims['l'] = 0
dims['q'] = []
dims['s'] = [sr]
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())[:, np.newaxis]
q = matrix(np.vstack((w, np.zeros((N, 1)))))
#ipdb.set_trace()
for i in xrange(maxiter):
w = Bf.dot(W.flatten())[:, np.newaxis]
sol = cvxsolvers.coneqp(P, q, G=Gaug, h=h, dims=dims, A=A_aug, b=b)
sol['x'] = sol['x'][0:D]
return sol
def __init__(self, prob=QP(), Free=[], **kwargs):
'Initialize the PDASc'
# Initialize the superclass
super(PDASc, self).__init__(prob, **kwargs)
# Add additional data
self.F = Free
nf = len(self.F)
m = self.QP.numeq
Q1 = sparse([self.QP.H[self.F, self.F], self.QP.Aeq[:, self.F]])
Q2 = sparse([self.QP.Aeq[:, self.F].T, spmatrix([], [], [], (m, m))])
self.ipiv = matrix(1, (nf + m, 1))
self.Q = matrix([[Q1], [Q2]])
lapack.sytrf(self.Q, self.ipiv)
# LDL factorization of Q, dense, option 1
# lapack.sytrf(self.Q,self.ipiv)
# self.DiagQi = spdiag([self.Q[i,i] for i in range(nf+m)])
# self.LQ = copy(self.Q)
# for i in range(nf+m):
# self.LQ[i,i] = 1
# for j in range(i+1,nf+m):
# self.LQ[i,j] = 0
# LDL factorization of Q, dense, option 2 by calling Matlab library
# mlab = Matlab()
# mlab.start()
filename = 'temp/' + ''.join([
random.choice(string.ascii_letters + string.digits)
for n in xrange(32)
])
s = dict()
s['A'] = sparse([[Q1], [Q2]])
write(filename + '.mat', s)
d = '/home/zhh210/workspace/pypdas/numeric/control/'
# Memory failure when size is large
# output = mlab.run_func('getldp.m',{'arg1':filename})
# self.LQ = matrix(output['result']['L'])
# self.Dinv = matrix(output['result']['Dinv'])
# self.P = matrix(output['result']['P'])
# output = mlab.run_func('saveldp.m',{'arg1':filename})
# Execute shell command
cmd = 'matlab -r ' + '"' + "saveldp('" + filename + "');quit" + '"'
print cmd
os.system(cmd)
data = read(filename + '.mat')
self.LQ = matrix(data['L'])
self.Dinv = data['Dinv']
self.P = data['P']
def setup_dynamics_constraints(self):
# implements dynamics constraints (6) in implementation notes
Tu = sparse(matrix(0.0, (self.ns * self.Tp, self.nu * self.Tp)))
for t1 in range(1, self.Tp + 1):
for t2 in range(1, t1 + 1):
Tu[t1 * self.ns - self.ns:t1 * self.ns,
t2 * self.nu - self.nu:t2 * self.nu] = self.B
# identity matrix
I = spmatrix(1.0, range(self.ns * self.Tp), range(self.ns * self.Tp))
# self.I = np.eye(self.ns*self.Tp)
# finally, construct dynamics matrix
A_dyn = sparse([[I], [-1.0 * Tu]])
return A_dyn
def solve(x0, targetalt, targetvel, amax, dt=0.5, t_max=12 * 60):
r = np.array([0, targetalt, targetvel, 0, 0, 0, 0, 9.81])
t = np.arange(0, t_max + dt, dt)
# Inequality Constraints
G = spmatrix(np.zeros([1, 6 * len(t)]), range(6 * len(t)),
range(6 * len(t)), (6 * len(t), 8 * len(t)))
h = np.zeros([6 * len(t), 1])
for i in range(len(t)):
G[2 * i:2 * i + 2, 8 * i:8 * i + 2] = -np.eye(2)
G[2 * len(t) + 4 * i:2 * len(t) + 4 * i + 4,
8 * i + 6:8 * i + 8] = np.array([[1, 0], [-1, 0], [0, 1], [0, -1]])
if i < len(amax):
h[2 * len(t) + 4 * i:2 * len(t) + 4 * i + 4,
0] = np.ones(4) * amax[i]
else:
h[2 * len(t) + 4 * i:2 * len(t) + 4 * i + 4, 0] = np.zeros(4)
h = matrix(h)
# Equality Constraints
A = np.array([[1, 0, dt, 0, 0, 0], [0, 1, 0, dt, 0, 0],
[0, 0, 1, 0, dt, 0], [0, 0, 0, 1, 0, dt], [0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0]])
B = np.array([[0, 0], [0, 0], [0, 0], [0, 0], [1, 0], [0, 1]])
Aeq = spmatrix(np.zeros([1, 6 * len(t)]), range(6 * len(t)),
range(6 * len(t)), (6 * len(t), 8 * len(t)))
Aeq[:6, :6] = sparse(matrix(np.eye(6)))
c = sparse(matrix(np.concatenate((-A, -B, np.eye(6)), axis=1)))
for i in range(len(t) - 1):
Aeq[6 * i + 6:6 * i + 12, 8 * i:8 * i + 14] = c
beq = np.zeros([6 * len(t), 1])
beq[:6, 0] = x0
for i in range(len(t)):
beq[6 * i + 5, 0] = -9.81
beq = matrix(beq)
# Objective
Q = matrix(2 * np.eye(8))
Q[0, 0] = 0
QQ = sparse(Q)
for i in range(len(t) - 1):
QQ = spdiag([QQ, Q])
p = -r.T.dot(Q)
pp = matrix(np.kron(np.ones([1, len(t)]), p).T)
sol = solvers.qp(QQ, pp, G, h, Aeq, beq)
x = np.array(sol['x']).reshape((-1, 8))
return t, x
def minimum_risk_subject_to_target_return():
# minimum expected return threshold
r_min = 0.04
P = covs
q = matrix(numpy.zeros((n, 1)), tc='d')
# inequality constraints Gx <= h
# captures the constraints (avg_ret'x >= r_min) and (x >= 0)
G = matrix(-numpy.transpose(numpy.array(avg_ret)))
h = matrix(-numpy.ones((1, 1)) * r_min)
# equality constraint Ax = b; captures the constraint sum(x) == 1
A = matrix(1.0, (1, n))
b = matrix(1.0)
groups = rets.groupby(axis=1, level=0, sort=False,
group_keys=False).count().ix[-1].values
num_group = len(groups)
num_asset = np.sum(groups)
G_sparse_list = []
for i in range(num_group):
for j in range(groups[i]):
G_sparse_list.append(i)
Group_sub = spmatrix(1.0, G_sparse_list, range(num_asset))
asset_sub = matrix(np.eye(n))
# asset_sub is for asset weight limit, Group sub is for group weight
# constraint.
G = matrix(sparse([G, asset_sub, -asset_sub, Group_sub, -Group_sub]))
b_asset = tuple((0.01, 1.0) for i in rets.columns)
b_asset_upper_bound = np.array([x[-1] for x in b_asset])
b_asset_lower_bound = np.array([x[0] for x in b_asset])
b_asset_matrix = matrix(
numpy.concatenate((b_asset_upper_bound, -b_asset_lower_bound), 0))
b_group = [(.05, .41), (.2, .66), (0.01, .16)]
b_group_upper_bound = np.array([x[-1] for x in b_group])
b_group_lower_bound = np.array([x[0] for x in b_group])
b_group_matrix = matrix(
numpy.concatenate((b_group_upper_bound, -b_group_lower_bound), 0))
h = matrix(sparse([h, b_asset_matrix, b_group_matrix]))
# solve minimum risk for maximum return above target .
sol = solvers.qp(P, q, G, h, A, b)
print(minimum_risk_subject_to_target_return.__name__)
print(sol['x'])
print(statistics(sol['x']))
def test_minres_scipy():
H = sprandsym(10)
v = sp_rand(10, 3, 0.8)
A = sparse([[H], [v]])
vrow = sparse([[v.T], [spmatrix([], [], [], (3, 3))]])
A = sparse([A, vrow])
b = sp_rand(13, 1, 0.8)
As = cvxopt_to_numpy_matrix(A)
bs = cvxopt_to_numpy_matrix(matrix(b))
result = minres(
As,
bs,
)
x = numpy_to_cvxopt_matrix(result[0])
print nrm2(A * x - b)
def generateG(num_traj, num_states, size_each_state, size_each_control): num_controls = num_states - 1 num_ctrl_ineq = 2 * (size_each_control * num_controls) X_block = sparseZero(num_ctrl_ineq, num_traj*(num_states-2)*size_each_state) U_upper_block = sparseIdentity(num_controls*size_each_control) U_lower_block = -1*sparseIdentity(num_controls*size_each_control) U_block = sparse([U_upper_block, U_lower_block]) L_block = sparseZero(num_ctrl_ineq, num_traj*(num_states-1)*size_each_state) E_block = sparseZero(num_ctrl_ineq, (num_controls-1)*size_each_control) D_block = sparseZero(num_ctrl_ineq, num_traj*(num_states-2)*size_each_state) XI_block = sparseZero(num_ctrl_ineq, num_traj*(num_states-2)*size_each_state) UI_block = sparseZero(num_ctrl_ineq, num_controls*size_each_control) B_block = sparseZero(num_ctrl_ineq, num_traj*(num_states-1)*size_each_state) ctrl_ineq = sparse([[X_block], [U_block], [L_block], [E_block], [D_block], [XI_block], [UI_block], [B_block]]) num_state_ineq = 2*(num_traj*(num_states-2)*size_each_state) X_upper_block = sparseIdentity(num_traj*(num_states-2)*size_each_state) X_lower_block = -1 * sparseIdentity(num_traj*(num_states-2)*size_each_state) X_block = sparse([X_upper_block, X_lower_block]) L_block = sparseZero(num_state_ineq, num_traj*(num_states-1)*size_each_state) U_block = sparseZero(num_state_ineq, num_controls*size_each_control) E_block = sparseZero(num_state_ineq, (num_controls-1)*size_each_control) D_block = sparseZero(num_state_ineq, num_traj*(num_states-2)*size_each_state) XI_upper_block = -1 * sparseIdentity(num_traj*(num_states-2)*size_each_state) XI_lower_block = sparseIdentity(num_traj*(num_states-2)*size_each_state) XI_block = sparse([XI_upper_block, XI_lower_block]) UI_block = sparseZero(num_state_ineq, num_controls*size_each_control) B_block = sparseZero(num_state_ineq, num_traj*(num_states-1)*size_each_state) state_ineq = sparse([[X_block], [U_block], [L_block], [E_block], [D_block], [XI_block], [UI_block], [B_block]]) num_ctrl_init_ineq = 2 * (size_each_control * num_controls) X_block = sparseZero(num_ctrl_init_ineq, num_traj*(num_states-2)*size_each_state) U_upper_block = sparseIdentity(num_controls*size_each_control) U_lower_block = -1*sparseIdentity(num_controls*size_each_control) U_block = sparse([U_upper_block, U_lower_block]) L_block = sparseZero(num_ctrl_init_ineq, num_traj*(num_states-1)*size_each_state) E_block = sparseZero(num_ctrl_init_ineq, (num_controls-1)*size_each_control) D_block = sparseZero(num_ctrl_init_ineq, num_traj*(num_states-2)*size_each_state) XI_block = sparseZero(num_ctrl_init_ineq, num_traj*(num_states-2)*size_each_state) UI_upper_block = -1 * sparseIdentity(num_controls*size_each_control) UI_lower_block = sparseIdentity(num_controls*size_each_control) UI_block = sparse([UI_upper_block, UI_lower_block]) B_block = sparseZero(num_ctrl_init_ineq, num_traj*(num_states-1)*size_each_state) ctrl_init_ineq = sparse([[X_block], [U_block], [L_block], [E_block], [D_block], [XI_block], [UI_block], [B_block]]) G = sparse([ctrl_ineq, state_ineq, ctrl_init_ineq]) return G
def lst_l1(A, y, integer=False, xmax=1000):
"""
Returns the solution of least L1-norm problem
minimize || x ||_1.
subject to A'x == y
x can be float or integer.
Return None if no optimal/feasible solution found.
This problem can be converted into a linear programming problem
by setting v = |u|, x = [u' v']',
minimize [0]' [u]
[1] [v]
subject to [ I -I] [ 0 ]
[-I -I] [u] = [ 0 ]
[ 0 -I] [v] [ 0 ]
[ I 0] [xmax]
[A]' [u] = [y]
[0] [v]
"""
m, n = A.size
c = matrix(0.0, (2*n,1))
c[n:] = 1.0
# inequality constraint
I = spmatrix(1.0, range(n), range(n))
O = matrix(0.0, (n,n))
G = sparse(matrix([[I, -I, O, I], [-I, -I, -I, O]]))
h = matrix(0.0, (4*n,1))
h[3*n:] = xmax
# equality constraint
Al = sparse(matrix([[A], [matrix(0.0, (m,n))]]))
bl = y
# solve the linear programming problem
if integer:
(status, x) = glpk.ilp(c, G, h, Al, bl)[0:2]
else:
(status, x) = glpk.ilp(c, G, h, Al, bl)[0:2]
return x
def solve_generalized_mom_conelp(MM, constraints, W=None, absslack=1e-4, totalslack=1e-2, maxiter = 1):
"""
solve using iterative GMM using the cone linear program
W is a specific weight matrix
we give generous bound for each constraint, and then harsh bound for
g'Wg
@params
constraints - E[g(x,X)] = f(x) - phi(X) that are supposed to be 0
Eggt - the function handle takes current f(x) and estimates
E[g(x,X)g(x,X)'] \in \Re^{n \times n}, the information matrix
maxiter - times to run the iterative GMM
"""
N = len(constraints)
D = len(MM.matrix_monos)
sr = len(MM.row_monos)
A,b = MM.get_Ab(constraints, cvxoptmode = False)
# augumented constraint matrix introduces slack variables g
A_aug = sparse(matrix(sc.hstack((A, 1*sc.eye(N+1)[:,:-1]))))
P = spdiag([matrix(0*np.eye(D)), matrix(np.eye(N))])
b = matrix(b)
indicatorlist = MM.get_LMI_coefficients()
G = sparse(indicatorlist).trans()
V,I,J = G.V, G.I, G.J,
Gaug = sparse(spmatrix(V,I,J,size=(sr*sr, N + D)))
h = matrix(np.zeros((sr*sr,1)))
dims = {}
dims['l'] = 0
dims['q'] = []
dims['s'] = [sr]
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())[:,np.newaxis]
q = matrix(np.vstack( (w,np.zeros((N,1))) ))
#ipdb.set_trace()
for i in xrange(maxiter):
w = Bf.dot(W.flatten())[:,np.newaxis]
sol = cvxsolvers.coneqp(P, q, G=Gaug, h=h, dims=dims, A=A_aug, b=b)
sol['x'] = sol['x'][0:D]
return sol
def constraints(self):
# construct the constraints for the attack routing problem
N = self.N
u = np.tile(range(N), N)
v = np.repeat(range(N),N)
w = np.array(range(N*N))
# build constraint matrix
A1 = spmatrix(np.repeat(self.nu, N), u, w, (N, N*N))
A2 = -spmatrix(np.repeat(self.nu + self.phi, N), v, w, (N, N*N))
I = np.array(range(N))
J = I + np.array(range(N)) * N
A3 = spmatrix(self.phi, I, J, (N, N*N))
tmp = np.dot(np.diag(self.phi), self.delta).transpose()
A4 = matrix(np.repeat(tmp, N, axis=1))
A5 = -spmatrix(tmp.flatten(), v, np.tile(J, N), (N, N*N))
A6 = A1 + A2 + A3 + A4 + A5
I = np.array([0]*(N-1))
J = np.array(range(self.k)+range((self.k+1),N)) + N * self.k
A7 = spmatrix(1., I, J, (1, N*N))
A = sparse([[A6, -A6, A7, -A7, -spdiag([1.]*(N*N))]])
tmp = np.zeros(2*N + 2 + N*N)
tmp[2*N] = 1.
tmp[2*N + 1] = -1.
b = matrix(tmp)
return b, A
def np2spmatrix(nparray):
"""
Convert a numpy ndarray to sparse cvxopt matrix
:param nparray: numpy ndarray
:return: cvxopt sparse matrix
"""
return sparse(matrix(nparray))
def build_matrix(A,b,b_height, params,vec_sizes,start_idxs,total_width):
h_cum = height_of(b_height, vec_sizes)
h_vec = o.matrix(0, (h_cum,1), 'd')
G_vals = []
G_I, G_J = [], []
idx = 0
for row, coeff, size in zip(A,b,b_height):
row_height = size.row_value(vec_sizes)
h_vec[idx:idx+row_height] = eval_coeff(coeff, params, row_height)
for k,v in row.iteritems(): # repeated function
# we ignore constant coefficients
if k != '1':
col_width = vec_sizes[k]
result_mat = o.sparse(eval_matrix_coeff(v, params, row_height, col_width))
# set the row
G_I += (result_mat.I + idx)
# set the column
G_J += (result_mat.J + start_idxs[k])
# set the values
G_vals += result_mat.V
idx += row_height
Gl_mat = o.spmatrix(G_vals, G_I, G_J, (h_cum, total_width))
hl_vec = h_vec
return (Gl_mat, hl_vec)
def solve_LP_problem(self):
(f_coef_matrix, f_column_vector) = self.build_function_coef_matrix_and_column_vector()
(d_coef_matrix, d_column_vector) = self.build_derivative_coef_matrix_and_column_vector()
# Solve the LP problem by combining constraints for both function and derivative info.
objective_function_vector = matrix(list(itertools.repeat(1.0, self.no_vars)))
coef_matrix = sparse([f_coef_matrix, d_coef_matrix])
column_vector = matrix([f_column_vector, d_column_vector])
min_sol = solvers.lp(objective_function_vector, coef_matrix, column_vector)
is_consistent = min_sol['x'] is not None
# Print the LP problem for debugging purposes.
if self.verbose:
self.display_LP_problem(coef_matrix, column_vector)
if is_consistent:
self.min_heights = np.array(min_sol['x']).reshape(self.no_points_per_axis)
print np.around(self.min_heights, decimals=2)
# Since consistency has been established, solve the converse LP problem to get the
# maximal bounding surface.
max_sol = solvers.lp(-objective_function_vector, coef_matrix, column_vector)
self.max_heights = np.array(max_sol['x']).reshape(self.no_points_per_axis)
print np.around(self.max_heights, decimals=2)
if self.plot_surfaces:
self.plot_3D_objects_for_2D_case()
else:
print 'No witness for consistency found.'
return is_consistent
def train_dual(self):
"""Trains an one-class svm in dual with kernel."""
if (self.samples<1):
print('Invalid training data.')
return SVDD.MSG_ERROR
# number of training examples
N = self.samples
C = self.C
# generate a kernel matrix
P = self.kernel
# this is the diagonal of the <kernel matrix
q = matrix([0.5*P[i,i] for i in range(N)], (N,1))
# sum_i alpha_i = A alpha = b = 1.0
A = matrix(1.0, (1,N))
b = matrix(1.0, (1,1))
# 0 <= alpha_i <= h = C
G1 = spmatrix(1.0, range(N), range(N))
G = sparse([G1,-G1])
h1 = matrix(C, (N,1))
h2 = matrix(0.0, (N,1))
h = matrix([h1,h2])
sol = qp(P,-q,G,h,A,b)
# mark dual as solved
self.isDualTrained = True
# store solution
self.alphas = sol['x']
self.obj_primal = sol['primal objective']
self.obj_dual = sol['dual objective']
# find support vectors
self.svs = []
for i in range(N):
if self.alphas[i]>SVDD.PRECISION:
self.svs.append(i)
# find support vectors with alpha < C for threshold calculation
self.threshold = 10**8
flag = False
for i in self.svs:
if self.alphas[i]<(C-SVDD.PRECISION) and flag==False:
(self.threshold, MSG) = self.apply_dual(self.kernel[i,self.svs],self.norms[i])
flag=True
break
# no threshold set yet?
if (flag==False):
(thres, MSG) = self.apply_dual(self.kernel[self.svs,self.svs],self.norms[self.svs])
self.threshold = matrix(min(thres))
print('Threshold is {0}'.format(self.threshold))
return SVDD.MSG_OK
def test_get_B_matrix_nes_on_gap():
x = np.array([0, 2, 3, 5])
period = 3
B_nes = mu.get_B_matrix_nes(x, period)
expected_matrix = [[1, 0, 0], [0, 0, 1], [1, 0, 0], [0, 0, 1]]
expected_result = cvxopt.sparse(cvxopt.matrix(expected_matrix).T)
assert max(B_nes - expected_result) < 1e-13
def objective_hyper(x, z, ks, p):
"""Objective function of UE program with hyperbolic delay functions
f(x) = sum_i f_i(v_i) with v = sum_w x_w
f_i(u) = ks[i,0]*u - ks[i,1]*log(ks[i,2]-u)
Parameters
----------
x,z: variables for the F(x,z) function for cvxopt.solvers.cp
ks: matrix of size (n,3)
p: number of destinations
(we use multiple-sources single-sink node-arc formulation)
"""
n = ks.size[0]
if x is None: return 0, matrix(1.0/p, (p*n,1))
l = matrix(0.0, (n,1))
for k in range(p): l += x[k*n:(k+1)*n]
f, Df, H = 0.0, matrix(0.0, (1,n)), matrix(0.0, (n,1))
for i in range(n):
tmp = 1.0/(ks[i,2]-l[i])
f += ks[i,0]*l[i] - ks[i,1]*np.log(max(ks[i,2]-l[i], 1e-13))
Df[i] = ks[i,0] + ks[i,1]*tmp
H[i] = ks[i,1]*tmp**2
Df = matrix([[Df]]*p)
if z is None: return f, Df
return f, Df, sparse([[spdiag(z[0] * H)]*p]*p)
def train(self, test_users=None):
self.model = {}
iset = set(range(self.n_items))
if test_users is None:
test_users = range(self.n_users)
for i, u in enumerate(test_users):
#if (i+1) % 100 == 0:
print "%d/%d" %(i+1, len(test_users))
Xp_set = self.data.get_items(u)
Xn_set = iset - Xp_set
Z = co.spdiag([1.0 if i in Xp_set else -1.0 for i in iset])
K = 2 * (Z * self.X.T * self.X * Z)
I = co.spdiag([self.lambda_p if i in Xp_set else self.lambda_n for i in iset])
P = K + I
o = utc.zeroes_vec(self.n_items)
G = -utc.identity(self.n_items)
A = co.matrix([[1.0 if i in Xp_set else 0.0 for i in iset],
[1.0 if j in Xn_set else 0.0 for j in iset]]).T
b = co.matrix([1.0, 1.0])
P = co.sparse(P)
solver.options['show_progress'] = False
sol = solver.qp(P, o, G, o, A, b)
self.sol[u] = sol
self.model[u] = self.X.T * self.X * Z * sol['x']
# endfor
return self
def solve(problem, sparse=False, **kwargs):
"""Minimize w*x subject to ||Ax + b|| <= c*x + d."""
gs = []
hs = []
for constraint in problem.constraints:
a = constraint.a
b = constraint.b
c = constraint.c
d = constraint.d
g = np.vstack((-c, -a))
hs.append(cx.matrix(np.hstack((d, b))))
if sparse:
gs.append(cx.sparse(cx.matrix(g)))
else:
gs.append(cx.matrix(g))
begin = time.clock()
cx.solvers.options.update(kwargs)
cx.solvers.options['MOSEK'] = {mosek.iparam.log: 100, mosek.iparam.intpnt_max_iterations: 50000}
solution = cx.solvers.socp(cx.matrix(problem.objective), Gq=gs, hq=hs, solver='mosek')
duration = time.clock() - begin
# duration = solution['duration']
timings['last_solve'] = duration
print 'SOCP duration: %.3f' % duration
print 'Total duration (including python wrappers): %.3f' % duration
print 'Solver exited with status "%s"' % solution['status']
return solution
def Norm_inf1 (X,W):
# X, W are two matrix of shape respectively f*n et r*n
f,n = X.size
r = W.size[0]
print f,n,r
P = matrix(W).trans()
F = matrix(1.0,(f,r))
onesn = matrix(1.0,(n,1))
Idn = spmatrix(1.0, range(n),range(n))
Idr = spmatrix(1.0, range(r),range(r))
Zrn = spmatrix(0,[r-1],[n-1])
Zr = spmatrix(0,[r-1],[0])
#Zn = spmatrix(0,[n-1],[0])
A = sparse([ [P,-P,-Idr], [-Idn,-Idn,Zrn] ])
for i in range(f):
V = X[i,:].trans()
C = matrix([[V,-V,Zr]])
e = matrix([ [Zr, onesn] ])
solution = solvers.lp(e,A,C)['x']
F[i,:] = solution[range(r)].trans()
return F
def split_linear_constraints(A, l, u):
""" Returns the linear equality and inequality constraints.
"""
ieq = []
igt = []
ilt = []
ibx = []
for i in range(len(l)):
if abs(u[i] - l[i]) <= EPS:
ieq.append(i)
elif (u[i] > 1e10) and (l[i] > -1e10):
igt.append(i)
elif (l[i] <= -1e10) and (u[i] < 1e10):
ilt.append(i)
elif (abs(u[i] - l[i]) > EPS) and (u[i] < 1e10) and (l[i] > -1e10):
ibx.append(i)
else:
raise ValueError
Ae = A[ieq, :]
Ai = sparse([A[ilt, :], -A[igt, :], A[ibx, :], -A[ibx, :]])
be = u[ieq, :]
bi = matrix([u[ilt], -l[igt], u[ibx], -l[ibx]])
return Ae, be, Ai, bi
def f(aff_exp):
V, I, J = [], [], []
vert_offset = expr_offsets[str(aff_exp)]
coefficients = aff_exp.coefficients()
for var, blocks in coefficients.items():
# Constant is not in var_offsets.
horiz_offset = var_offsets.get(var)
for col, block in enumerate(blocks):
vert_start = vert_offset + col*aff_exp.size[0]
vert_end = vert_start + aff_exp.size[0]
if var is s.CONSTANT:
pass
#const_vec[vert_start:vert_end, :] = block
else:
if isinstance(block, numbers.Number):
V.append(block)
I.append(vert_start)
J.append(horiz_offset)
else: # Block is a matrix or spmatrix.
if isinstance(block, cvxopt.matrix):
block = cvxopt.sparse(block)
V.extend(block.V)
I.extend(block.I + vert_start)
J.extend(block.J + horiz_offset)
return (V, I, J)
def __init__(self,prob = QP(),Free=[],**kwargs):
'Initialize the PDASc'
# Initialize the superclass
super(PDASc,self).__init__(prob,**kwargs)
# Add additional data
self.F = Free
nf = len(self.F)
m = self.QP.numeq
Q1 = sparse([self.QP.H[self.F,self.F], self.QP.Aeq[:,self.F] ])
Q2 = sparse([self.QP.Aeq[:,self.F].T, spmatrix([],[],[],(m,m)) ])
self.ipiv = matrix(1,(nf+m,1))
self.Q = matrix([[Q1],[Q2]])
lapack.sytrf(self.Q,self.ipiv)
# LDL factorization of Q, dense, option 1
# lapack.sytrf(self.Q,self.ipiv)
# self.DiagQi = spdiag([self.Q[i,i] for i in range(nf+m)])
# self.LQ = copy(self.Q)
# for i in range(nf+m):
# self.LQ[i,i] = 1
# for j in range(i+1,nf+m):
# self.LQ[i,j] = 0
# LDL factorization of Q, dense, option 2 by calling Matlab library
# mlab = Matlab()
# mlab.start()
filename = 'temp/'+''.join([random.choice(string.ascii_letters + string.digits) for n in xrange(32)])
s = dict()
s['A'] = sparse([[Q1],[Q2]])
write(filename+'.mat',s)
d = '/home/zhh210/workspace/pypdas/numeric/control/'
# Memory failure when size is large
# output = mlab.run_func('getldp.m',{'arg1':filename})
# self.LQ = matrix(output['result']['L'])
# self.Dinv = matrix(output['result']['Dinv'])
# self.P = matrix(output['result']['P'])
# output = mlab.run_func('saveldp.m',{'arg1':filename})
# Execute shell command
cmd = 'matlab -r '+'"'+ "saveldp('"+filename + "');quit" + '"'
print cmd
os.system(cmd)
data = read(filename+'.mat')
self.LQ = matrix(data['L'])
self.Dinv = data['Dinv']
self.P = data['P']
def test_basic_complex(self):
import cvxopt
a = cvxopt.matrix([1,-2,3])
b = cvxopt.matrix([1.0,-2.0,3.0])
c = cvxopt.matrix([1.0+2j,1-2j,0+1j])
d = cvxopt.spmatrix([complex(1.0,0.0), complex(0.0,1.0), complex(2.0,-1.0)],[0,1,3],[0,2,3],(4,4))
e = cvxopt.spmatrix([complex(1.0,0.0), complex(0.0,1.0), complex(2.0,-1.0)],[2,3,3],[1,2,3],(4,4))
self.assertAlmostEqualLists(list(cvxopt.div(b,c)),[0.2-0.4j,-0.4-0.8j,-3j])
self.assertAlmostEqualLists(list(cvxopt.div(b,2.0j)),[-0.5j,1j,-1.5j])
self.assertAlmostEqualLists(list(cvxopt.div(a,c)),[0.2-0.4j,-0.4-0.8j,-3j])
self.assertAlmostEqualLists(list(cvxopt.div(c,a)),[(1+2j),(-0.5+1j),0.3333333333333333j])
self.assertAlmostEqualLists(list(cvxopt.div(c,c)),[1.0,1.0,1.0])
self.assertAlmostEqualLists(list(cvxopt.div(a,2.0j)),[-0.5j,1j,-1.5j])
self.assertAlmostEqualLists(list(cvxopt.div(c,1.0j)),[2-1j,-2-1j,1+0j])
self.assertAlmostEqualLists(list(cvxopt.div(1j,c)),[0.4+0.2j,-0.4+0.2j,1+0j])
self.assertTrue(len(d)+len(e)==len(cvxopt.sparse([d,e])))
self.assertTrue(len(d)+len(e)==len(cvxopt.sparse([[d],[e]])))
def pinvert(spmat):
"""
:param spmat: a cvx sparse matrix
:return: the pseudo-inverse matrix as sparse
"""
arr = spmatrix2np(spmat)
arr_inv = np.linalg.pinv(arr)
return sparse(matrix(arr_inv))