def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
#x = arg_objs[0]
t = lu.create_var(size)
# log(1 + exp(x)) <= t <=> exp(-t) + exp(x - t) <= 1
'''
obj0, constr0 = exp.graph_implementation([lu.neg_expr(t)], size)
obj1, constr1 = exp.graph_implementation([lu.sub_expr(x, t)], size)
lhs = lu.sum_expr([obj0, obj1])
ones = lu.create_const(np.mat(np.ones(size)), size)
constr = constr0 + constr1 + [lu.create_leq(lhs, ones)]
'''
a = arg_objs[0]
b = arg_objs[1]
e_a, e_a_cons = exp.graph_implementation([lu.neg_expr(a)], (1,1))
e_b, e_b_cons = exp.graph_implementation([lu.neg_expr(b)], (1,1))
obj0, constr0 = log.graph_implementation(
[lu.sub_expr(e_b,e_a)],
(1,1)
)
lhs = obj0
constr = constr0 + e_a_cons + e_b_cons + [lu.create_leq(lhs, t)] + [lu.create_leq(e_a, e_b)]
return (t, constr)
def graph_implementation(arg_objs, shape, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
shape : tuple
The shape of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
x = arg_objs[0]
ones = lu.create_const(np.mat(np.ones(x.shape)), x.shape)
xp1 = lu.sum_expr([x, ones])
return log.graph_implementation([xp1], shape, data)
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
x = arg_objs[0]
ones = lu.create_const(np.mat(np.ones(x.size)), x.size)
xp1 = lu.sum_expr([x, ones])
return log.graph_implementation([xp1], size, data)
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Creates the equivalent problem::
maximize sum(log(D[i, i]))
subject to: D diagonal
diag(D) = diag(Z)
Z is upper triangular.
[D Z; Z.T A] is positive semidefinite
The problem computes the LDL factorization:
.. math::
A = (Z^TD^{-1})D(D^{-1}Z)
This follows from the inequality:
.. math::
\det(A) >= \det(D) + \det([D, Z; Z^T, A])/\det(D)
>= \det(D)
because (Z^TD^{-1})D(D^{-1}Z) is a feasible D, Z that achieves
det(A) = det(D) and the objective maximizes det(D).
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
A = arg_objs[0] # n by n matrix.
n, _ = A.size
X = lu.create_var((2 * n, 2 * n))
Z = lu.create_var((n, n))
D = lu.create_var((n, n))
# Require that X is symmetric (which implies
# A is symmetric).
# X == X.T
obj, constraints = transpose.graph_implementation([X], (n, n))
constraints.append(lu.create_eq(X, obj))
# Require that X and A are PSD.
constraints += [SDP(X), SDP(A)]
# Fix Z as upper triangular, D as diagonal,
# and diag(D) as diag(Z).
for i in xrange(n):
for j in xrange(n):
if i == j:
# D[i, j] == Z[i, j]
Dij = index.get_index(D, constraints, i, j)
Zij = index.get_index(Z, constraints, i, j)
constraints.append(lu.create_eq(Dij, Zij))
if i != j:
# D[i, j] == 0
Dij = index.get_index(D, constraints, i, j)
constraints.append(lu.create_eq(Dij))
if i > j:
# Z[i, j] == 0
Zij = index.get_index(Z, constraints, i, j)
constraints.append(lu.create_eq(Zij))
# Fix X using the fact that A must be affine by the DCP rules.
# X[0:n, 0:n] == D
index.block_eq(X, D, constraints, 0, n, 0, n)
# X[0:n, n:2*n] == Z,
index.block_eq(X, Z, constraints, 0, n, n, 2 * n)
# X[n:2*n, n:2*n] == A
index.block_eq(X, A, constraints, n, 2 * n, n, 2 * n)
# Add the objective sum(log(D[i, i])
log_diag = []
for i in xrange(n):
Dii = index.get_index(D, constraints, i, i)
obj, constr = log.graph_implementation([Dii], (1, 1))
constraints += constr
log_diag.append(obj)
obj = lu.sum_expr(log_diag)
return (obj, constraints)
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Creates the equivalent problem::
maximize sum(log(D[i, i]))
subject to: D diagonal
diag(D) = diag(Z)
Z is upper triangular.
[D Z; Z.T A] is positive semidefinite
The problem computes the LDL factorization:
.. math::
A = (Z^TD^{-1})D(D^{-1}Z)
This follows from the inequality:
.. math::
\det(A) >= \det(D) + \det([D, Z; Z^T, A])/\det(D)
>= \det(D)
because (Z^TD^{-1})D(D^{-1}Z) is a feasible D, Z that achieves
det(A) = det(D) and the objective maximizes det(D).
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
A = arg_objs[0] # n by n matrix.
n, _ = A.size
X = lu.create_var((2 * n, 2 * n))
X, constraints = Semidef(2 * n).canonical_form
Z = lu.create_var((n, n))
D = lu.create_var((n, 1))
# Require that X and A are PSD.
constraints += [SDP(A)]
# Fix Z as upper triangular, D as diagonal,
# and diag(D) as diag(Z).
Z_lower_tri = lu.upper_tri(lu.transpose(Z))
constraints.append(lu.create_eq(Z_lower_tri))
# D[i, i] = Z[i, i]
constraints.append(lu.create_eq(D, lu.diag_mat(Z)))
# Fix X using the fact that A must be affine by the DCP rules.
# X[0:n, 0:n] == D
index.block_eq(X, lu.diag_vec(D), constraints, 0, n, 0, n)
# X[0:n, n:2*n] == Z,
index.block_eq(X, Z, constraints, 0, n, n, 2 * n)
# X[n:2*n, n:2*n] == A
index.block_eq(X, A, constraints, n, 2 * n, n, 2 * n)
# Add the objective sum(log(D[i, i])
obj, constr = log.graph_implementation([D], (n, 1))
return (lu.sum_entries(obj), constraints + constr)
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Creates the equivalent problem::
maximize sum(log(D[i, i]))
subject to: D diagonal
diag(D) = diag(Z)
Z is upper triangular.
[D Z; Z.T A] is positive semidefinite
The problem computes the LDL factorization:
.. math::
A = (Z^TD^{-1})D(D^{-1}Z)
This follows from the inequality:
.. math::
\det(A) >= \det(D) + \det([D, Z; Z^T, A])/\det(D)
>= \det(D)
because (Z^TD^{-1})D(D^{-1}Z) is a feasible D, Z that achieves
det(A) = det(D) and the objective maximizes det(D).
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
A = arg_objs[0] # n by n matrix.
n, _ = A.size
X = lu.create_var((2*n, 2*n))
Z = lu.create_var((n, n))
D = lu.create_var((n, 1))
# Require that X and A are PSD.
constraints = [SDP(X), SDP(A)]
# Fix Z as upper triangular, D as diagonal,
# and diag(D) as diag(Z).
Z_lower_tri = lu.upper_tri(lu.transpose(Z))
constraints.append(lu.create_eq(Z_lower_tri))
# D[i, i] = Z[i, i]
constraints.append(lu.create_eq(D, lu.diag_mat(Z)))
# Fix X using the fact that A must be affine by the DCP rules.
# X[0:n, 0:n] == D
index.block_eq(X, lu.diag_vec(D), constraints, 0, n, 0, n)
# X[0:n, n:2*n] == Z,
index.block_eq(X, Z, constraints, 0, n, n, 2*n)
# X[n:2*n, n:2*n] == A
index.block_eq(X, A, constraints, n, 2*n, n, 2*n)
# Add the objective sum(log(D[i, i])
obj, constr = log.graph_implementation([D], (n, 1))
return (lu.sum_entries(obj), constraints + constr)
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Creates the equivalent problem::
maximize sum(log(D[i, i]))
subject to: D diagonal
diag(D) = diag(Z)
Z is upper triangular.
[D Z; Z.T A] is positive semidefinite
The problem computes the LDL factorization:
.. math::
A = (Z^TD^{-1})D(D^{-1}Z)
This follows from the inequality:
.. math::
\det(A) >= \det(D) + \det([D, Z; Z^T, A])/\det(D)
>= \det(D)
because (Z^TD^{-1})D(D^{-1}Z) is a feasible D, Z that achieves
det(A) = det(D) and the objective maximizes det(D).
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
A = arg_objs[0] # n by n matrix.
n, _ = A.size
X = lu.create_var((2 * n, 2 * n))
Z = lu.create_var((n, n))
D = lu.create_var((n, n))
# Require that X is symmetric (which implies
# A is symmetric).
# X == X.T
obj, constraints = transpose.graph_implementation([X], (n, n))
constraints.append(lu.create_eq(X, obj))
# Require that X and A are PSD.
constraints += [SDP(X), SDP(A)]
# Fix Z as upper triangular, D as diagonal,
# and diag(D) as diag(Z).
for i in xrange(n):
for j in xrange(n):
if i == j:
# D[i, j] == Z[i, j]
Dij = index.get_index(D, constraints, i, j)
Zij = index.get_index(Z, constraints, i, j)
constraints.append(lu.create_eq(Dij, Zij))
if i != j:
# D[i, j] == 0
Dij = index.get_index(D, constraints, i, j)
constraints.append(lu.create_eq(Dij))
if i > j:
# Z[i, j] == 0
Zij = index.get_index(Z, constraints, i, j)
constraints.append(lu.create_eq(Zij))
# Fix X using the fact that A must be affine by the DCP rules.
# X[0:n, 0:n] == D
index.block_eq(X, D, constraints, 0, n, 0, n)
# X[0:n, n:2*n] == Z,
index.block_eq(X, Z, constraints, 0, n, n, 2 * n)
# X[n:2*n, n:2*n] == A
index.block_eq(X, A, constraints, n, 2 * n, n, 2 * n)
# Add the objective sum(log(D[i, i])
log_diag = []
for i in xrange(n):
Dii = index.get_index(D, constraints, i, i)
obj, constr = log.graph_implementation([Dii], (1, 1))
constraints += constr
log_diag.append(obj)
obj = lu.sum_expr(log_diag)
return (obj, constraints)
