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)
"""
A = arg_objs[0]
n, _ = A.size
# Requires that A is symmetric.
obj, constraints = transpose.graph_implementation([A], (n, n))
# A == A.T
constraints.append(lu.create_eq(A, obj))
# SDP constraint.
t = lu.create_var((1, 1))
prom_t = lu.promote(t, (n, 1))
# I*t - A
expr = lu.sub_expr(A, lu.diag_vec(prom_t))
return (t, [SDP(expr)] + constraints)
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)
"""
A = arg_objs[0]
n, _ = A.size
# Requires that A is symmetric.
# A == A.T
obj, constraints = transpose.graph_implementation([A], (n, n))
constraints.append(lu.create_eq(A, obj))
# SDP constraint.
t = lu.create_var((1, 1))
prom_t = lu.promote(t, (n, 1))
# I*t - A
expr = lu.sub_expr(lu.diag_vec(prom_t), A)
return (t, [SDP(expr)] + constraints)
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)
"""
A = arg_objs[0] # m by n matrix.
n, m = A.size
# Create a matrix with Schur complement I*t - (1/t)*A.T*A.
X = lu.create_var((n+m, n+m))
t = lu.create_var((1, 1))
I_n = lu.create_const(sp.eye(n), (n, n))
I_m = lu.create_const(sp.eye(m), (m, m))
# Expand A.T.
obj, constraints = transpose.graph_implementation([A], (m, n))
# Fix X using the fact that A must be affine by the DCP rules.
# X[0:n, 0:n] == I_n*t
index.block_eq(X, lu.mul_expr(I_n, t, (n, n)), constraints,
0, n, 0, n)
# X[0:n, n:n+m] == A
index.block_eq(X, A, constraints,
0, n, n, n+m)
# X[n:n+m, 0:n] == obj
index.block_eq(X, obj, constraints,
n, n+m, 0, n)
# X[n:n+m, n:n+m] == I_m*t
index.block_eq(X, lu.mul_expr(I_m, t, (m, m)), constraints,
n, n+m, n, n+m)
# Add SDP constraint.
return (t, constraints + [SDP(X)])
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)
"""
A = arg_objs[0]
rows, cols = A.size
# Create the equivalent problem:
# minimize (trace(U) + trace(V))/2
# subject to:
# [U A; A.T V] is positive semidefinite
X = lu.create_var((rows+cols, rows+cols))
# Expand A.T.
obj, constraints = transpose.graph_implementation([A], (cols, rows))
# Fix X using the fact that A must be affine by the DCP rules.
# X[0:rows,rows:rows+cols] == A
index.block_eq(X, A, constraints,
0, rows, rows, rows+cols)
# X[rows:rows+cols,0:rows] == A.T
index.block_eq(X, obj, constraints,
rows, rows+cols, 0, rows)
diag = [index.get_index(X, constraints, i, i) for i in range(rows+cols)]
half = lu.create_const(0.5, (1, 1))
trace = lu.mul_expr(half, lu.sum_expr(diag), (1, 1))
# Add SDP constraint.
return (trace, [SDP(X)] + constraints)
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)
"""
A = arg_objs[0]
rows, cols = A.size
# Create the equivalent problem:
# minimize (trace(U) + trace(V))/2
# subject to:
# [U A; A.T V] is positive semidefinite
X = lu.create_var((rows + cols, rows + cols))
# Expand A.T.
obj, constraints = transpose.graph_implementation([A], (cols, rows))
# Fix X using the fact that A must be affine by the DCP rules.
# X[0:rows,rows:rows+cols] == A
index.block_eq(X, A, constraints, 0, rows, rows, rows + cols)
# X[rows:rows+cols,0:rows] == A.T
index.block_eq(X, obj, constraints, rows, rows + cols, 0, rows)
diag = [
index.get_index(X, constraints, i, i) for i in range(rows + cols)
]
half = lu.create_const(0.5, (1, 1))
trace = lu.mul_expr(half, lu.sum_expr(diag), (1, 1))
# Add SDP constraint.
return (trace, [SDP(X)] + 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))
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))
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)