def __format(self):
"""Internal version of format with cached results.
Returns
-------
tuple
(equality constraints, inequality constraints)
"""
upper_tri = lu.upper_tri(self.A)
lower_tri = lu.upper_tri(lu.transpose(self.A))
eq_constr = lu.create_eq(upper_tri, lower_tri)
leq_constr = lu.create_geq(self.A)
return ([eq_constr], [leq_constr])
def __format(self):
"""Internal version of format with cached results.
Returns
-------
tuple
(equality constraints, inequality constraints)
"""
upper_tri = lu.upper_tri(self.A)
lower_tri = lu.upper_tri(lu.transpose(self.A))
eq_constr = lu.create_eq(upper_tri, lower_tri)
leq_constr = lu.create_geq(self.A)
return ([eq_constr], [leq_constr])
def graph_implementation(arg_objs, size, data=None):
"""Vectorized strictly upper triagonal entries.
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)
"""
return (lu.upper_tri(arg_objs[0]), [])
def graph_implementation(arg_objs, size, data=None):
"""Vectorized strictly upper triagonal entries.
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)
"""
return (lu.upper_tri(arg_objs[0]), [])
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 _get_eq_constr(self):
"""Returns the equality constraints for the SDP constraint.
"""
upper_tri = lu.upper_tri(self.A)
lower_tri = lu.upper_tri(lu.transpose(self.A))
return lu.create_eq(upper_tri, lower_tri)
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)