def log_det_canon(expr, args):
"""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
----------
expr : log_det
args : list
The arguments for the expression
Returns
-------
tuple
(Variable for objective, list of constraints)
"""
A = args[0] # n by n matrix.
n, _ = A.shape
z = Variable(shape=(n * (n + 1) // 2, ))
Z = vec_to_upper_tri(z, strict=False)
d = diag_mat(Z) # a vector
D = diag_vec(d) # a matrix
X = bmat([[D, Z], [Z.T, A]])
constraints = [PSD(X)]
log_expr = log(d)
obj, constr = log_canon(log_expr, log_expr.args)
constraints += constr
return sum(obj), constraints
def log_det_canon(expr, args):
"""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
----------
expr : log_det
args : list
The arguments for the expression
Returns
-------
tuple
(Variable for objective, list of constraints)
"""
A = args[0] # n by n matrix.
n, _ = A.shape
# Require that X and A are PSD.
X = Variable((2 * n, 2 * n), PSD=True)
constraints = [PSD(A)]
# Fix Z as upper triangular
# TODO represent Z as upper tri vector.
Z = Variable((n, n))
Z_lower_tri = upper_tri(transpose(Z))
constraints.append(Z_lower_tri == 0)
# Fix diag(D) = diag(Z): D[i, i] = Z[i, i]
D = Variable(n)
constraints.append(D == diag_mat(Z))
# Fix X using the fact that A must be affine by the DCP rules.
# X[0:n, 0:n] == D
constraints.append(X[0:n, 0:n] == diag_vec(D))
# X[0:n, n:2*n] == Z,
constraints.append(X[0:n, n:2 * n] == Z)
# X[n:2*n, n:2*n] == A
constraints.append(X[n:2 * n, n:2 * n] == A)
# Add the objective sum(log(D[i, i])
log_expr = log(D)
obj, constr = log_canon(log_expr, log_expr.args)
constraints += constr
return sum(obj), constraints