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 by m 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))
constraints = []
# Fix X using the fact that A must be affine by the DCP rules.
# X[0:n, 0:n] == I_n*t
prom_t = lu.promote(t, (n, 1))
index.block_eq(X, lu.diag_vec(prom_t), 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, n:n+m] == I_m*t
prom_t = lu.promote(t, (m, 1))
# prom_t = lu.promote(lu.create_const(1, (1,1)), (m, 1))
index.block_eq(X, lu.diag_vec(prom_t), 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]
n, _ = A.size
# 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)])
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.
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 by m 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))
# 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
prom_t = lu.promote(t, (n, 1))
index.block_eq(X, lu.diag_vec(prom_t), 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
prom_t = lu.promote(t, (m, 1))
index.block_eq(X, lu.diag_vec(prom_t), constraints,
n, n+m, n, n+m)
# Add SDP constraint.
return (t, constraints + [SDP(X)])
def graph_implementation(arg_objs, size, data=None):
"""Convolve two vectors.
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.diag_vec(arg_objs[0]), [])
def graph_implementation(self,
arg_objs,
shape: Tuple[int, ...],
data=None) -> Tuple[lo.LinOp, List[Constraint]]:
"""Convolve two vectors.
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)
"""
return (lu.diag_vec(arg_objs[0]), [])
def graph_implementation(arg_objs, size, data=None):
"""Multiply the linear expressions.
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)
"""
# Promote the right hand side to a diagonal matrix if necessary.
if size[1] != 1 and arg_objs[1].size == (1, 1):
arg = lu.promote(arg_objs[1], (size[1], 1))
arg_objs[1] = lu.diag_vec(arg)
return (lu.mul_expr(arg_objs[0], arg_objs[1], size), [])
def graph_implementation(arg_objs, size, data=None):
"""Multiply the linear expressions.
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)
"""
# Promote the left hand side to a diagonal matrix if necessary.
if size[0] != 1 and arg_objs[0].size == (1, 1):
arg = lu.promote(arg_objs[0], (size[0], 1))
arg_objs[0] = lu.diag_vec(arg)
return (lu.rmul_expr(arg_objs[0], arg_objs[1], size), [])
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)
