def graph_implementation(self, arg_objs, shape: Tuple[int, ...], data=None):
"""Reshape
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)
"""
arg = arg_objs[0]
if data[1] == 'F':
return (lu.reshape(arg, shape), [])
else: # 'C':
arg = lu.transpose(arg)
if len(shape) <= 1:
return (lu.reshape(arg, shape), [])
else:
result = lu.reshape(arg, (shape[1], shape[0]))
return (lu.transpose(result), [])
def canonicalize(self):
obj, constraints = super(Assign, self).canonicalize()
shape = (self.size[1], 1)
one_row_vec = lu.create_const(np.ones(shape), shape)
shape = (1, self.size[0])
one_col_vec = lu.create_const(np.ones(shape), shape)
# Row sum <= 1
row_sum = lu.rmul_expr(obj, one_row_vec, (self.size[0], 1))
constraints += [lu.create_leq(row_sum, lu.transpose(one_col_vec))]
# Col sum == 1.
col_sum = lu.mul_expr(one_col_vec, obj, (1, self.size[1]))
constraints += [lu.create_eq(col_sum, lu.transpose(one_row_vec))]
return (obj, constraints)
def canonicalize(self):
obj, constraints = super(Assign, self).canonicalize()
shape = (self.size[1], 1)
one_row_vec = lu.create_const(np.ones(shape), shape)
shape = (1, self.size[0])
one_col_vec = lu.create_const(np.ones(shape), shape)
# Row sum <= 1
row_sum = lu.rmul_expr(obj, one_row_vec, (self.size[0], 1))
constraints += [lu.create_leq(row_sum, lu.transpose(one_col_vec))]
# Col sum == 1.
col_sum = lu.mul_expr(one_col_vec, obj, (1, self.size[1]))
constraints += [lu.create_eq(col_sum, lu.transpose(one_row_vec))]
return (obj, constraints)
def __format(self):
"""Internal version of format with cached results.
Returns
-------
tuple
(equality constraints, inequality constraints)
"""
eq_constr = lu.create_eq(self.A, lu.transpose(self.A))
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)
"""
eq_constr = lu.create_eq(self.A, lu.transpose(self.A))
leq_constr = lu.create_geq(self.A)
return ([eq_constr], [leq_constr])
def format_axis(t, X, axis):
"""Formats all the row/column cones for the solver.
Parameters
----------
t: The scalar part of the second-order constraint.
X: A matrix whose rows/columns are each a cone.
axis: Slice by column 0 or row 1.
Returns
-------
list
A list of LinLeqConstr that represent all the elementwise cones.
"""
# Reduce to norms of columns.
if axis == 1:
X = lu.transpose(X)
# Create matrices Tmat, Xmat such that Tmat*t + Xmat*X
# gives the format for the elementwise cone constraints.
num_cones = t.size[0]
cone_size = 1 + X.size[0]
terms = []
# Make t_mat
mat_size = (cone_size, 1)
prod_size = (cone_size, t.size[0])
t_mat = sp.coo_matrix(([1.0], ([0], [0])), mat_size).tocsc()
t_mat = lu.create_const(t_mat, mat_size, sparse=True)
terms += [lu.mul_expr(t_mat, lu.transpose(t), prod_size)]
# Make X_mat
mat_size = (cone_size, X.size[0])
prod_size = (cone_size, X.size[1])
val_arr = (cone_size - 1)*[1.0]
row_arr = range(1, cone_size)
col_arr = range(cone_size-1)
X_mat = sp.coo_matrix((val_arr, (row_arr, col_arr)), mat_size).tocsc()
X_mat = lu.create_const(X_mat, mat_size, sparse=True)
terms += [lu.mul_expr(X_mat, X, prod_size)]
return [lu.create_geq(lu.sum_expr(terms))]
def format_axis(t, X, axis):
"""Formats all the row/column cones for the solver.
Parameters
----------
t: The scalar part of the second-order constraint.
X: A matrix whose rows/columns are each a cone.
axis: Slice by column 0 or row 1.
Returns
-------
list
A list of LinLeqConstr that represent all the elementwise cones.
"""
# Reduce to norms of columns.
if axis == 1:
X = lu.transpose(X)
# Create matrices Tmat, Xmat such that Tmat*t + Xmat*X
# gives the format for the elementwise cone constraints.
num_cones = t.size[0]
cone_size = 1 + X.size[0]
terms = []
# Make t_mat
mat_size = (cone_size, 1)
prod_size = (cone_size, t.size[0])
t_mat = sp.coo_matrix(([1.0], ([0], [0])), mat_size).tocsc()
t_mat = lu.create_const(t_mat, mat_size, sparse=True)
terms += [lu.mul_expr(t_mat, lu.transpose(t), prod_size)]
# Make X_mat
mat_size = (cone_size, X.size[0])
prod_size = (cone_size, X.size[1])
val_arr = (cone_size - 1) * [1.0]
row_arr = range(1, cone_size)
col_arr = range(cone_size - 1)
X_mat = sp.coo_matrix((val_arr, (row_arr, col_arr)), mat_size).tocsc()
X_mat = lu.create_const(X_mat, mat_size, sparse=True)
terms += [lu.mul_expr(X_mat, X, prod_size)]
return [lu.create_geq(lu.sum_expr(terms))]
def canonicalize(self):
"""Returns the graph implementation of the object.
Marks the top level constraint as the dual_holder,
so the dual value will be saved to the EqConstraint.
Returns:
A tuple of (affine expression, [constraints]).
"""
obj, constraints = self._expr.canonical_form
half = lu.create_const(0.5, (1, 1))
symm = lu.mul_expr(half, lu.sum_expr([obj, lu.transpose(obj)]),
obj.size)
dual_holder = SDP(symm, enforce_sym=False, constr_id=self.id)
return (None, constraints + [dual_holder])
def canonicalize(self):
"""Returns the graph implementation of the object.
Marks the top level constraint as the dual_holder,
so the dual value will be saved to the EqConstraint.
Returns:
A tuple of (affine expression, [constraints]).
"""
obj, constraints = self._expr.canonical_form
half = lu.create_const(0.5, (1,1))
symm = lu.mul_expr(half, lu.sum_expr([obj, lu.transpose(obj)]),
obj.size)
dual_holder = SDP(symm, enforce_sym=False, constr_id=self.id)
return (None, constraints + [dual_holder])
def format_axis(t, X, axis):
"""Formats all the row/column cones for the solver.
Parameters
----------
t: The scalar part of the second-order constraint.
X: A matrix whose rows/columns are each a cone.
axis: Slice by column 0 or row 1.
Returns
-------
list
A list of LinLeqConstr that represent all the elementwise cones.
"""
# Reduce to norms of columns.
if axis == 1:
X = lu.transpose(X)
# Create matrices Tmat, Xmat such that Tmat*t + Xmat*X
# gives the format for the elementwise cone constraints.
cone_size = 1 + X.shape[0]
terms = []
# Make t_mat
mat_shape = (cone_size, 1)
t_mat = sp.csc_matrix(([1.0], ([0], [0])), mat_shape)
t_mat = lu.create_const(t_mat, mat_shape, sparse=True)
t_vec = t
if not t.shape:
# t is scalar
t_vec = lu.reshape(t, (1, 1))
else:
# t is 1D
t_vec = lu.reshape(t, (1, t.shape[0]))
mul_shape = (cone_size, t_vec.shape[1])
terms += [lu.mul_expr(t_mat, t_vec, mul_shape)]
# Make X_mat
if len(X.shape) == 1:
X = lu.reshape(X, (X.shape[0], 1))
mat_shape = (cone_size, X.shape[0])
val_arr = (cone_size - 1) * [1.0]
row_arr = list(range(1, cone_size))
col_arr = list(range(cone_size - 1))
X_mat = sp.csc_matrix((val_arr, (row_arr, col_arr)), mat_shape)
X_mat = lu.create_const(X_mat, mat_shape, sparse=True)
mul_shape = (cone_size, X.shape[1])
terms += [lu.mul_expr(X_mat, X, mul_shape)]
return [lu.create_geq(lu.sum_expr(terms))]
def graph_implementation(arg_objs, size, data=None):
"""Create a new variable equal to the argument transposed.
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.transpose(arg_objs[0]), [])
def graph_implementation(arg_objs, size, data=None):
"""Create a new variable equal to the argument transposed.
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.transpose(arg_objs[0])
def graph_implementation(self, arg_objs, shape, data=None):
"""Create a new variable equal to the argument transposed.
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)
"""
# TODO(akshakya): This will need to be updated when we add support
# for >2D ararys.
return (lu.transpose(arg_objs[0]), [])
def format(self, eq_constr, leq_constr, dims, solver):
"""Formats SDP constraints as inequalities for the solver.
Parameters
----------
eq_constr : list
A list of the equality constraints in the canonical problem.
leq_constr : list
A list of the inequality constraints in the canonical problem.
dims : dict
A dict with the dimensions of the conic constraints.
solver : str
The solver being called.
"""
# A == A.T
eq_constr.append(lu.create_eq(self.A, lu.transpose(self.A)))
# 0 <= A
leq_constr.append(lu.create_geq(self.A))
# Update dims.
dims[s.EQ_DIM] += self.size[0]*self.size[1]
dims[s.SDP_DIM].append(self.size[0])
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)
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)
