# Source code for cvxpy.constraints.psd

```
"""
Copyright 2013 Steven Diamond
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
"""
import cvxpy.settings as s
import cvxpy.lin_ops.lin_utils as lu
import cvxpy.utilities.performance_utils as pu
from cvxpy.expressions import cvxtypes
from cvxpy.constraints.constraint import Constraint
[docs]class PSD(Constraint):
"""A constraint of the form :math:`\\frac{1}{2}(X + X^T) \\succcurlyeq_{S_n^+} 0`
Applying a ``PSD`` constraint to a two-dimensional expression ``X``
constrains its symmetric part to be positive semidefinite: i.e.,
it constrains ``X`` to be such that
.. math::
z^T(X + X^T)z \\geq 0,
for all :math:`z`.
The preferred way of creating a ``PSD`` constraint is through operator
overloading. To constrain an expression ``X`` to be PSD, write
``X >> 0``; to constrain it to be negative semidefinite, write
``X << 0``. Strict definiteness constraints are not provided,
as they do not make sense in a numerical setting.
Parameters
----------
expr : Expression.
The expression to constrain; *must* be two-dimensional.
constr_id : int
A unique id for the constraint.
"""
def __init__(self, expr, constr_id=None):
# Argument must be square matrix.
if len(expr.shape) != 2 or expr.shape[0] != expr.shape[1]:
raise ValueError(
"Non-square matrix in positive definite constraint."
)
super(PSD, self).__init__([expr], constr_id)
def name(self):
return "%s >> 0" % self.args[0]
[docs] def is_dcp(self):
"""A PSD constraint is DCP if the constrained expression is affine.
"""
return self.args[0].is_affine()
def is_dgp(self):
return False
@property
def residual(self):
"""The residual of the constraint.
Returns
-------
NumPy.ndarray
"""
if self.expr.value is None:
return None
min_eig = cvxtypes.lambda_min()(self.args[0] + self.args[0].T)/2
return cvxtypes.neg()(min_eig).value
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 Zero.
Returns:
A tuple of (affine expression, [constraints]).
"""
obj, constraints = self.args[0].canonical_form
dual_holder = PSD(obj, constr_id=self.id)
return (None, constraints + [dual_holder])
def format(self, eq_constr, leq_constr, dims, solver):
"""Formats PSD 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.
"""
new_leq_constr = self.__format
# 0 <= A
leq_constr += new_leq_constr
# Update dims.
dims[s.PSD_DIM].append(self.shape[0])
@pu.lazyprop
def __format(self):
"""Internal version of format with cached results.
Returns
-------
tuple
(equality constraints, inequality constraints)
"""
leq_constr = lu.create_geq(self.expr, constr_id=self.constr_id)
return [leq_constr]
```