# ------------------------------------------------------------------------------
# Copyright (C) 2018-2019 Maximilian Stahlberg
#
# This file is part of PICOS.
#
# PICOS is free software: you can redistribute it and/or modify it under the
# terms of the GNU General Public License as published by the Free Software
# Foundation, either version 3 of the License, or (at your option) any later
# version.
#
# PICOS is distributed in the hope that it will be useful, but WITHOUT ANY
# WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR
# A PARTICULAR PURPOSE. See the GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License along with
# this program. If not, see <http://www.gnu.org/licenses/>.
# ------------------------------------------------------------------------------
"""Linear matrix inequalities."""
from collections import namedtuple
from .. import glyphs, settings
from ..apidoc import api_end, api_start
from ..caching import cached_property
from .constraint import ConicConstraint, ConstraintConversion
_API_START = api_start(globals())
# -------------------------------
[docs]class LMIConstraint(ConicConstraint):
"""Linear matrix inequality.
An inequality with respect to the positive semidefinite cone, also known as
a Linear Matrix Inequality (LMI) or an SDP constraint.
"""
[docs] def __init__(self, lhs, relation, rhs, customString=None):
"""Construct a :class:`LMIConstraint`.
:param ~picos.expressions.AffineExpression lhs:
Left hand side expression.
:param str relation:
Constraint relation symbol.
:param ~picos.expressions.AffineExpression rhs:
Right hand side expression.
:param str customString:
Optional string description.
"""
from ..expressions import AffineExpression
required_type = self._required_type()
assert isinstance(lhs, required_type)
assert isinstance(rhs, required_type)
assert relation in self.LE + self.GE
if lhs.shape != rhs.shape:
raise ValueError("Failed to form a constraint: "
"Expressions have incompatible dimensions.")
if lhs.shape[0] != lhs.shape[1]:
raise ValueError("Failed to form a constraint: "
"LMI expressions are not square.")
self.lhs = lhs
self.rhs = rhs
self.relation = relation
psd = self.psd
if not psd.hermitian:
needed = "symmetric" if required_type is AffineExpression \
else "hermitian"
raise ValueError("Failed to form a constraint: {} is not "
"necessarily {}. Consider a constraint on {} instead.".format(
psd.string, needed, glyphs.Tr("{}.hermitianized")(psd.string)))
super(LMIConstraint, self).__init__(
self._get_type_term(), customString, printSize=True)
def _get_type_term(self):
return "LMI"
def _required_type(self):
from ..expressions import AffineExpression
return AffineExpression
[docs] @cached_property
def semidefVar(self):
"""The variable posed positive semidefinite, or :obj:`None`."""
from ..expressions import HermitianVariable, SymmetricVariable
from ..expressions.data import cvxopt_equals
psd = self.psd
if len(psd._linear_coefs) == 1 and not psd._constant_coef:
var, coef = next(iter(psd._linear_coefs.items()))
if isinstance(var, (SymmetricVariable, HermitianVariable)) \
and cvxopt_equals(coef, var._vec.identity,
relTol=settings.RELATIVE_HERMITIANNESS_TOLERANCE):
return var
return None
@property
def smaller(self):
"""The smaller-or-equal side expression."""
return self.rhs if self.relation == self.GE else self.lhs
@property
def greater(self):
"""The greater-or-equal side expression."""
return self.lhs if self.relation == self.GE else self.rhs
[docs] @cached_property
def psd(self):
"""The matrix expression posed to be positive semidefinite."""
if self.relation == self.GE:
return self.lhs - self.rhs
else:
return self.rhs - self.lhs
[docs] @cached_property
def nsd(self):
"""The matrix expression posed to be negative semidefinite."""
if self.relation == self.GE:
return self.rhs - self.lhs
else:
return self.lhs - self.rhs
nnd = psd
npd = nsd
Subtype = namedtuple("Subtype", ("diag",))
def _subtype(self):
return self.Subtype(self.lhs.shape[0])
@classmethod
def _cost(cls, subtype):
n = subtype.diag
return n*(n + 1)//2
def _expression_names(self):
yield "lhs"
yield "rhs"
def _str(self):
if self.relation == self.LE:
return glyphs.psdle(self.lhs.string, self.rhs.string)
else:
return glyphs.psdge(self.lhs.string, self.rhs.string)
def _get_size(self):
return self.lhs.shape
def _get_slack(self):
return self.psd.safe_value
[docs]class ComplexLMIConstraint(LMIConstraint):
"""Complex linear matrix inequality."""
[docs] class RealConversion(ConstraintConversion):
"""Complex LMI to real LMI conversion."""
[docs] @classmethod
def predict(cls, subtype, options):
"""Implement :meth:`~.constraint.ConstraintConversion.predict`."""
n = subtype.diag
yield ("con", LMIConstraint.make_type(diag=2*n), 1)
[docs] @classmethod
def convert(cls, con, options):
"""Implement :meth:`~.constraint.ConstraintConversion.convert`."""
from ..expressions.algebra import block
from ..modeling import Problem
P = Problem()
Z = con.psd
P.add_constraint(block([[Z.real, -Z.imag], [Z.imag, Z.real]]) >> 0)
return P
[docs] @classmethod
def dual(cls, auxVarPrimals, auxConDuals, options):
"""Implement :meth:`~.constraint.ConstraintConversion.dual`."""
assert len(auxConDuals) == 1
auxConDual = auxConDuals[0]
if auxConDual is None:
return None
else:
assert auxConDual.size[0] == auxConDual.size[1]
n = auxConDual.size[0] // 2
assert 2*n == auxConDual.size[0]
A = auxConDual[:n, :n]
B = auxConDual[:n, n:]
D = auxConDual[n:, n:]
return (A + 1j*B) + (D + 1j*B).H
def _get_type_term(self):
return "Complex LMI"
def _required_type(self):
from ..expressions import ComplexAffineExpression
return ComplexAffineExpression
@classmethod
def _cost(cls, subtype):
return subtype.diag**2
# --------------------------------------
__all__ = api_end(_API_START, globals())