Source code for picos.constraints.con_lmi

# ------------------------------------------------------------------------------
# Copyright (C) 2018-2019 Maximilian Stahlberg
#
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"""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, HermitianVariable, SymmetricVariable) from ..expressions.vectorizations import (HermitianVectorization, SymmetricVectorization) from ..expressions.data import cvxopt_equals 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))) # Check if the constraint simply poses positive semidefiniteness on a # matrix variable, as certain solvers can handle this more efficiently # than a general linear matrix inequality. self.semidefVar = None if len(psd._linear_coefs) == 1 and not psd._constant_coef: var, coef = list(psd._linear_coefs.items())[0] if isinstance(var, SymmetricVariable) and cvxopt_equals( coef, SymmetricVectorization(psd.shape).identity, relTol=settings.RELATIVE_HERMITIANNESS_TOLERANCE): self.semidefVar = var elif isinstance(var, HermitianVariable) and cvxopt_equals( coef, HermitianVectorization(psd.shape).identity, relTol=settings.RELATIVE_HERMITIANNESS_TOLERANCE): self.semidefVar = var 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 @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
[docs] @cached_property def conic_membership_form(self): """Implement for :class:`~.constraint.ConicConstraint`.""" from ..expressions import PositiveSemidefiniteCone element = self.psd.svec return element, PositiveSemidefiniteCone(dim=len(element))
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())