# Source code for picos.expressions.exp_detrootn

# ------------------------------------------------------------------------------
# Copyright (C) 2019 Maximilian Stahlberg
# Based on the original picos.expressions module by Guillaume Sagnol.
#
# This file is part of PICOS.
#
# PICOS is free software: you can redistribute it and/or modify it under the
# 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/>.
# ------------------------------------------------------------------------------

"""Implements :class:DetRootN."""

import operator
from collections import namedtuple

import cvxopt
import numpy

from .. import glyphs
from ..apidoc import api_end, api_start
from ..constraints import DetRootNConstraint
from .data import convert_and_refine_arguments, convert_operands, cvx2np
from .exp_affine import AffineExpression, ComplexAffineExpression
from .expression import Expression, refine_operands, validate_prediction

_API_START = api_start(globals())
# -------------------------------

[docs]class DetRootN(Expression): r"""The :math:n-th root of the determinant of an :math:n\times n matrix. :Definition: For an :math:n \times n positive semidefinite hermitian matrix :math:X, this is .. math:: \sqrt[n]{\det X}. .. warning:: When you pose a lower bound on the :math:n-th root of a determinant of the matrix :math:X, then PICOS enforces positive semidefiniteness :math:X \succeq 0 through an auxiliary constraint during solution search. """ # -------------------------------------------------------------------------- # Initialization and factory methods. # --------------------------------------------------------------------------
[docs] @convert_and_refine_arguments("x") def __init__(self, x): """Construct a :class:DetRootN. :param x: The matrix concerned. Must be hermitian by definition. :type x: ~picos.expressions.ComplexAffineExpression """ if not isinstance(x, ComplexAffineExpression): raise TypeError("Can only form the determinant of an affine " "expression, not of {}.".format(type(x).__name__)) elif not x.square: raise TypeError("Can't take the determinant of non-square {0}." .format(x.string)) elif not x.hermitian: raise NotImplementedError("Taking the n-th root of the determinant " "of {0} is not supported as {0} is not necessarily hermitian." .format(x.string)) self._x = x Expression.__init__(self, "n-th Root of a Determinant", glyphs.power(glyphs.det(x.string), glyphs.div(1, x.shape[0])))
# -------------------------------------------------------------------------- # Abstract method implementations and method overridings, except _predict. # -------------------------------------------------------------------------- def _get_refined(self): if self._x.constant: return AffineExpression.from_constant(self.value, 1, self._symbStr) elif len(self._x) == 1: return self._x.renamed(self._symbStr) else: return self Subtype = namedtuple("Subtype", ("diag", "complex")) def _get_subtype(self): return self.Subtype(self.n, self._x.complex) def _get_value(self): value = self._x._get_value() det = numpy.linalg.det(cvx2np(value)) if det < 0: raise ArithmeticError("Cannot evaluate {}: {} is negative." .format(self.string, glyphs.eq(glyphs.det(self.x.string), det))) return cvxopt.matrix(det**(1.0 / self._x.shape[0])) def _get_mutables(self): return self._x._get_mutables() def _is_convex(self): return False def _is_concave(self): return True def _replace_mutables(self, mapping): return self.__class__(self._x._replace_mutables(mapping)) def _freeze_mutables(self, freeze): return self.__class__(self._x._freeze_mutables(freeze)) # -------------------------------------------------------------------------- # Python special method implementations, except constraint-creating ones. # -------------------------------------------------------------------------- @classmethod def _mul(cls, self, other, forward): if isinstance(other, AffineExpression) and other.constant: factor = other.safe_value if not factor: return AffineExpression.zero() elif factor == 1: return self elif factor > 0: if forward: string = glyphs.clever_mul(self.string, other.string) else: string = glyphs.clever_mul(other.string, self.string) result = cls(other*self._x) result._typeStr = "Scaled " + result._typeStr result._symbStr = string return result if forward: return Expression.__mul__(self, other) else: return Expression.__rmul__(self, other)
[docs] @convert_operands(scalarRHS=True) @refine_operands() def __mul__(self, other): return DetRootN._mul(self, other, True)
[docs] @convert_operands(scalarRHS=True) @refine_operands() def __rmul__(self, other): return DetRootN._mul(self, other, False)
# -------------------------------------------------------------------------- # Methods and properties that return modified copies. # -------------------------------------------------------------------------- @property def x(self): """The matrix concerned.""" return self._x # -------------------------------------------------------------------------- # Methods and properties that describe the expression. # -------------------------------------------------------------------------- @property def n(self): """Diagonal length of :attr:x.""" return self._x.shape[0] # -------------------------------------------------------------------------- # Constraint-creating operators, and _predict. # -------------------------------------------------------------------------- @classmethod def _predict(cls, subtype, relation, other): assert isinstance(subtype, cls.Subtype) if relation == operator.__ge__: if issubclass(other.clstype, AffineExpression) \ and other.subtype.dim == 1: return DetRootNConstraint.make_type( diag=subtype.diag, complex=subtype.complex) return NotImplemented
[docs] @convert_operands(scalarRHS=True) @validate_prediction @refine_operands() def __ge__(self, other): if isinstance(other, AffineExpression): return DetRootNConstraint(self, other) else: return NotImplemented
# -------------------------------------- __all__ = api_end(_API_START, globals())