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1# ------------------------------------------------------------------------------

2# Copyright (C) 2019 Maximilian Stahlberg

3# Based on the original picos.expressions module by Guillaume Sagnol.

4#

5# This file is part of PICOS.

6#

7# PICOS is free software: you can redistribute it and/or modify it under the

9# Foundation, either version 3 of the License, or (at your option) any later

10# version.

11#

12# PICOS is distributed in the hope that it will be useful, but WITHOUT ANY

13# WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR

14# A PARTICULAR PURPOSE. See the GNU General Public License for more details.

15#

16# You should have received a copy of the GNU General Public License along with

17# this program. If not, see <http://www.gnu.org/licenses/>.

18# ------------------------------------------------------------------------------

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

22import operator

23from collections import namedtuple

25import cvxopt

26import numpy

28from .. import glyphs

29from ..apidoc import api_end, api_start

30from ..constraints import DetRootNConstraint

31from .data import convert_and_refine_arguments, convert_operands, cvx2np

32from .exp_affine import AffineExpression, ComplexAffineExpression

33from .expression import Expression, refine_operands, validate_prediction

35_API_START = api_start(globals())

36# -------------------------------

39class DetRootN(Expression):

40 r"""The :math:n-th root of the determinant of an :math:n\times n matrix.

42 :Definition:

44 For an :math:n \times n positive semidefinite hermitian matrix :math:X,

45 this is

47 .. math::

49 \sqrt[n]{\det X}.

51 .. warning::

53 When you pose a lower bound on the :math:n-th root of a determinant of

54 the matrix :math:X, then PICOS enforces positive semidefiniteness

55 :math:X \succeq 0 through an auxiliary constraint during solution

56 search.

57 """

59 # --------------------------------------------------------------------------

60 # Initialization and factory methods.

61 # --------------------------------------------------------------------------

63 @convert_and_refine_arguments("x")

64 def __init__(self, x):

65 """Construct a :class:DetRootN.

67 :param x: The matrix concerned. Must be hermitian by definition.

68 :type x: ~picos.expressions.ComplexAffineExpression

69 """

70 if not isinstance(x, ComplexAffineExpression):

71 raise TypeError("Can only form the determinant of an affine "

72 "expression, not of {}.".format(type(x).__name__))

73 elif not x.square:

74 raise TypeError("Can't take the determinant of non-square {0}."

75 .format(x.string))

76 elif not x.hermitian:

77 raise NotImplementedError("Taking the n-th root of the determinant "

78 "of {0} is not supported as {0} is not necessarily hermitian."

79 .format(x.string))

81 self._x = x

83 Expression.__init__(self, "n-th Root of a Determinant",

84 glyphs.power(glyphs.det(x.string), glyphs.div(1, x.shape[0])))

86 # --------------------------------------------------------------------------

87 # Abstract method implementations and method overridings, except _predict.

88 # --------------------------------------------------------------------------

90 def _get_refined(self):

91 if self._x.constant:

92 return AffineExpression.from_constant(self.value, 1, self._symbStr)

93 elif len(self._x) == 1:

94 return self._x.renamed(self._symbStr)

95 else:

96 return self

98 Subtype = namedtuple("Subtype", ("diag", "complex"))

100 def _get_subtype(self):

101 return self.Subtype(self.n, self._x.complex)

103 def _get_value(self):

104 value = self._x._get_value()

106 det = numpy.linalg.det(cvx2np(value))

108 if det < 0:

109 raise ArithmeticError("Cannot evaluate {}: {} is negative."

110 .format(self.string, glyphs.eq(glyphs.det(self.x.string), det)))

112 return cvxopt.matrix(det**(1.0 / self._x.shape[0]))

114 def _get_mutables(self):

115 return self._x._get_mutables()

117 def _is_convex(self):

118 return False

120 def _is_concave(self):

121 return True

123 def _replace_mutables(self, mapping):

124 return self.__class__(self._x._replace_mutables(mapping))

126 def _freeze_mutables(self, freeze):

127 return self.__class__(self._x._freeze_mutables(freeze))

129 # --------------------------------------------------------------------------

130 # Python special method implementations, except constraint-creating ones.

131 # --------------------------------------------------------------------------

133 @convert_operands(scalarRHS=True)

134 @refine_operands()

135 def __mul__(self, other):

136 if isinstance(other, AffineExpression):

137 if not other.constant:

138 raise NotImplementedError("You may only multiply a nonconstant "

139 "PICOS n-th root of a determinant with a constant term.")

141 root = DetRootN(other.value * self._x)

142 root._typeStr = "Scaled " + root._typeStr

143 root._symbStr = glyphs.clever_mul(self.string, other.string)

144 return root

145 else:

146 return NotImplemented

148 @convert_operands(scalarRHS=True)

149 @refine_operands()

150 def __rmul__(self, other):

151 if isinstance(other, AffineExpression):

152 mean = self.__mul__(other)

153 # NOTE: __mul__ always creates a fresh expression.

154 mean._symbStr = glyphs.clever_mul(other.string, self.string)

155 return mean

156 else:

157 return NotImplemented

159 # --------------------------------------------------------------------------

160 # Methods and properties that return modified copies.

161 # --------------------------------------------------------------------------

163 @property

164 def x(self):

165 """The matrix concerned."""

166 return self._x

168 # --------------------------------------------------------------------------

169 # Methods and properties that describe the expression.

170 # --------------------------------------------------------------------------

172 @property

173 def n(self):

174 """Diagonal length of :attr:x."""

175 return self._x.shape[0]

177 # --------------------------------------------------------------------------

178 # Constraint-creating operators, and _predict.

179 # --------------------------------------------------------------------------

181 @classmethod

182 def _predict(cls, subtype, relation, other):

183 assert isinstance(subtype, cls.Subtype)

185 if relation == operator.__ge__:

186 if issubclass(other.clstype, AffineExpression) \

187 and other.subtype.dim == 1:

188 return DetRootNConstraint.make_type(

189 diag=subtype.diag, complex=subtype.complex)

191 return NotImplemented

193 @convert_operands(scalarRHS=True)

194 @validate_prediction

195 @refine_operands()

196 def __ge__(self, other):

197 if isinstance(other, AffineExpression):

198 return DetRootNConstraint(self, other)

199 else:

200 return NotImplemented

203# --------------------------------------

204__all__ = api_end(_API_START, globals())