# ------------------------------------------------------------------------------
# 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
# 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/>.
# ------------------------------------------------------------------------------
"""Implements :class:`PowerTrace`."""
import operator
from collections import namedtuple
import cvxopt
import numpy
from .. import glyphs
from ..apidoc import api_end, api_start
from ..constraints import Constraint, PowerTraceConstraint
from .data import (convert_and_refine_arguments, convert_operands, cvx2np,
cvxopt_hpsd, make_fraction)
from .exp_affine import AffineExpression, ComplexAffineExpression, Constant
from .expression import Expression, refine_operands, validate_prediction
_API_START = api_start(globals())
# -------------------------------
[docs]class PowerTrace(Expression):
r"""The trace of the :math:`p`-th power of a hermitian matrix.
:Definition:
Let :math:`p \in \mathbb{Q}`.
1. If the base expressions is a real scalar :math:`x` and no additional
constant :math:`m` is given, then this is the power :math:`x^p`.
2. If the base expressions is a real scalar :math:`x`,
:math:`p \in [0, 1]`, and a positive scalar constant :math:`m` is given,
then this is the scaled power :math:`m x^p`.
3. If the base expression is a hermitian matrix :math:`X` and no additional
constant :math:`M` is given, then this is the trace of power
:math:`\operatorname{tr}(X^p)`.
4. If the base expression is a hermitian matrix :math:`X`,
:math:`p \in [0, 1]`, and a hermitian positive semidefinite constant
matrix :math:`M` of same shape as :math:`X` is given, then this is the
trace of a scaled power :math:`\operatorname{tr}(M X^p)`.
No other case is supported. In particular, if :math:`p \not\in [0, 1]`, then
:math:`m`/:math:`M` must be undefined (:obj:`None`).
.. warning::
1. For a constraint of the form :math:`x^p \leq t` with :math:`p < 1`
and :math:`p \neq 0`, PICOS enforces :math:`x \geq 0` during solution
search.
2. For a constraint of the form :math:`\operatorname{tr}(X^p) \leq t` or
:math:`\operatorname{tr}(M X^p) \leq t` with :math:`p < 1` and
:math:`p \neq 0`, PICOS enforces :math:`X \succeq 0` during solution
search.
3. For a constraint of the form :math:`\operatorname{tr}(X^p) \leq t`
or :math:`\operatorname{tr}(M X^p) \leq t` with :math:`p > 1`, PICOS
enforces :math:`t \geq 0` during solution search.
"""
# --------------------------------------------------------------------------
# Initialization and factory methods.
# --------------------------------------------------------------------------
[docs] @convert_and_refine_arguments("x")
def __init__(self, x, p, m=None, denominator_limit=1000):
"""Construct a :class:`PowerTrace`.
:param x: The scalar or symmetric matrix to form a power of.
:type x: ~picos.expressions.AffineExpression
:param float p: The value for :math:`p`, which is cast to a limited
precision fraction.
:param m: An additional positive semidefinite constant to multiply the
power with.
:type m: :class:`~picos.expressions.AffineExpression` or anything
recognized by :func:`~picos.expressions.data.load_data`
:param int denominator_limit: The largest allowed denominator when
casting :math:`p` to a fraction. Higher values can yield a greater
precision at reduced performance.
"""
# Validate x.
if not isinstance(x, ComplexAffineExpression):
raise TypeError("Can only form the power of an affine expression, "
"not of {}.".format(x.string))
elif not x.square:
raise TypeError(
"Can't form the power of non-square {}.".format(x.string))
elif not x.hermitian:
raise NotImplementedError("Taking {} to a power is not supported "
"as it is not necessarily hermitian.".format(x.string))
# Load p.
pNum, pDen, p, pStr = make_fraction(p, denominator_limit)
# Load m.
if m is not None:
mStr = "m" if len(x) == 1 else "M"
if p < 0 or p > 1:
raise ValueError(
"p-th power with an additional factor {} requires {}."
.format(mStr, glyphs.le(0, glyphs.le("p", 1))))
if not isinstance(m, ComplexAffineExpression):
try:
m = Constant(mStr, m, x.shape)
except Exception as error:
raise TypeError(
"Failed to load the additional factor {} as a matrix of"
" same shape as {}.".format(mStr, x.string)) from error
else:
m = m.refined
if not m.constant:
raise TypeError("The additional factor {} is not constant."
.format(m.string))
elif not cvxopt_hpsd(m.safe_value_as_matrix):
raise ValueError("The additional factor {} is not hermitian "
"positive semidefinite.".format(m.string))
self._x = x
self._num = pNum
self._den = pDen
self._m = m
self._limit = denominator_limit
if m is None:
if len(x) == 1:
typeStr = "Power"
if p == 2:
symbStr = glyphs.squared(x.string)
elif p == 3:
symbStr = glyphs.cubed(x.string)
else:
symbStr = glyphs.power(x.string, pStr)
else:
typeStr = "Trace of Power"
symbStr = glyphs.trace(glyphs.power(x.string, pStr))
else:
if len(x) == 1:
typeStr = "Scaled Power"
symbStr = glyphs.mul(m.string, glyphs.power(x.string, pStr))
else:
typeStr = "Trace of Scaled Power"
symbStr = glyphs.trace(glyphs.mul(
m.string, glyphs.power(x.string, pStr)))
Expression.__init__(self, typeStr, symbStr)
# --------------------------------------------------------------------------
# Abstract method implementations and method overridings, except _predict.
# --------------------------------------------------------------------------
def _get_refined(self):
if self._x.constant:
return Constant(self._symbStr, self.value)
elif self.p == 0:
if self._m is not None:
return self._m.tr
else:
return Constant(
glyphs.Fn("diaglen({})")(self._x.string), self._x.shape[0])
elif self.p == 1:
if self._m is not None:
# NOTE: No hermitian transpose as both m and x are hermitian.
return (self._m | self._x)
else:
return self._x.tr
elif self.p == 2 and self._x.scalar and self._m is None:
return (self._x | self._x)
else:
return self
Subtype = namedtuple("Subtype", ("diag", "num", "den", "hasM", "complex"))
def _get_subtype(self):
return self.Subtype(
self._x.shape[0], self._num, self._den, self._m is not None,
self._x.complex)
def _get_value(self):
x = cvx2np(self._x._get_value())
p = self.p
eigenvalues = numpy.linalg.eigvalsh(x)
if p != int(p) and any(value < 0 for value in eigenvalues):
raise ArithmeticError("Cannot evaluate {}: {} is not positive "
"semidefinite and the exponent is fractional."
.format(self.string, self._x.string))
if self._m is None:
trace = sum([value**p for value in eigenvalues])
else:
m = cvx2np(self._m._get_value())
U, S, V = numpy.linalg.svd(x)
power = U*numpy.diag(S**p)*V
trace = numpy.trace(m * power)
return cvxopt.matrix(trace)
def _get_mutables(self):
return self._x._get_mutables()
def _is_convex(self):
return self.p >= 1 or self.p <= 0
def _is_concave(self):
return self.p >= 0 and self.p <= 1
def _replace_mutables(self, mapping):
return self.__class__(
self._x._replace_mutables(mapping), self.p, self._m, self._limit)
def _freeze_mutables(self, freeze):
return self.__class__(
self._x._freeze_mutables(freeze), self.p, self._m, self._limit)
# --------------------------------------------------------------------------
# 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 and self.p >= 0 and self.p <= 1:
if self._m is None:
m = other.dupdiag(self.n).renamed(other.string)
else:
m = other*self._m
return cls(self._x, self.p, m, self._limit)
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 PowerTrace._mul(self, other, True)
[docs] @convert_operands(scalarRHS=True)
@refine_operands()
def __rmul__(self, other):
return PowerTrace._mul(self, other, False)
# --------------------------------------------------------------------------
# Methods and properties that return expressions.
# --------------------------------------------------------------------------
@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]
@property
def p(self):
"""The parameter :math:`p`.
This is a limited precision version of the parameter used when the
expression was constructed.
"""
return float(self._num) / float(self._den)
@property
def num(self):
"""The limited precision fraction numerator of :math:`p`."""
return self._num
@property
def den(self):
"""The limited precision fraction denominator of :math:`p`."""
return self._den
@property
def m(self):
"""An additional factor to multiply the power with."""
return self._m
# --------------------------------------------------------------------------
# Constraint-creating operators, and _predict.
# --------------------------------------------------------------------------
@classmethod
def _predict(cls, subtype, relation, other):
assert isinstance(subtype, cls.Subtype)
p = float(subtype.num) / float(subtype.den)
if relation == operator.__le__:
if p > 0 and p < 1: # Not convex.
return NotImplemented
if issubclass(other.clstype, AffineExpression) \
and other.subtype.dim == 1:
return PowerTraceConstraint.make_type(*subtype)
elif relation == operator.__ge__:
if p < 0 or p > 1: # Not concave.
return NotImplemented
if issubclass(other.clstype, AffineExpression) \
and other.subtype.dim == 1:
return PowerTraceConstraint.make_type(*subtype)
return NotImplemented
[docs] @convert_operands(scalarRHS=True)
@validate_prediction
@refine_operands()
def __le__(self, other):
if not self.convex:
raise TypeError("Cannot upper-bound a nonconvex (trace of) power.")
if isinstance(other, AffineExpression):
return PowerTraceConstraint(self, Constraint.LE, other)
else:
return NotImplemented
[docs] @convert_operands(scalarRHS=True)
@validate_prediction
@refine_operands()
def __ge__(self, other):
if not self.concave:
raise TypeError("Cannot lower-bound a nonconcave (trace of) power.")
if isinstance(other, AffineExpression):
return PowerTraceConstraint(self, Constraint.GE, other)
else:
return NotImplemented
# --------------------------------------
__all__ = api_end(_API_START, globals())