# Source code for picos.expressions.exp_powtrace

# ------------------------------------------------------------------------------
# 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: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))

pNum, pDen, p, pStr = make_fraction(p, denominator_limit)

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(
" 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())