# ------------------------------------------------------------------------------
# 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:`SumExponentials`."""
import math
import operator
from collections import namedtuple
import cvxopt
import numpy
from .. import glyphs
from ..apidoc import api_end, api_start
from ..caching import cached_property, cached_unary_operator
from ..constraints import LogSumExpConstraint, SumExponentialsConstraint
from .data import convert_and_refine_arguments, convert_operands, cvx2np
from .exp_affine import AffineExpression
from .expression import Expression, refine_operands, validate_prediction
_API_START = api_start(globals())
# -------------------------------
[docs]class SumExponentials(Expression):
r"""Sum of elementwise exponentials of an affine expression.
:Definition:
Let :math:`x` be an :math:`n`-dimensional real affine expression.
1. If no additional expression :math:`y` is given, this is the sum of
elementwise exponentials
.. math::
\sum_{i = 1}^n \exp(\operatorname{vec}(x)_i).
2. If an additional affine expression :math:`y` of same shape as :math:`x`
is given, this is the sum of elementwise perspectives of exponentials
.. math::
\sum_{i = 1}^n \operatorname{vec}(y)_i \exp\left(
\frac{\operatorname{vec}(x)_i}{\operatorname{vec}(y)_i}\right).
.. warning::
When you pose an upper bound :math:`t` on a sum of elementwise
exponentials, then PICOS enforces :math:`t \geq 0` through an auxiliary
constraint during solution search. When an additional expression
:math:`y` is given, PICOS enforces :math:`y \geq 0` as well.
"""
# --------------------------------------------------------------------------
# Initialization and factory methods.
# --------------------------------------------------------------------------
[docs] @convert_and_refine_arguments("x", "y", allowNone=True)
def __init__(self, x, y=None):
"""Construct a :class:`SumExponentials`.
:param x: The affine expression :math:`x`.
:type x: ~picos.expressions.AffineExpression
:param y: An additional affine expression :math:`y`. If necessary, PICOS
will attempt to reshape or broadcast it to the shape of :math:`x`.
:type y: ~picos.expressions.AffineExpression
"""
if not isinstance(x, AffineExpression):
raise TypeError("Can only sum the elementwise exponentials of a "
"real affine expression, not of {}.".format(x.string))
if y is not None:
if not isinstance(y, AffineExpression):
raise TypeError("The additional parameter y must be a real "
"affine expression, not {}.".format(y.string))
elif x.shape != y.shape:
y = y.reshaped_or_broadcasted(x.shape)
if y.is1:
y = None
self._x = x
self._y = y
if len(x) == 1:
if y is None:
typeStr = "Exponential"
symbStr = glyphs.exp(x.string)
else:
typeStr = "Exponential Perspective"
symbStr = glyphs.mul(
y.string, glyphs.exp(glyphs.div(x.string, y.string)))
else:
if y is None:
typeStr = "Sum of Exponentials"
symbStr = glyphs.make_function("sum", "exp")(x.string)
else:
typeStr = "Sum of Exponential Perspectives"
symbStr = glyphs.sum(glyphs.mul(glyphs.slice(y.string, "i"),
glyphs.exp(glyphs.div(glyphs.slice(x.string, "i"),
glyphs.slice(y.string, "i")))))
Expression.__init__(self, typeStr, symbStr)
# --------------------------------------------------------------------------
# Abstract method implementations and method overridings, except _predict.
# --------------------------------------------------------------------------
def _get_refined(self):
if self._x.constant and (self._y is None or self._y.constant):
return AffineExpression.from_constant(self.value, 1, self._symbStr)
else:
return self
Subtype = namedtuple("Subtype", ("argdim", "y"))
def _get_subtype(self):
return self.Subtype(len(self._x), self._y is not None)
def _get_value(self):
x = numpy.ravel(cvx2np(self._x._get_value()))
if self._y is None:
s = numpy.sum(numpy.exp(x))
else:
y = numpy.ravel(cvx2np(self._y._get_value()))
s = y.dot(numpy.exp(x / y))
return cvxopt.matrix(s)
@cached_unary_operator
def _get_mutables(self):
if self._y is None:
return self._x._get_mutables()
else:
return self._x._get_mutables().union(self._y.mutables)
def _is_convex(self):
return True
def _is_concave(self):
return False
def _replace_mutables(self, mapping):
return self.__class__(self._x._replace_mutables(mapping),
None if self._y is None else self._y._replace_mutables(mapping))
def _freeze_mutables(self, freeze):
return self.__class__(self._x._freeze_mutables(freeze),
None if self._y is None else self._y._freeze_mutables(freeze))
# --------------------------------------------------------------------------
# Python special method implementations, except constraint-creating ones.
# --------------------------------------------------------------------------
@convert_operands(scalarRHS=True)
@refine_operands()
def __add__(self, other):
if isinstance(other, AffineExpression):
if not other.constant:
raise NotImplementedError("You may only add a constant term to "
"a nonconstant PICOS sum of exponentials.")
value = other.value
if value < 0:
raise NotImplementedError("You may only add a nonnegative term "
"to a nonconstant PICOS sum of exponentials.")
if value == 0:
# NOTE: We could return self here, but this is more consistent
# with other expressions' __add__ methods.
sumexp = SumExponentials(self._x)
elif self._y is None:
sumexp = SumExponentials(self._x // math.log(value))
else:
sumexp = SumExponentials(self._x // value, self._y // 1)
sumexp._typeStr = "Offset " + sumexp._typeStr
sumexp._symbStr = glyphs.clever_add(self.string, other.string)
return sumexp
elif isinstance(other, SumExponentials):
if self._y is None and other._y is None:
sumexp = SumExponentials(self._x.vec // other._x.vec)
elif self._y is not None and other._y is None:
one = AffineExpression.from_constant(1.0, (other.n, 1))
sumexp = SumExponentials(
self._x.vec // other._x.vec, self._y.vec // one)
elif self._y is None and other._y is not None:
one = AffineExpression.from_constant(1.0, (self.n, 1))
sumexp = SumExponentials(
self._x.vec // other._x.vec, one // other._y.vec)
else:
sumexp = SumExponentials(
self._x.vec // other._x.vec, self._y.vec // other._y.vec)
sumexp._symbStr = glyphs.clever_add(self.string, other.string)
return sumexp
else:
return NotImplemented
@convert_operands(scalarRHS=True)
@refine_operands()
def __radd__(self, other):
if isinstance(other, (AffineExpression, SumExponentials)):
sumexp = self.__add__(other)
# NOTE: __add__ always creates a fresh expression.
sumexp._symbStr = glyphs.clever_add(other.string, self.string)
return sumexp
else:
return NotImplemented
@convert_operands(scalarRHS=True)
@refine_operands()
def __mul__(self, other):
if isinstance(other, AffineExpression):
if not other.constant:
raise NotImplementedError("You may only multiply a nonconstant "
"PICOS sum of exponentials with a constant term.")
value = other.value
if value < 0:
raise NotImplementedError("You may only multiply a nonconstant "
"PICOS sum of exponential with a nonnegative term.")
if value == 0:
return AffineExpression.zero()
if self._y is None:
sumexp = SumExponentials(self._x + math.log(value))
else:
sumexp = SumExponentials(self._x * value, self._y * value)
sumexp._typeStr = "Scaled " + sumexp._typeStr
sumexp._symbStr = glyphs.clever_mul(self.string, other.string)
return sumexp
else:
return NotImplemented
@convert_operands(scalarRHS=True)
@refine_operands()
def __rmul__(self, other):
if isinstance(other, AffineExpression):
sumexp = self.__mul__(other)
# NOTE: __mul__ always creates a fresh expression.
sumexp._symbStr = glyphs.clever_mul(other.string, self.string)
return sumexp
else:
return NotImplemented
@convert_operands(scalarRHS=True)
@refine_operands()
def __truediv__(self, other):
if isinstance(other, AffineExpression):
if not other.constant:
raise NotImplementedError("You may only divide a nonconstant "
"PICOS sum of exponentials by a constant term.")
value = other.value
if value <= 0:
raise NotImplementedError("You may only divide a nonconstant "
"PICOS sum of exponential by a positive term.")
sumexp = self * (1.0 / value)
# NOTE: __mul__ always creates a fresh expression.
sumexp._symbStr = glyphs.div(self.string, other.string)
else:
return NotImplemented
# --------------------------------------------------------------------------
# Methods and properties that return expressions.
# --------------------------------------------------------------------------
@property
def x(self):
"""The expression :math:`x`."""
return self._x
@property
def y(self):
"""The additional expression :math:`y`, or :obj:`None`."""
return self._y
[docs] @cached_property
def log(self):
"""The logarithm of the expression."""
from . import LogSumExp
if self._y is not None:
raise NotImplementedError("May only take the logarithm of a sum of"
" exponentials, not of a sum of exponential perspectives.")
return LogSumExp(self._x)
# --------------------------------------------------------------------------
# Methods and properties that describe the expression.
# --------------------------------------------------------------------------
@property
def n(self):
"""Length of :attr:`x`."""
return len(self._x)
# --------------------------------------------------------------------------
# Constraint-creating operators, and _predict.
# --------------------------------------------------------------------------
@classmethod
def _predict(cls, subtype, relation, other):
assert isinstance(subtype, cls.Subtype)
if relation == operator.__le__:
if issubclass(other.clstype, AffineExpression) \
and other.subtype.dim == 1:
return SumExponentialsConstraint.make_type(
argdim=subtype.argdim,
lse_representable=(not subtype.y and other.subtype.nonneg))
elif issubclass(other.clstype, SumExponentials):
if subtype.y or other.subtype.y:
return NotImplemented
if other.subtype.argdim != 1:
return NotImplemented
return LogSumExpConstraint.make_type(argdim=subtype.argdim)
return NotImplemented
@convert_operands(scalarRHS=True)
@validate_prediction
@refine_operands()
def __le__(self, other):
from . import LogSumExp
if isinstance(other, AffineExpression):
return SumExponentialsConstraint(self, other)
elif isinstance(other, SumExponentials):
if self._y is not None or other._y is not None:
raise NotImplementedError("Comparing two sums of exponentials "
"is not supported if either expression has the additional "
"perspectives parameter y set.")
if other.n != 1:
raise NotImplementedError("You may only upper bound a sum of "
"exponentials by a single exponential, not by another sum.")
return LogSumExp(self._x) <= other._x
else:
return NotImplemented
# --------------------------------------
__all__ = api_end(_API_START, globals())