# Source code for picos.expressions.exp_sumexp

# ------------------------------------------------------------------------------
# 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: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:

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.
# --------------------------------------------------------------------------

@classmethod
if isinstance(other, AffineExpression) and other.constant:
value = other.value

if not value:
return self
elif value > 0:
if self._y is None:
result = cls(self._x // math.log(value))
else:
result = cls(self._x // value, self._y // 1)

if forward:
else:

result._typeStr = "Offset " + result._typeStr
result._symbStr = string

return result
elif isinstance(other, cls):
assert forward, "Encountered __radd__ on equal types."

if self._y is None and other._y is None:
result = cls(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))
result = cls(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))
result = cls(self._x.vec // other._x.vec, one // other._y.vec)
else:
result = cls(
self._x.vec // other._x.vec, self._y.vec // other._y.vec)

return result

if forward:
else:

[docs]    @convert_operands(scalarRHS=True)
@refine_operands()

[docs]    @convert_operands(scalarRHS=True)
@refine_operands()

@classmethod
def _mul_div(cls, self, other, div, forward):
assert not div or forward

if isinstance(other, AffineExpression) and other.constant:
factor = other.safe_value

if not factor:
if div:
raise ZeroDivisionError(
"Cannot divide {} by zero.".format(self.string))
else:
return AffineExpression.zero()
elif factor == 1:
return self
elif factor > 0:
if div:
factor = 1 / factor
string = glyphs.div(self.string, other.string)
elif forward:
string = glyphs.clever_mul(self.string, other.string)
else:
string = glyphs.clever_mul(other.string, self.string)

if self._y is None:
result = cls(self._x + math.log(factor))
else:
result = cls(other*self._x, other*self._y)

result._typeStr = "Scaled " + result._typeStr
result._symbStr = string

return result

if div:
return Expression.__div__(self, other)
elif forward:
return Expression.__mul__(self, other)
else:
return Expression.__rmul__(self, other)

[docs]    @convert_operands(scalarRHS=True)
@refine_operands()
def __mul__(self, other):
"""Denote scaling from the right hand side."""
return SumExponentials._mul_div(self, other, div=False, forward=True)

[docs]    @convert_operands(scalarRHS=True)
@refine_operands()
def __rmul__(self, other):
"""Denote scaling from the left hand side."""
return SumExponentials._mul_div(self, other, div=False, forward=False)

[docs]    @convert_operands(scalarRHS=True)
@refine_operands()
def __truediv__(self, other):
"""Denote division by a constant scalar."""
return SumExponentials._mul_div(self, other, div=True, forward=True)

# --------------------------------------------------------------------------
# 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

[docs]    @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())