# Source code for picos.expressions.exp_sumxtr

# ------------------------------------------------------------------------------
# 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:SumExtremes."""

import operator
from collections import namedtuple

import cvxopt
import numpy

from .. import glyphs
from ..apidoc import api_end, api_start
from ..constraints import Constraint, SumExtremesConstraint
from .data import convert_and_refine_arguments, convert_operands, cvx2np
from .exp_affine import AffineExpression, ComplexAffineExpression
from .expression import Expression, refine_operands, validate_prediction

_API_START = api_start(globals())
# -------------------------------

[docs]class SumExtremes(Expression):
r"""Sum of the :math:k largest or smallest elements or eigenvalues.

:Definition:

Let :math:k \in \mathbb{Z}_{\geq 1}.

1.  If :math:x is an :math:n-dimensional real vector or matrix and
eigenvalues == False, then this is the sum of the :math:k \leq n
largest or smallest scalar elements of :math:x, depending on the truth
value of largest.

Special cases:

-   If :math:k = 1, this is either the largest element
:math:\max_{i = 1}^n \operatorname{vec}(x)_i or the smallest
element :math:\min_{i = 1}^n \operatorname{vec}(x)_i of :math:x.
-   If :math:k = n, this is the sum of all elements
:math:\langle x, 1 \rangle of :math:x.

2.  If :math:X is an :math:n \times n hermitian matrix and
eigenvalues == True, then this is the sum of the :math:k \leq n
largest or smallest eigenvalues of :math:X, depending on the truth
value of largest. Recall that the eigenvalues of a hermitian matrix
are real.

Special cases:

-   If :math:k = 1, this is either the largest eigenvalue
:math:\lambda_{\max}(X) or the smallest eigenvalue
:math:\lambda_{\min}(X) of :math:X.
-   If :math:k = n, this equals the trace
:math:\operatorname{tr}(X).

If the given :math:k exceeds the :math:n of either case, then :math:k
is silently clipped to :math:n.
"""

# --------------------------------------------------------------------------
# Initialization and factory methods.
# --------------------------------------------------------------------------

[docs]    @convert_and_refine_arguments("x")
def __init__(self, x, k, largest, eigenvalues=False):
"""Construct a :class:SumExtremes.

:param x: The affine expression to take a sum over.
:type x: ~picos.expressions.ComplexAffineExpression
:param int k: Number of summands.
:param bool largest: Whether to sum over the largest (eigen)values as
opposed to the smallest.
:param bool eigenvalues: Whether to sum eigenvalues instead of elements.
"""
largest     = bool(largest)
eigenvalues = bool(eigenvalues)

lStr = "largest" if largest else "smallest"
eStr = "eigenvalues" if eigenvalues else "scalar elements"
what = "{} {}".format(lStr, eStr)

# Validate x.
if not isinstance(x, ComplexAffineExpression):
raise TypeError("Can only sum {} of an affine expression, not of "
"{}.".format(what, type(x).__name__))

# Further validate x.
if eigenvalues:
if not x.square:
raise TypeError("Cannot sum {} of {} as its shape of {} is not "
"square.".format(what, x.string, glyphs.shape(x.shape)))
elif not x.hermitian:
raise NotImplementedError(
"Summing the {0} of {1} is not supported as {1} is not "
"necessarily hermitian.".format(what, x.string))
else:
if not isinstance(x, AffineExpression):
raise TypeError("Can only sum {} of a real-valued expression "
"but {} is properly complex.".format(what, x.string))

# Validate k.
if int(k) != k:
raise ValueError(
"Conversion of k = {} to an integer is ambiguous.".format(k))
k = int(k)
if k < 1:
raise ValueError(
"Number of {} to sum must be positive.".format(what))

# Clip k to be at most n.
k = min(k, x.shape[0]) if eigenvalues else min(k, len(x))

# Find out if all (eigen)values are summed.
full = k == x.shape[0] if eigenvalues else k == len(x)
assert len(x) != 1 or full

self._x           = x
self._k           = k
self._largest     = largest
self._eigenvalues = eigenvalues
self._full        = full

s, lbd = x.string, glyphs.lambda_()
if full:
if eigenvalues:
typeStr = "Sum of Eigenvalues"
symbStr = symbStr = glyphs.trace(s)
else:
typeStr = "Sum of Elements"
symbStr = glyphs.sum(s)
elif k > 1:
if eigenvalues and largest:
typeStr = "Sum of Largest Eigenvalues"
symbStr = glyphs.make_function(
"sum_{}_largest_{}".format(k, lbd))(s)
elif eigenvalues and not largest:
typeStr = "Sum of Smallest Eigenvalues"
symbStr = glyphs.make_function(
"sum_{}_smallest_{}".format(k, lbd))(s)
elif not eigenvalues and largest:
typeStr = "Sum of Largest Elements"
symbStr = glyphs.make_function("sum_{}_largest".format(k))(s)
else:
typeStr = "Sum of Smallest Elements"
symbStr = glyphs.make_function("sum_{}_smallest".format(k))(s)
else:
if eigenvalues and largest:
typeStr = "Largest Eigenvalue"
symbStr = glyphs.make_function("{}_max".format(lbd))(s)
elif eigenvalues and not largest:
typeStr = "Smallest Eigenvalue"
symbStr = glyphs.make_function("{}_min".format(lbd))(s)
elif not eigenvalues and largest:
typeStr = "Largest Element"
symbStr = glyphs.max(s)
else:
typeStr = "Smallest Element"
symbStr = glyphs.min(s)

Expression.__init__(self, typeStr, symbStr)

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

def _get_refined(self):
if self._x.constant:
return AffineExpression.from_constant(self.value, 1, self._symbStr)
elif self._full:
if len(self._x) == 1:
return self._x  # Don't carry the string for an identity.
if self._eigenvalues:
return self._x.tr  # Symbolic strings already match.
else:
return (1 | self._x).renamed(self._symbStr)
else:
return self

Subtype = namedtuple("Subtype",
("argdim", "k", "largest", "eigenvalues", "complex"))

def _get_subtype(self):
return self.Subtype(len(self._x), self._k, self._largest,
self._eigenvalues, self._x.complex)

def _get_value(self):
value = self._x._get_value()

if self._eigenvalues:
value = sorted(numpy.linalg.eigvalsh(cvx2np(value)))
else:
value = sorted(value)

value = sum(value[-self._k:] if self._largest else value[:self._k])
value = cvxopt.matrix(value)

return value

def _get_mutables(self):
return self._x._get_mutables()

def _is_convex(self):
return self._largest or self._full

def _is_concave(self):
return not self._largest or self._full

def _replace_mutables(self, mapping):
return self.__class__(self._x._replace_mutables(mapping),
self._k, self._largest, self._eigenvalues)

def _freeze_mutables(self, freeze):
return self.__class__(self._x._freeze_mutables(freeze),
self._k, self._largest, self._eigenvalues)

# --------------------------------------------------------------------------
# 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:
if forward:
string = glyphs.clever_mul(self.string, other.string)
else:
string = glyphs.clever_mul(other.string, self.string)

product = cls(
factor*self._x, self._k, self._largest, self._eigenvalues)
product._typeStr = "Scaled " + product._typeStr
product._symbStr = string

return product

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 SumExtremes._mul(self, other, True)

[docs]    @convert_operands(scalarRHS=True)
@refine_operands()
def __rmul__(self, other):
return SumExtremes._mul(self, other, False)

# --------------------------------------------------------------------------
# Methods and properties that return expressions.
# --------------------------------------------------------------------------

@property
def x(self):
"""The expression under the sum."""
return self._x

# --------------------------------------------------------------------------
# Methods and properties that describe the expression.
# --------------------------------------------------------------------------

@property
def k(self):
"""Number of (eigen)values to sum."""
return self._k

@property
def largest(self):
"""Whether the sum concerns largest values as opposed to smallest."""
return self._largest

@property
def eigenvalues(self):
"""Whether the sum concerns eigenvalues as opposed to elements."""
return self._eigenvalues

@property
def full(self):
"""Whether the sum concerns *all* (eigen)values of the expression."""
return self._full

# --------------------------------------------------------------------------
# Constraint-creating operators, and _predict.
# --------------------------------------------------------------------------

@classmethod
def _predict(cls, subtype, relation, other):
assert isinstance(subtype, cls.Subtype)

n = subtype.argdim
k = subtype.k
e = subtype.eigenvalues
c = subtype.complex

kmax = int(n**0.5) if e else n
full = k == kmax

convex  = subtype.largest or full
concave = not subtype.largest or full

if relation == operator.__le__:
if not convex:
return NotImplemented

if issubclass(other.clstype, AffineExpression) \
and other.subtype.dim == 1:
return SumExtremesConstraint.make_type(n, k, e, c)
elif relation == operator.__ge__:
if not concave:
return NotImplemented

if issubclass(other.clstype, AffineExpression) \
and other.subtype.dim == 1:
return SumExtremesConstraint.make_type(n, k, e, c)

return NotImplemented

[docs]    @convert_operands(scalarRHS=True)
@validate_prediction
@refine_operands()
def __le__(self, other):
if not self.convex:
raise TypeError("Cannot upper-bound the nonconvex expression {}."
.format(self._symbStr))

if isinstance(other, AffineExpression):
return SumExtremesConstraint(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 upper-bound the nonconcave expression {}."
.format(self._symbStr))

if isinstance(other, AffineExpression):
return SumExtremesConstraint(self, Constraint.GE, other)
else:
return NotImplemented

# --------------------------------------
__all__ = api_end(_API_START, globals())