Source code for picos.expressions.exp_sumxtr

# coding: utf-8

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

import operator
from collections import namedtuple

import cvxopt
import numpy

from .. import glyphs
from ..apidoc import api_end, api_start
from ..caching import cached_unary_operator
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 @cached_unary_operator def _get_variables(self): return self._x.variables def _is_convex(self): return self._largest or self._full def _is_concave(self): return not self._largest or self._full def _replace_variables(self, var_map): return self.__class__(self._x._replace_variables(var_map), self._k, self._largest, self._eigenvalues) # -------------------------------------------------------------------------- # Python special method implementations, except constraint-creating ones. # -------------------------------------------------------------------------- @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 extremes with a constant term.") if other.value < 0: raise NotImplementedError("You may only multiply a nonconstant " "PICOS sum of extremes with a nonnegative term.") product = SumExtremes( other.value*self._x, self._k, self._largest, self._eigenvalues) product._typeStr = "Scaled " + product._typeStr product._symbStr = glyphs.clever_mul(self.string, other.string) return product else: return NotImplemented @convert_operands(scalarRHS=True) @refine_operands() def __rmul__(self, other): if isinstance(other, AffineExpression): product = self.__mul__(other) # NOTE: __mul__ always creates a fresh expression. product._symbStr = glyphs.clever_mul(other.string, self.string) return product else: return NotImplemented # -------------------------------------------------------------------------- # 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 @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 @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())