# ------------------------------------------------------------------------------
# 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:`RotatedSecondOrderCone`."""
import operator
from collections import namedtuple
from .. import glyphs
from ..apidoc import api_end, api_start
from ..caching import cached_property
from ..constraints import RSOCConstraint
from .cone import Cone
from .exp_affine import AffineExpression
_API_START = api_start(globals())
# -------------------------------
[docs]class RotatedSecondOrderCone(Cone):
r"""The (narrowed or widened) rotated second order cone.
.. _rotatedcone:
For :math:`n \in \mathbb{Z}_{\geq 3}` and :math:`p \in \mathbb{R}_{> 0}`,
represents the convex cone
.. math::
\mathcal{R}_{p}^n = \left\{
x \in \mathbb{R}^n
~\middle|~
p x_1 x_2 \geq \sum_{i = 3}^n x_i^2 \land x_1, x_2 \geq 0
\right\}.
For :math:`p = 2`, this is the standard rotated second order cone
:math:`\mathcal{R}^n` obtained by rotating the
:class:`second order cone <picos.SecondOrderCone>` :math:`\mathcal{Q}^n`
by :math:`\frac{\pi}{4}` in the :math:`(x_1, x_2)` plane.
The default instance of this class has :math:`p = 1`, which can be
understood as a narrowed version of the standard cone. This is more
convenient for defining the primal problem but it should be noted that
:math:`\mathcal{R}_{1}^n` is not self-dual, so working with
:math:`p = 2` may seem more natural when the dual problem is of interest.
:Dual cone:
The dual cone is
.. math::
\left(\mathcal{R}_{p}^n\right)^* = \left\{
x \in \mathbb{R}^n
~\middle|~
\frac{4}{p} x_1 x_2 \geq \sum_{i = 2}^n x_i^2 \land x_1, x_2 \geq 0
\right\}=\mathcal{R}_{4/p}^n.
The cone is thus self-dual for :math:`p = 2`.
"""
[docs] def __init__(self, p=1, dim=None):
"""Construct a rotated second order cone.
:param float p:
The positive factor :math:`p` in the definition.
"""
try:
p = float(p)
except Exception as error:
raise TypeError("Failed to load the parameter 'p' as a float.") \
from error
if p <= 0:
raise ValueError("The parameter 'p' must be positive.")
self._p = p
if dim and dim < 3:
raise ValueError("The minimal dimension for {} is {}."
.format(self.__class__.__name__, 3))
typeStr = "Rotated Second Order Cone"
if p < 2:
typeStr = "Narrowed " + typeStr
elif p > 2:
typeStr = "Widened " + typeStr
symbStr = glyphs.set(glyphs.sep(
glyphs.col_vectorize("u", "v", "x"), glyphs.and_(
glyphs.le(
glyphs.squared(glyphs.norm("x")),
glyphs.clever_mul(glyphs.scalar(p), glyphs.mul("u", "v"))),
glyphs.ge("u", 0))))
Cone.__init__(self, dim, typeStr, symbStr)
@property
def p(self):
"""A narrowing (:math:`p < 2`) or widening (:math:`p > 2`) factor."""
return self._p
def _get_mutables(self):
return frozenset()
def _replace_mutables(self):
return self
Subtype = namedtuple("Subtype", ("dim",))
def _get_subtype(self):
return self.Subtype(self.dim)
@classmethod
def _predict(cls, subtype, relation, other):
assert isinstance(subtype, cls.Subtype)
if relation == operator.__rshift__:
if issubclass(other.clstype, AffineExpression) \
and not subtype.dim or subtype.dim == other.subtype.dim \
and other.subtype.dim >= 3:
return RSOCConstraint.make_type(other.subtype.dim - 2)
return Cone._predict_base(cls, subtype, relation, other)
def _rshift_implementation(self, element):
if isinstance(element, AffineExpression):
self._check_dimension(element)
if len(element) < 3:
raise TypeError("Elements of the rotated second order cone must"
" be at least three-dimensional.")
element = element.vec
return RSOCConstraint(element[2:], self.p * element[0], element[1])
# Handle scenario uncertainty for all cones.
return Cone._rshift_base(self, element)
[docs] @cached_property
def dual_cone(self):
"""Implement :attr:`.cone.Cone.dual_cone`."""
return self.__class__(p=(4.0/self._p), dim=self.dim)
# --------------------------------------
__all__ = api_end(_API_START, globals())