Source code for picos.expressions.set_rsoc

# 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:`RotatedSecondOrderCone`."""

import operator
from collections import namedtuple

from .. import glyphs
from ..apidoc import api_end, api_start
from ..constraints import RSOCConstraint
from .data import convert_operands
from .exp_affine import AffineExpression
from .expression import refine_operands, validate_prediction
from .set import Set

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


[docs]class RotatedSecondOrderCone(Set): 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): """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: {}" .format(error)) if p <= 0: raise ValueError("The parameter 'p' must be positive.") self._p = p 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)))) Set.__init__(self, typeStr, symbStr)
@property def p(self): """A narrowing (:math:`p < 2`) or widening (:math:`p > 2`) factor.""" return self._p def _get_variables(self): return set() def _replace_variables(self): return self Subtype = namedtuple("Subtype", ()) def _get_subtype(self): return self.Subtype() @classmethod def _predict(cls, subtype, relation, other): assert isinstance(subtype, cls.Subtype) if relation == operator.__rshift__: if issubclass(other.clstype, AffineExpression): if other.subtype.dim >= 3: return RSOCConstraint.make_type(other.subtype.dim - 2) return NotImplemented @convert_operands() @validate_prediction @refine_operands() def __rshift__(self, element): if isinstance(element, AffineExpression): 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]) else: return NotImplemented
# -------------------------------------- __all__ = api_end(_API_START, globals())