Source code for picos.expressions.exp_powtrace

# 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:`PowerTrace`."""

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, PowerTraceConstraint
from .data import (convert_and_refine_arguments, convert_operands, cvx2np,
                   cvxopt_hpsd, make_fraction)
from .exp_affine import AffineExpression, ComplexAffineExpression, Constant
from .expression import Expression, refine_operands, validate_prediction

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


[docs]class PowerTrace(Expression): r"""The trace of the :math:`p`-th power of a hermitian matrix. :Definition: Let :math:`p \in \mathbb{Q}`. 1. If the base expressions is a real scalar :math:`x` and no additional constant :math:`m` is given, then this is the power :math:`x^p`. 2. If the base expressions is a real scalar :math:`x`, :math:`p \in [0, 1]`, and a positive scalar constant :math:`m` is given, then this is the scaled power :math:`m x^p`. 3. If the base expression is a hermitian matrix :math:`X` and no additional constant :math:`M` is given, then this is the trace of power :math:`\operatorname{tr}(X^p)`. 4. If the base expression is a hermitian matrix :math:`X`, :math:`p \in [0, 1]`, and a hermitian positive semidefinite constant matrix :math:`M` of same shape as :math:`X` is given, then this is the trace of a scaled power :math:`\operatorname{tr}(M X^p)`. No other case is supported. In particular, if :math:`p \not\in [0, 1]`, then :math:`m`/:math:`M` must be undefined (:obj:`None`). .. warning:: 1. For a constraint of the form :math:`x^p \leq t` with :math:`p < 1` and :math:`p \neq 0`, PICOS enforces :math:`x \geq 0` during solution search. 2. For a constraint of the form :math:`\operatorname{tr}(X^p) \leq t` or :math:`\operatorname{tr}(M X^p) \leq t` with :math:`p < 1` and :math:`p \neq 0`, PICOS enforces :math:`X \succeq 0` during solution search. 3. For a constraint of the form :math:`\operatorname{tr}(X^p) \leq t` or :math:`\operatorname{tr}(M X^p) \leq t` with :math:`p > 1`, PICOS enforces :math:`t \geq 0` during solution search. """ # -------------------------------------------------------------------------- # Initialization and factory methods. # --------------------------------------------------------------------------
[docs] @convert_and_refine_arguments("x") def __init__(self, x, p, m=None, denominator_limit=1000): """Construct a :class:`PowerTrace`. :param x: The scalar or symmetric matrix to form a power of. :type x: ~picos.expressions.AffineExpression :param float p: The value for :math:`p`, which is cast to a limited precision fraction. :param m: An additional positive semidefinite constant to multiply the power with. :type m: :class:`~picos.expressions.AffineExpression` or anything recognized by :func:`~picos.expressions.data.load_data` :param int denominator_limit: The largest allowed denominator when casting :math:`p` to a fraction. Higher values can yield a greater precision at reduced performance. """ # Validate x. if not isinstance(x, ComplexAffineExpression): raise TypeError("Can only form the power of an affine expression, " "not of {}.".format(x.string)) elif not x.square: raise TypeError( "Can't form the power of non-square {}.".format(x.string)) elif not x.hermitian: raise NotImplementedError("Taking {} to a power is not supported " "as it is not necessarily hermitian.".format(x.string)) # Load p. pNum, pDen, p, pStr = make_fraction(p, denominator_limit) # Load m. if m is not None: mStr = "m" if len(x) == 1 else "M" if p < 0 or p > 1: raise ValueError( "p-th power with an additional factor {} requires {}." .format(mStr, glyphs.le(0, glyphs.le("p", 1)))) if not isinstance(m, ComplexAffineExpression): try: m = Constant(mStr, m, x.shape) except Exception as error: raise TypeError("Failed to load the additional factor {} " "as a matrix of same shape as {}: {}" .format(mStr, x.string, error)) else: m = m.refined if not m.constant: raise TypeError("The additional factor {} is not constant." .format(m.string)) elif not cvxopt_hpsd(m.value): raise ValueError("The additional factor {} is not hermitian " "positive semidefinite.".format(m.string)) self._x = x self._num = pNum self._den = pDen self._m = m self._limit = denominator_limit if m is None: if len(x) == 1: typeStr = "Power" if p == 2: symbStr = glyphs.squared(x.string) elif p == 3: symbStr = glyphs.cubed(x.string) else: symbStr = glyphs.power(x.string, pStr) else: typeStr = "Trace of Power" symbStr = glyphs.trace(glyphs.power(x.string, pStr)) else: if len(x) == 1: typeStr = "Scaled Power" symbStr = glyphs.mul(m.string, glyphs.power(x.string, pStr)) else: typeStr = "Trace of Scaled Power" symbStr = glyphs.trace(glyphs.mul( m.string, glyphs.power(x.string, pStr))) Expression.__init__(self, typeStr, symbStr)
# -------------------------------------------------------------------------- # Abstract method implementations and method overridings, except _predict. # -------------------------------------------------------------------------- def _get_refined(self): if self._x.constant: return Constant(self._symbStr, self.value) elif self.p == 0: if self._m is not None: return self._m.tr else: return Constant( glyphs.Fn("diaglen({})")(self._x.string), self._x.shape[0]) elif self.p == 1: if self._m is not None: # NOTE: No hermitian transpose as both m and x are hermitian. return (self._m | self._x) else: return self._x.tr elif self.p == 2 and self._x.scalar and self._m is None: return (self._x | self._x) else: return self Subtype = namedtuple("Subtype", ("diag", "num", "den", "hasM", "complex")) def _get_subtype(self): return self.Subtype( self._x.shape[0], self._num, self._den, self._m is not None, self._x.complex) def _get_value(self): x = cvx2np(self._x._get_value()) p = self.p eigenvalues = numpy.linalg.eigvalsh(x) if p != int(p) and any(value < 0 for value in eigenvalues): raise ArithmeticError("Cannot evaluate {}: {} is not positive " "semidefinite and the exponent is fractional." .format(self.string, self._x.string)) if self._m is None: trace = sum([value**p for value in eigenvalues]) else: m = cvx2np(self._m._get_value()) U, S, V = numpy.linalg.svd(x) power = U*numpy.diag(S**p)*V trace = numpy.trace(m * power) return cvxopt.matrix(trace) @cached_unary_operator def _get_variables(self): return self._x.variables def _is_convex(self): return self.p >= 1 or self.p <= 0 def _is_concave(self): return self.p >= 0 and self.p <= 1 def _replace_variables(self, var_map): return self.__class__( self._x._replace_variables(var_map), self.p, self._m, self._limit) # -------------------------------------------------------------------------- # 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 trace of a power with a constant term.") value = other.value if value < 0: raise NotImplementedError("You may only multiply a nonconstant " "PICOS trace of a power with a nonnegative term.") if value == 1: return self if self._m is None: m = other.dupdiag(self.n).renamed(other.string) else: m = other*self._m return PowerTrace(self._x, self.p, m, self._limit) else: return NotImplemented @convert_operands(scalarRHS=True) @refine_operands() def __rmul__(self, other): if isinstance(other, AffineExpression): return self.__mul__(other) else: return NotImplemented # -------------------------------------------------------------------------- # Methods and properties that return expressions. # -------------------------------------------------------------------------- @property def x(self): """The matrix concerned.""" return self._x # -------------------------------------------------------------------------- # Methods and properties that describe the expression. # -------------------------------------------------------------------------- @property def n(self): """Diagonal length of :attr:`x`.""" return self._x.shape[0] @property def p(self): """The parameter :math:`p`. This is a limited precision version of the paramter used when the expression was constructed. """ return float(self._num) / float(self._den) @property def num(self): """The limited precision fraction numerator of :math:`p`.""" return self._num @property def den(self): """The limited precision fraction denominator of :math:`p`.""" return self._den @property def m(self): """An additional factor to multiply the power with.""" return self._m # -------------------------------------------------------------------------- # Constraint-creating operators, and _predict. # -------------------------------------------------------------------------- @classmethod def _predict(cls, subtype, relation, other): assert isinstance(subtype, cls.Subtype) p = float(subtype.num) / float(subtype.den) if relation == operator.__le__: if p > 0 and p < 1: # Not convex. return NotImplemented if issubclass(other.clstype, AffineExpression) \ and other.subtype.dim == 1: return PowerTraceConstraint.make_type(*subtype) elif relation == operator.__ge__: if p < 0 or p > 1: # Not concave. return NotImplemented if issubclass(other.clstype, AffineExpression) \ and other.subtype.dim == 1: return PowerTraceConstraint.make_type(*subtype) return NotImplemented @convert_operands(scalarRHS=True) @validate_prediction @refine_operands() def __le__(self, other): if not self.convex: raise TypeError("Cannot upper-bound a nonconvex (trace of) power.") if isinstance(other, AffineExpression): return PowerTraceConstraint(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 lower-bound a nonconcave (trace of) power.") if isinstance(other, AffineExpression): return PowerTraceConstraint(self, Constraint.GE, other) else: return NotImplemented
# -------------------------------------- __all__ = api_end(_API_START, globals())