"""
Copyright 2013 Steven Diamond
This file is part of CVXPY.
CVXPY 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.
CVXPY 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 CVXPY. If not, see <http://www.gnu.org/licenses/>.
"""
import cvxpy.utilities as u
import cvxpy.lin_ops.lin_utils as lu
from cvxpy.atoms.elementwise.elementwise import Elementwise
import numpy as np
from cvxpy.utilities.power_tools import (is_power2, gm_constrs, pow_mid,
pow_high, pow_neg)
[docs]class power(Elementwise):
r""" Elementwise power function :math:`f(x) = x^p`.
If ``expr`` is a CVXPY expression, then ``expr**p``
is equivalent to ``power(expr, p)``.
Specifically, the atom is given by the cases
.. math::
\begin{array}{ccl}
p = 0 & f(x) = 1 & \text{constant, positive} \\
p = 1 & f(x) = x & \text{affine, increasing, same sign as $x$} \\
p = 2,4,8,\ldots &f(x) = |x|^p & \text{convex, signed monotonicity, positive} \\
p < 0 & f(x) = \begin{cases} x^p & x > 0 \\ +\infty & x \leq 0 \end{cases} & \text{convex, decreasing, positive} \\
0 < p < 1 & f(x) = \begin{cases} x^p & x \geq 0 \\ -\infty & x < 0 \end{cases} & \text{concave, increasing, positive} \\
p > 1,\ p \neq 2,4,8,\ldots & f(x) = \begin{cases} x^p & x \geq 0 \\ +\infty & x < 0 \end{cases} & \text{convex, increasing, positive}.
\end{array}
.. note::
Generally, ``p`` cannot be represented exactly, so a rational,
i.e., fractional, **approximation** must be made.
Internally, ``power`` computes a rational approximation
to ``p`` with a denominator up to ``max_denom``. The resulting
approximation can be found through the attribute ``power.p``.
The approximation error is given by the attribute ``power.approx_error``.
Increasing ``max_denom`` can give better approximations.
When ``p`` is an ``int`` or ``Fraction`` object, the approximation
is usually **exact**.
.. note::
The final domain, sign, monotonicity, and curvature of the ``power`` atom
are determined by the rational approximation to ``p``, **not** the input parameter ``p``.
For example,
>>> from cvxpy import Variable, power
>>> x = Variable()
>>> g = power(x, 1.001)
>>> g.p
Fraction(1001, 1000)
>>> g
Expression(CONVEX, POSITIVE, (1, 1))
results in a convex atom with implicit constraint :math:`x \geq 0`, while
>>> g = power(x, 1.0001)
>>> g.p
1
>>> g
Expression(AFFINE, UNKNOWN, (1, 1))
results in an affine atom with no constraint on ``x``.
- When :math:`p > 1` and ``p`` is not a power of two, the monotonically increasing version
of the function with full domain,
.. math::
f(x) = \begin{cases} x^p & x \geq 0 \\ 0 & x < 0 \end{cases}
can be formed with the composition ``power(pos(x), p)``.
- The symmetric version with full domain,
.. math::
f(x) = |x|^p
can be formed with the composition ``power(abs(x), p)``.
Parameters
----------
x : cvx.Variable
p : int, float, or Fraction
Scalar power.
max_denom : int
The maximum denominator considered in forming a rational approximation of ``p``.
"""
def __init__(self, x, p, max_denom=1024):
p_old = p
# how we convert p to a rational depends on the branch of the function
if p > 1:
p, w = pow_high(p, max_denom)
elif 0 < p < 1:
p, w = pow_mid(p, max_denom)
elif p < 0:
p, w = pow_neg(p, max_denom)
# note: if, after making the rational approximation, p ends up being 0 or 1,
# we default to using the 0 or 1 behavior of the atom, which affects the curvature, domain, etc...
# maybe unexpected behavior to the user if they put in 1.00001?
if p == 1:
# in case p is a fraction equivalent to 1
p = 1
w = None
if p == 0:
p = 0
w = None
self.p, self.w = p, w
self.approx_error = float(abs(self.p - p_old))
super(power, self).__init__(x)
@Elementwise.numpy_numeric
def numeric(self, values):
if self.p == 0:
return np.ones(self.size)
else:
return np.power(values[0], self.p)
def sign_from_args(self):
if self.p == 1:
# same sign as input
return self.args[0]._dcp_attr.sign
else:
return u.Sign.POSITIVE
def func_curvature(self):
if self.p == 0:
return u.Curvature.CONSTANT
elif self.p == 1:
return u.Curvature.AFFINE
elif self.p < 0 or self.p > 1:
return u.Curvature.CONVEX
elif 0 < self.p < 1:
return u.Curvature.CONCAVE
def monotonicity(self):
if self.p == 0:
return [u.monotonicity.INCREASING]
if self.p == 1:
return [u.monotonicity.INCREASING]
if self.p < 0:
return [u.monotonicity.DECREASING]
if 0 < self.p < 1:
return [u.monotonicity.INCREASING]
if self.p > 1:
if is_power2(self.p):
return [u.monotonicity.SIGNED]
else:
return [u.monotonicity.INCREASING]
def validate_arguments(self):
pass
def get_data(self):
return self.p, self.w
@staticmethod
def graph_implementation(arg_objs, size, data=None):
"""Reduces the atom to an affine expression and list of constraints.
Parameters
----------
arg_objs : list
LinExpr for each argument.
size : tuple
The size of the resulting expression.
data :
Additional data required by the atom.
Returns
-------
tuple
(LinOp for objective, list of constraints)
"""
x = arg_objs[0]
p, w = data
if p == 1:
return x, []
else:
one = lu.create_const(np.mat(np.ones(size)), size)
if p == 0:
return one, []
else:
t = lu.create_var(size)
if 0 < p < 1:
return t, gm_constrs(t, [x, one], w)
elif p > 1:
return t, gm_constrs(x, [t, one], w)
elif p < 0:
return t, gm_constrs(one, [x, t], w)
else:
raise NotImplementedError('this power is not yet supported.')
def name(self):
return "%s(%s, %s)" % (self.__class__.__name__,
self.args[0].name(),
self.p)
