"""
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/>.
"""
from cvxpy.atoms.atom import Atom
import cvxpy.utilities as u
import cvxpy.lin_ops.lin_utils as lu
import numpy as np
from ..utilities.power_tools import pow_high, pow_mid, pow_neg, gm_constrs
from cvxpy.constraints.second_order import SOC
from fractions import Fraction
[docs]class pnorm(Atom):
r"""The vector p-norm.
If given a matrix variable, ``pnorm`` will treat it as a vector, and compute the p-norm
of the concatenated columns.
For :math:`p \geq 1`, the p-norm is given by
.. math::
\|x\|_p = \left(\sum_i |x_i|^p \right)^{1/p},
with domain :math:`x \in \mathbf{R}^n`.
For :math:`p < 1,\ p \neq 0`, the p-norm is given by
.. math::
\|x\|_p = \left(\sum_i x_i^p \right)^{1/p},
with domain :math:`x \in \mathbf{R}^n_+`.
- Note that the "p-norm" is actually a **norm** only when
:math:`p \geq 1` or :math:`p = +\infty`. For these cases,
it is convex.
- The expression is not defined when :math:`p = 0`.
- Otherwise, when :math:`p < 1`, the expression is
concave, but it is not a true norm.
.. note::
Generally, ``p`` cannot be represented exactly, so a rational,
i.e., fractional, **approximation** must be made.
Internally, ``pnorm`` computes a rational approximation
to the reciprocal :math:`1/p` with a denominator up to ``max_denom``.
The resulting
approximation can be found through the attribute ``pnorm.p``.
The approximation error is given by the attribute ``pnorm.approx_error``.
Increasing ``max_denom`` can give better approximations.
When ``p`` is an ``int`` or ``Fraction`` object, the approximation
is usually **exact**.
Parameters
----------
x : cvxpy.Variable
The value to take the norm of.
p : int, float, Fraction, or string
If ``p`` is an ``int``, ``float``, or ``Fraction`` then we must have :math:`p \geq 1`.
The only other valid inputs are ``numpy.inf``, ``float('inf')``, ``float('Inf')``, or
the strings ``"inf"`` or ``"inf"``, all of which are equivalent and give the infinity norm.
max_denom : int
The maximum denominator considered in forming a rational approximation for ``p``.
Returns
-------
Expression
An Expression representing the norm.
"""
def __init__(self, x, p=2, max_denom=1024):
p_old = p
if p in ('inf', 'Inf', np.inf):
self.p = np.inf
elif p < 0:
self.p, _ = pow_neg(p, max_denom)
elif 0 < p < 1:
self.p, _ = pow_mid(p, max_denom)
elif p > 1:
self.p, _ = pow_high(p, max_denom)
elif p == 1:
self.p = 1
else:
raise ValueError('Invalid p: {}'.format(p))
super(pnorm, self).__init__(x)
if self.p == np.inf:
self.approx_error = 0
else:
self.approx_error = float(abs(self.p - p_old))
@Atom.numpy_numeric
def numeric(self, values):
"""Returns the p-norm of x.
"""
values = np.array(values[0]).flatten()
if self.p < 1 and np.any(values < 0):
return -np.inf
if self.p < 0 and np.any(values == 0):
return 0.0
return np.linalg.norm(values, self.p)
def shape_from_args(self):
"""Resolves to a scalar.
"""
return u.Shape(1, 1)
def sign_from_args(self):
"""Always positive.
"""
return u.Sign.POSITIVE
def func_curvature(self):
"""Default curvature is convex.
"""
if self.p >= 1:
return u.Curvature.CONVEX
else:
return u.Curvature.CONCAVE
def monotonicity(self):
"""Increasing for positive arguments and decreasing for negative.
"""
if self.p >= 1:
return [u.monotonicity.SIGNED]
else:
return [u.monotonicity.INCREASING]
def get_data(self):
return self.p
def name(self):
return "%s(%s, %s)" % (self.__class__.__name__,
self.args[0].name(),
self.p)
@staticmethod
[docs] def graph_implementation(arg_objs, size, data=None):
r"""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)
Notes
-----
Implementation notes.
- For general :math:`p \geq 1`, the inequality :math:`\|x\|_p \leq t`
is equivalent to the following convex inequalities:
.. math::
|x_i| &\leq r_i^{1/p} t^{1 - 1/p}\\
\sum_i r_i &= t.
These inequalities happen to also be correct for :math:`p = +\infty`,
if we interpret :math:`1/\infty` as :math:`0`.
- For general :math:`0 < p < 1`, the inequality :math:`\|x\|_p \geq t`
is equivalent to the following convex inequalities:
.. math::
r_i &\leq x_i^{p} t^{1 - p}\\
\sum_i r_i &= t.
- For general :math:`p < 0`, the inequality :math:`\|x\|_p \geq t`
is equivalent to the following convex inequalities:
.. math::
t &\leq x_i^{-p/(1-p)} r_i^{1/(1 - p)}\\
\sum_i r_i &= t.
Although the inequalities above are correct, for a few special cases, we can represent the p-norm
more efficiently and with fewer variables and inequalities.
- For :math:`p = 1`, we use the representation
.. math::
x_i &\leq r_i\\
-x_i &\leq r_i\\
\sum_i r_i &= t
- For :math:`p = \infty`, we use the representation
.. math::
x_i &\leq t\\
-x_i &\leq t
Note that we don't need the :math:`r` variable or the sum inequality.
- For :math:`p = 2`, we use the natural second-order cone representation
.. math::
\|x\|_2 \leq t
Note that we could have used the set of inequalities given above if we wanted an alternate decomposition
of a large second-order cone into into several smaller inequalities.
"""
p = data
x = arg_objs[0]
t = lu.create_var((1, 1))
constraints = []
# first, take care of the special cases of p = 2, inf, and 1
if p == 2:
return t, [SOC(t, [x])]
if p == np.inf:
t_ = lu.promote(t, x.size)
return t, [lu.create_leq(x, t_), lu.create_geq(lu.sum_expr([x, t_]))]
# we need an absolute value constraint for the symmetric convex branches (p >= 1)
# we alias |x| as x from this point forward to make the code pretty :)
if p >= 1:
absx = lu.create_var(x.size)
constraints += [lu.create_leq(x, absx), lu.create_geq(lu.sum_expr([x, absx]))]
x = absx
if p == 1:
return lu.sum_entries(x), constraints
# now, we take care of the remaining convex and concave branches
# to create the rational powers, we need a new variable, r, and
# the constraint sum(r) == t
r = lu.create_var(x.size)
t_ = lu.promote(t, x.size)
constraints += [lu.create_eq(lu.sum_entries(r), t)]
# make p a fraction so that the input weight to gm_constrs
# is a nice tuple of fractions.
p = Fraction(p)
if p < 0:
constraints += gm_constrs(t_, [x, r], (-p/(1-p), 1/(1-p)))
if 0 < p < 1:
constraints += gm_constrs(r, [x, t_], (p, 1-p))
if p > 1:
constraints += gm_constrs(x, [r, t_], (1/p, 1-1/p))
return t, constraints
# todo: no need to run gm_constr to form the tree each time. we only need to form the tree once
