""" Copyright 2013 Steven Diamond Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. """ from typing import List, Tuple, Union import numpy as np import scipy.sparse as sp from cvxpy.atoms.axis_atom import AxisAtom from cvxpy.atoms.norm1 import norm1 from cvxpy.atoms.norm_inf import norm_inf from cvxpy.constraints.constraint import Constraint from cvxpy.utilities.power_tools import pow_high, pow_mid, pow_neg def pnorm(x, p: Union[int, str] = 2, axis=None, keepdims: bool = False, max_denom: int = 1024): """Factory function for a mathematical p-norm. Parameters ---------- p : numeric type or string The type of norm to construct; set this to np.inf or 'inf' to construct an infinity norm. Returns ------- Atom A norm1, norm_inf, or Pnorm object. """ if p == 1: return norm1(x, axis=axis, keepdims=keepdims) elif p in [np.inf, 'inf', 'Inf']: return norm_inf(x, axis=axis, keepdims=keepdims) else: return Pnorm(x, p=p, axis=axis, keepdims=keepdims, max_denom=max_denom) class Pnorm(AxisAtom): r"""The vector p-norm, for p not equal to 1 or infinity. If given a matrix variable, ``pnorm`` will treat it as a vector, and compute the p-norm of the concatenated columns. Only accepts p values that are not equal to 1 or infinity; the norm1 and norm_inf classes handle those norms. For :math:`p > 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 > 1`. 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, or Fraction We require that :math:`p > 1`, but :math:`p \neq \infty`. See the norm1 and norm_inf classes for these norms, or use the pnorm function wrapper to instantiate them. max_denom : int The maximum denominator considered in forming a rational approximation for ``p``. axis : 0 or 1 The axis to apply the norm to. Returns ------- Expression An Expression representing the norm. """ _allow_complex = True def __init__(self, x, p: int = 2, axis=None, keepdims: bool = False, max_denom: int = 1024) -> None: if p < 0: # TODO(akshayka): Why do we accept 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: raise ValueError('Use the norm1 class to instantiate a one norm.') elif p == 'inf' or p == 'Inf' or p == np.inf: raise ValueError('Use the norm_inf class to instantiate an ' 'infinity norm.') else: raise ValueError('Invalid p: {}'.format(p)) self.approx_error = float(abs(self.p - p)) self.original_p = p super(Pnorm, self).__init__(x, axis=axis, keepdims=keepdims) def numeric(self, values): """Returns the p-norm of x. """ if self.axis is None: values = np.array(values[0]).flatten() else: values = np.array(values[0]) 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, float(self.p), axis=self.axis, keepdims=self.keepdims) def validate_arguments(self) -> None: super(Pnorm, self).validate_arguments() # TODO(akshayka): Why is axis not supported for other norms? if self.axis is not None and self.p != 2: raise ValueError( "The axis parameter is only supported for p=2.") if self.p < 1 and self.args[0].is_complex(): raise ValueError("pnorm(x, p) cannot have x complex for p < 1.") def sign_from_args(self) -> Tuple[bool, bool]: """Returns sign (is positive, is negative) of the expression. """ # Always positive. return (True, False) def is_atom_convex(self) -> bool: """Is the atom convex? """ return self.p > 1 def is_atom_concave(self) -> bool: """Is the atom concave? """ return self.p < 1 def is_atom_log_log_convex(self) -> bool: """Is the atom log-log convex? """ return True def is_atom_log_log_concave(self) -> bool: """Is the atom log-log concave? """ return False def is_incr(self, idx) -> bool: """Is the composition non-decreasing in argument idx? """ return self.p < 1 or (self.p > 1 and self.args[0].is_nonneg()) def is_decr(self, idx) -> bool: """Is the composition non-increasing in argument idx? """ return self.p > 1 and self.args[0].is_nonpos() def is_pwl(self) -> bool: """Is the atom piecewise linear? """ return False def get_data(self): return [self.p, self.axis] def name(self) -> str: return "%s(%s, %s)" % (self.__class__.__name__, self.args[0].name(), self.p) def _domain(self) -> List[Constraint]: """Returns constraints describing the domain of the node. """ if self.p < 1 and self.p != 0: return [self.args[0] >= 0] else: return [] def _grad(self, values): """Gives the (sub/super)gradient of the atom w.r.t. each argument. Matrix expressions are vectorized, so the gradient is a matrix. Args: values: A list of numeric values for the arguments. Returns: A list of SciPy CSC sparse matrices or None. """ return self._axis_grad(values) def _column_grad(self, value): """Gives the (sub/super)gradient of the atom w.r.t. a column argument. Matrix expressions are vectorized, so the gradient is a matrix. Args: value: A numeric value for a column. Returns: A NumPy ndarray or None. """ rows = value.size # Outside domain. if self.p < 1 and np.any(value <= 0): return None D_null = sp.csc_array((rows, 1), dtype='float64') # Ensure vector semantics and consistent column-major vectorization. value = np.asarray(value).ravel(order='F') denominator = np.linalg.norm(value, float(self.p)) exp = float(self.p - 1) # cast to float to avoid dtype=object with Fraction exponents denominator = np.power(denominator, exp) # Subgrad is 0 when denom is 0 (or undefined). if denominator == 0: if self.p > 1: return D_null.todense() else: return None if self.p > 1: # nominator = sign(value) * |value|^(p-1) nominator = np.sign(value) * np.power(np.abs(value), exp) else: nominator = np.power(value, exp) frac = np.divide(nominator, denominator) return np.reshape(frac, (frac.size, 1))