"""
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 __future__ import annotations
import abc
from typing import TYPE_CHECKING, Iterable
if TYPE_CHECKING:
from cvxpy import Constant, Parameter, Variable
from cvxpy.atoms.atom import Atom
import numbers
import numpy as np
import numpy.linalg as LA
import scipy.sparse as sp
import cvxpy.interface as intf
from cvxpy.constraints.constraint import Constraint
from cvxpy.expressions import expression
from cvxpy.settings import (
GENERAL_PROJECTION_TOL,
PSD_NSD_PROJECTION_TOL,
SPARSE_PROJECTION_TOL,
)
[docs]class Leaf(expression.Expression):
"""
A leaf node of an expression tree; i.e., a Variable, Constant, or Parameter.
A leaf may carry *attributes* that constrain the set values permissible
for it. Leafs can have no more than one attribute, with the exception
that a leaf may be both ``nonpos`` and ``nonneg`` or both ``boolean``
in some indices and ``integer`` in others.
An error is raised if a leaf is assigned a value that contradicts
one or more of its attributes. See the ``project`` method for a convenient
way to project a value onto a leaf's domain.
Parameters
----------
shape : Iterable of ints or int
The leaf dimensions. Either an integer n for a 1D shape, or an
iterable where the semantics are the same as NumPy ndarray shapes.
**Shapes cannot be more than 2D**.
value : numeric type
A value to assign to the leaf.
nonneg : bool
Is the variable constrained to be nonnegative?
nonpos : bool
Is the variable constrained to be nonpositive?
complex : bool
Is the variable complex valued?
symmetric : bool
Is the variable symmetric?
diag : bool
Is the variable diagonal?
PSD : bool
Is the variable constrained to be positive semidefinite?
NSD : bool
Is the variable constrained to be negative semidefinite?
Hermitian : bool
Is the variable Hermitian?
boolean : bool or list of tuple
Is the variable boolean? True, which constrains
the entire Variable to be boolean, False, or a list of
indices which should be constrained as boolean, where each
index is a tuple of length exactly equal to the
length of shape.
integer : bool or list of tuple
Is the variable integer? The semantics are the same as the
boolean argument.
sparsity : list of tuplewith
Fixed sparsity pattern for the variable.
pos : bool
Is the variable positive?
neg : bool
Is the variable negative?
bounds : Iterable
An iterable of length two specifying lower and upper bounds.
"""
__metaclass__ = abc.ABCMeta
def __init__(
self, shape: int | Iterable[int, ...], value=None, nonneg: bool = False,
nonpos: bool = False, complex: bool = False, imag: bool = False,
symmetric: bool = False, diag: bool = False, PSD: bool = False,
NSD: bool = False, hermitian: bool = False,
boolean: bool = False, integer: bool = False,
sparsity=None, pos: bool = False, neg: bool = False, bounds: Iterable | None=None
) -> None:
if isinstance(shape, numbers.Integral):
shape = (int(shape),)
elif len(shape) > 2:
raise ValueError("Expressions of dimension greater than 2 "
"are not supported.")
for d in shape:
if not isinstance(d, numbers.Integral) or d <= 0:
raise ValueError("Invalid dimensions %s." % (shape,))
shape = tuple(np.int32(d) for d in shape)
self._shape = shape
if (PSD or NSD or symmetric or diag or hermitian) and (len(shape) != 2
or shape[0] != shape[1]):
raise ValueError("Invalid dimensions %s. Must be a square matrix."
% (shape,))
# Process attributes.
self.attributes = {'nonneg': nonneg, 'nonpos': nonpos,
'pos': pos, 'neg': neg,
'complex': complex, 'imag': imag,
'symmetric': symmetric, 'diag': diag,
'PSD': PSD, 'NSD': NSD,
'hermitian': hermitian, 'boolean': bool(boolean),
'integer': integer, 'sparsity': sparsity, 'bounds': bounds}
if boolean:
self.boolean_idx = boolean if not isinstance(boolean, bool) else list(
np.ndindex(max(shape, (1,))))
else:
self.boolean_idx = []
if integer:
self.integer_idx = integer if not isinstance(integer, bool) else list(
np.ndindex(max(shape, (1,))))
else:
self.integer_idx = []
# Only one attribute be True (except can be boolean and integer).
true_attr = sum(1 for k, v in self.attributes.items() if v)
# HACK we should remove this feature or allow multiple attributes in general.
if boolean and integer:
true_attr -= 1
if true_attr > 1:
raise ValueError("Cannot set more than one special attribute in %s."
% self.__class__.__name__)
if value is not None:
self.value = value
self.args = []
self.bounds = bounds
def _get_attr_str(self) -> str:
"""Get a string representing the attributes.
"""
attr_str = ""
for attr, val in self.attributes.items():
if attr != 'real' and val:
attr_str += ", %s=%s" % (attr, val)
return attr_str
def copy(self, args=None, id_objects=None):
"""Returns a shallow copy of the object.
Used to reconstruct an object tree.
Parameters
----------
args : list, optional
The arguments to reconstruct the object. If args=None, use the
current args of the object.
Returns
-------
Expression
"""
id_objects = {} if id_objects is None else id_objects
if id(self) in id_objects:
return id_objects[id(self)]
return self # Leaves are not deep copied.
def get_data(self) -> None:
"""Leaves are not copied.
"""
@property
def shape(self) -> tuple[int, ...]:
""" tuple : The dimensions of the expression.
"""
return self._shape
def variables(self) -> list[Variable]:
"""Default is empty list of Variables.
"""
return []
def parameters(self) -> list[Parameter]:
"""Default is empty list of Parameters.
"""
return []
def constants(self) -> list[Constant]:
"""Default is empty list of Constants.
"""
return []
def is_convex(self) -> bool:
"""Is the expression convex?
"""
return True
def is_concave(self) -> bool:
"""Is the expression concave?
"""
return True
def is_log_log_convex(self) -> bool:
"""Is the expression log-log convex?
"""
return self.is_pos()
def is_log_log_concave(self) -> bool:
"""Is the expression log-log concave?
"""
return self.is_pos()
def is_nonneg(self) -> bool:
"""Is the expression nonnegative?
"""
return (self.attributes['nonneg'] or self.attributes['pos'] or
self.attributes['boolean'])
def is_nonpos(self) -> bool:
"""Is the expression nonpositive?
"""
return self.attributes['nonpos'] or self.attributes['neg']
def is_pos(self) -> bool:
"""Is the expression positive?
"""
return self.attributes['pos']
def is_neg(self) -> bool:
"""Is the expression negative?
"""
return self.attributes['neg']
def is_hermitian(self) -> bool:
"""Is the Leaf hermitian?
"""
return (self.is_real() and self.is_symmetric()) or \
self.attributes['hermitian'] or self.is_psd() or self.is_nsd()
def is_symmetric(self) -> bool:
"""Is the Leaf symmetric?
"""
return self.is_scalar() or \
any(self.attributes[key] for key in ['diag', 'symmetric', 'PSD', 'NSD'])
def is_imag(self) -> bool:
"""Is the Leaf imaginary?
"""
return self.attributes['imag']
def is_complex(self) -> bool:
"""Is the Leaf complex valued?
"""
return self.attributes['complex'] or self.is_imag() or self.attributes['hermitian']
def _has_lower_bounds(self) -> bool:
"""Does the variable have lower bounds?"""
if self.is_nonneg():
return True
elif self.attributes['bounds'] is not None:
lower_bound = self.attributes['bounds'][0]
if np.isscalar(lower_bound):
return lower_bound != -np.inf
else:
return np.any(lower_bound != -np.inf)
else:
return False
def _has_upper_bounds(self) -> bool:
"""Does the variable have upper bounds?"""
if self.is_nonpos():
return True
elif self.attributes['bounds'] is not None:
upper_bound = self.attributes['bounds'][1]
if np.isscalar(upper_bound):
return upper_bound != np.inf
else:
return np.any(upper_bound != np.inf)
else:
return False
def _bound_domain(self, term: expression.Expression, constraints: list[Constraint]) -> None:
"""A utility function to append constraints from lower and upper bounds.
Parameters
----------
term: The term to encode in the constraints.
constraints: An existing list of constraitns to append to.
"""
if self.attributes['nonneg'] or self.attributes['pos']:
constraints.append(term >= 0)
elif self.attributes['nonpos'] or self.attributes['neg']:
constraints.append(term <= 0)
elif self.attributes['bounds']:
bounds = self.bounds
lower_bounds, upper_bounds = bounds
# Create masks if -inf or inf is present in the bounds
lower_bound_mask = (lower_bounds != -np.inf)
upper_bound_mask = (upper_bounds != np.inf)
if np.any(lower_bound_mask):
# At least one valid lower bound,
# so we apply the constraint only to those entries
if self.ndim > 0:
constraints.append(term[lower_bound_mask] >= lower_bounds[lower_bound_mask])
else:
constraints.append(term >= lower_bounds)
if np.any(upper_bound_mask):
# At least one valid upper bound,
# so we apply the constraint only to those entries
if self.ndim > 0:
constraints.append(term[upper_bound_mask] <= upper_bounds[upper_bound_mask])
else:
constraints.append(term <= upper_bounds)
@property
def domain(self) -> list[Constraint]:
"""A list of constraints describing the closure of the region
where the expression is finite.
"""
# Default is full domain.
domain = []
# Add constraints from bounds.
self._bound_domain(self, domain)
# Add positive/negative semidefiniteness constraints.
if self.attributes['PSD']:
domain.append(self >> 0)
elif self.attributes['NSD']:
domain.append(self << 0)
return domain
[docs] def project(self, val):
"""Project value onto the attribute set of the leaf.
A sensible idiom is ``leaf.value = leaf.project(val)``.
Parameters
----------
val : numeric type
The value assigned.
Returns
-------
numeric type
The value rounded to the attribute type.
"""
# Only one attribute can be active at once (besides real,
# nonpos/nonneg, and bool/int).
if not self.is_complex():
val = np.real(val)
if self.attributes['nonpos'] and self.attributes['nonneg']:
return 0*val
elif self.attributes['nonpos'] or self.attributes['neg']:
return np.minimum(val, 0.)
elif self.attributes['nonneg'] or self.attributes['pos']:
return np.maximum(val, 0.)
elif self.attributes['bounds']:
return np.clip(val, self.bounds[0], self.bounds[1])
elif self.attributes['imag']:
return np.imag(val)*1j
elif self.attributes['complex']:
return val.astype(complex)
elif self.attributes['boolean']:
# TODO(akshayka): respect the boolean indices.
return np.round(np.clip(val, 0., 1.))
elif self.attributes['integer']:
# TODO(akshayka): respect the integer indices.
# also, a variable may be integer in some indices and
# boolean in others.
return np.round(val)
elif self.attributes['diag']:
if intf.is_sparse(val):
val = val.diagonal()
else:
val = np.diag(val)
return sp.diags([val], [0])
elif self.attributes['hermitian']:
return (val + np.conj(val).T)/2.
elif any([self.attributes[key] for
key in ['symmetric', 'PSD', 'NSD']]):
if val.dtype.kind in 'ib':
val = val.astype(float)
val = val + val.T
val /= 2.
if self.attributes['symmetric']:
return val
w, V = LA.eigh(val)
if self.attributes['PSD']:
bad = w < 0
if not bad.any():
return val
w[bad] = 0
else: # NSD
bad = w > 0
if not bad.any():
return val
w[bad] = 0
return (V * w).dot(V.T)
else:
return val
# Getter and setter for parameter value.
def save_value(self, val) -> None:
self._value = val
@property
def value(self):
"""NumPy.ndarray or None: The numeric value of the parameter.
"""
return self._value
@value.setter
def value(self, val) -> None:
self.save_value(self._validate_value(val))
[docs] def project_and_assign(self, val) -> None:
"""Project and assign a value to the variable.
"""
self.save_value(self.project(val))
def _validate_value(self, val):
"""Check that the value satisfies the leaf's symbolic attributes.
Parameters
----------
val : numeric type
The value assigned.
Returns
-------
numeric type
The value converted to the proper matrix type.
"""
if val is not None:
# Convert val to ndarray or sparse matrix.
val = intf.convert(val)
if intf.shape(val) != self.shape:
raise ValueError(
"Invalid dimensions %s for %s value." %
(intf.shape(val), self.__class__.__name__)
)
projection = self.project(val)
# ^ might be a numpy array, or sparse scipy matrix.
delta = np.abs(val - projection)
# ^ might be a numpy array, scipy matrix, or sparse scipy matrix.
if intf.is_sparse(delta):
# ^ based on current implementation of project(...),
# is is not possible for this Leaf to be PSD/NSD *and*
# a sparse matrix.
close_enough = np.allclose(delta.data, 0,
atol=SPARSE_PROJECTION_TOL)
# ^ only check for near-equality on nonzero values.
else:
# the data could be a scipy matrix, or a numpy array.
# First we convert to a numpy array.
delta = np.array(delta)
# Now that we have the residual, we need to measure it
# in some canonical way.
if self.attributes['PSD'] or self.attributes['NSD']:
# For PSD/NSD Leafs, we use the largest-singular-value norm.
close_enough = LA.norm(delta, ord=2) <= PSD_NSD_PROJECTION_TOL
else:
# For all other Leafs we use the infinity norm on
# the vectorized Leaf.
close_enough = np.allclose(delta, 0,
atol=GENERAL_PROJECTION_TOL)
if not close_enough:
if self.attributes['nonneg']:
attr_str = 'nonnegative'
elif self.attributes['pos']:
attr_str = 'positive'
elif self.attributes['nonpos']:
attr_str = 'nonpositive'
elif self.attributes['neg']:
attr_str = 'negative'
elif self.attributes['diag']:
attr_str = 'diagonal'
elif self.attributes['PSD']:
attr_str = 'positive semidefinite'
elif self.attributes['NSD']:
attr_str = 'negative semidefinite'
elif self.attributes['imag']:
attr_str = 'imaginary'
elif self.attributes['bounds']:
attr_str = 'in bounds'
else:
attr_str = ([k for (k, v) in self.attributes.items() if v] + ['real'])[0]
raise ValueError(
"%s value must be %s." % (self.__class__.__name__, attr_str)
)
return val
def is_psd(self) -> bool:
"""Is the expression a positive semidefinite matrix?
"""
return self.attributes['PSD']
def is_nsd(self) -> bool:
"""Is the expression a negative semidefinite matrix?
"""
return self.attributes['NSD']
def is_diag(self) -> bool:
"""Is the expression a diagonal matrix?
"""
return self.attributes['diag']
def is_quadratic(self) -> bool:
"""Leaf nodes are always quadratic.
"""
return True
def has_quadratic_term(self) -> bool:
"""Leaf nodes are not quadratic terms.
"""
return False
def is_pwl(self) -> bool:
"""Leaf nodes are always piecewise linear.
"""
return True
def is_dpp(self, context: str = 'dcp') -> bool:
"""The expression is a disciplined parameterized expression.
context: dcp or dgp
"""
return True
def atoms(self) -> list[Atom]:
return []
@property
def bounds(self):
return self._bounds
@bounds.setter
def bounds(self, value):
# In case for a constant or no bounds
if value is None:
self._bounds = None
return
# Check that bounds is an iterable of two items
if not isinstance(value, Iterable) or len(value) != 2:
raise ValueError("Bounds should be a list of two items.")
# Check that bounds contains two scalars or two arrays with matching shapes.
for val in value:
valid_array = isinstance(val, np.ndarray) and val.shape == self.shape
if not (val is None or np.isscalar(val) or valid_array):
raise ValueError(
"Bounds should be None, scalars, or arrays with the "
"same dimensions as the variable/parameter."
)
# Promote upper and lower bounds to arrays.
none_bounds = [-np.inf, np.inf]
for idx, val in enumerate(value):
if val is None:
value[idx] = np.full(self.shape, none_bounds[idx])
elif np.isscalar(val):
value[idx] = np.full(self.shape, val)
# Check that upper_bound >= lower_bound
if np.any(value[0] > value[1]):
raise ValueError("Invalid bounds: some upper bounds are less "
"than corresponding lower bounds.")
if np.any(np.isnan(value[0])) or np.any(np.isnan(value[1])):
raise ValueError("np.nan is not feasible as lower "
"or upper bound.")
# Upper bound cannot be -np.inf.
if np.any(value[1] == -np.inf):
raise ValueError("-np.inf is not feasible as an upper bound.")
# Lower bound cannot be np.inf.
if np.any(value[0] == np.inf):
raise ValueError("np.inf is not feasible as a lower bound.")
self._bounds = value