Source code for cvxpy.expressions.leaf

"""
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.
"""

import abc
from cvxpy.expressions import expression
from cvxpy.settings import (GENERAL_PROJECTION_TOL,
                            PSD_NSD_PROJECTION_TOL,
                            SPARSE_PROJECTION_TOL)
import cvxpy.interface as intf
import numbers
import numpy as np
import numpy.linalg as LA
import scipy.sparse as sp


[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 : tuple or int The leaf dimensions. Either an integer n for a 1D shape, or a tuple 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? """ __metaclass__ = abc.ABCMeta def __init__(self, shape, value=None, nonneg=False, nonpos=False, complex=False, imag=False, symmetric=False, diag=False, PSD=False, NSD=False, hermitian=False, boolean=False, integer=False, sparsity=None, pos=False, neg=False): 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,)) self._shape = tuple(np.int32(d) for d in 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} if boolean: self.boolean_idx = boolean if not type(boolean) == bool else list( np.ndindex(max(shape, (1,)))) else: self.boolean_idx = [] if integer: self.integer_idx = integer if not type(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) 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 = [] def _get_attr_str(self): """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={}): """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 """ if id(self) in id_objects: return id_objects[id(self)] return self # Leaves are not deep copied. def get_data(self): """Leaves are not copied. """ pass @property def shape(self): """ tuple : The dimensions of the expression. """ return self._shape def variables(self): """Default is empty list of Variables. """ return [] def parameters(self): """Default is empty list of Parameters. """ return [] def constants(self): """Default is empty list of Constants. """ return [] def is_convex(self): """Is the expression convex? """ return True def is_concave(self): """Is the expression concave? """ return True def is_log_log_convex(self): """Is the expression log-log convex? """ return self.is_pos() def is_log_log_concave(self): """Is the expression log-log concave? """ return self.is_pos() def is_nonneg(self): """Is the expression nonnegative? """ return (self.attributes['nonneg'] or self.attributes['pos'] or self.attributes['boolean']) def is_nonpos(self): """Is the expression nonpositive? """ return self.attributes['nonpos'] or self.attributes['neg'] def is_pos(self): """Is the expression positive? """ return self.attributes['pos'] def is_neg(self): """Is the expression negative? """ return self.attributes['neg'] def is_hermitian(self): """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): """Is the Leaf symmetric? """ return self.is_scalar() or \ any(self.attributes[key] for key in ['diag', 'symmetric', 'PSD', 'NSD']) def is_imag(self): """Is the Leaf imaginary? """ return self.attributes['imag'] def is_complex(self): """Is the Leaf complex valued? """ return self.attributes['complex'] or self.is_imag() or self.attributes['hermitian'] @property def domain(self): """A list of constraints describing the closure of the region where the expression is finite. """ # Default is full domain. domain = [] if self.attributes['nonneg']: domain.append(self >= 0) elif self.attributes['nonpos']: domain.append(self >= 0) elif self.attributes['PSD']: domain.append(self >> 0) elif self.attributes['NSD']: domain.append(self << 0) return []
[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['imag']: return np.imag(val) elif self.attributes['complex']: return val.astype(np.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(np.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): 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): self.save_value(self._validate_value(val))
[docs] def project_and_assign(self, val): """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' 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): """Is the expression a positive semidefinite matrix? """ return self.attributes['PSD'] def is_nsd(self): """Is the expression a negative semidefinite matrix? """ return self.attributes['NSD'] def is_quadratic(self): """Leaf nodes are always quadratic. """ return True def is_pwl(self): """Leaf nodes are always piecewise linear. """ return True def atoms(self): return []