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Source code for qiskit_optimization.problems.quadratic_expression

# This code is part of Qiskit.
#
# (C) Copyright IBM 2019, 2021.
#
# This code is licensed under the Apache License, Version 2.0. You may
# obtain a copy of this license in the LICENSE.txt file in the root directory
# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
#
# Any modifications or derivative works of this code must retain this
# copyright notice, and modified files need to carry a notice indicating
# that they have been altered from the originals.

"""Quadratic expression interface."""

from typing import List, Union, Dict, Tuple, Any

import numpy as np
from numpy import ndarray
from scipy.sparse import spmatrix, dok_matrix, tril, triu

from .quadratic_program_element import QuadraticProgramElement
from ..exceptions import QiskitOptimizationError
from ..infinity import INFINITY
from .linear_expression import ExpressionBounds


[docs]class QuadraticExpression(QuadraticProgramElement): """Representation of a quadratic expression by its coefficients.""" def __init__( self, quadratic_program: Any, coefficients: Union[ ndarray, spmatrix, List[List[float]], Dict[Tuple[Union[int, str], Union[int, str]], float], ], ) -> None: """Creates a new quadratic expression. The quadratic expression can be defined via an array, a list, a sparse matrix, or a dictionary that uses variable names or indices as keys and stores the values internally as a dok_matrix. We stores values in a compressed way, i.e., values at symmetric positions are summed up in the upper triangle. For example, {(0, 1): 1, (1, 0): 2} -> {(0, 1): 3}. Args: quadratic_program: The parent QuadraticProgram. coefficients: The (sparse) representation of the coefficients. """ super().__init__(quadratic_program) self.coefficients = coefficients
[docs] def __getitem__(self, key: Tuple[Union[int, str], Union[int, str]]) -> float: """Returns the coefficient where i, j can be a variable names or indices. Args: key: The tuple of indices or names of the variables corresponding to the coefficient. Returns: The coefficient corresponding to the addressed variables. """ i, j = key if isinstance(i, str): i = self.quadratic_program.variables_index[i] if isinstance(j, str): j = self.quadratic_program.variables_index[j] return self.coefficients[min(i, j), max(i, j)]
def __setitem__(self, key: Tuple[Union[int, str], Union[int, str]], value: float) -> None: """Sets the coefficient where i, j can be a variable names or indices. Args: key: The tuple of indices or names of the variables corresponding to the coefficient. value: The coefficient corresponding to the addressed variables. """ i, j = key if isinstance(i, str): i = self.quadratic_program.variables_index[i] if isinstance(j, str): j = self.quadratic_program.variables_index[j] self.coefficients[min(i, j), max(i, j)] = value def _coeffs_to_dok_matrix( self, coefficients: Union[ ndarray, spmatrix, List[List[float]], Dict[Tuple[Union[int, str], Union[int, str]], float], ], ) -> dok_matrix: """Maps given coefficients to a dok_matrix. Args: coefficients: The coefficients to be mapped. Returns: The given coefficients as a dok_matrix Raises: QiskitOptimizationError: if coefficients are given in unsupported format. """ if isinstance(coefficients, (list, ndarray, spmatrix)): coefficients = dok_matrix(coefficients) elif isinstance(coefficients, dict): n = self.quadratic_program.get_num_vars() coeffs = dok_matrix((n, n)) for (i, j), value in coefficients.items(): if isinstance(i, str): i = self.quadratic_program.variables_index[i] if isinstance(j, str): j = self.quadratic_program.variables_index[j] coeffs[i, j] = value coefficients = coeffs else: raise QiskitOptimizationError( "Unsupported format for coefficients: {}".format(coefficients) ) return self._triangle_matrix(coefficients) @staticmethod def _triangle_matrix(mat: dok_matrix) -> dok_matrix: lower = tril(mat, -1, format="dok") # `todok` is necessary because subtraction results in other format return (mat + lower.transpose() - lower).todok() @staticmethod def _symmetric_matrix(mat: dok_matrix) -> dok_matrix: upper = triu(mat, 1, format="dok") / 2 # `todok` is necessary because subtraction results in other format return (mat + upper.transpose() - upper).todok() @property def coefficients(self) -> dok_matrix: """Returns the coefficients of the quadratic expression. Returns: The coefficients of the quadratic expression. """ return self._coefficients @coefficients.setter def coefficients( self, coefficients: Union[ ndarray, spmatrix, List[List[float]], Dict[Tuple[Union[int, str], Union[int, str]], float], ], ) -> None: """Sets the coefficients of the quadratic expression. Args: coefficients: The coefficients of the quadratic expression. """ self._coefficients = self._coeffs_to_dok_matrix(coefficients)
[docs] def to_array(self, symmetric: bool = False) -> ndarray: """Returns the coefficients of the quadratic expression as array. Args: symmetric: Determines whether the output is in a symmetric form or not. Returns: An array with the coefficients corresponding to the quadratic expression. """ coeffs = self._symmetric_matrix(self._coefficients) if symmetric else self._coefficients return coeffs.toarray()
[docs] def to_dict( self, symmetric: bool = False, use_name: bool = False ) -> Dict[Union[Tuple[int, int], Tuple[str, str]], float]: """Returns the coefficients of the quadratic expression as dictionary, either using tuples of variable names or indices as keys. Args: symmetric: Determines whether the output is in a symmetric form or not. use_name: Determines whether to use index or names to refer to variables. Returns: An dictionary with the coefficients corresponding to the quadratic expression. """ coeffs = self._symmetric_matrix(self._coefficients) if symmetric else self._coefficients if use_name: return { ( self.quadratic_program.variables[i].name, self.quadratic_program.variables[j].name, ): v for (i, j), v in coeffs.items() } else: return {(int(i), int(j)): v for (i, j), v in coeffs.items()}
[docs] def evaluate(self, x: Union[ndarray, List, Dict[Union[int, str], float]]) -> float: """Evaluate the quadratic expression for given variables: x * Q * x. Args: x: The values of the variables to be evaluated. Returns: The value of the quadratic expression given the variable values. """ x = self._cast_as_array(x) # compute x * Q * x for the quadratic expression val = x @ self.coefficients @ x # return the result return val
[docs] def evaluate_gradient(self, x: Union[ndarray, List, Dict[Union[int, str], float]]) -> ndarray: """Evaluate the gradient of the quadratic expression for given variables. Args: x: The values of the variables to be evaluated. Returns: The value of the gradient quadratic expression given the variable values. """ x = self._cast_as_array(x) # compute (Q' + Q) * x for the quadratic expression val = (self.coefficients.transpose() + self.coefficients) @ x # return the result return val
def _cast_as_array( self, x: Union[ndarray, List, Dict[Union[int, str], float]] ) -> Union[dok_matrix, np.ndarray]: """Converts input to an array if it is a dictionary or list.""" if isinstance(x, dict): x_aux = np.zeros(self.quadratic_program.get_num_vars()) for i, v in x.items(): if isinstance(i, str): i = self.quadratic_program.variables_index[i] x_aux[i] = v x = x_aux if isinstance(x, list): x = np.array(x) return x @property def bounds(self) -> ExpressionBounds: """Returns the lower bound and the upper bound of the quadratic expression Returns: The lower bound and the upper bound of the quadratic expression Raises: QiskitOptimizationError: if the quadratic expression contains any unbounded variable """ l_b = u_b = 0.0 for (ind1, ind2), coeff in self.to_dict().items(): x = self.quadratic_program.get_variable(ind1) if x.lowerbound == -INFINITY or x.upperbound == INFINITY: raise QiskitOptimizationError( f"Quadratic expression contains an unbounded variable: {x.name}" ) y = self.quadratic_program.get_variable(ind2) if y.lowerbound == -INFINITY or y.upperbound == INFINITY: raise QiskitOptimizationError( f"Quadratic expression contains an unbounded variable: {y.name}" ) lst = [] if ind1 == ind2: if x.lowerbound * x.upperbound <= 0.0: # lower bound and upper bound have different signs lst.append(0.0) lst.extend([x.lowerbound ** 2, x.upperbound ** 2]) else: lst.extend( [ x.lowerbound * y.lowerbound, x.lowerbound * y.upperbound, x.upperbound * y.lowerbound, x.upperbound * y.upperbound, ] ) lst2 = [coeff * val for val in lst] l_b += min(lst2) u_b += max(lst2) return ExpressionBounds(lowerbound=l_b, upperbound=u_b)

© Copyright 2018, 2021, Qiskit Optimization Development Team.

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