Source code for qiskit_optimization.problems.quadratic_expression

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# (C) Copyright IBM 2019, 2023.
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"""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 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(f"Unsupported format for coefficients: {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) def __repr__(self): # pylint: disable=cyclic-import from ..translators.prettyprint import expr2str, DEFAULT_TRUNCATE return f"<{self.__class__.__name__}: {expr2str(quadratic=self, truncate=DEFAULT_TRUNCATE)}>" def __str__(self): # pylint: disable=cyclic-import from ..translators.prettyprint import expr2str return f"{expr2str(quadratic=self)}"