# 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 Program."""
from typing import cast, List, Union, Dict, Optional, Tuple
import logging
from collections import defaultdict
from enum import Enum
from math import fsum, isclose
import warnings
import numpy as np
from numpy import (ndarray, zeros)
from scipy.sparse import spmatrix
from docplex.mp.constr import (LinearConstraint as DocplexLinearConstraint,
QuadraticConstraint as DocplexQuadraticConstraint,
NotEqualConstraint)
try:
# new location for 2.16.196 (Nov 2020) or newer
from docplex.mp.dvar import Var
except ImportError:
# old location for 2.15.194 (Jul 2020) or before
from docplex.mp.linear import Var
from docplex.mp.model import Model
from docplex.mp.model_reader import ModelReader
from docplex.mp.quad import QuadExpr
from docplex.mp.vartype import ContinuousVarType, BinaryVarType, IntegerVarType
from qiskit.aqua import MissingOptionalLibraryError
from qiskit.aqua.operators import I, OperatorBase, PauliOp, WeightedPauliOperator, SummedOp, ListOp
from qiskit.quantum_info import Pauli
from .constraint import Constraint, ConstraintSense
from .linear_constraint import LinearConstraint
from .linear_expression import LinearExpression
from .quadratic_constraint import QuadraticConstraint
from .quadratic_expression import QuadraticExpression
from .quadratic_objective import QuadraticObjective
from .variable import Variable, VarType
from ..exceptions import QiskitOptimizationError
from ..infinity import INFINITY
logger = logging.getLogger(__name__)
class QuadraticProgramStatus(Enum):
"""Status of QuadraticProgram"""
VALID = 0
INFEASIBLE = 1
[docs]class QuadraticProgram:
"""Quadratically Constrained Quadratic Program representation.
This representation supports inequality and equality constraints,
as well as continuous, binary, and integer variables.
"""
Status = QuadraticProgramStatus
[docs] def __init__(self, name: str = '') -> None:
"""
Args:
name: The name of the quadratic program.
"""
self._name = name
self._status = QuadraticProgram.Status.VALID
self._variables = [] # type: List[Variable]
self._variables_index = {} # type: Dict[str, int]
self._linear_constraints = [] # type: List[LinearConstraint]
self._linear_constraints_index = {} # type: Dict[str, int]
self._quadratic_constraints = [] # type: List[QuadraticConstraint]
self._quadratic_constraints_index = {} # type: Dict[str, int]
self._objective = QuadraticObjective(self)
def __repr__(self) -> str:
return self.to_docplex().export_as_lp_string()
[docs] def clear(self) -> None:
"""Clears the quadratic program, i.e., deletes all variables, constraints, the
objective function as well as the name.
"""
self._name = ''
self._status = QuadraticProgram.Status.VALID
self._variables.clear()
self._variables_index.clear()
self._linear_constraints.clear()
self._linear_constraints_index.clear()
self._quadratic_constraints.clear()
self._quadratic_constraints_index.clear()
self._objective = QuadraticObjective(self)
@property
def name(self) -> str:
"""Returns the name of the quadratic program.
Returns:
The name of the quadratic program.
"""
return self._name
@name.setter
def name(self, name: str) -> None:
"""Sets the name of the quadratic program.
Args:
name: The name of the quadratic program.
"""
self._name = name
@property
def status(self) -> QuadraticProgramStatus:
"""Status of the quadratic program.
It can be infeasible due to variable substitution.
Returns:
The status of the quadratic program
"""
return self._status
@property
def variables(self) -> List[Variable]:
"""Returns the list of variables of the quadratic program.
Returns:
List of variables.
"""
return self._variables
@property
def variables_index(self) -> Dict[str, int]:
"""Returns the dictionary that maps the name of a variable to its index.
Returns:
The variable index dictionary.
"""
return self._variables_index
def _add_variable(self,
lowerbound: Union[float, int] = 0,
upperbound: Union[float, int] = INFINITY,
vartype: VarType = VarType.CONTINUOUS,
name: Optional[str] = None) -> Variable:
"""Checks whether a variable name is already taken and adds the variable to list and index
if not.
Args:
lowerbound: The lowerbound of the variable.
upperbound: The upperbound of the variable.
vartype: The type of the variable.
name: The name of the variable.
Returns:
The added variable.
Raises:
QiskitOptimizationError: if the variable name is already taken.
"""
if name:
if name in self._variables_index:
raise QiskitOptimizationError("Variable name already exists: {}".format(name))
else:
k = self.get_num_vars()
while 'x{}'.format(k) in self._variables_index:
k += 1
name = 'x{}'.format(k)
self.variables_index[name] = len(self.variables)
variable = Variable(self, name, lowerbound, upperbound, vartype)
self.variables.append(variable)
return variable
[docs] def continuous_var(self, lowerbound: Union[float, int] = 0,
upperbound: Union[float, int] = INFINITY,
name: Optional[str] = None) -> Variable:
"""Adds a continuous variable to the quadratic program.
Args:
lowerbound: The lowerbound of the variable.
upperbound: The upperbound of the variable.
name: The name of the variable.
Returns:
The added variable.
Raises:
QiskitOptimizationError: if the variable name is already occupied.
"""
return self._add_variable(lowerbound, upperbound, Variable.Type.CONTINUOUS, name)
[docs] def binary_var(self, name: Optional[str] = None) -> Variable:
"""Adds a binary variable to the quadratic program.
Args:
name: The name of the variable.
Returns:
The added variable.
Raises:
QiskitOptimizationError: if the variable name is already occupied.
"""
return self._add_variable(0, 1, Variable.Type.BINARY, name)
[docs] def integer_var(self, lowerbound: Union[float, int] = 0,
upperbound: Union[float, int] = INFINITY,
name: Optional[str] = None) -> Variable:
"""Adds an integer variable to the quadratic program.
Args:
lowerbound: The lowerbound of the variable.
upperbound: The upperbound of the variable.
name: The name of the variable.
Returns:
The added variable.
Raises:
QiskitOptimizationError: if the variable name is already occupied.
"""
return self._add_variable(lowerbound, upperbound, Variable.Type.INTEGER, name)
[docs] def get_variable(self, i: Union[int, str]) -> Variable:
"""Returns a variable for a given name or index.
Args:
i: the index or name of the variable.
Returns:
The corresponding variable.
"""
if isinstance(i, int):
return self.variables[i]
else:
return self.variables[self._variables_index[i]]
[docs] def get_num_vars(self, vartype: Optional[VarType] = None) -> int:
"""Returns the total number of variables or the number of variables of the specified type.
Args:
vartype: The type to be filtered on. All variables are counted if None.
Returns:
The total number of variables.
"""
if vartype:
return sum(variable.vartype == vartype for variable in self._variables)
else:
return len(self._variables)
[docs] def get_num_continuous_vars(self) -> int:
"""Returns the total number of continuous variables.
Returns:
The total number of continuous variables.
"""
return self.get_num_vars(Variable.Type.CONTINUOUS)
[docs] def get_num_binary_vars(self) -> int:
"""Returns the total number of binary variables.
Returns:
The total number of binary variables.
"""
return self.get_num_vars(Variable.Type.BINARY)
[docs] def get_num_integer_vars(self) -> int:
"""Returns the total number of integer variables.
Returns:
The total number of integer variables.
"""
return self.get_num_vars(Variable.Type.INTEGER)
@property
def linear_constraints(self) -> List[LinearConstraint]:
"""Returns the list of linear constraints of the quadratic program.
Returns:
List of linear constraints.
"""
return self._linear_constraints
@property
def linear_constraints_index(self) -> Dict[str, int]:
"""Returns the dictionary that maps the name of a linear constraint to its index.
Returns:
The linear constraint index dictionary.
"""
return self._linear_constraints_index
[docs] def linear_constraint(self,
linear: Union[ndarray, spmatrix, List[float],
Dict[Union[int, str], float]] = None,
sense: Union[str, ConstraintSense] = '<=',
rhs: float = 0.0, name: Optional[str] = None) -> LinearConstraint:
"""Adds a linear equality constraint to the quadratic program of the form:
linear * x sense rhs.
Args:
linear: The linear coefficients of the left-hand-side of the constraint.
sense: The sense of the constraint,
- '==', '=', 'E', and 'EQ' denote 'equal to'.
- '>=', '>', 'G', and 'GE' denote 'greater-than-or-equal-to'.
- '<=', '<', 'L', and 'LE' denote 'less-than-or-equal-to'.
rhs: The right hand side of the constraint.
name: The name of the constraint.
Returns:
The added constraint.
Raises:
QiskitOptimizationError: if the constraint name already exists or the sense is not
valid.
"""
if name:
if name in self.linear_constraints_index:
raise QiskitOptimizationError(
"Linear constraint's name already exists: {}".format(name))
else:
k = self.get_num_linear_constraints()
while 'c{}'.format(k) in self.linear_constraints_index:
k += 1
name = 'c{}'.format(k)
self.linear_constraints_index[name] = len(self.linear_constraints)
if linear is None:
linear = {}
constraint = LinearConstraint(self, name, linear, Constraint.Sense.convert(sense), rhs)
self.linear_constraints.append(constraint)
return constraint
[docs] def get_linear_constraint(self, i: Union[int, str]) -> LinearConstraint:
"""Returns a linear constraint for a given name or index.
Args:
i: the index or name of the constraint.
Returns:
The corresponding constraint.
Raises:
IndexError: if the index is out of the list size
KeyError: if the name does not exist
"""
if isinstance(i, int):
return self._linear_constraints[i]
else:
return self._linear_constraints[self._linear_constraints_index[i]]
[docs] def get_num_linear_constraints(self) -> int:
"""Returns the number of linear constraints.
Returns:
The number of linear constraints.
"""
return len(self._linear_constraints)
@property
def quadratic_constraints(self) -> List[QuadraticConstraint]:
"""Returns the list of quadratic constraints of the quadratic program.
Returns:
List of quadratic constraints.
"""
return self._quadratic_constraints
@property
def quadratic_constraints_index(self) -> Dict[str, int]:
"""Returns the dictionary that maps the name of a quadratic constraint to its index.
Returns:
The quadratic constraint index dictionary.
"""
return self._quadratic_constraints_index
[docs] def quadratic_constraint(self,
linear: Union[ndarray, spmatrix, List[float],
Dict[Union[int, str], float]] = None,
quadratic: Union[ndarray, spmatrix, List[List[float]],
Dict[Tuple[Union[int, str],
Union[int, str]], float]] = None,
sense: Union[str, ConstraintSense] = '<=',
rhs: float = 0.0, name: Optional[str] = None) -> QuadraticConstraint:
"""Adds a quadratic equality constraint to the quadratic program of the form:
x * Q * x <= rhs.
Args:
linear: The linear coefficients of the constraint.
quadratic: The quadratic coefficients of the constraint.
sense: The sense of the constraint,
- '==', '=', 'E', and 'EQ' denote 'equal to'.
- '>=', '>', 'G', and 'GE' denote 'greater-than-or-equal-to'.
- '<=', '<', 'L', and 'LE' denote 'less-than-or-equal-to'.
rhs: The right hand side of the constraint.
name: The name of the constraint.
Returns:
The added constraint.
Raises:
QiskitOptimizationError: if the constraint name already exists.
"""
if name:
if name in self.quadratic_constraints_index:
raise QiskitOptimizationError(
"Quadratic constraint name already exists: {}".format(name))
else:
k = self.get_num_quadratic_constraints()
while 'q{}'.format(k) in self.quadratic_constraints_index:
k += 1
name = 'q{}'.format(k)
self.quadratic_constraints_index[name] = len(self.quadratic_constraints)
if linear is None:
linear = {}
if quadratic is None:
quadratic = {}
constraint = QuadraticConstraint(self, name, linear, quadratic,
Constraint.Sense.convert(sense), rhs)
self.quadratic_constraints.append(constraint)
return constraint
[docs] def get_quadratic_constraint(self, i: Union[int, str]) -> QuadraticConstraint:
"""Returns a quadratic constraint for a given name or index.
Args:
i: the index or name of the constraint.
Returns:
The corresponding constraint.
Raises:
IndexError: if the index is out of the list size
KeyError: if the name does not exist
"""
if isinstance(i, int):
return self._quadratic_constraints[i]
else:
return self._quadratic_constraints[self._quadratic_constraints_index[i]]
[docs] def get_num_quadratic_constraints(self) -> int:
"""Returns the number of quadratic constraints.
Returns:
The number of quadratic constraints.
"""
return len(self._quadratic_constraints)
[docs] def remove_linear_constraint(self, i: Union[str, int]) -> None:
"""Remove a linear constraint
Args:
i: an index or a name of a linear constraint
Raises:
KeyError: if name does not exist
IndexError: if index is out of range
"""
if isinstance(i, str):
i = self._linear_constraints_index[i]
del self._linear_constraints[i]
self._linear_constraints_index = {cst.name: j for j, cst in
enumerate(self._linear_constraints)}
[docs] def remove_quadratic_constraint(self, i: Union[str, int]) -> None:
"""Remove a quadratic constraint
Args:
i: an index or a name of a quadratic constraint
Raises:
KeyError: if name does not exist
IndexError: if index is out of range
"""
if isinstance(i, str):
i = self._quadratic_constraints_index[i]
del self._quadratic_constraints[i]
self._quadratic_constraints_index = {cst.name: j for j, cst in
enumerate(self._quadratic_constraints)}
@property
def objective(self) -> QuadraticObjective:
"""Returns the quadratic objective.
Returns:
The quadratic objective.
"""
return self._objective
[docs] def minimize(self,
constant: float = 0.0,
linear: Union[ndarray, spmatrix, List[float], Dict[Union[str, int], float]] = None,
quadratic: Union[ndarray, spmatrix, List[List[float]],
Dict[Tuple[Union[int, str], Union[int, str]], float]] = None
) -> None:
"""Sets a quadratic objective to be minimized.
Args:
constant: the constant offset of the objective.
linear: the coefficients of the linear part of the objective.
quadratic: the coefficients of the quadratic part of the objective.
Returns:
The created quadratic objective.
"""
self._objective = QuadraticObjective(self, constant, linear, quadratic,
QuadraticObjective.Sense.MINIMIZE)
[docs] def maximize(self,
constant: float = 0.0,
linear: Union[ndarray, spmatrix, List[float], Dict[Union[str, int], float]] = None,
quadratic: Union[ndarray, spmatrix, List[List[float]],
Dict[Tuple[Union[int, str], Union[int, str]], float]] = None
) -> None:
"""Sets a quadratic objective to be maximized.
Args:
constant: the constant offset of the objective.
linear: the coefficients of the linear part of the objective.
quadratic: the coefficients of the quadratic part of the objective.
Returns:
The created quadratic objective.
"""
self._objective = QuadraticObjective(self, constant, linear, quadratic,
QuadraticObjective.Sense.MAXIMIZE)
[docs] def from_docplex(self, model: Model) -> None:
"""Loads this quadratic program from a docplex model.
Note that this supports only basic functions of docplex as follows:
- quadratic objective function
- linear / quadratic constraints
- binary / integer / continuous variables
Args:
model: The docplex model to be loaded.
Raises:
QiskitOptimizationError: if the model contains unsupported elements.
"""
# clear current problem
self.clear()
# get name
self.name = model.name
# get variables
# keep track of names separately, since docplex allows to have None names.
var_names = {}
for x in model.iter_variables():
if isinstance(x.vartype, ContinuousVarType):
x_new = self.continuous_var(x.lb, x.ub, x.name)
elif isinstance(x.vartype, BinaryVarType):
x_new = self.binary_var(x.name)
elif isinstance(x.vartype, IntegerVarType):
x_new = self.integer_var(x.lb, x.ub, x.name)
else:
raise QiskitOptimizationError(
"Unsupported variable type: {} {}".format(x.name, x.vartype))
var_names[x] = x_new.name
# objective sense
minimize = model.objective_sense.is_minimize()
# make sure objective expression is linear or quadratic and not a variable
if isinstance(model.objective_expr, Var):
model.objective_expr = model.objective_expr + 0
# get objective offset
constant = model.objective_expr.constant
# get linear part of objective
linear = {}
linear_part = model.objective_expr.get_linear_part()
for x in linear_part.iter_variables():
linear[var_names[x]] = linear_part.get_coef(x)
# get quadratic part of objective
quadratic = {}
if isinstance(model.objective_expr, QuadExpr):
for quad_triplet in model.objective_expr.iter_quad_triplets():
i = var_names[quad_triplet[0]]
j = var_names[quad_triplet[1]]
v = quad_triplet[2]
quadratic[i, j] = v
# set objective
if minimize:
self.minimize(constant, linear, quadratic)
else:
self.maximize(constant, linear, quadratic)
# get linear constraints
for constraint in model.iter_constraints():
if isinstance(constraint, DocplexQuadraticConstraint):
# ignore quadratic constraints here and process them later
continue
if not isinstance(constraint, DocplexLinearConstraint) or \
isinstance(constraint, NotEqualConstraint):
# If any constraint is not linear/quadratic constraints, it raises an error.
# Notice that NotEqualConstraint is a subclass of Docplex's LinearConstraint,
# but it cannot be handled by Aqua optimization.
raise QiskitOptimizationError(
'Unsupported constraint: {}'.format(constraint))
name = constraint.name
sense = constraint.sense
left_expr = constraint.get_left_expr()
right_expr = constraint.get_right_expr()
# for linear constraints we may get an instance of Var instead of expression,
# e.g. x + y = z
if isinstance(left_expr, Var):
left_expr = left_expr + 0
if isinstance(right_expr, Var):
right_expr = right_expr + 0
rhs = right_expr.constant - left_expr.constant
lhs = {}
for x in left_expr.iter_variables():
lhs[var_names[x]] = left_expr.get_coef(x)
for x in right_expr.iter_variables():
lhs[var_names[x]] = lhs.get(var_names[x], 0.0) - right_expr.get_coef(x)
if sense == sense.EQ:
self.linear_constraint(lhs, '==', rhs, name)
elif sense == sense.GE:
self.linear_constraint(lhs, '>=', rhs, name)
elif sense == sense.LE:
self.linear_constraint(lhs, '<=', rhs, name)
else:
raise QiskitOptimizationError(
"Unsupported constraint sense: {}".format(constraint))
# get quadratic constraints
for constraint in model.iter_quadratic_constraints():
name = constraint.name
sense = constraint.sense
left_expr = constraint.get_left_expr()
right_expr = constraint.get_right_expr()
rhs = right_expr.constant - left_expr.constant
linear = {}
quadratic = {}
if left_expr.is_quad_expr():
for x in left_expr.linear_part.iter_variables():
linear[var_names[x]] = left_expr.linear_part.get_coef(x)
for quad_triplet in left_expr.iter_quad_triplets():
i = var_names[quad_triplet[0]]
j = var_names[quad_triplet[1]]
v = quad_triplet[2]
quadratic[i, j] = v
else:
for x in left_expr.iter_variables():
linear[var_names[x]] = left_expr.get_coef(x)
if right_expr.is_quad_expr():
for x in right_expr.linear_part.iter_variables():
linear[var_names[x]] = linear.get(var_names[x], 0.0) - \
right_expr.linear_part.get_coef(x)
for quad_triplet in right_expr.iter_quad_triplets():
i = var_names[quad_triplet[0]]
j = var_names[quad_triplet[1]]
v = quad_triplet[2]
quadratic[i, j] = quadratic.get((i, j), 0.0) - v
else:
for x in right_expr.iter_variables():
linear[var_names[x]] = linear.get(var_names[x], 0.0) - right_expr.get_coef(x)
if sense == sense.EQ:
self.quadratic_constraint(linear, quadratic, '==', rhs, name)
elif sense == sense.GE:
self.quadratic_constraint(linear, quadratic, '>=', rhs, name)
elif sense == sense.LE:
self.quadratic_constraint(linear, quadratic, '<=', rhs, name)
else:
raise QiskitOptimizationError(
"Unsupported constraint sense: {}".format(constraint))
[docs] def to_docplex(self) -> Model:
"""Returns a docplex model corresponding to this quadratic program.
Returns:
The docplex model corresponding to this quadratic program.
Raises:
QiskitOptimizationError: if non-supported elements (should never happen).
"""
# initialize model
mdl = Model(self.name)
# add variables
var = {}
for idx, x in enumerate(self.variables):
if x.vartype == Variable.Type.CONTINUOUS:
var[idx] = mdl.continuous_var(lb=x.lowerbound, ub=x.upperbound, name=x.name)
elif x.vartype == Variable.Type.BINARY:
var[idx] = mdl.binary_var(name=x.name)
elif x.vartype == Variable.Type.INTEGER:
var[idx] = mdl.integer_var(lb=x.lowerbound, ub=x.upperbound, name=x.name)
else:
# should never happen
raise QiskitOptimizationError('Unsupported variable type: {}'.format(x.vartype))
# add objective
objective = self.objective.constant
for i, v in self.objective.linear.to_dict().items():
objective += v * var[cast(int, i)]
for (i, j), v in self.objective.quadratic.to_dict().items():
objective += v * var[cast(int, i)] * var[cast(int, j)]
if self.objective.sense == QuadraticObjective.Sense.MINIMIZE:
mdl.minimize(objective)
else:
mdl.maximize(objective)
# add linear constraints
for i, l_constraint in enumerate(self.linear_constraints):
name = l_constraint.name
rhs = l_constraint.rhs
if rhs == 0 and l_constraint.linear.coefficients.nnz == 0:
continue
linear_expr = 0
for j, v in l_constraint.linear.to_dict().items():
linear_expr += v * var[cast(int, j)]
sense = l_constraint.sense
if sense == Constraint.Sense.EQ:
mdl.add_constraint(linear_expr == rhs, ctname=name)
elif sense == Constraint.Sense.GE:
mdl.add_constraint(linear_expr >= rhs, ctname=name)
elif sense == Constraint.Sense.LE:
mdl.add_constraint(linear_expr <= rhs, ctname=name)
else:
# should never happen
raise QiskitOptimizationError("Unsupported constraint sense: {}".format(sense))
# add quadratic constraints
for i, q_constraint in enumerate(self.quadratic_constraints):
name = q_constraint.name
rhs = q_constraint.rhs
if rhs == 0 \
and q_constraint.linear.coefficients.nnz == 0 \
and q_constraint.quadratic.coefficients.nnz == 0:
continue
quadratic_expr = 0
for j, v in q_constraint.linear.to_dict().items():
quadratic_expr += v * var[cast(int, j)]
for (j, k), v in q_constraint.quadratic.to_dict().items():
quadratic_expr += v * var[cast(int, j)] * var[cast(int, k)]
sense = q_constraint.sense
if sense == Constraint.Sense.EQ:
mdl.add_constraint(quadratic_expr == rhs, ctname=name)
elif sense == Constraint.Sense.GE:
mdl.add_constraint(quadratic_expr >= rhs, ctname=name)
elif sense == Constraint.Sense.LE:
mdl.add_constraint(quadratic_expr <= rhs, ctname=name)
else:
# should never happen
raise QiskitOptimizationError("Unsupported constraint sense: {}".format(sense))
return mdl
[docs] def export_as_lp_string(self) -> str:
"""Returns the quadratic program as a string of LP format.
Returns:
A string representing the quadratic program.
"""
return self.to_docplex().export_as_lp_string()
[docs] def pprint_as_string(self) -> str:
"""DEPRECATED Returns the quadratic program as a string in Docplex's pretty print format.
Returns:
A string representing the quadratic program.
"""
warnings.warn("The pprint_as_string method is deprecated and will be "
"removed in a future release. Instead use the"
"to_docplex() method and run pprint_as_string() on that "
"output", DeprecationWarning)
return self.to_docplex().pprint_as_string()
[docs] def prettyprint(self, out: Optional[str] = None) -> None:
"""DEPRECATED Pretty prints the quadratic program to a given output stream (None = default).
Args:
out: The output stream or file name to print to.
if you specify a file name, the output file name is has '.mod' as suffix.
"""
warnings.warn("The prettyprint method is deprecated and will be "
"removed in a future release. Instead use the"
"to_docplex() method and run prettyprint() on that "
"output", DeprecationWarning)
self.to_docplex().prettyprint(out)
[docs] def read_from_lp_file(self, filename: str) -> None:
"""Loads the quadratic program from a LP file.
Args:
filename: The filename of the file to be loaded.
Raises:
FileNotFoundError: If the file does not exist.
MissingOptionalLibraryError: If CPLEX is not installed.
Note:
This method requires CPLEX to be installed and present in ``PYTHONPATH``.
"""
try:
import cplex # pylint: disable=unused-import
except ImportError as ex:
raise MissingOptionalLibraryError(
libname='CPLEX',
name='QuadraticProgram.read_from_lp_file',
pip_install='pip install qiskit-aqua[cplex]') from ex
def _parse_problem_name(filename: str) -> str:
# Because docplex model reader uses the base name as model name,
# we parse the model name in the LP file manually.
# https://ibmdecisionoptimization.github.io/docplex-doc/mp/docplex.mp.model_reader.html
prefix = '\\Problem name:'
model_name = ''
with open(filename) as file:
for line in file:
if line.startswith(prefix):
model_name = line[len(prefix):].strip()
if not line.startswith('\\'):
break
return model_name
model_reader = ModelReader()
model = model_reader.read(filename, model_name=_parse_problem_name(filename))
self.from_docplex(model)
[docs] def write_to_lp_file(self, filename: str) -> None:
"""Writes the quadratic program to an LP file.
Args:
filename: The filename of the file the model is written to.
If filename is a directory, file name 'my_problem.lp' is appended.
If filename does not end with '.lp', suffix '.lp' is appended.
Raises:
OSError: If this cannot open a file.
DOcplexException: If filename is an empty string
"""
self.to_docplex().export_as_lp(filename)
[docs] def substitute_variables(
self, constants: Optional[Dict[Union[str, int], float]] = None,
variables: Optional[Dict[Union[str, int], Tuple[Union[str, int], float]]] = None) \
-> 'QuadraticProgram':
"""Substitutes variables with constants or other variables.
Args:
constants: replace variable by constant
e.g., {'x': 2} means 'x' is substituted with 2
variables: replace variables by weighted other variable
need to copy everything using name reference to make sure that indices are matched
correctly. The lower and upper bounds are updated accordingly.
e.g., {'x': ('y', 2)} means 'x' is substituted with 'y' * 2
Returns:
An optimization problem by substituting variables with constants or other variables.
If the substitution is valid, `QuadraticProgram.status` is still
`QuadraticProgram.Status.VALIAD`.
Otherwise, it gets `QuadraticProgram.Status.INFEASIBLE`.
Raises:
QiskitOptimizationError: if the substitution is invalid as follows.
- Same variable is substituted multiple times.
- Coefficient of variable substitution is zero.
"""
return SubstituteVariables().substitute_variables(self, constants, variables)
[docs] def to_ising(self) -> Tuple[OperatorBase, float]:
"""Return the Ising Hamiltonian of this problem.
Returns:
qubit_op: The qubit operator for the problem
offset: The constant value in the Ising Hamiltonian.
Raises:
QiskitOptimizationError: If a variable type is not binary.
QiskitOptimizationError: If constraints exist in the problem.
"""
# if problem has variables that are not binary, raise an error
if self.get_num_vars() > self.get_num_binary_vars():
raise QiskitOptimizationError('The type of variable must be a binary variable. '
'Use a QuadraticProgramToQubo converter to convert '
'integer variables to binary variables. '
'If the problem contains continuous variables, '
'currently we can not apply VQE/QAOA directly. '
'you might want to use an ADMM optimizer '
'for the problem. ')
# if constraints exist, raise an error
if self.linear_constraints \
or self.quadratic_constraints:
raise QiskitOptimizationError('An constraint exists. '
'The method supports only model with no constraints. '
'Use a QuadraticProgramToQubo converter. '
'It converts inequality constraints to equality '
'constraints, and then, it converters equality '
'constraints to penalty terms of the object function.')
# initialize Hamiltonian.
num_nodes = self.get_num_vars()
pauli_list = []
offset = 0
zero = zeros(num_nodes, dtype=bool)
# set a sign corresponding to a maximized or minimized problem.
# sign == 1 is for minimized problem. sign == -1 is for maximized problem.
sense = self.objective.sense.value
# convert a constant part of the object function into Hamiltonian.
offset += self.objective.constant * sense # type: ignore
# convert linear parts of the object function into Hamiltonian.
for idx, coef in self.objective.linear.to_dict().items():
z_p = zeros(num_nodes, dtype=bool)
weight = coef * sense / 2
z_p[idx] = True
pauli_list.append([-weight, Pauli(z_p, zero)])
offset += weight # type: ignore
# convert quadratic parts of the object function into Hamiltonian.
# first merge coefficients (i, j) and (j, i)
coeffs = {} # type: Dict
for (i, j), coeff in self.objective.quadratic.to_dict().items():
if j < i: # type: ignore
coeffs[(j, i)] = coeffs.get((j, i), 0.0) + coeff
else:
coeffs[(i, j)] = coeffs.get((i, j), 0.0) + coeff
# create Pauli terms
for (i, j), coeff in coeffs.items():
weight = coeff * sense / 4
if i == j:
offset += weight # type: ignore
else:
z_p = zeros(num_nodes, dtype=bool)
z_p[i] = True
z_p[j] = True
pauli_list.append([weight, Pauli(z_p, zero)])
z_p = zeros(num_nodes, dtype=bool)
z_p[i] = True
pauli_list.append([-weight, Pauli(z_p, zero)])
z_p = zeros(num_nodes, dtype=bool)
z_p[j] = True
pauli_list.append([-weight, Pauli(z_p, zero)])
offset += weight # type: ignore
# Remove paulis whose coefficients are zeros.
qubit_op = sum(PauliOp(pauli, coeff=coeff) for coeff, pauli in pauli_list)
# qubit_op could be the integer 0, in this case return an identity operator of
# appropriate size
if isinstance(qubit_op, OperatorBase):
qubit_op = qubit_op.reduce()
else:
qubit_op = I ^ num_nodes
return qubit_op, offset
[docs] def from_ising(self,
qubit_op: Union[OperatorBase, WeightedPauliOperator],
offset: float = 0.0, linear: bool = False) -> None:
r"""Create a quadratic program from a qubit operator and a shift value.
Args:
qubit_op: The qubit operator of the problem.
offset: The constant value in the Ising Hamiltonian.
linear: If linear is True, :math:`x^2` is treated as a linear term
since :math:`x^2 = x` for :math:`x \in \{0,1\}`.
Else, :math:`x^2` is treat as a quadratic term.
The default value is False.
Raises:
QiskitOptimizationError: If there are Pauli Xs in any Pauli term
QiskitOptimizationError: If there are more than 2 Pauli Zs in any Pauli term
NotImplementedError: If the input operator is a ListOp
"""
if isinstance(qubit_op, WeightedPauliOperator):
qubit_op = qubit_op.to_opflow()
# No support for ListOp yet, this can be added in future
# pylint: disable=unidiomatic-typecheck
if type(qubit_op) == ListOp:
raise NotImplementedError(
'Conversion of a ListOp is not supported, convert each '
'operator in the ListOp separately.'
)
# add binary variables
for i in range(qubit_op.num_qubits):
self.binary_var(name='x_{0}'.format(i))
# Create a QUBO matrix
# The Qubo matrix is an upper triangular matrix.
# Diagonal elements in the QUBO matrix are for linear terms of the qubit operator.
# The other elements in the QUBO matrix are for quadratic terms of the qubit operator.
qubo_matrix = zeros((qubit_op.num_qubits, qubit_op.num_qubits))
if not isinstance(qubit_op, SummedOp):
oplist = [qubit_op.to_pauli_op()]
else:
oplist = qubit_op.to_pauli_op().oplist
for pauli_op in oplist:
pauli = pauli_op.primitive
coeff = pauli_op.coeff
# Count the number of Pauli Zs in a Pauli term
lst_z = pauli.z.tolist()
z_index = [i for i, z in enumerate(lst_z) if z is True]
num_z = len(z_index)
# Add its weight of the Pauli term to the corresponding element of QUBO matrix
if num_z == 1:
qubo_matrix[z_index[0], z_index[0]] = coeff.real
elif num_z == 2:
qubo_matrix[z_index[0], z_index[1]] = coeff.real
else:
raise QiskitOptimizationError(
'There are more than 2 Pauli Zs in the Pauli term {}'.format(pauli.z)
)
# If there are Pauli Xs in the Pauli term, raise an error
lst_x = pauli.x.tolist()
x_index = [i for i, x in enumerate(lst_x) if x is True]
if len(x_index) > 0:
raise QiskitOptimizationError('Pauli Xs exist in the Pauli {}'.format(pauli.x))
# Initialize dicts for linear terms and quadratic terms
linear_terms = {}
quadratic_terms = {}
# For quadratic pauli terms of operator
# x_i * x_ j = (1 - Z_i - Z_j + Z_i * Z_j)/4
for i, row in enumerate(qubo_matrix):
for j, weight in enumerate(row):
# Focus on the upper triangular matrix
if j <= i:
continue
# Add a quadratic term to the object function of `QuadraticProgram`
# The coefficient of the quadratic term in `QuadraticProgram` is
# 4 * weight of the pauli
coef = weight * 4
quadratic_terms[i, j] = coef
# Sub the weight of the quadratic pauli term from the QUBO matrix
qubo_matrix[i, j] -= weight
# Sub the weight of the linear pauli term from the QUBO matrix
qubo_matrix[i, i] += weight
qubo_matrix[j, j] += weight
# Sub the weight from offset
offset -= weight
# After processing quadratic pauli terms, only linear paulis are left
# x_i = (1 - Z_i)/2
for i in range(qubit_op.num_qubits):
weight = qubo_matrix[i, i]
# Add a linear term to the object function of `QuadraticProgram`
# The coefficient of the linear term in `QuadraticProgram` is
# 2 * weight of the pauli
coef = weight * 2
if linear:
# If the linear option is True, add it into linear_terms
linear_terms[i] = -coef
else:
# Else, add it into quadratic_terms as a diagonal element.
quadratic_terms[i, i] = -coef
# Sub the weight of the linear pauli term from the QUBO matrix
qubo_matrix[i, i] -= weight
offset += weight
# Set the objective function
self.minimize(constant=offset, linear=linear_terms, quadratic=quadratic_terms)
offset -= offset
[docs] def get_feasibility_info(self, x: Union[List[float], np.ndarray]) \
-> Tuple[bool, List[Variable], List[Constraint]]:
"""Returns whether a solution is feasible or not along with the violations.
Args:
x: a solution value, such as returned in an optimizer result.
Returns:
feasible: Whether the solution provided is feasible or not.
List[Variable]: List of variables which are violated.
List[Constraint]: List of constraints which are violated.
Raises:
QiskitOptimizationError: If the input `x` is not same len as total vars
"""
# if input `x` is not the same len as the total vars, raise an error
if len(x) != self.get_num_vars():
raise QiskitOptimizationError(
'The size of solution `x`: {}, does not match the number of problem variables: {}'
.format(len(x), self.get_num_vars())
)
# check whether the input satisfy the bounds of the problem
violated_variables = [] # type: List[Variable]
for i, val in enumerate(x):
variable = self.get_variable(i)
if val < variable.lowerbound or variable.upperbound < val:
violated_variables.append(variable)
# check whether the input satisfy the constraints of the problem
violated_constraints = [] # type: List[Constraint]
for constraint in cast(List[Constraint], self._linear_constraints) + \
cast(List[Constraint], self._quadratic_constraints):
lhs = constraint.evaluate(x)
if constraint.sense == ConstraintSense.LE and lhs > constraint.rhs:
violated_constraints.append(constraint)
elif constraint.sense == ConstraintSense.GE and lhs < constraint.rhs:
violated_constraints.append(constraint)
elif constraint.sense == ConstraintSense.EQ and not isclose(lhs, constraint.rhs):
violated_constraints.append(constraint)
feasible = not violated_variables and not violated_constraints
return feasible, violated_variables, violated_constraints
[docs] def is_feasible(self, x: Union[List[float], np.ndarray]) -> bool:
"""Returns whether a solution is feasible or not.
Args:
x: a solution value, such as returned in an optimizer result.
Returns:
``True`` if the solution provided is feasible otherwise ``False``.
"""
feasible, _, _ = self.get_feasibility_info(x)
return feasible
class SubstituteVariables:
"""A class to substitute variables of an optimization problem with constants for other
variables"""
CONST = '__CONSTANT__'
def __init__(self):
self._src = None # type: Optional[QuadraticProgram]
self._dst = None # type: Optional[QuadraticProgram]
self._subs = {} # type: Dict[Union[int, str], Tuple[str, float]]
def substitute_variables(
self, src: QuadraticProgram,
constants: Optional[Dict[Union[str, int], float]] = None,
variables: Optional[Dict[Union[str, int], Tuple[Union[str, int], float]]] = None) \
-> QuadraticProgram:
"""Substitutes variables with constants or other variables.
Args:
src: a quadratic program to be substituted.
constants: replace variable by constant
e.g., {'x': 2} means 'x' is substituted with 2
variables: replace variables by weighted other variable
need to copy everything using name reference to make sure that indices are matched
correctly. The lower and upper bounds are updated accordingly.
e.g., {'x': ('y', 2)} means 'x' is substituted with 'y' * 2
Returns:
An optimization problem by substituting variables with constants or other variables.
If the substitution is valid, `QuadraticProgram.status` is still
`QuadraticProgram.Status.VALIAD`.
Otherwise, it gets `QuadraticProgram.Status.INFEASIBLE`.
Raises:
QiskitOptimizationError: if the substitution is invalid as follows.
- Same variable is substituted multiple times.
- Coefficient of variable substitution is zero.
"""
self._src = src
self._dst = QuadraticProgram(src.name)
self._subs_dict(constants, variables)
results = [
self._variables(),
self._objective(),
self._linear_constraints(),
self._quadratic_constraints(),
]
if any(not r for r in results):
self._dst._status = QuadraticProgram.Status.INFEASIBLE
return self._dst
@staticmethod
def _feasible(sense: ConstraintSense, rhs: float) -> bool:
"""Checks feasibility of the following condition
0 `sense` rhs
"""
# I use the following pylint option because `rhs` should come to right
# pylint: disable=misplaced-comparison-constant
if sense == Constraint.Sense.EQ:
if 0 == rhs:
return True
elif sense == Constraint.Sense.LE:
if 0 <= rhs:
return True
elif sense == Constraint.Sense.GE:
if 0 >= rhs:
return True
return False
@staticmethod
def _replace_dict_keys_with_names(op, dic):
key = []
val = []
for k in sorted(dic.keys()):
key.append(op.variables.get_names(k))
val.append(dic[k])
return key, val
def _subs_dict(self, constants, variables):
# guarantee that there is no overlap between variables to be replaced and combine input
subs = {} # type: Dict[Union[int, str], Tuple[str, float]]
if constants is not None:
for i, v in constants.items():
# substitute i <- v
i_2 = self._src.get_variable(i).name
if i_2 in subs:
raise QiskitOptimizationError(
'Cannot substitute the same variable twice: {} <- {}'.format(i, v))
subs[i_2] = (self.CONST, v)
if variables is not None:
for i, (j, v) in variables.items():
if v == 0:
raise QiskitOptimizationError(
'coefficient must be non-zero: {} {} {}'.format(i, j, v))
# substitute i <- j * v
i_2 = self._src.get_variable(i).name
j_2 = self._src.get_variable(j).name
if i_2 == j_2:
raise QiskitOptimizationError(
'Cannot substitute the same variable: {} <- {} {}'.format(i, j, v))
if i_2 in subs:
raise QiskitOptimizationError(
'Cannot substitute the same variable twice: {} <- {} {}'.format(i, j, v))
if j_2 in subs:
raise QiskitOptimizationError(
'Cannot substitute by variable that gets substituted itself: '
'{} <- {} {}'.format(i, j, v))
subs[i_2] = (j_2, v)
self._subs = subs
def _variables(self) -> bool:
# copy variables that are not replaced
feasible = True
for var in self._src.variables:
name = var.name
vartype = var.vartype
lowerbound = var.lowerbound
upperbound = var.upperbound
if name not in self._subs:
self._dst._add_variable(lowerbound, upperbound, vartype, name)
for i, (j, v) in self._subs.items():
lb_i = self._src.get_variable(i).lowerbound
ub_i = self._src.get_variable(i).upperbound
if j == self.CONST:
if not lb_i <= v <= ub_i:
logger.warning(
'Infeasible substitution for variable: %s', i)
feasible = False
else:
# substitute i <- j * v
# lb_i <= i <= ub_i --> lb_i / v <= j <= ub_i / v if v > 0
# ub_i / v <= j <= lb_i / v if v < 0
if v == 0:
raise QiskitOptimizationError(
'Coefficient of variable substitution should be nonzero: '
'{} {} {}'.format(i, j, v))
if abs(lb_i) < INFINITY:
new_lb_i = lb_i / v
else:
new_lb_i = lb_i if v > 0 else -lb_i
if abs(ub_i) < INFINITY:
new_ub_i = ub_i / v
else:
new_ub_i = ub_i if v > 0 else -ub_i
var_j = self._dst.get_variable(j)
lb_j = var_j.lowerbound
ub_j = var_j.upperbound
if v > 0:
var_j.lowerbound = max(lb_j, new_lb_i)
var_j.upperbound = min(ub_j, new_ub_i)
else:
var_j.lowerbound = max(lb_j, new_ub_i)
var_j.upperbound = min(ub_j, new_lb_i)
for var in self._dst.variables:
if var.lowerbound > var.upperbound:
logger.warning(
'Infeasible lower and upper bound: %s %f %f', var, var.lowerbound,
var.upperbound)
feasible = False
return feasible
def _linear_expression(self, lin_expr: LinearExpression) \
-> Tuple[List[float], LinearExpression]:
const = []
lin_dict = defaultdict(float) # type: Dict[Union[int, str], float]
for i, w_i in lin_expr.to_dict(use_name=True).items():
repl_i = self._subs[i] if i in self._subs else (i, 1)
prod = w_i * repl_i[1]
if repl_i[0] == self.CONST:
const.append(prod)
else:
k = repl_i[0]
lin_dict[k] += prod
new_lin = LinearExpression(quadratic_program=self._dst,
coefficients=lin_dict if lin_dict else {})
return const, new_lin
def _quadratic_expression(self, quad_expr: QuadraticExpression) \
-> Tuple[List[float], Optional[LinearExpression], Optional[QuadraticExpression]]:
const = []
lin_dict = defaultdict(float) # type: Dict[Union[int, str], float]
quad_dict = defaultdict(float) # type: Dict[Tuple[Union[int, str], Union[int, str]], float]
for (i, j), w_ij in quad_expr.to_dict(use_name=True).items():
repl_i = self._subs[i] if i in self._subs else (i, 1)
repl_j = self._subs[j] if j in self._subs else (j, 1)
idx = tuple(x for x, _ in [repl_i, repl_j] if x != self.CONST)
prod = w_ij * repl_i[1] * repl_j[1]
if len(idx) == 2:
quad_dict[idx] += prod # type: ignore
elif len(idx) == 1:
lin_dict[idx[0]] += prod
else:
const.append(prod)
new_lin = LinearExpression(quadratic_program=self._dst,
coefficients=lin_dict if lin_dict else {})
new_quad = QuadraticExpression(quadratic_program=self._dst,
coefficients=quad_dict if quad_dict else {})
return const, new_lin, new_quad
def _objective(self) -> bool:
obj = self._src.objective
const1, lin1 = self._linear_expression(obj.linear)
const2, lin2, quadratic = self._quadratic_expression(obj.quadratic)
constant = fsum([obj.constant] + const1 + const2)
linear = lin1.coefficients + lin2.coefficients
if obj.sense == obj.sense.MINIMIZE:
self._dst.minimize(constant=constant, linear=linear, quadratic=quadratic.coefficients)
else:
self._dst.maximize(constant=constant, linear=linear, quadratic=quadratic.coefficients)
return True
def _linear_constraints(self) -> bool:
feasible = True
for lin_cst in self._src.linear_constraints:
constant, linear = self._linear_expression(lin_cst.linear)
rhs = -fsum([-lin_cst.rhs] + constant)
if linear.coefficients.nnz > 0:
self._dst.linear_constraint(name=lin_cst.name, linear=linear.coefficients,
sense=lin_cst.sense, rhs=rhs)
else:
if not self._feasible(lin_cst.sense, rhs):
logger.warning('constraint %s is infeasible due to substitution', lin_cst.name)
feasible = False
return feasible
def _quadratic_constraints(self) -> bool:
feasible = True
for quad_cst in self._src.quadratic_constraints:
const1, lin1 = self._linear_expression(quad_cst.linear)
const2, lin2, quadratic = self._quadratic_expression(quad_cst.quadratic)
rhs = -fsum([-quad_cst.rhs] + const1 + const2)
linear = lin1.coefficients + lin2.coefficients
if quadratic.coefficients.nnz > 0:
self._dst.quadratic_constraint(name=quad_cst.name, linear=linear,
quadratic=quadratic.coefficients,
sense=quad_cst.sense, rhs=rhs)
elif linear.nnz > 0:
name = quad_cst.name
lin_names = set(lin.name for lin in self._dst.linear_constraints)
while name in lin_names:
name = '_' + name
self._dst.linear_constraint(name=name, linear=linear, sense=quad_cst.sense, rhs=rhs)
else:
if not self._feasible(quad_cst.sense, rhs):
logger.warning('constraint %s is infeasible due to substitution', quad_cst.name)
feasible = False
return feasible