01 - Quadratic Programs

# This code is from:
# https://qiskit-community.github.io/qiskit-optimization/tutorials/01_quadratic_program.html

from qiskit_optimization import QuadraticProgram
from qiskit_optimization.translators import from_docplex_mp

# Make a Docplex model
from docplex.mp.model import Model

mdl = Model("docplex model")
x = mdl.binary_var("x")
y = mdl.integer_var(lb=-1, ub=5, name="y")
mdl.minimize(x + 2 * y)
mdl.add_constraint(x - y == 3)
mdl.add_constraint((x + y) * (x - y) <= 1)
print(mdl.export_as_lp_string())

\ This file has been generated by DOcplex
\ ENCODING=ISO-8859-1
\Problem name: docplex model

Minimize
 obj: x + 2 y
Subject To
 c1: x - y = 3
 qc1: [ x^2 - y^2 ] <= 1

Bounds
 0 <= x <= 1
 -1 <= y <= 5

Binaries
 x

Generals
 y
End

# load from a Docplex model
mod = from_docplex_mp(mdl)
print(type(mod))
print()
print(mod.prettyprint())

<class 'qiskit_optimization.problems.quadratic_program.QuadraticProgram'>

Problem name: docplex model

Minimize
  x + 2*y

Subject to
  Linear constraints (1)
    x - y == 3  'c0'

  Quadratic constraints (1)
    x^2 - y^2 <= 1  'q0'

  Integer variables (1)
    -1 <= y <= 5

  Binary variables (1)
    x

# make an empty problem
mod = QuadraticProgram("my problem")
print(mod.prettyprint())

Problem name: my problem

Minimize
  0

Subject to
  No constraints

  No variables

# Add variables
mod.binary_var(name="x")
mod.integer_var(name="y", lowerbound=-1, upperbound=5)
mod.continuous_var(name="z", lowerbound=-1, upperbound=5)
print(mod.prettyprint())

Problem name: my problem

Minimize
  0

Subject to
  No constraints

  Integer variables (1)
    -1 <= y <= 5

  Continuous variables (1)
    -1 <= z <= 5

  Binary variables (1)
    x

# Add objective function using dictionaries
mod.minimize(constant=3, linear={"x": 1}, quadratic={("x", "y"): 2, ("z", "z"): -1})
print(mod.prettyprint())

Problem name: my problem

Minimize
  2*x*y - z^2 + x + 3

Subject to
  No constraints

  Integer variables (1)
    -1 <= y <= 5

  Continuous variables (1)
    -1 <= z <= 5

  Binary variables (1)
    x

# Add objective function using lists/arrays
mod.minimize(constant=3, linear=[1, 0, 0], quadratic=[[0, 1, 0], [1, 0, 0], [0, 0, -1]])
print(mod.prettyprint())

Problem name: my problem

Minimize
  2*x*y - z^2 + x + 3

Subject to
  No constraints

  Integer variables (1)
    -1 <= y <= 5

  Continuous variables (1)
    -1 <= z <= 5

  Binary variables (1)
    x

print("constant:\t\t\t", mod.objective.constant)
print("linear dict:\t\t\t", mod.objective.linear.to_dict())
print("linear array:\t\t\t", mod.objective.linear.to_array())
print("linear array as sparse matrix:\n", mod.objective.linear.coefficients, "\n")
print("quadratic dict w/ index:\t", mod.objective.quadratic.to_dict())
print("quadratic dict w/ name:\t\t", mod.objective.quadratic.to_dict(use_name=True))
print(
    "symmetric quadratic dict w/ name:\t",
    mod.objective.quadratic.to_dict(use_name=True, symmetric=True),
)
print("quadratic matrix:\n", mod.objective.quadratic.to_array(), "\n")
print("symmetric quadratic matrix:\n", mod.objective.quadratic.to_array(symmetric=True), "\n")
print("quadratic matrix as sparse matrix:\n", mod.objective.quadratic.coefficients)

constant:                        3
linear dict:                     {np.int32(0): np.int64(1)}
linear array:                    [1 0 0]
linear array as sparse matrix:
 <Dictionary Of Keys sparse matrix of dtype 'int64'
        with 1 stored elements and shape (1, 3)>
  Coords        Values
  (0, 0)        1

quadratic dict w/ index:         {(0, 1): np.int64(2), (2, 2): np.int64(-1)}
quadratic dict w/ name:          {('x', 'y'): np.int64(2), ('z', 'z'): np.int64(-1)}
symmetric quadratic dict w/ name:        {('x', 'y'): np.int64(1), ('y', 'x'): np.int64(1), ('z', 'z'): np.int64(-1)}
quadratic matrix:
 [[ 0  2  0]
 [ 0  0  0]
 [ 0  0 -1]]

symmetric quadratic matrix:
 [[ 0  1  0]
 [ 1  0  0]
 [ 0  0 -1]]

quadratic matrix as sparse matrix:
 <Dictionary Of Keys sparse matrix of dtype 'int64'
        with 2 stored elements and shape (3, 3)>
  Coords        Values
  (0, 1)        2
  (2, 2)        -1

# Add linear constraints
mod.linear_constraint(linear={"x": 1, "y": 2}, sense="==", rhs=3, name="lin_eq")
mod.linear_constraint(linear={"x": 1, "y": 2}, sense="<=", rhs=3, name="lin_leq")
mod.linear_constraint(linear={"x": 1, "y": 2}, sense=">=", rhs=3, name="lin_geq")
print(mod.prettyprint())

Problem name: my problem

Minimize
  2*x*y - z^2 + x + 3

Subject to
  Linear constraints (3)
    x + 2*y == 3  'lin_eq'
    x + 2*y <= 3  'lin_leq'
    x + 2*y >= 3  'lin_geq'

  Integer variables (1)
    -1 <= y <= 5

  Continuous variables (1)
    -1 <= z <= 5

  Binary variables (1)
    x

# Add quadratic constraints
mod.quadratic_constraint(
    linear={"x": 1, "y": 1},
    quadratic={("x", "x"): 1, ("y", "z"): -1},
    sense="==",
    rhs=1,
    name="quad_eq",
)
mod.quadratic_constraint(
    linear={"x": 1, "y": 1},
    quadratic={("x", "x"): 1, ("y", "z"): -1},
    sense="<=",
    rhs=1,
    name="quad_leq",
)
mod.quadratic_constraint(
    linear={"x": 1, "y": 1},
    quadratic={("x", "x"): 1, ("y", "z"): -1},
    sense=">=",
    rhs=1,
    name="quad_geq",
)
print(mod.prettyprint())

Problem name: my problem

Minimize
  2*x*y - z^2 + x + 3

Subject to
  Linear constraints (3)
    x + 2*y == 3  'lin_eq'
    x + 2*y <= 3  'lin_leq'
    x + 2*y >= 3  'lin_geq'

  Quadratic constraints (3)
    x^2 - y*z + x + y == 1  'quad_eq'
    x^2 - y*z + x + y <= 1  'quad_leq'
    x^2 - y*z + x + y >= 1  'quad_geq'

  Integer variables (1)
    -1 <= y <= 5

  Continuous variables (1)
    -1 <= z <= 5

  Binary variables (1)
    x

lin_geq = mod.get_linear_constraint("lin_geq")
print("lin_geq:", lin_geq.linear.to_dict(use_name=True), lin_geq.sense, lin_geq.rhs)
quad_geq = mod.get_quadratic_constraint("quad_geq")
print(
    "quad_geq:",
    quad_geq.linear.to_dict(use_name=True),
    quad_geq.quadratic.to_dict(use_name=True),
    quad_geq.sense,
    lin_geq.rhs,
)

lin_geq: {'x': np.float64(1.0), 'y': np.float64(2.0)} ConstraintSense.GE 3
quad_geq: {'x': np.float64(1.0), 'y': np.float64(1.0)} {('x', 'x'): np.float64(1.0), ('y', 'z'): np.float64(-1.0)} ConstraintSense.GE 3

# Remove constraints
mod.remove_linear_constraint("lin_eq")
mod.remove_quadratic_constraint("quad_leq")
print(mod.prettyprint())

Problem name: my problem

Minimize
  2*x*y - z^2 + x + 3

Subject to
  Linear constraints (2)
    x + 2*y <= 3  'lin_leq'
    x + 2*y >= 3  'lin_geq'

  Quadratic constraints (2)
    x^2 - y*z + x + y == 1  'quad_eq'
    x^2 - y*z + x + y >= 1  'quad_geq'

  Integer variables (1)
    -1 <= y <= 5

  Continuous variables (1)
    -1 <= z <= 5

  Binary variables (1)
    x

sub = mod.substitute_variables(constants={"x": 0}, variables={"y": ("z", -1)})
print(sub.prettyprint())

Problem name: my problem

Minimize
  -z^2 + 3

Subject to
  Linear constraints (2)
    -2*z <= 3  'lin_leq'
    -2*z >= 3  'lin_geq'

  Quadratic constraints (2)
    z^2 - z == 1  'quad_eq'
    z^2 - z >= 1  'quad_geq'

  Continuous variables (1)
    -1 <= z <= 1

sub = mod.substitute_variables(constants={"x": -1})
print(sub.status)

Infeasible substitution for variable: x

QuadraticProgramStatus.INFEASIBLE

from qiskit_optimization import QiskitOptimizationError

try:
    sub = mod.substitute_variables(constants={"x": -1}, variables={"y": ("x", 1)})
except QiskitOptimizationError as e:
    print("Error: {}".format(e))

Error: 'Cannot substitute by variable that gets substituted itself: y <- x 1'

mod = QuadraticProgram()
mod.binary_var(name="e")
mod.binary_var(name="f")
mod.continuous_var(name="g")
mod.minimize(linear=[1, 2, 3])
print(mod.export_as_lp_string())

\ This file has been generated by DOcplex
\ ENCODING=ISO-8859-1
\Problem name: CPLEX

Minimize
 obj: _e + 2 f + 3 g
Subject To

Bounds
 0 <= _e <= 1
 0 <= f <= 1

Binaries
 _e f
End