PPL Backend

AUTHORS:

• Risan (2012-02): initial implementation

• Jeroen Demeyer (2014-08-04) allow rational coefficients for constraints and objective function (github issue #16755)

class sage.numerical.backends.ppl_backend.PPLBackend

Bases: GenericBackend

MIP Backend that uses the exact MIP solver from the Parma Polyhedra Library.

add_col(indices, coeffs)

Add a column.

INPUT:

• indices (list of integers) – this list contains the indices of the constraints in which the variable’s coefficient is nonzero

• coeffs (list of real values) – associates a coefficient to the variable in each of the constraints in which it appears. Namely, the ith entry of coeffs corresponds to the coefficient of the variable in the constraint represented by the ith entry in indices.

Note

indices and coeffs are expected to be of the same length.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.ncols()
0
sage: p.nrows()
0
sage: p.add_linear_constraints(5, 0, None)
sage: p.add_col(list(range(5)), list(range(5)))
sage: p.nrows()
5

add_linear_constraint(coefficients, lower_bound, upper_bound, name=None)

Add a linear constraint.

INPUT:

• coefficients – an iterable with (c,v) pairs where c is a variable index (integer) and v is a value (real value).

• lower_bound – a lower bound, either a real value or None

• upper_bound – an upper bound, either a real value or None

• name – an optional name for this row (default: None)

EXAMPLES:

sage: p = MixedIntegerLinearProgram(solver="PPL")
sage: x = p.new_variable(nonnegative=True)
sage: p.add_constraint(x[0]/2 + x[1]/3 <= 2/5)
sage: p.set_objective(x[1])
sage: p.solve()
6/5
sage: p.add_constraint(x[0] - x[1] >= 1/10)
sage: p.solve()
21/50
sage: p.set_max(x[0], 1/2)
sage: p.set_min(x[1], 3/8)
sage: p.solve()
2/5

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.add_variables(5)
4
sage: p.add_linear_constraint(zip(range(5), range(5)), 2.0, 2.0)
sage: p.row(0)
([1, 2, 3, 4], [1, 2, 3, 4])
sage: p.row_bounds(0)
(2.00000000000000, 2.00000000000000)
sage: p.add_linear_constraint( zip(range(5), range(5)), 1.0, 1.0, name='foo')
sage: p.row_name(-1)
'foo'

add_linear_constraints(number, lower_bound, upper_bound, names=None)

Add constraints.

INPUT:

• number (integer) – the number of constraints to add.

• lower_bound – a lower bound, either a real value or None

• upper_bound – an upper bound, either a real value or None

• names – an optional list of names (default: None)

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.add_variables(5)
4
sage: p.add_linear_constraints(5, None, 2)
sage: p.row(4)
([], [])
sage: p.row_bounds(4)
(None, 2)

add_variable(lower_bound=0, upper_bound=None, binary=False, continuous=False, integer=False, obj=0, name=None)

Add a variable.

This amounts to adding a new column to the matrix. By default, the variable is both positive and real.

It has not been implemented for selecting the variable type yet.

INPUT:

• lower_bound – the lower bound of the variable (default: 0)

• upper_bound – the upper bound of the variable (default: None)

• binary – True if the variable is binary (default: False).

• continuous – True if the variable is continuous (default: True).

• integer – True if the variable is integral (default: False).

• obj – (optional) coefficient of this variable in the objective function (default: 0)

• name – an optional name for the newly added variable (default: None).

OUTPUT: The index of the newly created variable

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.ncols()
0
sage: p.add_variable()
0
sage: p.ncols()
1
sage: p.add_variable(lower_bound=-2)
1
sage: p.add_variable(name='x',obj=2/3)
2
sage: p.col_name(2)
'x'
sage: p.objective_coefficient(2)
2/3
sage: p.add_variable(integer=True)
3

add_variables(n, lower_bound=0, upper_bound=None, binary=False, continuous=True, integer=False, obj=0, names=None)

Add n variables.

This amounts to adding new columns to the matrix. By default, the variables are both positive and real.

It has not been implemented for selecting the variable type yet.

INPUT:

• n – the number of new variables (must be > 0)

• lower_bound – the lower bound of the variable (default: 0)

• upper_bound – the upper bound of the variable (default: None)

• binary – True if the variable is binary (default: False).

• continuous – True if the variable is continuous (default: True).

• integer – True if the variable is integral (default: False).

• obj – (optional) coefficient of all variables in the objective function (default: 0)

• names – optional list of names (default: None)

OUTPUT: The index of the variable created last.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.ncols()
0
sage: p.add_variables(5)
4
sage: p.ncols()
5
sage: p.add_variables(2, lower_bound=-2.0, obj=42.0, names=['a','b'])
6

base_ring()

col_bounds(index)

Return the bounds of a specific variable.

INPUT:

• index (integer) – the variable’s id.

OUTPUT:

A pair (lower_bound, upper_bound). Each of them can be set to None if the variable is not bounded in the corresponding direction, and is a real value otherwise.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.add_variable()
0
sage: p.col_bounds(0)
(0, None)
sage: p.variable_upper_bound(0, 5)
sage: p.col_bounds(0)
(0, 5)

col_name(index)

Return the index th col name

INPUT:

• index (integer) – the col’s id

• name (char *) – its name. When set to NULL (default), the method returns the current name.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.add_variable(name="I am a variable")
0
sage: p.col_name(0)
'I am a variable'

get_objective_value()

Return the exact value of the objective function.

Note

Behaviour is undefined unless solve has been called before.

EXAMPLES:

sage: p = MixedIntegerLinearProgram(solver="PPL")
sage: x = p.new_variable(nonnegative=True)
sage: p.add_constraint(5/13*x[0] + x[1]/2 == 8/7)
sage: p.set_objective(5/13*x[0] + x[1]/2)
sage: p.solve()
8/7

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.add_variables(2)
1
sage: p.add_linear_constraint([(0,1), (1,2)], None, 3)
sage: p.set_objective([2, 5])
sage: p.solve()
0
sage: p.get_objective_value()
15/2
sage: p.get_variable_value(0)
0
sage: p.get_variable_value(1)
3/2

get_variable_value(variable)

Return the value of a variable given by the solver.

Note

Behaviour is undefined unless solve has been called before.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.add_variables(2)
1
sage: p.add_linear_constraint([(0,1), (1, 2)], None, 3)
sage: p.set_objective([2, 5])
sage: p.solve()
0
sage: p.get_objective_value()
15/2
sage: p.get_variable_value(0)
0
sage: p.get_variable_value(1)
3/2

init_mip()

Converting the matrix form of the MIP Problem to PPL MIP_Problem.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver="PPL")
sage: p.base_ring()
Rational Field
sage: type(p.zero())
<class 'sage.rings.rational.Rational'>
sage: p.init_mip()

is_maximization()

Test whether the problem is a maximization

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.is_maximization()
True
sage: p.set_sense(-1)
sage: p.is_maximization()
False

is_variable_binary(index)

Test whether the given variable is of binary type.

INPUT:

• index (integer) – the variable’s id

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.ncols()
0
sage: p.add_variable()
0
sage: p.is_variable_binary(0)
False

is_variable_continuous(index)

Test whether the given variable is of continuous/real type.

INPUT:

• index (integer) – the variable’s id

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.ncols()
0
sage: p.add_variable()
0
sage: p.is_variable_continuous(0)
True

is_variable_integer(index)

Test whether the given variable is of integer type.

INPUT:

• index (integer) – the variable’s id

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.ncols()
0
sage: p.add_variable()
0
sage: p.is_variable_integer(0)
False

ncols()

Return the number of columns/variables.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.ncols()
0
sage: p.add_variables(2)
1
sage: p.ncols()
2

nrows()

Return the number of rows/constraints.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.nrows()
0
sage: p.add_linear_constraints(2, 2.0, None)
sage: p.nrows()
2

objective_coefficient(variable, coeff=None)

Set or get the coefficient of a variable in the objective function

INPUT:

• variable (integer) – the variable’s id

• coeff (integer) – its coefficient

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.add_variable()
0
sage: p.objective_coefficient(0)
0
sage: p.objective_coefficient(0,2)
sage: p.objective_coefficient(0)
2

problem_name(name=None)

Return or define the problem’s name

INPUT:

• name (str) – the problem’s name. When set to None (default), the method returns the problem’s name.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.problem_name("There once was a french fry")
sage: print(p.problem_name())
There once was a french fry

row(i)

Return a row

INPUT:

• index (integer) – the constraint’s id.

OUTPUT:

A pair (indices, coeffs) where indices lists the entries whose coefficient is nonzero, and to which coeffs associates their coefficient on the model of the add_linear_constraint method.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.add_variables(5)
4
sage: p.add_linear_constraint(zip(range(5), range(5)), 2, 2)
sage: p.row(0)
([1, 2, 3, 4], [1, 2, 3, 4])
sage: p.row_bounds(0)
(2, 2)

row_bounds(index)

Return the bounds of a specific constraint.

INPUT:

• index (integer) – the constraint’s id.

OUTPUT:

A pair (lower_bound, upper_bound). Each of them can be set to None if the constraint is not bounded in the corresponding direction, and is a real value otherwise.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.add_variables(5)
4
sage: p.add_linear_constraint(zip(range(5), range(5)), 2, 2)
sage: p.row(0)
([1, 2, 3, 4], [1, 2, 3, 4])
sage: p.row_bounds(0)
(2, 2)

row_name(index)

Return the index th row name

INPUT:

• index (integer) – the row’s id

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.add_linear_constraints(1, 2, None, names=["Empty constraint 1"])
sage: p.row_name(0)
'Empty constraint 1'

set_objective(coeff, d=0)

Set the objective function.

INPUT:

• coeff – a list of real values, whose ith element is the coefficient of the ith variable in the objective function.

EXAMPLES:

sage: p = MixedIntegerLinearProgram(solver="PPL")
sage: x = p.new_variable(nonnegative=True)
sage: p.add_constraint(x[0]*5 + x[1]/11 <= 6)
sage: p.set_objective(x[0])
sage: p.solve()
6/5
sage: p.set_objective(x[0]/2 + 1)
sage: p.show()
Maximization:
  1/2 x_0 + 1

Constraints:
  constraint_0: 5 x_0 + 1/11 x_1 <= 6
Variables:
  x_0 is a continuous variable (min=0, max=+oo)
  x_1 is a continuous variable (min=0, max=+oo)
sage: p.solve()
8/5

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.add_variables(5)
4
sage: p.set_objective([1, 1, 2, 1, 3])
sage: [p.objective_coefficient(x) for x in range(5)]
[1, 1, 2, 1, 3]

set_sense(sense)

Set the direction (maximization/minimization).

INPUT:

• sense (integer) :

  • +1 => Maximization

  • -1 => Minimization

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.is_maximization()
True
sage: p.set_sense(-1)
sage: p.is_maximization()
False

set_variable_type(variable, vtype)

Set the type of a variable.

INPUT:

• variable (integer) – the variable’s id

• vtype (integer) :

  • 1 Integer

  • 0 Binary

  • -1

    Continuous

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.add_variables(5)
4
sage: p.set_variable_type(0,1)
sage: p.is_variable_integer(0)
True
sage: p.set_variable_type(3,0)
sage: p.is_variable_integer(3) or p.is_variable_binary(3)
True
sage: p.col_bounds(3) # tol 1e-6
(0, 1)
sage: p.set_variable_type(3, -1)
sage: p.is_variable_continuous(3)
True

set_verbosity(level)

Set the log (verbosity) level. Not Implemented.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.set_verbosity(0)

solve()

Solve the problem.

Note

This method raises MIPSolverException exceptions when the solution cannot be computed for any reason (none exists, or the solver was not able to find it, etc…)

EXAMPLES:

A linear optimization problem:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.add_linear_constraints(5, 0, None)
sage: p.add_col(list(range(5)), list(range(5)))
sage: p.solve()
0

An unbounded problem:

sage: p.objective_coefficient(0,1)
sage: p.solve()
Traceback (most recent call last):
...
MIPSolverException: ...

An integer optimization problem:

sage: p = MixedIntegerLinearProgram(solver='PPL')
sage: x = p.new_variable(integer=True, nonnegative=True)
sage: p.add_constraint(2*x[0] + 3*x[1], max = 6)
sage: p.add_constraint(3*x[0] + 2*x[1], max = 6)
sage: p.set_objective(x[0] + x[1] + 7)
sage: p.solve()
9

variable_lower_bound(index, value=False)

Return or define the lower bound on a variable

INPUT:

• index (integer) – the variable’s id

• value – real value, or None to mean that the variable has not lower bound. When set to False (default), the method returns the current value.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.add_variable()
0
sage: p.col_bounds(0)
(0, None)
sage: p.variable_lower_bound(0, 5)
sage: p.col_bounds(0)
(5, None)
sage: p.variable_lower_bound(0, None)
sage: p.col_bounds(0)
(None, None)

variable_upper_bound(index, value=False)

Return or define the upper bound on a variable

INPUT:

• index (integer) – the variable’s id

• value – real value, or None to mean that the variable has not upper bound. When set to False (default), the method returns the current value.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "PPL")
sage: p.add_variable()
0
sage: p.col_bounds(0)
(0, None)
sage: p.variable_upper_bound(0, 5)
sage: p.col_bounds(0)
(0, 5)
sage: p.variable_upper_bound(0, None)
sage: p.col_bounds(0)
(0, None)

zero()