# InteractiveLP Backend¶

AUTHORS:

• Nathann Cohen (2010-10) : generic_backend template

• Matthias Koeppe (2016-03) : this backend

class sage.numerical.backends.interactivelp_backend.InteractiveLPBackend

MIP Backend that works with InteractiveLPProblem.

This backend should be used only for linear programs over general fields, or for educational purposes. For fast computations with floating point arithmetic, use one of the numerical backends. For exact computations with rational numbers, use backend ‘PPL’.

There is no support for integer variables.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "InteractiveLP")


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 i-th entry of coeffs corresponds to the coefficient of the variable in the constraint represented by the i-th 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 = "InteractiveLP")
sage: p.ncols()
0
sage: p.nrows()
0
sage: p.nrows()
5


INPUT:

• coefficients – an iterable of pairs (i, v). In each pair, i is a variable index (integer) and v is a value (element of base_ring()).

• lower_bound – element of base_ring() or None. The lower bound.

• upper_bound – element of base_ring() or None. The upper bound.

• name – string or None. Optional name for this row.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "InteractiveLP")
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)
sage: p.add_linear_constraint( zip(range(5), range(5)), 1, 1, name='foo')
sage: p.row_name(1)
'foo'

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

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

In this backend, variables are always continuous (real). If integer variables are requested via the parameters binary and integer, an error will be raised.

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 binary (default: True).

• integer - True if the variable is binary (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).

• coefficients – (optional) an iterable of pairs (i, v). In each pair, i is a variable index (integer) and v is a value (element of base_ring()).

OUTPUT: The index of the newly created variable

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "InteractiveLP")
sage: p.ncols()
0
0
sage: p.ncols()
1
Traceback (most recent call last):
...
ValueError: ...
1
sage: p.col_name(1)
'x'
sage: p.objective_coefficient(1)
1

base_ring()

Return the base ring.

OUTPUT:

A ring. The coefficients that the chosen solver supports.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "InteractiveLP")
sage: p.base_ring()
Rational Field

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 = "InteractiveLP")
0
sage: p.col_bounds(0)
(None, None)
sage: p.variable_lower_bound(0, 0)
sage: p.col_bounds(0)
(0, None)

col_name(index)

Return the index-th column name

INPUT:

• index (integer) – the column 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 = "InteractiveLP")
0
sage: p.col_name(0)
'I_am_a_variable'

dictionary()

Return a dictionary representing the current basis.

EXAMPLES:

sage: p = MixedIntegerLinearProgram(maximization=True,                                                solver="InteractiveLP")
sage: x = p.new_variable(nonnegative=True)
sage: p.add_constraint(-x[0] + x[1] <= 2)
sage: p.add_constraint(8 * x[0] + 2 * x[1] <= 17)
sage: p.set_objective(11/2 * x[0] - 3 * x[1])
sage: b = p.get_backend()
sage: # Backend-specific commands to instruct solver to use simplex method here
sage: b.solve()
0
sage: d = b.dictionary(); d
LP problem dictionary ...
sage: set(d.basic_variables())
{x1, x3}
sage: d.basic_solution()
(17/8, 0)

get_objective_value()

Return the value of the objective function.

Note

Behavior is undefined unless solve has been called before.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "InteractiveLP")
1
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

Behavior is undefined unless solve has been called before.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "InteractiveLP")
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

interactive_lp_problem()

Return the InteractiveLPProblem object associated with this backend.

EXAMPLES:

sage: p = MixedIntegerLinearProgram(maximization=True,                                                solver="InteractiveLP")
sage: x = p.new_variable(nonnegative=True)
sage: p.add_constraint(-x[0] + x[1] <= 2)
sage: p.add_constraint(8 * x[0] + 2 * x[1] <= 17)
sage: p.set_objective(11/2 * x[0] - 3 * x[1])
sage: b = p.get_backend()
sage: b.interactive_lp_problem()
LP problem ...

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 = "InteractiveLP")
sage: p.is_maximization()
True
sage: p.set_sense(-1)
sage: p.is_maximization()
False

is_slack_variable_basic(index)

Test whether the slack variable of the given row is basic.

This assumes that the problem has been solved with the simplex method and a basis is available. Otherwise an exception will be raised.

INPUT:

• index (integer) – the variable’s id

EXAMPLES:

sage: p = MixedIntegerLinearProgram(maximization=True,                                                solver="InteractiveLP")
sage: x = p.new_variable(nonnegative=True)
sage: p.add_constraint(-x[0] + x[1] <= 2)
sage: p.add_constraint(8 * x[0] + 2 * x[1] <= 17)
sage: p.set_objective(11/2 * x[0] - 3 * x[1])
sage: b = p.get_backend()
sage: # Backend-specific commands to instruct solver to use simplex method here
sage: b.solve()
0
sage: b.is_slack_variable_basic(0)
True
sage: b.is_slack_variable_basic(1)
False

is_slack_variable_nonbasic_at_lower_bound(index)

Test whether the given variable is nonbasic at lower bound.

This assumes that the problem has been solved with the simplex method and a basis is available. Otherwise an exception will be raised.

INPUT:

• index (integer) – the variable’s id

EXAMPLES:

sage: p = MixedIntegerLinearProgram(maximization=True,                                                solver="InteractiveLP")
sage: x = p.new_variable(nonnegative=True)
sage: p.add_constraint(-x[0] + x[1] <= 2)
sage: p.add_constraint(8 * x[0] + 2 * x[1] <= 17)
sage: p.set_objective(11/2 * x[0] - 3 * x[1])
sage: b = p.get_backend()
sage: # Backend-specific commands to instruct solver to use simplex method here
sage: b.solve()
0
sage: b.is_slack_variable_nonbasic_at_lower_bound(0)
False
sage: b.is_slack_variable_nonbasic_at_lower_bound(1)
True

is_variable_basic(index)

Test whether the given variable is basic.

This assumes that the problem has been solved with the simplex method and a basis is available. Otherwise an exception will be raised.

INPUT:

• index (integer) – the variable’s id

EXAMPLES:

sage: p = MixedIntegerLinearProgram(maximization=True,                                                solver="InteractiveLP")
sage: x = p.new_variable(nonnegative=True)
sage: p.add_constraint(-x[0] + x[1] <= 2)
sage: p.add_constraint(8 * x[0] + 2 * x[1] <= 17)
sage: p.set_objective(11/2 * x[0] - 3 * x[1])
sage: b = p.get_backend()
sage: # Backend-specific commands to instruct solver to use simplex method here
sage: b.solve()
0
sage: b.is_variable_basic(0)
True
sage: b.is_variable_basic(1)
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 = "InteractiveLP")
sage: p.ncols()
0
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 = "InteractiveLP")
sage: p.ncols()
0
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 = "InteractiveLP")
sage: p.ncols()
0
0
sage: p.is_variable_integer(0)
False

is_variable_nonbasic_at_lower_bound(index)

Test whether the given variable is nonbasic at lower bound.

This assumes that the problem has been solved with the simplex method and a basis is available. Otherwise an exception will be raised.

INPUT:

• index (integer) – the variable’s id

EXAMPLES:

sage: p = MixedIntegerLinearProgram(maximization=True,                                                solver="InteractiveLP")
sage: x = p.new_variable(nonnegative=True)
sage: p.add_constraint(-x[0] + x[1] <= 2)
sage: p.add_constraint(8 * x[0] + 2 * x[1] <= 17)
sage: p.set_objective(11/2 * x[0] - 3 * x[1])
sage: b = p.get_backend()
sage: # Backend-specific commands to instruct solver to use simplex method here
sage: b.solve()
0
sage: b.is_variable_nonbasic_at_lower_bound(0)
False
sage: b.is_variable_nonbasic_at_lower_bound(1)
True

ncols()

Return the number of columns/variables.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "InteractiveLP")
sage: p.ncols()
0
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 = "InteractiveLP")
sage: p.nrows()
0
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 (double) – its coefficient

EXAMPLES:

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

objective_constant_term(d=None)

Set or get the constant term in the objective function

INPUT:

• d (double) – its coefficient. If $$None$$ (default), return the current value.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "InteractiveLP")
sage: p.objective_constant_term()
0
sage: p.objective_constant_term(42)
sage: p.objective_constant_term()
42

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 = "InteractiveLP")
sage: p.problem_name("There_once_was_a_french_fry")
sage: print(p.problem_name())
There_once_was_a_french_fry

remove_constraint(i)

Remove a constraint.

INPUT:

• i – index of the constraint to remove.

EXAMPLES:

sage: p = MixedIntegerLinearProgram(solver="InteractiveLP")
sage: v = p.new_variable(nonnegative=True)
sage: x,y = v[0], v[1]
sage: p.add_constraint(2*x + 3*y, max = 6)
sage: p.add_constraint(3*x + 2*y, max = 6)
sage: p.set_objective(x + y + 7)
sage: p.solve()
47/5
sage: p.remove_constraint(0)
sage: p.solve()
10
sage: p.get_values([x,y])
[0, 3]

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 = "InteractiveLP")
4
sage: p.row(0)
([1, 2, 3, 4], [1, 2, 3, 4])

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 = "InteractiveLP")
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 = "InteractiveLP")
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 i-th element is the coefficient of the i-th variable in the objective function.

• d (real) – the constant term in the linear function (set to $$0$$ by default)

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "InteractiveLP")
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]


Constants in the objective function are respected:

sage: p = MixedIntegerLinearProgram(solver='InteractiveLP')
sage: x,y = p[0], p[1]
sage: p.add_constraint(2*x + 3*y, max = 6)
sage: p.add_constraint(3*x + 2*y, max = 6)
sage: p.set_objective(x + y + 7)
sage: p.solve()
47/5

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 = "InteractiveLP")
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.

In this backend, variables are always continuous (real). If integer or binary variables are requested via the parameter vtype, an error will be raised.

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 = "InteractiveLP")
sage: p.ncols()
0
0
sage: p.set_variable_type(0,-1)
sage: p.is_variable_continuous(0)
True

set_verbosity(level)

Set the log (verbosity) level

INPUT:

• level (integer) – From 0 (no verbosity) to 3.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "InteractiveLP")
sage: p.set_verbosity(2)

solve()

Solve the problem.

Note

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

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "InteractiveLP")
sage: p.solve()
0
sage: p.objective_coefficient(0,1)
sage: p.solve()
Traceback (most recent call last):
...
MIPSolverException: ...

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 no lower bound. When set to None (default), the method returns the current value.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "InteractiveLP")
0
sage: p.col_bounds(0)
(None, None)
sage: p.variable_lower_bound(0) is None
True
sage: p.variable_lower_bound(0, 0)
sage: p.col_bounds(0)
(0, None)
sage: p.variable_lower_bound(0)
0
sage: p.variable_lower_bound(0, None)
sage: p.variable_lower_bound(0) is None
True

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 None (default), the method returns the current value.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "InteractiveLP")
0
sage: p.col_bounds(0)
(None, None)
sage: p.variable_upper_bound(0) is None
True
sage: p.variable_upper_bound(0, 0)
sage: p.col_bounds(0)
(None, 0)
sage: p.variable_upper_bound(0)
0
sage: p.variable_upper_bound(0, None)
sage: p.variable_upper_bound(0) is None
True