GLPK Backend

AUTHORS:

  • Nathann Cohen (2010-10): initial implementation
  • John Perry (2012-01): glp_simplex preprocessing
  • John Perry and Raniere Gaia Silva (2012-03): solver parameters
  • Christian Kuper (2012-10): Additions for sensitivity analysis
class sage.numerical.backends.glpk_backend.GLPKBackend

Bases: sage.numerical.backends.generic_backend.GenericBackend

MIP Backend that uses the GLPK solver.

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 = "GLPK")
sage: p.ncols()
0
sage: p.nrows()
0
sage: p.add_linear_constraints(5, 0, None)
sage: p.add_col(range(5), 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: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "GLPK")
sage: p.add_variables(5)
4
sage: p.add_linear_constraint(zip(range(5), range(5)), 2.0, 2.0)
sage: p.row(0)
([4, 3, 2, 1], [4.0, 3.0, 2.0, 1.0])
sage: p.row_bounds(0)
(2.0, 2.0)
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 'number linear 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 = "GLPK")
sage: p.add_variables(5)
4
sage: p.add_linear_constraints(5, None, 2)
sage: p.row(4)
([], [])
sage: p.row_bounds(4)
(None, 2.0)
sage: p.add_linear_constraints(2, None, 2, names=['foo','bar'])
add_variable(lower_bound=0.0, upper_bound=None, binary=False, continuous=False, integer=False, obj=0.0, name=None)

Add a variable.

This amounts to adding a new column to the matrix. By default, the variable is both positive, real and the coefficient in the objective function is 0.0.

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: 0.0)
  • 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 = "GLPK")
sage: p.ncols()
0
sage: p.add_variable()
0
sage: p.ncols()
1
sage: p.add_variable(binary=True)
1
sage: p.add_variable(lower_bound=-2.0, integer=True)
2
sage: p.add_variable(continuous=True, integer=True)
Traceback (most recent call last):
...
ValueError: ...
sage: p.add_variable(name='x', obj=1.0)
3
sage: p.col_name(3)
'x'
sage: p.objective_coefficient(3)
1.0
add_variables(number, lower_bound=0.0, upper_bound=None, binary=False, continuous=False, integer=False, obj=0.0, names=None)

Add number new variables.

This amounts to adding new columns to the matrix. By default, the variables are both positive, real and their coefficient in the objective function is 0.0.

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 binary (default: True).
  • integer - True if the variable is binary (default: False).
  • obj - (optional) coefficient of all variables in the objective function (default: 0.0)
  • 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 = "GLPK")
sage: p.ncols()
0
sage: p.add_variables(5)
4
sage: p.ncols()
5
sage: p.add_variables(2, lower_bound=-2.0, integer=True, obj=42.0, names=['a','b'])
6
best_known_objective_bound()

Return the value of the currently best known bound.

This method returns the current best upper (resp. lower) bound on the optimal value of the objective function in a maximization (resp. minimization) problem. It is equal to the output of get_objective_value() if the MILP found an optimal solution, but it can differ if it was interrupted manually or after a time limit (cf solver_parameter()).

Note

Has no meaning unless solve has been called before.

EXAMPLES:

sage: g = graphs.CubeGraph(9)
sage: p = MixedIntegerLinearProgram(solver="GLPK")
sage: p.solver_parameter("mip_gap_tolerance",100)
sage: b = p.new_variable(binary=True)
sage: p.set_objective(p.sum(b[v] for v in g))
sage: for v in g:
....:     p.add_constraint(b[v]+p.sum(b[u] for u in g.neighbors(v)) <= 1)
sage: p.add_constraint(b[v] == 1) # Force an easy non-0 solution
sage: p.solve() # rel tol 100
1.0
sage: backend = p.get_backend()
sage: backend.best_known_objective_bound() # random
48.0
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 = "GLPK")
sage: p.add_variable()
0
sage: p.col_bounds(0)
(0.0, None)
sage: p.variable_upper_bound(0, 5)
sage: p.col_bounds(0)
(0.0, 5.0)
col_name(index)

Return the index th col name

INPUT:

  • index (integer) – the col’s id

EXAMPLES:

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

Computes a column of the current simplex tableau.

A (column) corresponds to some non-basic variable specified by the parameter k as follows:

  • if \(0 \leq k \leq m-1\), the non-basic variable is \(k\)-th auxiliary variable,
  • if \(m \leq k \leq m+n-1\), the non-basic variable is \((k-m)\)-th structural variable,

where \(m\) is the number of rows and \(n\) is the number of columns in the specified problem object.

Note

The basis factorization must exist. Otherwise a MIPSolverException will be raised.

INPUT:

  • k (integer) – the id of the non-basic variable.

OUTPUT:

A pair (indices, coeffs) where indices lists the entries whose coefficient is nonzero, and to which coeffs associates their coefficient in the computed column of the current simplex tableau.

Note

Elements in indices have the same sense as index \(k\). All these variables are basic by definition.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: lp = get_solver(solver = "GLPK")
sage: lp.add_variables(3)
2
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [8, 6, 1])), None, 48)
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [4, 2, 1.5])), None, 20)
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [2, 1.5, 0.5])), None, 8)
sage: lp.set_objective([60, 30, 20])
sage: import sage.numerical.backends.glpk_backend as backend
sage: lp.solver_parameter(backend.glp_simplex_or_intopt, backend.glp_simplex_only)
sage: lp.eval_tab_col(1)
Traceback (most recent call last):
...
MIPSolverException: ...
sage: lp.solve()
0
sage: lp.eval_tab_col(1)
([0, 5, 3], [-2.0, 2.0, -0.5])
sage: lp.eval_tab_col(2)
([0, 5, 3], [8.0, -4.0, 1.5])
sage: lp.eval_tab_col(4)
([0, 5, 3], [-2.0, 2.0, -1.25])
sage: lp.eval_tab_col(0)
Traceback (most recent call last):
...
MIPSolverException: ...
sage: lp.eval_tab_col(-1)
Traceback (most recent call last):
...
ValueError: ...
eval_tab_row(k)

Computes a row of the current simplex tableau.

A row corresponds to some basic variable specified by the parameter k as follows:

  • if \(0 \leq k \leq m-1\), the basic variable is \(k\)-th auxiliary variable,
  • if \(m \leq k \leq m+n-1\), the basic variable is \((k-m)\)-th structural variable,

where \(m\) is the number of rows and \(n\) is the number of columns in the specified problem object.

Note

The basis factorization must exist. Otherwise, a MIPSolverException will be raised.

INPUT:

  • k (integer) – the id of the basic variable.

OUTPUT:

A pair (indices, coeffs) where indices lists the entries whose coefficient is nonzero, and to which coeffs associates their coefficient in the computed row of the current simplex tableau.

Note

Elements in indices have the same sense as index k. All these variables are non-basic by definition.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: lp = get_solver(solver = "GLPK")
sage: lp.add_variables(3)
2
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [8, 6, 1])), None, 48)
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [4, 2, 1.5])), None, 20)
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [2, 1.5, 0.5])), None, 8)
sage: lp.set_objective([60, 30, 20])
sage: import sage.numerical.backends.glpk_backend as backend
sage: lp.solver_parameter(backend.glp_simplex_or_intopt, backend.glp_simplex_only)
sage: lp.eval_tab_row(0)
Traceback (most recent call last):
...
MIPSolverException: ...
sage: lp.solve()
0
sage: lp.eval_tab_row(0)
([1, 2, 4], [-2.0, 8.0, -2.0])
sage: lp.eval_tab_row(3)
([1, 2, 4], [-0.5, 1.5, -1.25])
sage: lp.eval_tab_row(5)
([1, 2, 4], [2.0, -4.0, 2.0])
sage: lp.eval_tab_row(1)
Traceback (most recent call last):
...
MIPSolverException: ...
sage: lp.eval_tab_row(-1)
Traceback (most recent call last):
...
ValueError: ...
get_col_dual(variable)

Returns the dual value (reduced cost) of a variable

The dual value is the reduced cost of a variable. The reduced cost is the amount by which the objective coefficient of a non basic variable has to change to become a basic variable.

INPUT:

  • variable – The number of the variable

Note

Behaviour is undefined unless solve has been called before. If the simplex algorithm has not been used for solving just a 0.0 will be returned.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "GLPK")
sage: p.add_variables(3)
2
sage: p.add_linear_constraint(list(zip([0, 1, 2], [8, 6, 1])), None, 48)
sage: p.add_linear_constraint(list(zip([0, 1, 2], [4, 2, 1.5])), None, 20)
sage: p.add_linear_constraint(list(zip([0, 1, 2], [2, 1.5, 0.5])), None, 8)
sage: p.set_objective([60, 30, 20])
sage: import sage.numerical.backends.glpk_backend as backend
sage: p.solver_parameter(backend.glp_simplex_or_intopt, backend.glp_simplex_only)
sage: p.solve()
0
sage: p.get_col_dual(1)
-5.0
get_col_stat(j)

Retrieve the status of a variable.

INPUT:

  • j – The index of the variable

OUTPUT:

  • Returns current status assigned to the structural variable associated with j-th column:

    • GLP_BS = 1 basic variable
    • GLP_NL = 2 non-basic variable on lower bound
    • GLP_NU = 3 non-basic variable on upper bound
    • GLP_NF = 4 non-basic free (unbounded) variable
    • GLP_NS = 5 non-basic fixed variable

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: lp = get_solver(solver = "GLPK")
sage: lp.add_variables(3)
2
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [8, 6, 1])), None, 48)
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [4, 2, 1.5])), None, 20)
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [2, 1.5, 0.5])), None, 8)
sage: lp.set_objective([60, 30, 20])
sage: import sage.numerical.backends.glpk_backend as backend
sage: lp.solver_parameter(backend.glp_simplex_or_intopt, backend.glp_simplex_only)
sage: lp.solve()
0
sage: lp.get_col_stat(0)
1
sage: lp.get_col_stat(1)
2
sage: lp.get_col_stat(100)
Traceback (most recent call last):
...
ValueError: The variable's index j must satisfy 0 <= j < number_of_variables
get_objective_value()

Returns the value of the objective function.

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 = "GLPK")
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()
7.5
sage: p.get_variable_value(0) # abs tol 1e-15
0.0
sage: p.get_variable_value(1)
1.5
get_relative_objective_gap()

Return the relative objective gap of the best known solution.

For a minimization problem, this value is computed by \((\texttt{bestinteger} - \texttt{bestobjective}) / (1e-10 + |\texttt{bestobjective}|)\), where bestinteger is the value returned by get_objective_value() and bestobjective is the value returned by best_known_objective_bound(). For a maximization problem, the value is computed by \((\texttt{bestobjective} - \texttt{bestinteger}) / (1e-10 + |\texttt{bestobjective}|)\).

Note

Has no meaning unless solve has been called before.

EXAMPLES:

sage: g = graphs.CubeGraph(9)
sage: p = MixedIntegerLinearProgram(solver="GLPK")
sage: p.solver_parameter("mip_gap_tolerance",100)
sage: b = p.new_variable(binary=True)
sage: p.set_objective(p.sum(b[v] for v in g))
sage: for v in g:
....:     p.add_constraint(b[v]+p.sum(b[u] for u in g.neighbors(v)) <= 1)
sage: p.add_constraint(b[v] == 1) # Force an easy non-0 solution
sage: p.solve() # rel tol 100
1.0
sage: backend = p.get_backend()
sage: backend.get_relative_objective_gap() # random
46.99999999999999
get_row_dual(variable)

Returns the dual value of a constraint.

The dual value of the ith row is also the value of the ith variable of the dual problem.

The dual value of a constraint is the shadow price of the constraint. The shadow price is the amount by which the objective value will change if the constraints bounds change by one unit under the precondition that the basis remains the same.

INPUT:

  • variable – The number of the constraint

Note

Behaviour is undefined unless solve has been called before. If the simplex algorithm has not been used for solving 0.0 will be returned.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: lp = get_solver(solver = "GLPK")
sage: lp.add_variables(3)
2
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [8, 6, 1])), None, 48)
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [4, 2, 1.5])), None, 20)
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [2, 1.5, 0.5])), None, 8)
sage: lp.set_objective([60, 30, 20])
sage: import sage.numerical.backends.glpk_backend as backend
sage: lp.solver_parameter(backend.glp_simplex_or_intopt, backend.glp_simplex_only)
sage: lp.solve()
0
sage: lp.get_row_dual(0)   # tolerance 0.00001
0.0
sage: lp.get_row_dual(1)   # tolerance 0.00001
10.0
get_row_prim(i)

Returns the value of the auxiliary variable associated with i-th row.

Note

Behaviour is undefined unless solve has been called before.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: lp = get_solver(solver = "GLPK")
sage: lp.add_variables(3)
2
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [8, 6, 1])), None, 48)
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [4, 2, 1.5])), None, 20)
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [2, 1.5, 0.5])), None, 8)
sage: lp.set_objective([60, 30, 20])
sage: import sage.numerical.backends.glpk_backend as backend
sage: lp.solver_parameter(backend.glp_simplex_or_intopt, backend.glp_simplex_only)
sage: lp.solve()
0
sage: lp.get_objective_value()
280.0
sage: lp.get_row_prim(0)
24.0
sage: lp.get_row_prim(1)
20.0
sage: lp.get_row_prim(2)
8.0
get_row_stat(i)

Retrieve the status of a constraint.

INPUT:

  • i – The index of the constraint

OUTPUT:

  • Returns current status assigned to the auxiliary variable associated with i-th row:

    • GLP_BS = 1 basic variable
    • GLP_NL = 2 non-basic variable on lower bound
    • GLP_NU = 3 non-basic variable on upper bound
    • GLP_NF = 4 non-basic free (unbounded) variable
    • GLP_NS = 5 non-basic fixed variable

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: lp = get_solver(solver = "GLPK")
sage: lp.add_variables(3)
2
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [8, 6, 1])), None, 48)
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [4, 2, 1.5])), None, 20)
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [2, 1.5, 0.5])), None, 8)
sage: lp.set_objective([60, 30, 20])
sage: import sage.numerical.backends.glpk_backend as backend
sage: lp.solver_parameter(backend.glp_simplex_or_intopt, backend.glp_simplex_only)
sage: lp.solve()
0
sage: lp.get_row_stat(0)
1
sage: lp.get_row_stat(1)
3
sage: lp.get_row_stat(-1)
Traceback (most recent call last):
...
ValueError: The constraint's index i must satisfy 0 <= i < number_of_constraints
get_variable_value(variable)

Returns 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 = "GLPK")
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()
7.5
sage: p.get_variable_value(0) # abs tol 1e-15
0.0
sage: p.get_variable_value(1)
1.5
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 = "GLPK")
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="GLPK")
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(5.5 * x[0] - 3 * x[1])
sage: b = p.get_backend()
sage: import sage.numerical.backends.glpk_backend as backend
sage: b.solver_parameter(backend.glp_simplex_or_intopt, backend.glp_simplex_only)
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 slack variable of the given row 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="GLPK")
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(5.5 * x[0] - 3 * x[1])
sage: b = p.get_backend()
sage: import sage.numerical.backends.glpk_backend as backend
sage: b.solver_parameter(backend.glp_simplex_or_intopt, backend.glp_simplex_only)
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="GLPK")
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(5.5 * x[0] - 3 * x[1])
sage: b = p.get_backend()
sage: import sage.numerical.backends.glpk_backend as backend
sage: b.solver_parameter(backend.glp_simplex_or_intopt, backend.glp_simplex_only)
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 = "GLPK")
sage: p.ncols()
0
sage: p.add_variable()
0
sage: p.set_variable_type(0,0)
sage: p.is_variable_binary(0)
True
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 = "GLPK")
sage: p.ncols()
0
sage: p.add_variable()
0
sage: p.is_variable_continuous(0)
True
sage: p.set_variable_type(0,1)
sage: p.is_variable_continuous(0)
False
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 = "GLPK")
sage: p.ncols()
0
sage: p.add_variable()
0
sage: p.set_variable_type(0,1)
sage: p.is_variable_integer(0)
True
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="GLPK")
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(5.5 * x[0] - 3 * x[1])
sage: b = p.get_backend()
sage: import sage.numerical.backends.glpk_backend as backend
sage: b.solver_parameter(backend.glp_simplex_or_intopt, backend.glp_simplex_only)
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 = "GLPK")
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 = "GLPK")
sage: p.nrows()
0
sage: p.add_linear_constraints(2, 2, 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 (double) – its coefficient or None for reading (default: None)

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "GLPK")
sage: p.add_variable()
0
sage: p.objective_coefficient(0)
0.0
sage: p.objective_coefficient(0,2)
sage: p.objective_coefficient(0)
2.0
print_ranges(filename=None)

Print results of a sensitivity analysis

If no filename is given as an input the results of the sensitivity analysis are displayed on the screen. If a filename is given they are written to a file.

INPUT:

  • filename – (optional) name of the file

OUTPUT:

Zero if the operations was successful otherwise nonzero.

Note

This method is only effective if an optimal solution has been found for the lp using the simplex algorithm. In all other cases an error message is printed.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "GLPK")
sage: p.add_variables(2)
1
sage: p.add_linear_constraint(list(zip([0, 1], [1, 2])), None, 3)
sage: p.set_objective([2, 5])
sage: import sage.numerical.backends.glpk_backend as backend
sage: p.solver_parameter(backend.glp_simplex_or_intopt, backend.glp_simplex_only)
sage: p.print_ranges()
glp_print_ranges: optimal basic solution required
1
sage: p.solve()
0
sage: p.print_ranges()
Write sensitivity analysis report to ...
GLPK ... - SENSITIVITY ANALYSIS REPORT                                                                         Page   1

Problem:
Objective:  7.5 (MAXimum)

   No. Row name     St      Activity         Slack   Lower bound       Activity      Obj coef  Obj value at Limiting
                                          Marginal   Upper bound          range         range   break point variable
------ ------------ -- ------------- ------------- -------------  ------------- ------------- ------------- ------------
     1              NU       3.00000        .               -Inf         .           -2.50000        .
                                           2.50000       3.00000           +Inf          +Inf          +Inf

GLPK ... - SENSITIVITY ANALYSIS REPORT                                                                         Page   2

Problem:
Objective:  7.5 (MAXimum)

   No. Column name  St      Activity      Obj coef   Lower bound       Activity      Obj coef  Obj value at Limiting
                                          Marginal   Upper bound          range         range   break point variable
------ ------------ -- ------------- ------------- -------------  ------------- ------------- ------------- ------------
     1              NL        .            2.00000        .                -Inf          -Inf          +Inf
                                           -.50000          +Inf        3.00000       2.50000       6.00000

     2              BS       1.50000       5.00000        .                -Inf       4.00000       6.00000
                                            .               +Inf        1.50000          +Inf          +Inf

End of report

0
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 = "GLPK")
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 from self.

INPUT:

  • i – index of the constraint to remove

EXAMPLES:

sage: p = MixedIntegerLinearProgram(solver='GLPK')
sage: x, y = p['x'], p['y']
sage: p.add_constraint(2*x + 3*y <= 6)
sage: p.add_constraint(3*x + 2*y <= 6)
sage: p.add_constraint(x >= 0)
sage: p.set_objective(x + y + 7)
sage: p.set_integer(x); p.set_integer(y)
sage: p.solve()
9.0
sage: p.remove_constraint(0)
sage: p.solve()
10.0

Removing fancy constraints does not make Sage crash:

sage: MixedIntegerLinearProgram(solver = "GLPK").remove_constraint(-2)
Traceback (most recent call last):
...
ValueError: The constraint's index i must satisfy 0 <= i < number_of_constraints
remove_constraints(constraints)

Remove several constraints.

INPUT:

  • constraints – an iterable containing the indices of the rows to remove.

EXAMPLES:

sage: p = MixedIntegerLinearProgram(solver='GLPK')
sage: x, y = p['x'], p['y']
sage: p.add_constraint(2*x + 3*y <= 6)
sage: p.add_constraint(3*x + 2*y <= 6)
sage: p.add_constraint(x >= 0)
sage: p.set_objective(x + y + 7)
sage: p.set_integer(x); p.set_integer(y)
sage: p.solve()
9.0
sage: p.remove_constraints([0])
sage: p.solve()
10.0
sage: p.get_values([x,y])
[0.0, 3.0]
row(index)

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 = "GLPK")
sage: p.add_variables(5)
4
sage: p.add_linear_constraint(list(zip(range(5), range(5))), 2, 2)
sage: p.row(0)
([4, 3, 2, 1], [4.0, 3.0, 2.0, 1.0])
sage: p.row_bounds(0)
(2.0, 2.0)
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 = "GLPK")
sage: p.add_variables(5)
4
sage: p.add_linear_constraint(list(zip(range(5), range(5))), 2, 2)
sage: p.row(0)
([4, 3, 2, 1], [4.0, 3.0, 2.0, 1.0])
sage: p.row_bounds(0)
(2.0, 2.0)
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 = "GLPK")
sage: p.add_linear_constraints(1, 2, None, names=['Empty constraint 1'])
sage: p.row_name(0)
'Empty constraint 1'
set_col_stat(j, stat)

Set the status of a variable.

INPUT:

  • j – The index of the constraint
  • stat – The status to set to

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: lp = get_solver(solver = "GLPK")
sage: lp.add_variables(3)
2
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [8, 6, 1])), None, 48)
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [4, 2, 1.5])), None, 20)
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [2, 1.5, 0.5])), None, 8)
sage: lp.set_objective([60, 30, 20])
sage: import sage.numerical.backends.glpk_backend as backend
sage: lp.solver_parameter(backend.glp_simplex_or_intopt, backend.glp_simplex_only)
sage: lp.solve()
0
sage: lp.get_col_stat(0)
1
sage: lp.set_col_stat(0, 2)
sage: lp.get_col_stat(0)
2
set_objective(coeff, d=0.0)

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.
  • d (double) – 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 = "GLPK")
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)]
[1.0, 1.0, 2.0, 1.0, 3.0]
set_row_stat(i, stat)

Set the status of a constraint.

INPUT:

  • i – The index of the constraint
  • stat – The status to set to

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: lp = get_solver(solver = "GLPK")
sage: lp.add_variables(3)
2
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [8, 6, 1])), None, 48)
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [4, 2, 1.5])), None, 20)
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [2, 1.5, 0.5])), None, 8)
sage: lp.set_objective([60, 30, 20])
sage: import sage.numerical.backends.glpk_backend as backend
sage: lp.solver_parameter(backend.glp_simplex_or_intopt, backend.glp_simplex_only)
sage: lp.solve()
0
sage: lp.get_row_stat(0)
1
sage: lp.set_row_stat(0, 3)
sage: lp.get_row_stat(0)
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 = "GLPK")
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 Real

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "GLPK")
sage: p.ncols()
0
sage: p.add_variable()
0
sage: p.set_variable_type(0,1)
sage: p.is_variable_integer(0)
True
set_verbosity(level)

Set the verbosity level

INPUT:

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

EXAMPLES:

sage: p.<x> = MixedIntegerLinearProgram(solver="GLPK")
sage: p.add_constraint(10 * x[0] <= 1)
sage: p.add_constraint(5 * x[1] <= 1)
sage: p.set_objective(x[0] + x[1])
sage: p.solve()
0.30000000000000004
sage: p.get_backend().set_verbosity(3)
sage: p.solve()
GLPK Integer Optimizer...
2 rows, 2 columns, 2 non-zeros
0 integer variables, none of which are binary
Preprocessing...
Objective value =   3.000000000e-01
INTEGER OPTIMAL SOLUTION FOUND BY MIP PREPROCESSOR
0.30000000000000004
sage: p.<x> = MixedIntegerLinearProgram(solver="GLPK/exact")
sage: p.add_constraint(10 * x[0] <= 1)
sage: p.add_constraint(5 * x[1] <= 1)
sage: p.set_objective(x[0] + x[1])
sage: p.solve()
0.3
sage: p.get_backend().set_verbosity(2)
sage: p.solve()
*     2:   objval =                    0.3   (0)
*     2:   objval =                    0.3   (0)
0.3
sage: p.get_backend().set_verbosity(3)
sage: p.solve()
glp_exact: 2 rows, 2 columns, 2 non-zeros
GNU MP bignum library is being used
*     2:   objval =                    0.3   (0)
*     2:   objval =                    0.3   (0)
OPTIMAL SOLUTION FOUND
0.3
solve()

Solve the problem.

Sage uses GLPK’s implementation of the branch-and-cut algorithm (glp_intopt) to solve the mixed-integer linear program. This algorithm can be requested explicitly by setting the solver parameter “simplex_or_intopt” to “intopt_only”. (If all variables are continuous, the algorithm reduces to solving the linear program by the simplex method.)

EXAMPLES:

sage: lp = MixedIntegerLinearProgram(solver = 'GLPK', maximization = False)
sage: x, y = lp[0], lp[1]
sage: lp.add_constraint(-2*x + y <= 1)
sage: lp.add_constraint(x - y <= 1)
sage: lp.add_constraint(x + y >= 2)
sage: lp.set_objective(x + y)
sage: lp.set_integer(x)
sage: lp.set_integer(y)
sage: lp.solve()
2.0
sage: lp.get_values([x, y])
[1.0, 1.0]

Note

This method raises MIPSolverException exceptions when the solution can not 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 = "GLPK")
sage: p.add_linear_constraints(5, 0, None)
sage: p.add_col(range(5), range(5))
sage: p.solve()
0
sage: p.objective_coefficient(0,1)
sage: p.solve()
Traceback (most recent call last):
...
MIPSolverException: ...

Warning

Sage uses GLPK’s glp_intopt to find solutions. This routine sometimes FAILS CATASTROPHICALLY when given a system it cannot solve. (trac ticket #12309.) Here, “catastrophic” can mean either “infinite loop” or segmentation fault. Upstream considers this behavior “essentially innate” to their design, and suggests preprocessing it with glp_simplex first. Thus, if you suspect that your system is infeasible, set the preprocessing option first.

EXAMPLES:

sage: lp = MixedIntegerLinearProgram(solver = "GLPK")
sage: v = lp.new_variable(nonnegative=True)
sage: lp.add_constraint(v[1] +v[2] -2.0 *v[3], max=-1.0)
sage: lp.add_constraint(v[0] -4.0/3 *v[1] +1.0/3 *v[2], max=-1.0/3)
sage: lp.add_constraint(v[0] +0.5 *v[1] -0.5 *v[2] +0.25 *v[3], max=-0.25)
sage: lp.solve()
0.0
sage: lp.add_constraint(v[0] +4.0 *v[1] -v[2] +v[3], max=-1.0)
sage: lp.solve()
Traceback (most recent call last):
...
GLPKError: Assertion failed: ...
sage: lp.solver_parameter("simplex_or_intopt", "simplex_then_intopt")
sage: lp.solve()
Traceback (most recent call last):
...
MIPSolverException: GLPK: Problem has no feasible solution

If we switch to “simplex_only”, the integrality constraints are ignored, and we get an optimal solution to the continuous relaxation.

EXAMPLES:

sage: lp = MixedIntegerLinearProgram(solver = 'GLPK', maximization = False)
sage: x, y = lp[0], lp[1]
sage: lp.add_constraint(-2*x + y <= 1)
sage: lp.add_constraint(x - y <= 1)
sage: lp.add_constraint(x + y >= 2)
sage: lp.set_objective(x + y)
sage: lp.set_integer(x)
sage: lp.set_integer(y)
sage: lp.solver_parameter("simplex_or_intopt", "simplex_only") # use simplex only
sage: lp.solve()
2.0
sage: lp.get_values([x, y])
[1.5, 0.5]

If one solves a linear program and wishes to access dual information (\(get_col_dual\) etc.) or tableau data (\(get_row_stat\) etc.), one needs to switch to “simplex_only” before solving.

GLPK also has an exact rational simplex solver. The only access to data is via double-precision floats, however. It reconstructs rationals from doubles and also provides results as doubles.

EXAMPLES:

sage: lp.solver_parameter("simplex_or_intopt", "exact_simplex_only") # use exact simplex only
sage: lp.solve()
2.0
sage: lp.get_values([x, y])
[1.5, 0.5]

If you need the rational solution, you need to retrieve the basis information via get_col_stat and get_row_stat and calculate the corresponding basic solution. Below we only test that the basis information is indeed available. Calculating the corresponding basic solution is left as an exercise.

EXAMPLES:

sage: lp.get_backend().get_row_stat(0)
1
sage: lp.get_backend().get_col_stat(0)
1

Below we test that integers that can be exactly represented by IEEE 754 double-precision floating point numbers survive the rational reconstruction done by glp_exact and the subsequent conversion to double-precision floating point numbers.

EXAMPLES:

sage: lp = MixedIntegerLinearProgram(solver = 'GLPK', maximization = True)
sage: test = 2^53 - 43
sage: lp.solver_parameter("simplex_or_intopt", "exact_simplex_only") # use exact simplex only
sage: x = lp[0]
sage: lp.add_constraint(x <= test)
sage: lp.set_objective(x)
sage: lp.solve() == test # yes, we want an exact comparison here
True
sage: lp.get_values(x) == test # yes, we want an exact comparison here
True

Below we test that GLPK backend can detect unboundedness in “simplex_only” mode (trac ticket #18838).

EXAMPLES:

sage: lp = MixedIntegerLinearProgram(maximization=True, solver = "GLPK")
sage: lp.set_objective(lp[0])
sage: lp.solver_parameter("simplex_or_intopt", "simplex_only")
sage: lp.solve()
Traceback (most recent call last):
...
MIPSolverException: GLPK: Problem has unbounded solution
sage: lp.set_objective(lp[1])
sage: lp.solver_parameter("primal_v_dual", "GLP_DUAL")
sage: lp.solve()
Traceback (most recent call last):
...
MIPSolverException: GLPK: Problem has unbounded solution
sage: lp.solver_parameter("simplex_or_intopt", "simplex_then_intopt")
sage: lp.solve()
Traceback (most recent call last):
...
MIPSolverException: GLPK: The LP (relaxation) problem has no dual feasible solution
sage: lp.solver_parameter("simplex_or_intopt", "intopt_only")
sage: lp.solve()
Traceback (most recent call last):
...
MIPSolverException: GLPK: The LP (relaxation) problem has no dual feasible solution
sage: lp.set_max(lp[1],5)
sage: lp.solve()
5.0

Solving a LP within the acceptable gap. No exception is raised, even if the result is not optimal. To do this, we try to compute the maximum number of disjoint balls (of diameter 1) in a hypercube:

sage: g = graphs.CubeGraph(9)
sage: p = MixedIntegerLinearProgram(solver="GLPK")
sage: p.solver_parameter("mip_gap_tolerance",100)
sage: b = p.new_variable(binary=True)
sage: p.set_objective(p.sum(b[v] for v in g))
sage: for v in g:
....:     p.add_constraint(b[v]+p.sum(b[u] for u in g.neighbors(v)) <= 1)
sage: p.add_constraint(b[v] == 1) # Force an easy non-0 solution
sage: p.solve() # rel tol 100
1

Same, now with a time limit:

sage: p.solver_parameter("mip_gap_tolerance",1)
sage: p.solver_parameter("timelimit",3.0)
sage: p.solve() # rel tol 100
1
solver_parameter(name, value=None)

Return or define a solver parameter

INPUT:

  • name (string) – the parameter
  • value – the parameter’s value if it is to be defined, or None (default) to obtain its current value.

You can supply the name of a parameter and its value using either a string or a glp_ constant (which are defined as Cython variables of this module).

In most cases, you can use the same name for a parameter as that given in the GLPK documentation, which is available by downloading GLPK from <http://www.gnu.org/software/glpk/>. The exceptions relate to parameters common to both methods; these require you to append _simplex or _intopt to the name to resolve ambiguity, since the interface allows access to both.

We have also provided more meaningful names, to assist readability.

Parameter names are specified in lower case. To use a constant instead of a string, prepend glp_ to the name. For example, both glp_gmi_cuts or "gmi_cuts" control whether to solve using Gomory cuts.

Parameter values are specified as strings in upper case, or as constants in lower case. For example, both glp_on and "GLP_ON" specify the same thing.

Naturally, you can use True and False in cases where glp_on and glp_off would be used.

A list of parameter names, with their possible values:

General-purpose parameters:

timelimit specify the time limit IN SECONDS. This affects both simplex and intopt.
timelimit_simplex and timelimit_intopt specify the time limit IN MILLISECONDS. (This is glpk’s default.)
simplex_or_intopt specify which of simplex, exact and intopt routines in GLPK to use. This is controlled by setting simplex_or_intopt to glp_simplex_only, glp_exact_simplex_only, glp_intopt_only and glp_simplex_then_intopt, respectively. The latter is useful to deal with a problem in GLPK where problems with no solution hang when using integer optimization; if you specify glp_simplex_then_intopt, sage will try simplex first, then perform integer optimization only if a solution of the LP relaxation exists.
verbosity_intopt and verbosity_simplex one of GLP_MSG_OFF, GLP_MSG_ERR, GLP_MSG_ON, or GLP_MSG_ALL. The default is GLP_MSG_OFF.
output_frequency_intopt and output_frequency_simplex the output frequency, in milliseconds. Default is 5000.
output_delay_intopt and output_delay_simplex the output delay, in milliseconds, regarding the use of the simplex method on the LP relaxation. Default is 10000.

intopt-specific parameters:

branching
  • GLP_BR_FFV first fractional variable
  • GLP_BR_LFV last fractional variable
  • GLP_BR_MFV most fractional variable
  • GLP_BR_DTH Driebeck-Tomlin heuristic (default)
  • GLP_BR_PCH hybrid pseudocost heuristic
backtracking
  • GLP_BT_DFS depth first search
  • GLP_BT_BFS breadth first search
  • GLP_BT_BLB best local bound (default)
  • GLP_BT_BPH best projection heuristic
preprocessing
  • GLP_PP_NONE
  • GLP_PP_ROOT preprocessing only at root level
  • GLP_PP_ALL (default)
feasibility_pump GLP_ON or GLP_OFF (default)
gomory_cuts GLP_ON or GLP_OFF (default)
mixed_int_rounding_cuts GLP_ON or GLP_OFF (default)
mixed_cover_cuts GLP_ON or GLP_OFF (default)
clique_cuts GLP_ON or GLP_OFF (default)
absolute_tolerance (double) used to check if optimal solution to LP relaxation is integer feasible. GLPK manual advises, “do not change… without detailed understanding of its purpose.”
relative_tolerance (double) used to check if objective value in LP relaxation is not better than best known integer solution. GLPK manual advises, “do not change… without detailed understanding of its purpose.”
mip_gap_tolerance (double) relative mip gap tolerance. Default is 0.0.
presolve_intopt GLP_ON (default) or GLP_OFF.
binarize GLP_ON or GLP_OFF (default)

simplex-specific parameters:

primal_v_dual
  • GLP_PRIMAL (default)
  • GLP_DUAL
  • GLP_DUALP
pricing
  • GLP_PT_STD standard (textbook)
  • GLP_PT_PSE projected steepest edge (default)
ratio_test
  • GLP_RT_STD standard (textbook)
  • GLP_RT_HAR Harris’ two-pass ratio test (default)
tolerance_primal (double) tolerance used to check if basic solution is primal feasible. GLPK manual advises, “do not change… without detailed understanding of its purpose.”
tolerance_dual (double) tolerance used to check if basic solution is dual feasible. GLPK manual advises, “do not change… without detailed understanding of its purpose.”
tolerance_pivot (double) tolerance used to choose pivot. GLPK manual advises, “do not change… without detailed understanding of its purpose.”
obj_lower_limit (double) lower limit of the objective function. The default is -DBL_MAX.
obj_upper_limit (double) upper limit of the objective function. The default is DBL_MAX.
iteration_limit (int) iteration limit of the simplex algorithm. The default is INT_MAX.
presolve_simplex GLP_ON or GLP_OFF (default).

Note

The coverage for GLPK’s control parameters for simplex and integer optimization is nearly complete. The only thing lacking is a wrapper for callback routines.

To date, no attempt has been made to expose the interior point methods.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "GLPK")
sage: p.solver_parameter("timelimit", 60)
sage: p.solver_parameter("timelimit")
60.0
  • Don’t forget the difference between timelimit and timelimit_intopt
sage: p.solver_parameter("timelimit_intopt")
60000

If you don’t care for an integer answer, you can ask for an LP relaxation instead. The default solver performs integer optimization, but you can switch to the standard simplex algorithm through the glp_simplex_or_intopt parameter.

EXAMPLES:

sage: lp = MixedIntegerLinearProgram(solver = 'GLPK', maximization = False)
sage: x, y = lp[0], lp[1]
sage: lp.add_constraint(-2*x + y <= 1)
sage: lp.add_constraint(x - y <= 1)
sage: lp.add_constraint(x + y >= 2)
sage: lp.set_integer(x); lp.set_integer(y)
sage: lp.set_objective(x + y)
sage: lp.solve()
2.0
sage: lp.get_values([x,y])
[1.0, 1.0]
sage: import sage.numerical.backends.glpk_backend as backend
sage: lp.solver_parameter(backend.glp_simplex_or_intopt, backend.glp_simplex_only)
sage: lp.solve()
2.0
sage: lp.get_values([x,y])
[1.5, 0.5]

You can get GLPK to spout all sorts of information at you. The default is to turn this off, but sometimes (debugging) it’s very useful:

sage: lp.solver_parameter(backend.glp_simplex_or_intopt, backend.glp_simplex_then_intopt)
sage: lp.solver_parameter(backend.glp_mir_cuts, backend.glp_on)
sage: lp.solver_parameter(backend.glp_msg_lev_intopt, backend.glp_msg_all)
sage: lp.solver_parameter(backend.glp_mir_cuts)
1

If you actually try to solve lp, you will get a lot of detailed information.

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 = "GLPK")
sage: p.add_variable()
0
sage: p.col_bounds(0)
(0.0, None)
sage: p.variable_lower_bound(0, 5)
sage: p.col_bounds(0)
(5.0, 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 = "GLPK")
sage: p.add_variable()
0
sage: p.col_bounds(0)
(0.0, None)
sage: p.variable_upper_bound(0, 5)
sage: p.col_bounds(0)
(0.0, 5.0)
warm_up()

Warm up the basis using current statuses assigned to rows and cols.

OUTPUT:

  • Returns the warming up status

    • 0 The operation has been successfully performed.
    • GLP_EBADB The basis matrix is invalid.
    • GLP_ESING The basis matrix is singular within the working precision.
    • GLP_ECOND The basis matrix is ill-conditioned.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: lp = get_solver(solver = "GLPK")
sage: lp.add_variables(3)
2
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [8, 6, 1])), None, 48)
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [4, 2, 1.5])), None, 20)
sage: lp.add_linear_constraint(list(zip([0, 1, 2], [2, 1.5, 0.5])), None, 8)
sage: lp.set_objective([60, 30, 20])
sage: import sage.numerical.backends.glpk_backend as backend
sage: lp.solver_parameter(backend.glp_simplex_or_intopt, backend.glp_simplex_only)
sage: lp.solve()
0
sage: lp.get_objective_value()
280.0
sage: lp.set_row_stat(0,3)
sage: lp.set_col_stat(1,1)
sage: lp.warm_up()
0
write_lp(filename)

Write the problem to a .lp file

INPUT:

  • filename (string)

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "GLPK")
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.write_lp(os.path.join(SAGE_TMP, "lp_problem.lp"))
Writing problem data to ...
9 lines were written
write_mps(filename, modern)

Write the problem to a .mps file

INPUT:

  • filename (string)

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "GLPK")
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.write_mps(os.path.join(SAGE_TMP, "lp_problem.mps"), 2)
Writing problem data to ...
17 records were written