Generic Backend for LP solvers#

This class only lists the methods that should be defined by any interface with a LP Solver. All these methods immediately raise NotImplementedError exceptions when called, and are obviously meant to be replaced by the solver-specific method. This file can also be used as a template to create a new interface : one would only need to replace the occurrences of "Nonexistent_LP_solver" by the solver’s name, and replace GenericBackend by SolverName(GenericBackend) so that the new solver extends this class.

AUTHORS:

Nathann Cohen (2010-10) : initial implementation
Risan (2012-02) : extension for PPL backend
Ingolfur Edvardsson (2014-06): extension for CVXOPT backend

class sage.numerical.backends.generic_backend.GenericBackend#

Bases: SageObject

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 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 = "Nonexistent_LP_solver")  # optional - Nonexistent_LP_solver
sage: p.ncols()                                       # optional - Nonexistent_LP_solver
0
sage: p.nrows()                                       # optional - Nonexistent_LP_solver
0
sage: p.add_linear_constraints(5, 0, None)            # optional - Nonexistent_LP_solver
sage: p.add_col(list(range(5)), list(range(5)))                   # optional - Nonexistent_LP_solver
sage: p.nrows()                                       # optional - Nonexistent_LP_solver
5

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

Add a linear constraint.

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 = "Nonexistent_LP_solver")             # optional - Nonexistent_LP_solver
sage: p.add_variables(5)                                           # optional - Nonexistent_LP_solver
4
sage: p.add_linear_constraint( zip(range(5), range(5)), 2.0, 2.0)  # optional - Nonexistent_LP_solver
sage: p.row(0)                                                     # optional - Nonexistent_LP_solver
([0, 1, 2, 3, 4], [0.0, 1.0, 2.0, 3.0, 4.0])
sage: p.row_bounds(0)                                              # optional - Nonexistent_LP_solver
(2.0, 2.0)
sage: p.add_linear_constraint( zip(range(5), range(5)), 1.0, 1.0, name='foo') # optional - Nonexistent_LP_solver
sage: p.row_name(1)                                                           # optional - Nonexistent_LP_solver
'foo'

add_linear_constraint_vector(degree, coefficients, lower_bound, upper_bound, name=None)#

Add a vector-valued linear constraint.

Note

This is the generic implementation, which will split the vector-valued constraint into components and add these individually. Backends are encouraged to replace it with their own optimized implementation.

INPUT:

degree – integer. The vector degree, that is, the number of new scalar constraints.
coefficients – an iterable of pairs (i, v). In each pair, i is a variable index (integer) and v is a vector (real and of length degree).
lower_bound – either a vector or None. The component-wise lower bound.
upper_bound – either a vector or None. The component-wise upper bound.
name – string or None. An optional name for all new rows.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "Nonexistent_LP_solver")  # optional - Nonexistent_LP_solver
sage: coeffs = ([0, vector([1, 2])], [1, vector([2, 3])])
sage: upper = vector([5, 5])
sage: lower = vector([0, 0])
sage: p.add_variables(2)  # optional - Nonexistent_LP_solver
1
sage: p.add_linear_constraint_vector(2, coeffs, lower, upper, 'foo')  # optional - Nonexistent_LP_solver

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 = "Nonexistent_LP_solver")   # optional - Nonexistent_LP_solver
sage: p.add_variables(5)                                # optional - Nonexistent_LP_solver
5
sage: p.add_linear_constraints(5, None, 2)          # optional - Nonexistent_LP_solver
sage: p.row(4)                                      # optional - Nonexistent_LP_solver
([], [])
sage: p.row_bounds(4)                               # optional - Nonexistent_LP_solver
(None, 2.0)

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

Add a variable.

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

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 = "Nonexistent_LP_solver")    # optional - Nonexistent_LP_solver
sage: p.ncols()                                           # optional - Nonexistent_LP_solver
0
sage: p.add_variable()                                    # optional - Nonexistent_LP_solver
0
sage: p.ncols()                                           # optional - Nonexistent_LP_solver
1
sage: p.add_variable(binary=True)                         # optional - Nonexistent_LP_solver
1
sage: p.add_variable(lower_bound=-2.0, integer=True)      # optional - Nonexistent_LP_solver
2
sage: p.add_variable(continuous=True, integer=True)       # optional - Nonexistent_LP_solver
Traceback (most recent call last):
...
ValueError: ...
sage: p.add_variable(name='x',obj=1.0)                    # optional - Nonexistent_LP_solver
3
sage: p.col_name(3)                                       # optional - Nonexistent_LP_solver
'x'
sage: p.objective_coefficient(3)                          # optional - Nonexistent_LP_solver
1.0

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

Add n variables.

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

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 = "Nonexistent_LP_solver")    # optional - Nonexistent_LP_solver
sage: p.ncols()                                           # optional - Nonexistent_LP_solver
0
sage: p.add_variables(5)                                  # optional - Nonexistent_LP_solver
4
sage: p.ncols()                                           # optional - Nonexistent_LP_solver
5
sage: p.add_variables(2, lower_bound=-2.0, integer=True, names=['a','b']) # optional - Nonexistent_LP_solver
6

base_ring()#

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: p = MixedIntegerLinearProgram(solver="Nonexistent_LP_solver") # optional - Nonexistent_LP_solver
sage: b = p.new_variable(binary=True)                      # optional - Nonexistent_LP_solver
sage: for u,v in graphs.CycleGraph(5).edges(labels=False): # optional - Nonexistent_LP_solver
....:     p.add_constraint(b[u]+b[v]<=1)                   # optional - Nonexistent_LP_solver
sage: p.set_objective(p.sum(b[x] for x in range(5)))       # optional - Nonexistent_LP_solver
sage: p.solve()                                            # optional - Nonexistent_LP_solver
2.0
sage: pb = p.get_backend()                                 # optional - Nonexistent_LP_solver
sage: pb.get_objective_value()                             # optional - Nonexistent_LP_solver
2.0
sage: pb.best_known_objective_bound()                      # optional - Nonexistent_LP_solver
2.0

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 = "Nonexistent_LP_solver")  # optional - Nonexistent_LP_solver
sage: p.add_variable()                                 # optional - Nonexistent_LP_solver
0
sage: p.col_bounds(0)                              # optional - Nonexistent_LP_solver
(0.0, None)
sage: p.variable_upper_bound(0, 5)                 # optional - Nonexistent_LP_solver
sage: p.col_bounds(0)                              # optional - Nonexistent_LP_solver
(0.0, 5.0)

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 = "Nonexistent_LP_solver")  # optional - Nonexistent_LP_solver
sage: p.add_variable(name="I am a variable")            # optional - Nonexistent_LP_solver
1
sage: p.col_name(0)                                     # optional - Nonexistent_LP_solver
'I am a variable'

copy()#

Returns a copy of self.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = MixedIntegerLinearProgram(solver = "Nonexistent_LP_solver") # optional - Nonexistent_LP_solver
sage: b = p.new_variable() # optional - Nonexistent_LP_solver
sage: p.add_constraint(b[1] + b[2] <= 6) # optional - Nonexistent_LP_solver
sage: p.set_objective(b[1] + b[2]) # optional - Nonexistent_LP_solver
sage: copy(p).solve() # optional - Nonexistent_LP_solver
6.0

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 = "Nonexistent_LP_solver") # optional - Nonexistent_LP_solver
sage: p.add_variables(2)                               # optional - Nonexistent_LP_solver
1
sage: p.add_linear_constraint([(0,1), (1,2)], None, 3) # optional - Nonexistent_LP_solver
sage: p.set_objective([2, 5])                          # optional - Nonexistent_LP_solver
sage: p.solve()                                        # optional - Nonexistent_LP_solver
0
sage: p.get_objective_value()                          # optional - Nonexistent_LP_solver
7.5
sage: p.get_variable_value(0)                          # optional - Nonexistent_LP_solver
0.0
sage: p.get_variable_value(1)                          # optional - Nonexistent_LP_solver
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: p = MixedIntegerLinearProgram(solver="Nonexistent_LP_solver") # optional - Nonexistent_LP_solver
sage: b = p.new_variable(binary=True)                      # optional - Nonexistent_LP_solver
sage: for u,v in graphs.CycleGraph(5).edges(labels=False): # optional - Nonexistent_LP_solver
....:     p.add_constraint(b[u]+b[v]<=1)                   # optional - Nonexistent_LP_solver
sage: p.set_objective(p.sum(b[x] for x in range(5)))       # optional - Nonexistent_LP_solver
sage: p.solve()                                            # optional - Nonexistent_LP_solver
2.0
sage: pb = p.get_backend()                                 # optional - Nonexistent_LP_solver
sage: pb.get_objective_value()                             # optional - Nonexistent_LP_solver
2.0
sage: pb.get_relative_objective_gap()                      # optional - Nonexistent_LP_solver
0.0

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 = "Nonexistent_LP_solver") # optional - Nonexistent_LP_solver
sage: p.add_variables(2)                              # optional - Nonexistent_LP_solver
1
sage: p.add_linear_constraint([(0,1), (1, 2)], None, 3) # optional - Nonexistent_LP_solver
sage: p.set_objective([2, 5])                         # optional - Nonexistent_LP_solver
sage: p.solve()                                       # optional - Nonexistent_LP_solver
0
sage: p.get_objective_value()                         # optional - Nonexistent_LP_solver
7.5
sage: p.get_variable_value(0)                         # optional - Nonexistent_LP_solver
0.0
sage: p.get_variable_value(1)                         # optional - Nonexistent_LP_solver
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 = "Nonexistent_LP_solver") # optional - Nonexistent_LP_solver
sage: p.is_maximization()                             # optional - Nonexistent_LP_solver
True
sage: p.set_sense(-1)                             # optional - Nonexistent_LP_solver
sage: p.is_maximization()                             # optional - Nonexistent_LP_solver
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="Nonexistent_LP_solver") # optional - Nonexistent_LP_solver
sage: x = p.new_variable(nonnegative=True)           # optional - Nonexistent_LP_solver
sage: p.add_constraint(-x[0] + x[1] <= 2)            # optional - Nonexistent_LP_solver
sage: p.add_constraint(8 * x[0] + 2 * x[1] <= 17)    # optional - Nonexistent_LP_solver
sage: p.set_objective(5.5 * x[0] - 3 * x[1])         # optional - Nonexistent_LP_solver
sage: b = p.get_backend()                            # optional - Nonexistent_LP_solver
sage: # Backend-specific commands to instruct solver to use simplex method here
sage: b.solve()                                      # optional - Nonexistent_LP_solver
0
sage: b.is_slack_variable_basic(0)                   # optional - Nonexistent_LP_solver
True
sage: b.is_slack_variable_basic(1)                   # optional - Nonexistent_LP_solver
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="Nonexistent_LP_solver") # optional - Nonexistent_LP_solver
sage: x = p.new_variable(nonnegative=True)           # optional - Nonexistent_LP_solver
sage: p.add_constraint(-x[0] + x[1] <= 2)            # optional - Nonexistent_LP_solver
sage: p.add_constraint(8 * x[0] + 2 * x[1] <= 17)    # optional - Nonexistent_LP_solver
sage: p.set_objective(5.5 * x[0] - 3 * x[1])         # optional - Nonexistent_LP_solver
sage: b = p.get_backend()                            # optional - Nonexistent_LP_solver
sage: # Backend-specific commands to instruct solver to use simplex method here
sage: b.solve()                                      # optional - Nonexistent_LP_solver
0
sage: b.is_slack_variable_nonbasic_at_lower_bound(0) # optional - Nonexistent_LP_solver
False
sage: b.is_slack_variable_nonbasic_at_lower_bound(1) # optional - Nonexistent_LP_solver
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="Nonexistent_LP_solver") # optional - Nonexistent_LP_solver
sage: x = p.new_variable(nonnegative=True)           # optional - Nonexistent_LP_solver
sage: p.add_constraint(-x[0] + x[1] <= 2)            # optional - Nonexistent_LP_solver
sage: p.add_constraint(8 * x[0] + 2 * x[1] <= 17)    # optional - Nonexistent_LP_solver
sage: p.set_objective(5.5 * x[0] - 3 * x[1])         # optional - Nonexistent_LP_solver
sage: b = p.get_backend()                            # optional - Nonexistent_LP_solver
sage: # Backend-specific commands to instruct solver to use simplex method here
sage: b.solve()                                      # optional - Nonexistent_LP_solver
0
sage: b.is_variable_basic(0)                         # optional - Nonexistent_LP_solver
True
sage: b.is_variable_basic(1)                         # optional - Nonexistent_LP_solver
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 = "Nonexistent_LP_solver")  # optional - Nonexistent_LP_solver
sage: p.ncols()                                       # optional - Nonexistent_LP_solver
0
sage: p.add_variable()                                 # optional - Nonexistent_LP_solver
0
sage: p.set_variable_type(0,0)                         # optional - Nonexistent_LP_solver
sage: p.is_variable_binary(0)                          # optional - Nonexistent_LP_solver
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 = "Nonexistent_LP_solver")  # optional - Nonexistent_LP_solver
sage: p.ncols()                                       # optional - Nonexistent_LP_solver
0
sage: p.add_variable()                                 # optional - Nonexistent_LP_solver
0
sage: p.is_variable_continuous(0)                      # optional - Nonexistent_LP_solver
True
sage: p.set_variable_type(0,1)                         # optional - Nonexistent_LP_solver
sage: p.is_variable_continuous(0)                      # optional - Nonexistent_LP_solver
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 = "Nonexistent_LP_solver")  # optional - Nonexistent_LP_solver
sage: p.ncols()                                       # optional - Nonexistent_LP_solver
0
sage: p.add_variable()                                 # optional - Nonexistent_LP_solver
0
sage: p.set_variable_type(0,1)                         # optional - Nonexistent_LP_solver
sage: p.is_variable_integer(0)                         # optional - Nonexistent_LP_solver
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="Nonexistent_LP_solver") # optional - Nonexistent_LP_solver
sage: x = p.new_variable(nonnegative=True)           # optional - Nonexistent_LP_solver
sage: p.add_constraint(-x[0] + x[1] <= 2)            # optional - Nonexistent_LP_solver
sage: p.add_constraint(8 * x[0] + 2 * x[1] <= 17)    # optional - Nonexistent_LP_solver
sage: p.set_objective(5.5 * x[0] - 3 * x[1])         # optional - Nonexistent_LP_solver
sage: b = p.get_backend()                            # optional - Nonexistent_LP_solver
sage: # Backend-specific commands to instruct solver to use simplex method here
sage: b.solve()                                      # optional - Nonexistent_LP_solver
0
sage: b.is_variable_nonbasic_at_lower_bound(0)       # optional - Nonexistent_LP_solver
False
sage: b.is_variable_nonbasic_at_lower_bound(1)       # optional - Nonexistent_LP_solver
True

ncols()#

Return the number of columns/variables.

EXAMPLES:

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

nrows()#

Return the number of rows/constraints.

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "Nonexistent_LP_solver") # optional - Nonexistent_LP_solver
sage: p.nrows()                                        # optional - Nonexistent_LP_solver
0
sage: p.add_linear_constraints(2, 2.0, None)         # optional - Nonexistent_LP_solver
sage: p.nrows()                                      # optional - Nonexistent_LP_solver
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 = "Nonexistent_LP_solver")  # optional - Nonexistent_LP_solver
sage: p.add_variable()                                 # optional - Nonexistent_LP_solver
0
sage: p.objective_coefficient(0)                         # optional - Nonexistent_LP_solver
0.0
sage: p.objective_coefficient(0,2)                       # optional - Nonexistent_LP_solver
sage: p.objective_coefficient(0)                         # optional - Nonexistent_LP_solver
2.0

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 = "Nonexistent_LP_solver")  # optional - Nonexistent_LP_solver
sage: p.objective_constant_term()                        # optional - Nonexistent_LP_solver
0.0
sage: p.objective_constant_term(42)                      # optional - Nonexistent_LP_solver
sage: p.objective_constant_term()                        # optional - Nonexistent_LP_solver
42.0

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 = "Nonexistent_LP_solver")   # optional - Nonexistent_LP_solver
sage: p.problem_name("There once was a french fry") # optional - Nonexistent_LP_solver
sage: print(p.problem_name())                       # optional - Nonexistent_LP_solver
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="Nonexistent_LP_solver")  # optional - Nonexistent_LP_solver
sage: v = p.new_variable(nonnegative=True)         # optional - Nonexistent_LP_solver
sage: x,y = v[0], v[1]                             # optional - Nonexistent_LP_solver
sage: p.add_constraint(2*x + 3*y, max = 6)         # optional - Nonexistent_LP_solver
sage: p.add_constraint(3*x + 2*y, max = 6)         # optional - Nonexistent_LP_solver
sage: p.set_objective(x + y + 7)                   # optional - Nonexistent_LP_solver
sage: p.set_integer(x); p.set_integer(y)           # optional - Nonexistent_LP_solver
sage: p.solve()                                    # optional - Nonexistent_LP_solver
9.0
sage: p.remove_constraint(0)                       # optional - Nonexistent_LP_solver
sage: p.solve()                                    # optional - Nonexistent_LP_solver
10.0
sage: p.get_values([x,y])                          # optional - Nonexistent_LP_solver
[0.0, 3.0]

remove_constraints(constraints)#

Remove several constraints.

INPUT:

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

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "Nonexistent_LP_solver")   # optional - Nonexistent_LP_solver
sage: p.add_variables(2)                                 # optional - Nonexistent_LP_solver
1
sage: p.add_linear_constraint([(0, 2), (1, 3)], None, 6) # optional - Nonexistent_LP_solver
sage: p.add_linear_constraint([(0, 3), (1, 2)], None, 6) # optional - Nonexistent_LP_solver
sage: p.remove_constraints([0, 1])                       # optional - Nonexistent_LP_solver

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 = "Nonexistent_LP_solver")  # optional - Nonexistent_LP_solver
sage: p.add_variables(5)                               # optional - Nonexistent_LP_solver
4
sage: p.add_linear_constraint(zip(range(5), range(5)), 2, 2) # optional - Nonexistent_LP_solver
sage: p.row(0)                                     # optional - Nonexistent_LP_solver
([4, 3, 2, 1], [4.0, 3.0, 2.0, 1.0]) ## FIXME: Why backwards?
sage: p.row_bounds(0)                              # optional - Nonexistent_LP_solver
(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 = "Nonexistent_LP_solver")  # optional - Nonexistent_LP_solver
sage: p.add_variables(5)                               # optional - Nonexistent_LP_solver
4
sage: p.add_linear_constraint(list(range(5)), list(range(5)), 2, 2) # optional - Nonexistent_LP_solver
sage: p.row(0)                                     # optional - Nonexistent_LP_solver
([4, 3, 2, 1], [4.0, 3.0, 2.0, 1.0]) ## FIXME: Why backwards?
sage: p.row_bounds(0)                              # optional - Nonexistent_LP_solver
(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 = "Nonexistent_LP_solver")  # optional - Nonexistent_LP_solver
sage: p.add_linear_constraints(1, 2, None, names=['Empty constraint 1'])  # optional - Nonexistent_LP_solver
sage: p.row_name(0)                                     # optional - Nonexistent_LP_solver
'Empty constraint 1'

set_objective(coeff, d=0.0)#

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 (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 = "Nonexistent_LP_solver")    # optional - Nonexistent_LP_solver
sage: p.add_variables(5)                                 # optional - Nonexistent_LP_solver
4
sage: p.set_objective([1, 1, 2, 1, 3])                   # optional - Nonexistent_LP_solver
sage: [p.objective_coefficient(x) for x in range(5)]  # optional - Nonexistent_LP_solver
[1.0, 1.0, 2.0, 1.0, 3.0]

Constants in the objective function are respected:

sage: p = MixedIntegerLinearProgram(solver='Nonexistent_LP_solver') # optional - Nonexistent_LP_solver
sage: x,y = p[0], p[1]                              # optional - Nonexistent_LP_solver
sage: p.add_constraint(2*x + 3*y, max = 6)          # optional - Nonexistent_LP_solver
sage: p.add_constraint(3*x + 2*y, max = 6)          # optional - Nonexistent_LP_solver
sage: p.set_objective(x + y + 7)                    # optional - Nonexistent_LP_solver
sage: p.set_integer(x); p.set_integer(y)            # optional - Nonexistent_LP_solver
sage: p.solve()                                     # optional - Nonexistent_LP_solver
9.0

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 = "Nonexistent_LP_solver")  # optional - Nonexistent_LP_solver
sage: p.is_maximization()                              # optional - Nonexistent_LP_solver
True
sage: p.set_sense(-1)                              # optional - Nonexistent_LP_solver
sage: p.is_maximization()                              # optional - Nonexistent_LP_solver
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 = "Nonexistent_LP_solver")   # optional - Nonexistent_LP_solver
sage: p.ncols()                                        # optional - Nonexistent_LP_solver
0
sage: p.add_variable()                                  # optional - Nonexistent_LP_solver
0
sage: p.set_variable_type(0,1)                          # optional - Nonexistent_LP_solver
sage: p.is_variable_integer(0)                          # optional - Nonexistent_LP_solver
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 = "Nonexistent_LP_solver")  # optional - Nonexistent_LP_solver
sage: p.set_verbosity(2)                                # optional - Nonexistent_LP_solver

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 = "Nonexistent_LP_solver") # optional - Nonexistent_LP_solver
sage: p.add_linear_constraints(5, 0, None)             # optional - Nonexistent_LP_solver
sage: p.add_col(list(range(5)), list(range(5)))                    # optional - Nonexistent_LP_solver
sage: p.solve()                                        # optional - Nonexistent_LP_solver
0
sage: p.objective_coefficient(0,1)                 # optional - Nonexistent_LP_solver
sage: p.solve()                                        # optional - Nonexistent_LP_solver
Traceback (most recent call last):
...
MIPSolverException: ...

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.

Note

The list of available parameters is available at solver_parameter().

EXAMPLES:

sage: from sage.numerical.backends.generic_backend import get_solver
sage: p = get_solver(solver = "Nonexistent_LP_solver")  # optional - Nonexistent_LP_solver
sage: p.solver_parameter("timelimit")                   # optional - Nonexistent_LP_solver
sage: p.solver_parameter("timelimit", 60)               # optional - Nonexistent_LP_solver
sage: p.solver_parameter("timelimit")                   # optional - Nonexistent_LP_solver

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 = "Nonexistent_LP_solver")  # optional - Nonexistent_LP_solver
sage: p.add_variable()                                 # optional - Nonexistent_LP_solver
0
sage: p.col_bounds(0)                              # optional - Nonexistent_LP_solver
(0.0, None)
sage: p.variable_lower_bound(0, 5)                 # optional - Nonexistent_LP_solver
sage: p.col_bounds(0)                              # optional - Nonexistent_LP_solver
(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 = "Nonexistent_LP_solver")  # optional - Nonexistent_LP_solver
sage: p.add_variable()                                 # optional - Nonexistent_LP_solver
0
sage: p.col_bounds(0)                              # optional - Nonexistent_LP_solver
(0.0, None)
sage: p.variable_upper_bound(0, 5)                 # optional - Nonexistent_LP_solver
sage: p.col_bounds(0)                              # optional - Nonexistent_LP_solver
(0.0, 5.0)

write_lp(name)#

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 = "Nonexistent_LP_solver")  # optional - Nonexistent_LP_solver
sage: p.add_variables(2)                               # optional - Nonexistent_LP_solver
2
sage: p.add_linear_constraint([(0, 1], (1, 2)], None, 3) # optional - Nonexistent_LP_solver
sage: p.set_objective([2, 5])                          # optional - Nonexistent_LP_solver
sage: from tempfile import NamedTemporaryFile  # optional - Nonexistent_LP_solver
sage: with NamedTemporaryFile(suffix=".lp") as f:  # optional - Nonexistent_LP_solver
....:     p.write_lp(f.name)

write_mps(name, 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 = "Nonexistent_LP_solver")  # optional - Nonexistent_LP_solver
sage: p.add_variables(2)                               # optional - Nonexistent_LP_solver
2
sage: p.add_linear_constraint([(0, 1), (1, 2)], None, 3) # optional - Nonexistent_LP_solver
sage: p.set_objective([2, 5])                          # optional - Nonexistent_LP_solver
sage: from tempfile import NamedTemporaryFile  # optional - Nonexistent_LP_solver
sage: with NamedTemporaryFile(suffix=".lp") as f:  # optional - Nonexistent_LP_solver
....:     p.write_lp(f.name)

zero()#

sage.numerical.backends.generic_backend.default_mip_solver(solver=None)#

Returns/sets the default MILP solver used by Sage

INPUT:

solver – one of the following:
- a string indicating one of the available solvers (see MixedIntegerLinearProgram);
- a callable (typically a subclass of sage.numerical.backends.generic_backend.GenericBackend);
- None (default), in which case the current default solver is returned; this is either a string or a callable.

OUTPUT:

This function returns the current default solver’s name if solver = None (default). Otherwise, it sets the default solver to the one given. If this solver does not exist, or is not available, a ValueError exception is raised.

EXAMPLES:

sage: former_solver = default_mip_solver()
sage: default_mip_solver("GLPK")
sage: default_mip_solver()
'Glpk'
sage: default_mip_solver("PPL")
sage: default_mip_solver()
'Ppl'
sage: default_mip_solver("GUROBI") # random
Traceback (most recent call last):
...
ValueError: Gurobi is not available. Please refer to the documentation to install it.
sage: default_mip_solver("Yeahhhhhhhhhhh")
Traceback (most recent call last):
...
ValueError: 'solver' should be set to ...
sage: default_mip_solver(former_solver)

sage.numerical.backends.generic_backend.get_solver(constraint_generation=False, solver=None, base_ring=None)#

Return a solver according to the given preferences

INPUT:

solver – one of the following:
- a string indicating one of the available solvers (see MixedIntegerLinearProgram);
- None (default), in which case the default solver is used (see default_mip_solver());
- or a callable (such as a class), in which case it is called, and its result is returned.
base_ring – If not None, request a solver that works over this
(ordered) field. If base_ring is not a field, its fraction field is used.

For example, is base_ring=ZZ is provided, the solver will work over the rational numbers. This is unrelated to whether variables are constrained to be integers or not.
constraint_generation – Only used when solver=None.
- When set to True, after solving the MixedIntegerLinearProgram, it is possible to add a constraint, and then solve it again. The effect is that solvers that do not support this feature will not be used. (Coin and SCIP are such solvers.)
- Defaults to False.