# Mixed Integer Linear Programming¶

A linear program (LP) is an optimization problem in the following form

$\max \{ c^T x \;|\; A x \leq b, x \geq 0 \}$

with given $$A \in \mathbb{R}^{m,n}$$, $$b \in \mathbb{R}^m$$, $$c \in \mathbb{R}^n$$ and unknown $$x \in \mathbb{R}^{n}$$. If some or all variables in the vector $$x$$ are restricted over the integers $$\mathbb{Z}$$, the problem is called mixed integer linear program (MILP). A wide variety of problems in optimization can be formulated in this standard form. Then, solvers are able to calculate a solution.

Imagine you want to solve the following linear system of three equations:

• $$w_0 + w_1 + w_2 - 14 w_3 = 0$$
• $$w_1 + 2 w_2 - 8 w_3 = 0$$
• $$2 w_2 - 3 w_3 = 0$$

• $$w_0 - w_1 - w_2 \geq 0$$

where all $$w_i \in \mathbb{Z}^+$$. You know that the trivial solution is $$w_i=0$$, but what is the first non-trivial one with $$w_3 \geq 1$$?

A mixed integer linear program can give you an answer:

1. You have to create an instance of MixedIntegerLinearProgram and – in our case – specify that it is a minimization.
2. Create an dictionary w of integer variables w via w = p.new_variable(integer=True) (note that by default all variables are non-negative, cf new_variable()).
4. Also add the inequality constraint.
5. Add an inequality constraint $$w_3 \geq 1$$ to exclude the trivial solution.
6. By default, all variables are non-negative. We remove that constraint via p.set_min(variable, None), see set_min.
7. Specify the objective function via set_objective. In our case that is just $$w_3$$. If it is a pure constraint satisfaction problem, specify it as None.
8. To check if everything is set up correctly, you can print the problem via show.
9. Solve it and print the solution.

The following example shows all these steps:

sage: p = MixedIntegerLinearProgram(maximization=False, solver = "GLPK")
sage: w = p.new_variable(integer=True, nonnegative=True)
sage: p.add_constraint(w[0] + w[1] + w[2] - 14*w[3] == 0)
sage: p.add_constraint(w[1] + 2*w[2] - 8*w[3] == 0)
sage: p.add_constraint(2*w[2] - 3*w[3] == 0)
sage: p.add_constraint(w[0] - w[1] - w[2] >= 0)
sage: _ = [ p.set_min(w[i], None) for i in range(1,4) ]
sage: p.set_objective(w[3])
sage: p.show()
Minimization:
x_3
Constraints:
0.0 <= x_0 + x_1 + x_2 - 14.0 x_3 <= 0.0
0.0 <= x_1 + 2.0 x_2 - 8.0 x_3 <= 0.0
0.0 <= 2.0 x_2 - 3.0 x_3 <= 0.0
- x_0 + x_1 + x_2 <= 0.0
- x_3 <= -1.0
Variables:
x_0 is an integer variable (min=0.0, max=+oo)
x_1 is an integer variable (min=-oo, max=+oo)
x_2 is an integer variable (min=-oo, max=+oo)
x_3 is an integer variable (min=-oo, max=+oo)
sage: print 'Objective Value:', p.solve()
Objective Value: 2.0
sage: for i, v in p.get_values(w).iteritems():
....:     print 'w_%s = %s' % (i, int(round(v)))
w_0 = 15
w_1 = 10
w_2 = 3
w_3 = 2


Different backends compute with different base fields, for example:

sage: p = MixedIntegerLinearProgram(solver='GLPK')
sage: p.base_ring()
Real Double Field
sage: x = p.new_variable(real=True, nonnegative=True)
sage: 0.5 + 3/2*x[1]
0.5 + 1.5*x_0

sage: p = MixedIntegerLinearProgram(solver='ppl')
sage: p.base_ring()
Rational Field
sage: x = p.new_variable(nonnegative=True)
sage: 0.5 + 3/2*x[1]
1/2 + 3/2*x_0


## Linear Variables and Expressions¶

The underlying linear programming backends always work with matrices where each column corresponds to a linear variable. These variables can be accessed using the MixedIntegerLinearProgram.gen() method or by calling with a dictionary variable index to coefficient:

sage: mip = MixedIntegerLinearProgram()
sage: 5 + mip.gen(0) + 2*mip.gen(1)
5 + x_0 + 2*x_1
sage: mip({-1:5, 0:1, 1:2})
5 + x_0 + 2*x_1


However, this alone is often not convenient to construct a linear program. To make your code more readable, you can construct MIPVariable objects that can be arbitrarily named and indexed. Internally, this is then translated back to the $$x_i$$ variables. For example:

sage: mip.<a,b> = MixedIntegerLinearProgram()
sage: a
MIPVariable of dimension 1.
sage: 5 + a[1] + 2*b[3]
5 + x_0 + 2*x_1


Indices can be any object, not necessarily integers. Multi-indices are also allowed:

sage: a[4, 'string', QQ]
x_2
sage: a[4, 'string', QQ] - 7*b[2]
x_2 - 7*x_3
sage: mip.show()
Maximization:

Constraints:
Variables:
a[1] = x_0 is a continuous variable (min=-oo, max=+oo)
b[3] = x_1 is a continuous variable (min=-oo, max=+oo)
a[(4, 'string', Rational Field)] = x_2 is a continuous variable (min=-oo, max=+oo)
b[2] = x_3 is a continuous variable (min=-oo, max=+oo)


## Index of functions and methods¶

Below are listed the methods of MixedIntegerLinearProgram. This module also implements the MIPSolverException exception, as well as the MIPVariable class.

 add_constraint() Adds a constraint to the MixedIntegerLinearProgram base_ring() Return the base ring constraints() Returns a list of constraints, as 3-tuples get_backend() Returns the backend instance used get_max() Returns the maximum value of a variable get_min() Returns the minimum value of a variable get_values() Return values found by the previous call to solve() is_binary() Tests whether the variable e is binary is_integer() Tests whether the variable is an integer is_real() Tests whether the variable is real linear_constraints_parent() Return the parent for all linear constraints linear_function() Construct a new linear function linear_functions_parent() Return the parent for all linear functions new_variable() Returns an instance of MIPVariable associated number_of_constraints() Returns the number of constraints assigned so far number_of_variables() Returns the number of variables used so far polyhedron() Returns the polyhedron defined by the Linear Program remove_constraint() Removes a constraint from self remove_constraints() Remove several constraints set_binary() Sets a variable or a MIPVariable as binary set_integer() Sets a variable or a MIPVariable as integer set_max() Sets the maximum value of a variable set_min() Sets the minimum value of a variable set_objective() Sets the objective of the MixedIntegerLinearProgram set_problem_name() Sets the name of the MixedIntegerLinearProgram set_real() Sets a variable or a MIPVariable as real show() Displays the MixedIntegerLinearProgram in a human-readable solve() Solves the MixedIntegerLinearProgram solver_parameter() Return or define a solver parameter sum() Efficiently computes the sum of a sequence of LinearFunction elements write_lp() Write the linear program as a LP file write_mps() Write the linear program as a MPS file

AUTHORS:

• Risan (2012/02): added extension for exact computation
exception sage.numerical.mip.MIPSolverException(value)

Bases: exceptions.RuntimeError

Exception raised when the solver fails.

class sage.numerical.mip.MIPVariable

MIPVariable is a variable used by the class MixedIntegerLinearProgram.

Warning

You should not instantiate this class directly. Instead, use MixedIntegerLinearProgram.new_variable().

depth()

Returns the current variable’s depth.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable(nonnegative=True)
sage: p.set_objective(v[0] + v[1])
sage: v.depth()
1

items()

Returns the pairs (keys,value) contained in the dictionary.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable(nonnegative=True)
sage: p.set_objective(v[0] + v[1])
sage: v.items()
[(0, x_0), (1, x_1)]

keys()

Returns the keys already defined in the dictionary.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable(nonnegative=True)
sage: p.set_objective(v[0] + v[1])
sage: v.keys()
[0, 1]

set_max(max)

Sets an upper bound on the variable.

INPUT:

• max – an upper bound, or None to mean that the variable is unbounded.

EXAMPLES:

sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable(real=True, nonnegative=True, dim=2)
sage: p.get_max(v)
sage: p.get_max(v[0])
sage: p.get_max(v[0][0])
sage: p.set_max(v,4)
sage: p.get_max(v)
4
sage: p.get_max(v[0])
4
sage: p.get_max(v[0][0])
4.0

set_min(min)

Sets a lower bound on the variable.

INPUT:

• min – a lower bound, or None to mean that the variable is unbounded.

EXAMPLES:

sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable(real=True, nonnegative=True, dim=2)
sage: p.get_min(v)
0
sage: p.get_min(v[0])
0
sage: p.get_min(v[0][0])
0.0
sage: p.set_min(v,4)
sage: p.get_min(v)
4
sage: p.get_min(v[0])
4
sage: p.get_min(v[0][0])
4.0

values()

Returns the symbolic variables associated to the current dictionary.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable(nonnegative=True)
sage: p.set_objective(v[0] + v[1])
sage: v.values()
[x_0, x_1]

class sage.numerical.mip.MIPVariableParent

Parent for MIPVariable.

Warning

This class is for internal use. You should not instantiate it yourself. Use MixedIntegerLinearProgram.new_variable() to generate mip variables.

Element

alias of MIPVariable

class sage.numerical.mip.MixedIntegerLinearProgram

The MixedIntegerLinearProgram class is the link between Sage, linear programming (LP) and mixed integer programming (MIP) solvers.

A Mixed Integer Linear Program (MILP) consists of variables, linear constraints on these variables, and an objective function which is to be maximised or minimised under these constraints.

See the Wikipedia article Linear_programming for further information on linear programming, and the MILP module for its use in Sage.

INPUT:

• solver – selects a solver:

• GLPK (solver="GLPK"). See the GLPK web site.

• COIN Branch and Cut (solver="Coin"). See the COIN-OR web site.

• CPLEX (solver="CPLEX"). See the CPLEX web site.

• Gurobi (solver="Gurobi"). See the Gurobi

web site.

• CVXOPT (solver="CVXOPT"). See the CVXOPT

web site.

• PPL (solver="PPL"). See the PPL

web site.

• If solver=None (default), the default solver is used (see default_mip_solver())

• maximization

• When set to True (default), the MixedIntegerLinearProgram is defined as a maximization.
• When set to False, the MixedIntegerLinearProgram is defined as a minimization.
• 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.
• Defaults to False.

Warning

All LP variables are non-negative by default (see new_variable() and set_min()).

EXAMPLES:

Computation of a maximum stable set in Petersen’s graph:

sage: g = graphs.PetersenGraph()
sage: p = MixedIntegerLinearProgram(maximization=True)
sage: b = p.new_variable(binary=True)
sage: p.set_objective(sum([b[v] for v in g]))
sage: for (u,v) in g.edges(labels=None):
sage: p.solve(objective_only=True)
4.0


TESTS:

Check that trac ticket #16497 is fixed:

sage: from sage.numerical.mip import MixedIntegerLinearProgram
sage: for type in ["binary", "integer"]:
....:     k = 3
....:     items = [1/5, 1/3, 2/3, 3/4, 5/7]
....:     maximum=1
....:     p=MixedIntegerLinearProgram()
....:     box=p.new_variable(nonnegative=True, **{type:True})
....:     for b in range(k):
....:          p.add_constraint(p.sum([items[i]*box[i,b] for i in range(len(items))]) <= maximum)
....:     for i in range(len(items)):
....:         p.add_constraint(p.sum([box[i,b] for b in range(k)]) == 1)
....:     p.set_objective(None)
....:     _ = p.solve()
....:     box=p.get_values(box)
....:     print(all(v in ZZ for v in box.values()))
True
True


Adds a constraint to the MixedIntegerLinearProgram.

INPUT:

• linear_function – Four different types of arguments are admissible:

• A linear function. In this case, one of the arguments min or max has to be specified.
• A linear constraint of the form A <= B, A >= B, A <= B <= C, A >= B >= C or A == B.
• A vector-valued linear function, see linear_tensor. In this case, one of the arguments min or max has to be specified.
• An (in)equality of vector-valued linear functions, that is, elements of the space of linear functions tensored with a vector space. See linear_tensor_constraints for details.
• max – constant or None (default). An upper bound on the linear function. This must be a numerical value for scalar linear functions, or a vector for vector-valued linear functions. Not allowed if the linear_function argument is a symbolic (in)-equality.

• min – constant or None (default). A lower bound on the linear function. This must be a numerical value for scalar linear functions, or a vector for vector-valued linear functions. Not allowed if the linear_function argument is a symbolic (in)-equality.

• name – A name for the constraint.

To set a lower and/or upper bound on the variables use the methods set_min and/or set_max of MixedIntegerLinearProgram.

EXAMPLE:

Consider the following linear program:

Maximize:
x + 5 * y
Constraints:
x + 0.2 y       <= 4
1.5 * x + 3 * y <= 4
Variables:
x is Real (min = 0, max = None)
y is Real (min = 0, max = None)


It can be solved as follows:

sage: p = MixedIntegerLinearProgram(maximization=True)
sage: x = p.new_variable(nonnegative=True)
sage: p.set_objective(x[0] + 5*x[1])
sage: p.solve()     # rel tol 1e-15
6.666666666666666


There are two different ways to add the constraint x[5] + 3*x[7] <= x[6] + 3 to a MixedIntegerLinearProgram.

The first one consists in giving add_constraint this very expression:

sage: p.add_constraint(x[5] + 3*x[7] <= x[6] + 3)


The second (slightly more efficient) one is to use the arguments min or max, which can only be numerical values:

sage: p.add_constraint(x[5] + 3*x[7] - x[6], max=3)


One can also define double-bounds or equality using symbols <=, >= and ==:

sage: p.add_constraint(x[5] + 3*x[7] == x[6] + 3)
sage: p.add_constraint(x[5] + 3*x[7] <= x[6] + 3 <= x[8] + 27)


Using this notation, the previous program can be written as:

sage: p = MixedIntegerLinearProgram(maximization=True)
sage: x = p.new_variable(nonnegative=True)
sage: p.set_objective(x[0] + 5*x[1])
sage: p.add_constraint(x[0] + 0.2*x[1] <= 4)
sage: p.add_constraint(1.5*x[0] + 3*x[1] <= 4)
sage: p.solve()     # rel tol 1e-15
6.666666666666666


The two constraints can alse be combined into a single vector-valued constraint:

sage: p = MixedIntegerLinearProgram(maximization=True)
sage: x = p.new_variable(nonnegative=True)
sage: p.set_objective(x[0] + 5*x[1])
sage: f_vec = vector([1, 1.5]) * x[0] + vector([0.2, 3]) * x[1];  f_vec
(1.0, 1.5)*x_0 + (0.2, 3.0)*x_1
sage: p.solve()     # rel tol 1e-15
6.666666666666666


Instead of specifying the maximum in the optional max argument, we can also use (in)equality notation for vector-valued linear functions:

sage: f_vec <= 4    # constant rhs becomes vector
(1.0, 1.5)*x_0 + (0.2, 3.0)*x_1 <= (4.0, 4.0)
sage: p = MixedIntegerLinearProgram(maximization=True)
sage: x = p.new_variable(nonnegative=True)
sage: p.set_objective(x[0] + 5*x[1])
sage: p.solve()     # rel tol 1e-15
6.666666666666666


Finally, one can use the matrix * MIPVariable notation to write vector-valued linear functions:

sage: m = matrix([[1.0, 0.2], [1.5, 3.0]]);  m
[ 1.00000000000000 0.200000000000000]
[ 1.50000000000000  3.00000000000000]
sage: p = MixedIntegerLinearProgram(maximization=True)
sage: x = p.new_variable(nonnegative=True)
sage: p.set_objective(x[0] + 5*x[1])
sage: p.add_constraint(m * x <= 4)
sage: p.solve()     # rel tol 1e-15
6.666666666666666


TESTS:

Complex constraints:

sage: p = MixedIntegerLinearProgram(solver = "GLPK")
sage: b = p.new_variable(nonnegative=True)
sage: p.add_constraint( b[8] - b[15] <= 3*b[8] + 9)
sage: p.show()
Maximization:

Constraints:
-2.0 x_0 - x_1 <= 9.0
Variables:
x_0 is a continuous variable (min=0.0, max=+oo)
x_1 is a continuous variable (min=0.0, max=+oo)


Empty constraint:

sage: p = MixedIntegerLinearProgram()


Min/Max are numerical

sage: v = p.new_variable(nonnegative=True)
sage: p.add_constraint(v[3] + v[5], min = v[6])
Traceback (most recent call last):
...
ValueError: min and max arguments are required to be constants
sage: p.add_constraint(v[3] + v[5], max = v[6])
Traceback (most recent call last):
...
ValueError: min and max arguments are required to be constants


Do not add redundant elements (notice only one copy of each constraint is added):

sage: lp = MixedIntegerLinearProgram(solver="GLPK", check_redundant=True)
sage: for each in xrange(10): lp.add_constraint(lp[0]-lp[1],min=1)
sage: lp.show()
Maximization:

Constraints:
1.0 <= x_0 - x_1
Variables:
x_0 is a continuous variable (min=0.0, max=+oo)
x_1 is a continuous variable (min=0.0, max=+oo)


We check for constant multiples of constraints as well:

sage: for each in xrange(10): lp.add_constraint(2*lp[0]-2*lp[1],min=2)
sage: lp.show()
Maximization:

Constraints:
1.0 <= x_0 - x_1
Variables:
x_0 is a continuous variable (min=0.0, max=+oo)
x_1 is a continuous variable (min=0.0, max=+oo)


But if the constant multiple is negative, we should add it anyway (once):

sage: for each in xrange(10): lp.add_constraint(-2*lp[0]+2*lp[1],min=-2)
sage: lp.show()
Maximization:

Constraints:
1.0 <= x_0 - x_1
-2.0 <= -2.0 x_0 + 2.0 x_1
Variables:
x_0 is a continuous variable (min=0.0, max=+oo)
x_1 is a continuous variable (min=0.0, max=+oo)


TESTS:

Catch True / False as INPUT (trac ticket #13646):

sage: p = MixedIntegerLinearProgram()
sage: x = p.new_variable(nonnegative=True)
Traceback (most recent call last):
...
ValueError: argument must be a linear function or constraint, got True

base_ring()

Return the base ring.

OUTPUT:

A ring. The coefficients that the chosen solver supports.

EXAMPLES:

sage: p = MixedIntegerLinearProgram(solver='GLPK')
sage: p.base_ring()
Real Double Field
sage: p = MixedIntegerLinearProgram(solver='ppl')
sage: p.base_ring()
Rational Field

constraints(indices=None)

Returns a list of constraints, as 3-tuples.

INPUT:

• indices – select which constraint(s) to return

• If indices = None, the method returns the list of all the constraints.
• If indices is an integer $$i$$, the method returns constraint $$i$$.
• If indices is a list of integers, the method returns the list of the corresponding constraints.

OUTPUT:

Each constraint is returned as a triple lower_bound, (indices, coefficients), upper_bound. For each of those entries, the corresponding linear function is the one associating to variable indices[i] the coefficient coefficients[i], and $$0$$ to all the others.

lower_bound and upper_bound are numerical values.

EXAMPLE:

First, let us define a small LP:

sage: p = MixedIntegerLinearProgram()
sage: p.add_constraint(p[0] - p[2], min = 1, max = 4)
sage: p.add_constraint(p[0] - 2*p[1], min = 1)


To obtain the list of all constraints:

sage: p.constraints()          # not tested
[(1.0, ([1, 0], [-1.0, 1.0]), 4.0), (1.0, ([2, 0], [-2.0, 1.0]), None)]


Or constraint $$0$$ only:

sage: p.constraints(0)         # not tested
(1.0, ([1, 0], [-1.0, 1.0]), 4.0)


A list of constraints containing only $$1$$:

sage: p.constraints([1])       # not tested
[(1.0, ([2, 0], [-2.0, 1.0]), None)]


TESTS:

As the ordering of the variables in each constraint depends on the solver used, we define a short function reordering it before it is printed. The output would look the same without this function applied:

sage: def reorder_constraint((lb,(ind,coef),ub)):
....:    d = dict(zip(ind, coef))
....:    ind.sort()
....:    return (lb, (ind, [d[i] for i in ind]), ub)


Running the examples from above, reordering applied:

sage: p = MixedIntegerLinearProgram(solver = "GLPK")
sage: p.add_constraint(p[0] - p[2], min = 1, max = 4)
sage: p.add_constraint(p[0] - 2*p[1], min = 1)
sage: sorted(map(reorder_constraint,p.constraints()))
[(1.0, ([0, 1], [1.0, -1.0]), 4.0), (1.0, ([0, 2], [1.0, -2.0]), None)]
sage: reorder_constraint(p.constraints(0))
(1.0, ([0, 1], [1.0, -1.0]), 4.0)
sage: sorted(map(reorder_constraint,p.constraints([1])))
[(1.0, ([0, 2], [1.0, -2.0]), None)]

gen(i)

Return the linear variable $$x_i$$.

OUTPUT:

sage: mip = MixedIntegerLinearProgram() sage: mip.gen(0) x_0 sage: [mip.gen(i) for i in range(10)] [x_0, x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9]
get_backend()

Returns the backend instance used.

This might be useful when acces to additional functions provided by the backend is needed.

EXAMPLE:

This example uses the simplex algorthm and prints information:

sage: p = MixedIntegerLinearProgram(solver="GLPK")
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: b = p.get_backend()
sage: b.solver_parameter("simplex_or_intopt", "simplex_only")
sage: b.solver_parameter("verbosity_simplex", "GLP_MSG_ALL")
sage: p.solve()  # rel tol 1e-5
GLPK Simplex Optimizer, v4.55
2 rows, 2 columns, 4 non-zeros
*     0: obj =   7.000000000e+00  infeas =  0.000e+00 (0)
*     2: obj =   9.400000000e+00  infeas =  0.000e+00 (0)
OPTIMAL LP SOLUTION FOUND
9.4

get_max(v)

Returns the maximum value of a variable.

INPUT:

• v – a variable.

OUTPUT:

Maximum value of the variable, or None if the variable has no upper bound.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable(nonnegative=True)
sage: p.set_objective(v[1])
sage: p.get_max(v[1])
sage: p.set_max(v[1],6)
sage: p.get_max(v[1])
6.0

get_min(v)

Returns the minimum value of a variable.

INPUT:

• v – a variable

OUTPUT:

Minimum value of the variable, or None if the variable has no lower bound.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable(nonnegative=True)
sage: p.set_objective(v[1])
sage: p.get_min(v[1])
0.0
sage: p.set_min(v[1],6)
sage: p.get_min(v[1])
6.0
sage: p.set_min(v[1], None)
sage: p.get_min(v[1])

get_values(*lists)

Return values found by the previous call to solve().

INPUT:

• Any instance of MIPVariable (or one of its elements), or lists of them.

OUTPUT:

• Each instance of MIPVariable is replaced by a dictionary containing the numerical values found for each corresponding variable in the instance.
• Each element of an instance of a MIPVariable is replaced by its corresponding numerical value.

Note

While a variable may be declared as binary or integer, its value as returned by the solver is of type float.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: x = p.new_variable(nonnegative=True)
sage: y = p.new_variable(nonnegative=True)
sage: p.set_objective(x[3] + 3*y[2,9] + x[5])
sage: p.add_constraint(x[3] + y[2,9] + 2*x[5], max=2)
sage: p.solve()
6.0


To return the optimal value of y[2,9]:

sage: p.get_values(y[2,9])
2.0


To get a dictionary identical to x containing optimal values for the corresponding variables

sage: x_sol = p.get_values(x)
sage: x_sol.keys()
[3, 5]


Obviously, it also works with variables of higher dimension:

sage: y_sol = p.get_values(y)


We could also have tried

sage: [x_sol, y_sol] = p.get_values(x, y)


Or:

sage: [x_sol, y_sol] = p.get_values([x, y])


TESTS:

When ‘dim’ will be removed, also remove from this function the code that uses it:

sage: p = MixedIntegerLinearProgram()
sage: b = p.new_variable(dim=2)
doctest:...: DeprecationWarning: The 'dim' argument will soon disappear. Fortunately variable[1,2] is easier to use than variable[1][2]
See http://trac.sagemath.org/15489 for details.
sage: p.add_constraint(b[1][2] +  b[2][3] == 0)
sage: _ = p.solve()
sage: _ = p.get_values(b)

is_binary(e)

Tests whether the variable e is binary. Variables are real by default.

INPUT:

• e – A variable (not a MIPVariable, but one of its elements.)

OUTPUT:

True if the variable e is binary; False otherwise.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable(nonnegative=True)
sage: p.set_objective(v[1])
sage: p.is_binary(v[1])
False
sage: p.set_binary(v[1])
sage: p.is_binary(v[1])
True

is_integer(e)

Tests whether the variable is an integer. Variables are real by default.

INPUT:

• e – A variable (not a MIPVariable, but one of its elements.)

OUTPUT:

True if the variable e is an integer; False otherwise.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable(nonnegative=True)
sage: p.set_objective(v[1])
sage: p.is_integer(v[1])
False
sage: p.set_integer(v[1])
sage: p.is_integer(v[1])
True

is_real(e)

Tests whether the variable is real.

INPUT:

• e – A variable (not a MIPVariable, but one of its elements.)

OUTPUT:

True if the variable is real; False otherwise.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable(nonnegative=True)
sage: p.set_objective(v[1])
sage: p.is_real(v[1])
True
sage: p.set_binary(v[1])
sage: p.is_real(v[1])
False
sage: p.set_real(v[1])
sage: p.is_real(v[1])
True

linear_constraints_parent()

Return the parent for all linear constraints

See linear_functions for more details.

EXAMPLES:

sage: p = MixedIntegerLinearProgram()
sage: p.linear_constraints_parent()
Linear constraints over Real Double Field

linear_function(x)

Construct a new linear function

EXAMPLES:

sage: p = MixedIntegerLinearProgram()
sage: p.linear_function({1:3, 4:5})
3*x_1 + 5*x_4


This is equivalent to:

sage: p({1:3, 4:5})
3*x_1 + 5*x_4

linear_functions_parent()

Return the parent for all linear functions

EXAMPLES:

sage: p = MixedIntegerLinearProgram()
sage: p.linear_functions_parent()
Linear functions over Real Double Field

new_variable(real=False, binary=False, integer=False, nonnegative=None, dim=1, name='')

Return a new MIPVariable

A new variable x is defined by:

sage: p = MixedIntegerLinearProgram()
sage: x = p.new_variable(nonnegative=True)


It behaves exactly as an usual dictionary would. It can use any key argument you may like, as x[5] or x["b"], and has methods items() and keys().

• set_min(),:meth:$$get_min$$ – set/get the lower bound of a variable. Note that by default, all variables are non-negative.
• set_max(),:meth:$$get_max$$ – set/get the upper bound of a variable.

INPUT:

• dim – integer. Defines the dimension of the dictionary. If x has dimension $$2$$, its fields will be of the form x[key1][key2]. Deprecated.
• binary, integer, real – boolean. Set one of these arguments to True to ensure that the variable gets the corresponding type.
• nonnegative – boolean. Whether the variable should be assumed to be nonnegative. Rather useless for the binary type.
• name – string. Associates a name to the variable. This is only useful when exporting the linear program to a file using write_mps or write_lp, and has no other effect.

OUTPUT:

A new instance of MIPVariable associated to the current MixedIntegerLinearProgram.

EXAMPLE:

   sage: p = MixedIntegerLinearProgram()

To define two dictionaries of variables, the first being
of real type, and the second of integer type ::

sage: x = p.new_variable(real=True, nonnegative=True)
sage: y = p.new_variable(integer=True, nonnegative=True)
sage: p.is_integer(x[2])
False
sage: p.is_integer(y[3,5])
True


An exception is raised when two types are supplied

sage: z = p.new_variable(real = True, integer = True)
Traceback (most recent call last):
...
ValueError: Exactly one of the available types has to be True


Unbounded variables:

sage: p = MixedIntegerLinearProgram()
sage: x = p.new_variable(real=True, nonnegative=False)
sage: y = p.new_variable(integer=True, nonnegative=False)
sage: p.show()
Maximization:

Constraints:
x_0 + x_1 <= 8.0
- x_2 + x_3 <= 0.0
Variables:
x_0 is a continuous variable (min=-oo, max=+oo)
x_1 is a continuous variable (min=-oo, max=+oo)
x_2 is an integer variable (min=-oo, max=+oo)
x_3 is an integer variable (min=-oo, max=+oo)


On the Sage command line, generator syntax is accepted as a shorthand for generating new variables with default (nonnegative=False) settings:

sage: mip.<x, y, z> = MixedIntegerLinearProgram()
sage: mip.add_constraint(x[0] + y[1] + z[2] <= 10)
sage: mip.show()
Maximization:

Constraints:
x[0] + y[1] + z[2] <= 10.0
Variables:
x[0] = x_0 is a continuous variable (min=-oo, max=+oo)
y[1] = x_1 is a continuous variable (min=-oo, max=+oo)
z[2] = x_2 is a continuous variable (min=-oo, max=+oo)


TESTS:

Default behaviour (trac ticket #15521):

sage: x = p.new_variable(nonnegative=True)
sage: p.get_min(x[0])
0.0

number_of_constraints()

Returns the number of constraints assigned so far.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: p.add_constraint(p[0] - p[2], min = 1, max = 4)
sage: p.add_constraint(p[0] - 2*p[1], min = 1)
sage: p.number_of_constraints()
2

number_of_variables()

Returns the number of variables used so far.

Note that this is backend-dependent, i.e. we count solver’s variables rather than user’s variables. An example of the latter can be seen below: Gurobi converts double inequalities, i.e. inequalities like $$m <= c^T x <= M$$, with $$m<M$$, into equations, by adding extra variables: $$c^T x + y = M$$, $$0 <= y <= M-m$$.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: p.add_constraint(p[0] - p[2], max = 4)
sage: p.number_of_variables()
2
sage: p.add_constraint(p[0] - 2*p[1], min = 1)
sage: p.number_of_variables()
3
sage: p = MixedIntegerLinearProgram(solver="glpk")
sage: p.add_constraint(p[0] - p[2], min = 1, max = 4)
sage: p.number_of_variables()
2
sage: p = MixedIntegerLinearProgram(solver="gurobi")   # optional - Gurobi
sage: p.add_constraint(p[0] - p[2], min = 1, max = 4)  # optional - Gurobi
sage: p.number_of_variables()                          # optional - Gurobi
3

polyhedron(**kwds)

Returns the polyhedron defined by the Linear Program.

INPUT:

All arguments given to this method are forwarded to the constructor of the Polyhedron() class.

OUTPUT:

A Polyhedron() object whose $$i$$-th variable represents the $$i$$-th variable of self.

Warning

The polyhedron is built from the variables stored by the LP solver (i.e. the output of show()). While they usually match the ones created explicitely when defining the LP, a solver like Gurobi has been known to introduce additional variables to store constraints of the type lower_bound <= linear_function <= upper bound. You should be fine if you did not install Gurobi or if you do not use it as a solver, but keep an eye on the number of variables in the polyhedron, or on the output of show(). Just in case.

EXAMPLES:

A LP on two variables:

sage: p = MixedIntegerLinearProgram()
sage: p.add_constraint(2*p['x'] + p['y'] <= 1)
sage: p.add_constraint(3*p['y'] + p['x'] <= 2)
sage: P = p.polyhedron(); P
A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 4 vertices


3-D Polyhedron:

sage: p = MixedIntegerLinearProgram()
sage: p.add_constraint(2*p['x'] + p['y'] + 3*p['z'] <= 1)
sage: p.add_constraint(2*p['y'] + p['z'] + 3*p['x'] <= 1)
sage: p.add_constraint(2*p['z'] + p['x'] + 3*p['y'] <= 1)
sage: P = p.polyhedron(); P
A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 8 vertices


An empty polyhedron:

sage: p = MixedIntegerLinearProgram()
sage: p.add_constraint(2*p['x'] + p['y'] + 3*p['z'] <= 1)
sage: p.add_constraint(2*p['y'] + p['z'] + 3*p['x'] <= 1)
sage: p.add_constraint(2*p['z'] + p['x'] + 3*p['y'] >= 2)
sage: P = p.polyhedron(); P
The empty polyhedron in QQ^3


An unbounded polyhedron:

sage: p = MixedIntegerLinearProgram()
sage: p.add_constraint(2*p['x'] + p['y'] - p['z'] <= 1)
sage: P = p.polyhedron(); P
A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 3 vertices and 3 rays


A square (see trac ticket #14395)

sage: p = MixedIntegerLinearProgram()
sage: x,y = p['x'], p['y']
sage: p.set_min(x,None)
sage: p.set_min(y,None)
sage: p.add_constraint( x <= 1 )
sage: p.add_constraint( x >= -1 )
sage: p.add_constraint( y <= 1 )
sage: p.add_constraint( y >= -1 )
sage: p.polyhedron()
A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 4 vertices

remove_constraint(i)

Removes a constraint from self.

INPUT:

• i – Index of the constraint to remove.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: x, y = p[0], p[1]
sage: p.add_constraint(x + y, max = 10)
sage: p.add_constraint(x - y, max = 0)
sage: p.show()
Maximization:

Constraints:
x_0 + x_1 <= 10.0
x_0 - x_1 <= 0.0
x_0 <= 4.0
...
sage: p.remove_constraint(1)
sage: p.show()
Maximization:

Constraints:
x_0 + x_1 <= 10.0
x_0 <= 4.0
...
sage: p.number_of_constraints()
2

remove_constraints(constraints)

Remove several constraints.

INPUT:

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

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: x, y = p[0], p[1]
sage: p.add_constraint(x + y, max = 10)
sage: p.add_constraint(x - y, max = 0)
sage: p.show()
Maximization:

Constraints:
x_0 + x_1 <= 10.0
x_0 - x_1 <= 0.0
x_0 <= 4.0
...
sage: p.remove_constraints([0, 1])
sage: p.show()
Maximization:

Constraints:
x_0 <= 4.0
...
sage: p.number_of_constraints()
1


When checking for redundant constraints, make sure you remove only the constraints that were actually added. Problems could arise if you have a function that builds lps non-interactively, but it fails to check whether adding a constraint actually increases the number of constraints. The function might later try to remove constraints that are not actually there:

sage: p = MixedIntegerLinearProgram(check_redundant=True)
sage: x, y = p[0], p[1]
sage: p.add_constraint(x + y, max = 10)
sage: for each in xrange(10): p.add_constraint(x - y, max = 10)
sage: p.number_of_constraints()
3
sage: p.remove_constraints(range(1,9))
Traceback (most recent call last):
...
IndexError: pop index out of range
sage: p.remove_constraint(1)
sage: p.number_of_constraints()
2


We should now be able to add the old constraint back in:

sage: for each in xrange(10): p.add_constraint(x - y, max = 10)
sage: p.number_of_constraints()
3

set_binary(ee)

Sets a variable or a MIPVariable as binary.

INPUT:

• ee – An instance of MIPVariable or one of its elements.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: x = p.new_variable(nonnegative=True)


With the following instruction, all the variables from x will be binary:

sage: p.set_binary(x)
sage: p.set_objective(x[0] + x[1])


It is still possible, though, to set one of these variables as integer while keeping the others as they are:

sage: p.set_integer(x[3])


TESTS:

When ‘dim’ will be removed, also remove all the is_* and set_* functions the code that uses it:

sage: p = MixedIntegerLinearProgram()
sage: b = p.new_variable(dim=2)
sage: p.add_constraint(b[1][2] +  b[2][3] == 0)
sage: p.set_binary(b)

set_integer(ee)

Sets a variable or a MIPVariable as integer.

INPUT:

• ee – An instance of MIPVariable or one of its elements.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: x = p.new_variable(nonnegative=True)


With the following instruction, all the variables from x will be integers:

sage: p.set_integer(x)
sage: p.set_objective(x[0] + x[1])


It is still possible, though, to set one of these variables as binary while keeping the others as they are:

sage: p.set_binary(x[3])

set_max(v, max)

Sets the maximum value of a variable.

INPUT

• v – a variable.
• max – the maximum value the variable can take. When max=None, the variable has no upper bound.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable(nonnegative=True)
sage: p.set_objective(v[1])
sage: p.get_max(v[1])
sage: p.set_max(v[1],6)
sage: p.get_max(v[1])
6.0


With a MIPVariable as an argument:

sage: vv = p.new_variable(real=True, nonnegative=False)
sage: p.get_max(vv)
sage: p.get_max(vv[0])
sage: p.set_max(vv,5)
sage: p.get_max(vv[0])
5.0
sage: p.get_max(vv[9])
5.0

set_min(v, min)

Sets the minimum value of a variable.

INPUT:

• v – a variable.
• min – the minimum value the variable can take. When min=None, the variable has no lower bound.

• get_min() – get the minimum value of a variable.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable(nonnegative=True)
sage: p.set_objective(v[1])
sage: p.get_min(v[1])
0.0
sage: p.set_min(v[1],6)
sage: p.get_min(v[1])
6.0
sage: p.set_min(v[1], None)
sage: p.get_min(v[1])


With a MIPVariable as an argument:

sage: vv = p.new_variable(real=True, nonnegative=False)
sage: p.get_min(vv)
sage: p.get_min(vv[0])
sage: p.set_min(vv,5)
sage: p.get_min(vv[0])
5.0
sage: p.get_min(vv[9])
5.0

set_objective(obj)

Sets the objective of the MixedIntegerLinearProgram.

INPUT:

• obj – A linear function to be optimized. ( can also be set to None or 0 when just looking for a feasible solution )

EXAMPLE:

Let’s solve the following linear program:

Maximize:
x + 5 * y
Constraints:
x + 0.2 y       <= 4
1.5 * x + 3 * y <= 4
Variables:
x is Real (min = 0, max = None)
y is Real (min = 0, max = None)


This linear program can be solved as follows:

sage: p = MixedIntegerLinearProgram(maximization=True)
sage: x = p.new_variable(nonnegative=True)
sage: p.set_objective(x[1] + 5*x[2])
sage: round(p.solve(),5)
6.66667
sage: p.set_objective(None)
sage: _ = p.solve()

set_problem_name(name)

Sets the name of the MixedIntegerLinearProgram.

INPUT:

• name – A string representing the name of the MixedIntegerLinearProgram.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: p.set_problem_name("Test program")
sage: p
Mixed Integer Program "Test program" ( maximization, 0 variables, 0 constraints )

set_real(ee)

Sets a variable or a MIPVariable as real.

INPUT:

• ee – An instance of MIPVariable or one of its elements.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: x = p.new_variable(nonnegative=True)


With the following instruction, all the variables from x will be real:

   sage: p.set_real(x)
sage: p.set_objective(x[0] + x[1])

It is still possible, though, to set one of these
variables as binary while keeping the others as they are::

sage: p.set_binary(x[3])

show()

Displays the MixedIntegerLinearProgram in a human-readable way.

EXAMPLES:

When constraints and variables have names

sage: p = MixedIntegerLinearProgram(solver="GLPK")
sage: x = p.new_variable(name="Hey")
sage: p.set_objective(x[1] + x[2])
sage: p.add_constraint(-3*x[1] + 2*x[2], max=2, name="Constraint_1")
sage: p.show()
Maximization:
Hey[1] + Hey[2]
Constraints:
Constraint_1: -3.0 Hey[1] + 2.0 Hey[2] <= 2.0
Variables:
Hey[1] = x_0 is a continuous variable (min=0.0, max=+oo)
Hey[2] = x_1 is a continuous variable (min=0.0, max=+oo)


Without any names

sage: p = MixedIntegerLinearProgram(solver="GLPK")
sage: x = p.new_variable(nonnegative=True)
sage: p.set_objective(x[1] + x[2])
sage: p.show()
Maximization:
x_0 + x_1
Constraints:
-3.0 x_0 + 2.0 x_1 <= 2.0
Variables:
x_0 is a continuous variable (min=0.0, max=+oo)
x_1 is a continuous variable (min=0.0, max=+oo)


With $$\QQ$$ coefficients:

sage: p = MixedIntegerLinearProgram(solver='ppl')
sage: x = p.new_variable(nonnegative=True)
sage: p.set_objective(x[1] + 1/2*x[2])
sage: p.show()
Maximization:
x_0 + 1/2 x_1
Constraints:
constraint_0: -3/5 x_0 + 2/7 x_1 <= 2/5
Variables:
x_0 is a continuous variable (min=0, max=+oo)
x_1 is a continuous variable (min=0, max=+oo)

solve(log=None, objective_only=False)

Solves the MixedIntegerLinearProgram.

INPUT:

• log – integer (default: None) The verbosity level. Indicates whether progress should be printed during computation. The solver is initialized to report no progress.
• objective_only – Boolean variable.
• When set to True, only the objective function is returned.
• When set to False (default), the optimal numerical values are stored (takes computational time).

OUTPUT:

The optimal value taken by the objective function.

Warning

By default, all variables of a LP are assumed to be non-negative. See set_min() to change it.

EXAMPLES:

Consider the following linear program:

Maximize:
x + 5 * y
Constraints:
x + 0.2 y       <= 4
1.5 * x + 3 * y <= 4
Variables:
x is Real (min = 0, max = None)
y is Real (min = 0, max = None)


This linear program can be solved as follows:

   sage: p = MixedIntegerLinearProgram(maximization=True)
sage: x = p.new_variable(nonnegative=True)
sage: p.set_objective(x[1] + 5*x[2])
sage: round(p.solve(),6)
6.666667
sage: x = p.get_values(x)
sage: round(x[1],6) # abs tol 1e-15
0.0
sage: round(x[2],6)
1.333333

Computation of a maximum stable set in Petersen's graph::

sage: g = graphs.PetersenGraph()
sage: p = MixedIntegerLinearProgram(maximization=True)
sage: b = p.new_variable(nonnegative=True)
sage: p.set_objective(sum([b[v] for v in g]))
sage: for (u,v) in g.edges(labels=None):
sage: p.set_binary(b)
sage: p.solve(objective_only=True)
4.0


Constraints in the objective function are respected:

sage: p = MixedIntegerLinearProgram()
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.set_integer(x); p.set_integer(y)
sage: p.solve()
9.0

solver_parameter(name, value=None)

Return or define a solver parameter

The solver parameters are by essence solver-specific, which means their meaning heavily depends on the solver used.

(If you do not know which solver you are using, then you use use GLPK).

Aliases:

Very common parameters have aliases making them solver-independent. For example, the following:

sage: p = MixedIntegerLinearProgram(solver = "GLPK")
sage: p.solver_parameter("timelimit", 60)


Sets the solver to stop its computations after 60 seconds, and works with GLPK, CPLEX and Gurobi.

• "timelimit" – defines the maximum time spent on a computation. Measured in seconds.

Solver-specific parameters:

• GLPK : We have implemented very close to comprehensive coverage of the GLPK solver parameters for the simplex and integer optimization methods. For details, see the documentation of GLPKBackend.solver_parameter.

• CPLEX’s parameters are identified by a string. Their list is available on ILOG’s website.

The command

sage: p = MixedIntegerLinearProgram(solver = "CPLEX") # optional - CPLEX
sage: p.solver_parameter("CPX_PARAM_TILIM", 60)       # optional - CPLEX


works as intended.

• Gurobi’s parameters should all be available through this method. Their list is available on Gurobi’s website http://www.gurobi.com/documentation/5.5/reference-manual/node798.

INPUT:

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

EXAMPLE:

sage: p = MixedIntegerLinearProgram(solver = "GLPK")
sage: p.solver_parameter("timelimit", 60)
sage: p.solver_parameter("timelimit")
60.0

sum(L)

Efficiently computes the sum of a sequence of LinearFunction elements

INPUT:

Note

The use of the regular sum function is not recommended as it is much less efficient than this one

EXAMPLES:

sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable(nonnegative=True)


The following command:

sage: s = p.sum([v[i] for i in xrange(90)])


is much more efficient than:

sage: s = sum([v[i] for i in xrange(90)])

write_lp(filename)

Write the linear program as a LP file.

This function export the problem as a LP file.

INPUT:

• filename – The file in which you want the problem to be written.

EXAMPLE:

sage: p = MixedIntegerLinearProgram(solver="GLPK")
sage: x = p.new_variable(nonnegative=True)
sage: p.set_objective(x[1] + x[2])
sage: p.write_lp(os.path.join(SAGE_TMP, "lp_problem.lp"))
Writing problem data to ...
9 lines were written


write_mps(filename, modern=True)

Write the linear program as a MPS file.

This function export the problem as a MPS file.

INPUT:

• filename – The file in which you want the problem to be written.

• modern – Lets you choose between Fixed MPS and Free MPS

• True – Outputs the problem in Free MPS
• False – Outputs the problem in Fixed MPS

EXAMPLE:

sage: p = MixedIntegerLinearProgram(solver="GLPK")
sage: x = p.new_variable(nonnegative=True)
sage: p.set_objective(x[1] + x[2])
sage: p.write_mps(os.path.join(SAGE_TMP, "lp_problem.mps"))
Writing problem data to ...
17 records were written


For information about the MPS file format : http://en.wikipedia.org/wiki/MPS_%28format%29