r"""
Mixed integer linear programming
A linear program (`LP <http://en.wikipedia.org/wiki/Linear_programming>`_)
is an `optimization problem <http://en.wikipedia.org/wiki/Optimization_%28mathematics%29>`_
in the following form
.. MATH::
\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 <http://en.wikipedia.org/wiki/Mixed_integer_linear_programming>`_).
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`
and this additional inequality:
- `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 \; \forall i`, but what is the first non-trivial one with
`w_3 \geq 1`?
A mixed integer linear program can give you an answer:
#. You have to create an instance of :class:`MixedIntegerLinearProgram` and
-- in our case -- specify that it is a minimization.
#. Create a variable vector ``w`` via ``w = p.new_variable(integer=True)`` and
tell the system that it is over the integers.
#. Add those three equations as equality constraints via
:meth:`add_constraint <sage.numerical.mip.MixedIntegerLinearProgram.add_constraint>`.
#. Also add the inequality constraint.
#. Add an inequality constraint `w_3 \geq 1` to exclude the trivial solution.
#. By default, all variables have a minimum of `0`. We remove that constraint
via ``p.set_min(variable, None)``, see :meth:`set_min <sage.numerical.mip.MixedIntegerLinearProgram.set_min>`.
#. Specify the objective function via :meth:`set_objective <sage.numerical.mip.MixedIntegerLinearProgram.set_objective>`.
In our case that is just `w_3`. If it
is a pure constraint satisfaction problem, specify it as ``None``.
#. To check if everything is set up correctly, you can print the problem via
:meth:`show <sage.numerical.mip.MixedIntegerLinearProgram.show>`.
#. :meth:`Solve <sage.numerical.mip.MixedIntegerLinearProgram.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)
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.add_constraint(w[3] >= 1)
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
"""
include "../ext/stdsage.pxi"
include "../ext/interrupt.pxi"
include "../ext/cdefs.pxi"
from copy import copy,deepcopy
cdef class MixedIntegerLinearProgram:
r"""
The ``MixedIntegerLinearProgram`` class is the link between Sage, linear
programming (LP) and mixed integer programming (MIP) solvers.
See the Wikipedia article on `linear programming
<http://en.wikipedia.org/wiki/Linear_programming>`_ for further information
on linear programming and the documentation of the :mod:`MILP module
<sage.numerical.mip>` for its use in Sage.
A mixed integer program consists of variables, linear constraints on these
variables, and an objective function which is to be maximised or minimised
under these constraints. An instance of ``MixedIntegerLinearProgram`` also
requires the information on the direction of the optimization.
INPUT:
- ``solver`` -- 4 solvers should be available through this class:
- GLPK (``solver="GLPK"``). See the `GLPK
<http://www.gnu.org/software/glpk/>`_ web site.
- COIN Branch and Cut (``solver="Coin"``). See the `COIN-OR
<http://www.coin-or.org>`_ web site.
- CPLEX (``solver="CPLEX"``). See the `CPLEX
<http://www.ilog.com/products/cplex/>`_ web site.
- Gurobi (``solver="Gurobi"``). See the `Gurobi <http://www.gurobi.com/>`_
web site.
- If ``solver=None`` (default), the default solver is used (see
``default_mip_solver`` method.
- ``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`` -- whether to require the returned solver to
support constraint generation (excludes Coin). ``False by default``.
.. SEEALSO::
- :func:`default_mip_solver` -- Returns/Sets the default MIP solver.
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()
sage: p.set_objective(sum([b[v] for v in g]))
sage: for (u,v) in g.edges(labels=None):
... p.add_constraint(b[u] + b[v], max=1)
sage: p.set_binary(b)
sage: p.solve(objective_only=True)
4.0
"""
def __init__(self, solver = None, maximization=True, constraint_generation = False, check_redundant = False):
r"""
Constructor for the ``MixedIntegerLinearProgram`` class.
INPUT:
- ``solver`` -- 4 solvers should be available through this class:
- GLPK (``solver="GLPK"``). See the `GLPK
<http://www.gnu.org/software/glpk/>`_ web site.
- COIN Branch and Cut (``solver="Coin"``). See the `COIN-OR
<http://www.coin-or.org>`_ web site.
- CPLEX (``solver="CPLEX"``). See the `CPLEX
<http://www.ilog.com/products/cplex/>`_ web site. An interface to
CPLEX is not yet implemented.
- Gurobi (``solver="Gurobi"``). See the `Gurobi
<http://www.gurobi.com/>`_ web site.
-If ``solver=None`` (default), the default solver is used (see
``default_mip_solver`` method.
- ``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`` -- whether to require the returned solver to
support constraint generation (excludes Coin). ``False by default``.
- ``check_redundant`` -- whether to check that constraints added to the
program are redundant with constraints already in the program.
Only obvious redundancies are checked: to be considered redundant,
either a constraint is equal to another constraint in the program,
or it is a constant multiple of the other. To make this search
effective and efficient, constraints are normalized; thus, the
constraint `-x_1 < 0` will be stored as `x_1 > 0`.
.. SEEALSO::
- :meth:`default_mip_solver` -- Returns/Sets the default MIP solver.
EXAMPLE::
sage: p = MixedIntegerLinearProgram(maximization=True)
TESTS:
Checks that the objects are deallocated without invoking the cyclic garbage
collector (cf. :trac:`12616`)::
sage: del p
sage: def just_create_variables():
... p = MixedIntegerLinearProgram()
... b = p.new_variable()
... p.add_constraint(b[3]+b[6] <= 2)
... p.solve()
sage: C = sage.numerical.mip.MixedIntegerLinearProgram
sage: import gc
sage: _ = gc.collect() # avoid side effects of other doc tests
sage: len([x for x in gc.get_objects() if isinstance(x,C)])
0
We now disable the cyclic garbage collector. Since :trac:`12616` avoids
a reference cycle, the mixed integer linear program created in
``just_create_variables()`` is removed even without the cyclic garbage
collection::
sage: gc.disable()
sage: just_create_variables()
sage: len([x for x in gc.get_objects() if isinstance(x,C)])
0
sage: gc.enable()
"""
from sage.numerical.backends.generic_backend import get_solver
self._backend = get_solver(solver=solver,
constraint_generation=constraint_generation)
if not maximization:
self._backend.set_sense(-1)
self.__BINARY = 0
self.__REAL = -1
self.__INTEGER = 1
self._variables = {}
self._check_redundant = check_redundant
if check_redundant:
self._constraints = list()
def __repr__(self):
r"""
Returns a short description of the ``MixedIntegerLinearProgram``.
EXAMPLE::
sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable()
sage: p.add_constraint(v[1] + v[2], max=2)
sage: print p
Mixed Integer Program ( maximization, 2 variables, 1 constraints )
"""
cdef GenericBackend b = self._backend
return ("Mixed Integer Program "+
( "\"" +self._backend.problem_name()+ "\""
if (str(self._backend.problem_name()) != "") else "")+
" ( " + ("maximization" if b.is_maximization() else "minimization" ) +
", " + str(b.ncols()) + " variables, " +
str(b.nrows()) + " constraints )")
def __copy__(self):
r"""
Returns a copy of the current ``MixedIntegerLinearProgram`` instance.
EXAMPLE::
sage: p = MixedIntegerLinearProgram()
sage: p.add_constraint(p[0] + p[1], max = 10)
sage: q = copy(p)
sage: q.number_of_constraints()
1
"""
cdef MixedIntegerLinearProgram p = MixedIntegerLinearProgram(solver="GLPK")
try:
p._variables = copy(self._variables)
except AttributeError:
pass
try:
p._default_mipvariable = self._default_mipvariable
except AttributeError:
pass
try:
p._check_redundant = self._check_redundant
p._constraints = copy(self._constraints)
except AttributeError:
pass
p._backend = (<GenericBackend> self._backend).copy()
return p
def __getitem__(self, v):
r"""
Returns the symbolic variable corresponding to the key
from a default dictionary.
It returns the element asked, and otherwise creates it.
If necessary, it also creates the default dictionary.
This method lets the user define LinearProgram without having to
define independent dictionaries when it is not necessary for him.
EXAMPLE::
sage: p = MixedIntegerLinearProgram()
sage: p.set_objective(p['x'] + p['z'])
sage: p['x']
x_0
"""
try:
return self._default_mipvariable[v]
except TypeError:
self._default_mipvariable = self.new_variable()
return self._default_mipvariable[v]
def set_problem_name(self,name):
r"""
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 )
"""
self._backend.problem_name(name)
def new_variable(self, real=False, binary=False, integer=False, dim=1,name=""):
r"""
Returns an instance of ``MIPVariable`` associated
to the current instance of ``MixedIntegerLinearProgram``.
A new variable ``x`` is defined by::
sage: p = MixedIntegerLinearProgram()
sage: x = p.new_variable()
It behaves exactly as a usual dictionary would. It can use any key
argument you may like, as ``x[5]`` or ``x["b"]``, and has methods
``items()`` and ``keys()``.
Any of its fields exists, and is uniquely defined.
By default, all ``x[i]`` are assumed to be non-negative
reals. They can be defined as binary through the parameter
``binary=True`` (or integer with ``integer=True``). Lower and
upper bounds can be defined or re-defined (for instance when you want
some variables to be negative) using ``MixedIntegerLinearProgram`` methods
``set_min`` and ``set_max``.
INPUT:
- ``dim`` (integer) -- Defines the dimension of the dictionary.
If ``x`` has dimension `2`, its fields will be of the form
``x[key1][key2]``.
- ``binary, integer, real`` (boolean) -- Set one of these arguments
to ``True`` to ensure that the variable gets the corresponding
type. The default type is ``real``.
- ``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.
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)
sage: y = p.new_variable(dim=2, integer=True)
sage: p.add_constraint(x[2] + y[3][5], max=2)
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
"""
if sum([real, binary, integer]) >= 2:
raise ValueError("Exactly one of the available types has to be True")
if binary:
vtype = self.__BINARY
elif integer:
vtype = self.__INTEGER
else:
vtype = self.__REAL
v=MIPVariable(self, vtype, dim=dim,name=name)
return v
cpdef int number_of_constraints(self):
r"""
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
"""
return self._backend.nrows()
def constraints(self, indices = None):
r"""
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)]
"""
from sage.rings.integer import Integer as Integer
cdef int i
cdef str s
cdef GenericBackend b = self._backend
result = list()
if indices == None:
indices = range(b.nrows())
if isinstance(indices, int) or isinstance(indices, Integer):
lb, ub = b.row_bounds(indices)
return (lb, b.row(indices), ub)
elif isinstance(indices, list):
for i in indices:
lb, ub = b.row_bounds(i)
result.append((lb, b.row(i), ub))
return result
else:
raise ValueError, "constraints() requires a list of integers, though it will accommodate None or an integer."
def show(self):
r"""
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] is a continuous variable (min=0.0, max=+oo)
Hey[2] is a continuous variable (min=0.0, max=+oo)
Without any names ::
sage: p = MixedIntegerLinearProgram(solver = "GLPK")
sage: x = p.new_variable()
sage: p.set_objective(x[1] + x[2])
sage: p.add_constraint(-3*x[1] + 2*x[2], max=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)
"""
cdef int i, j
cdef double c
cdef GenericBackend b = self._backend
inv_variables = {}
for (v, id) in self._variables.iteritems():
inv_variables[id]=v
varid_name = {}
for 0<= i < b.ncols():
s = b.col_name(i)
varid_name[i] = ("x_"+str(i)) if s == "" else s
print ("Maximization:" if b.is_maximization() else "Minimization:")
print " ",
first = True
for 0<= i< b.ncols():
c = b.objective_coefficient(i)
if c == 0:
continue
print (("+ " if (not first and c>0) else "") +
("" if c == 1 else ("- " if c == -1 else str(c)+" "))+varid_name[i]
),
first = False
if b.obj_constant_term > 0.0: print "+", b.obj_constant_term
elif b.obj_constant_term < 0.0: print "-", -b.obj_constant_term
print
print "Constraints:"
for 0<= i < b.nrows():
indices, values = b.row(i)
lb, ub = b.row_bounds(i)
print " ",
if b.row_name(i):
print b.row_name(i)+":",
if lb is not None:
print str(lb)+" <=",
first = True
for j,c in sorted(zip(indices, values)):
if c == 0:
continue
print (("+ " if (not first and c>0) else "") +
("" if c == 1 else ("- " if c == -1 else (str(c) + " " if first and c < 0 else ("- " + str(abs(c)) + " " if c < 0 else str(c) + " "))))+varid_name[j]
),
first = False
print ("<= "+str(ub) if ub!=None else "")
print "Variables:"
for 0<= i < b.ncols():
print " " + varid_name[i] + " is",
if b.is_variable_integer(i):
print "an integer variable",
elif b.is_variable_binary(i):
print "a boolean variable",
else:
print "a continuous variable",
lb, ub = b.col_bounds(i)
print "(min=" + ( str(lb) if lb != None else "-oo" )+",",
print "max=" + ( str(ub) if ub != None else "+oo" )+")"
def write_mps(self,filename,modern=True):
r"""
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()
sage: x = p.new_variable()
sage: p.set_objective(x[1] + x[2])
sage: p.add_constraint(-3*x[1] + 2*x[2], max=2,name="OneConstraint")
sage: p.write_mps(SAGE_TMP+"/lp_problem.mps")
For information about the MPS file format :
http://en.wikipedia.org/wiki/MPS_%28format%29
"""
self._backend.write_mps(filename, modern)
def write_lp(self,filename):
r"""
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()
sage: x = p.new_variable()
sage: p.set_objective(x[1] + x[2])
sage: p.add_constraint(-3*x[1] + 2*x[2], max=2)
sage: p.write_lp(SAGE_TMP+"/lp_problem.lp")
For more information about the LP file format :
http://lpsolve.sourceforge.net/5.5/lp-format.htm
"""
self._backend.write_lp(filename)
def get_values(self, *lists):
r"""
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()
sage: y = p.new_variable(dim=2)
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])
"""
val = []
for l in lists:
if isinstance(l, MIPVariable):
if l.depth() == 1:
c = {}
for (k,v) in l.items():
c[k] = self._backend.get_variable_value(self._variables[v])
val.append(c)
else:
c = {}
for (k,v) in l.items():
c[k] = self.get_values(v)
val.append(c)
elif isinstance(l, list):
if len(l) == 1:
val.append([self.get_values(l[0])])
else:
c = []
[c.append(self.get_values(ll)) for ll in l]
val.append(c)
elif self._variables.has_key(l):
val.append(self._backend.get_variable_value(self._variables[l]))
if len(lists) == 1:
return val[0]
else:
return val
def set_objective(self,obj):
r"""
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()
sage: p.set_objective(x[1] + 5*x[2])
sage: p.add_constraint(x[1] + 2/10*x[2], max=4)
sage: p.add_constraint(1.5*x[1]+3*x[2], max=4)
sage: round(p.solve(),5)
6.66667
sage: p.set_objective(None)
sage: _ = p.solve()
"""
cdef list values = []
cdef int i
if obj != None:
f = obj.dict()
else:
f = {-1 : 0}
cdef double d = f.pop(-1,0.0)
for i in range(self._backend.ncols()):
values.append(f.get(i,0.0))
self._backend.set_objective(values, d)
def add_constraint(self, linear_function, max=None, min=None, name=None):
r"""
Adds a constraint to the ``MixedIntegerLinearProgram``.
INPUT:
- ``linear_function`` -- Two different types of arguments are possible:
- A linear function. In this case, arguments ``min`` or ``max``
have to be specified.
- A linear constraint of the form ``A <= B``, ``A >= B``,
``A <= B <= C``, ``A >= B >= C`` or ``A == B``. In this
case, arguments ``min`` and ``max`` will be ignored.
- ``max`` -- An upper bound on the constraint (set to ``None``
by default). This must be a numerical value.
- ``min`` -- A lower bound on the constraint. This must be a
numerical value.
- ``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()
sage: p.set_objective(x[1] + 5*x[2])
sage: p.add_constraint(x[1] + 0.2*x[2], max=4)
sage: p.add_constraint(1.5*x[1] + 3*x[2], max=4)
sage: round(p.solve(),6)
6.666667
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 )
The previous program can be rewritten::
sage: p = MixedIntegerLinearProgram(maximization=True)
sage: x = p.new_variable()
sage: p.set_objective(x[1] + 5*x[2])
sage: p.add_constraint(x[1] + 0.2*x[2] <= 4)
sage: p.add_constraint(1.5*x[1] + 3*x[2] <= 4)
sage: round(p.solve(), 5)
6.66667
TESTS:
Complex constraints::
sage: p = MixedIntegerLinearProgram(solver = "GLPK")
sage: b = p.new_variable()
sage: p.add_constraint( b[8] - b[15] <= 3*b[8] + 9)
sage: p.show()
Maximization:
<BLANKLINE>
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()
sage: p.add_constraint(sum([]),min=2)
Min/Max are numerical ::
sage: v = p.new_variable()
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 numerical
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 numerical
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:
<BLANKLINE>
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:
<BLANKLINE>
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:
<BLANKLINE>
Constraints:
1.0 <= x_0 - x_1
x_0 - x_1 <= 1.0
Variables:
x_0 is a continuous variable (min=0.0, max=+oo)
x_1 is a continuous variable (min=0.0, max=+oo)
"""
if linear_function is None or linear_function is 0:
return None
from sage.rings.all import RR
if ((min is not None and min not in RR)
or (max is not None and max not in RR)):
raise ValueError("min and max arguments are required to be numerical")
if isinstance(linear_function, LinearFunction):
f = linear_function.dict()
constant_coefficient = f.get(-1,0)
max = (max - constant_coefficient) if max != None else None
min = (min - constant_coefficient) if min != None else None
indices = []
values = []
if self._check_redundant:
b = self._backend
from __builtin__ import min as min_function
i = min_function([v for (v,coeff) in f.iteritems() if coeff != 0])
c = f[i]
C = [(v,coeff/c) for (v,coeff) in f.iteritems() if v != -1]
if c > 0:
min = min/c if min != None else None
max = max/c if max != None else None
else:
tempmin = max/c if max != None else None
tempmax = min/c if min != None else None
min, max = tempmin, tempmax
if (tuple(C),min,max) in self._constraints:
return None
else:
self._constraints.append((tuple(C),min,max))
else:
C = [(v,coeff) for (v,coeff) in f.iteritems() if v != -1]
if min == None and max == None:
raise ValueError("Both max and min are set to None ? Weird!")
self._backend.add_linear_constraint(C, min, max, name)
elif isinstance(linear_function,LinearConstraint):
functions = linear_function.constraints
if linear_function.equality:
self.add_constraint(functions[0] - functions[1], min=0, max=0, name=name)
elif len(functions) == 2:
self.add_constraint(functions[0] - functions[1], max=0, name=name)
else:
self.add_constraint(functions[0] - functions[1], max=0, name=name)
self.add_constraint(functions[1] - functions[2], max=0, name=name)
def remove_constraint(self, int i):
r"""
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.add_constraint(x, max = 4)
sage: p.show()
Maximization:
<BLANKLINE>
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:
<BLANKLINE>
Constraints:
x_0 + x_1 <= 10.0
x_0 <= 4.0
...
sage: p.number_of_constraints()
2
"""
if self._check_redundant: self._constraints.pop(i)
self._backend.remove_constraint(i)
def remove_constraints(self, constraints):
r"""
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.add_constraint(x, max = 4)
sage: p.show()
Maximization:
<BLANKLINE>
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:
<BLANKLINE>
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.add_constraint(x, max = 4)
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
"""
if self._check_redundant:
for i in sorted(constraints,reverse=True):
self._constraints.pop(i)
self._backend.remove_constraints(constraints)
def set_binary(self, ee):
r"""
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()
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])
sage: p.add_constraint(-3*x[0] + 2*x[1], max=2)
It is still possible, though, to set one of these
variables as real while keeping the others as they are::
sage: p.set_real(x[3])
"""
cdef MIPVariable e
e = <MIPVariable> ee
if isinstance(e, MIPVariable):
e._vtype = self.__BINARY
if e.depth() == 1:
for v in e.values():
self._backend.set_variable_type(self._variables[v],self.__BINARY)
else:
for v in e.keys():
self.set_binary(e[v])
elif self._variables.has_key(e):
self._backend.set_variable_type(self._variables[e],self.__BINARY)
else:
raise ValueError("e must be an instance of MIPVariable or one of its elements.")
def is_binary(self, e):
r"""
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()
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
"""
return self._backend.is_variable_binary(self._variables[e])
def set_integer(self, ee):
r"""
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()
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])
sage: p.add_constraint(-3*x[0] + 2*x[1], max=2)
It is still possible, though, to set one of these
variables as real while keeping the others as they are::
sage: p.set_real(x[3])
"""
cdef MIPVariable e
e = <MIPVariable> ee
if isinstance(e, MIPVariable):
e._vtype = self.__INTEGER
if e.depth() == 1:
for v in e.values():
self._backend.set_variable_type(self._variables[v],self.__INTEGER)
else:
for v in e.keys():
self.set_integer(e[v])
elif self._variables.has_key(e):
self._backend.set_variable_type(self._variables[e],self.__INTEGER)
else:
raise ValueError("e must be an instance of MIPVariable or one of its elements.")
def is_integer(self, e):
r"""
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()
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
"""
return self._backend.is_variable_integer(self._variables[e])
def set_real(self,ee):
r"""
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()
With the following instruction, all the variables
from x will be real (they are by default, though)::
sage: p.set_real(x)
sage: p.set_objective(x[0] + x[1])
sage: p.add_constraint(-3*x[0] + 2*x[1], max=2)
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])
"""
cdef MIPVariable e
e = <MIPVariable> ee
if isinstance(e, MIPVariable):
e._vtype = self.__REAL
if e.depth() == 1:
for v in e.values():
self._backend.set_variable_type(self._variables[v],self.__REAL)
else:
for v in e.keys():
self.set_real(e[v])
elif self._variables.has_key(e):
self._backend.set_variable_type(self._variables[e],self.__REAL)
else:
raise ValueError("e must be an instance of MIPVariable or one of its elements.")
def is_real(self, e):
r"""
Tests whether the variable is real. Variables are real by default.
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()
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
"""
return self._backend.is_variable_continuous(self._variables[e])
def solve(self, solver=None, log=None, objective_only=False):
r"""
Solves the ``MixedIntegerLinearProgram``.
INPUT:
- ``solver`` -- DEPRECATED -- the solver now has to be set
when calling the class' constructor
- ``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.
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()
sage: p.set_objective(x[1] + 5*x[2])
sage: p.add_constraint(x[1] + 0.2*x[2], max=4)
sage: p.add_constraint(1.5*x[1] + 3*x[2], max=4)
sage: round(p.solve(),6)
6.666667
sage: x = p.get_values(x)
sage: round(x[1],6)
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()
sage: p.set_objective(sum([b[v] for v in g]))
sage: for (u,v) in g.edges(labels=None):
... p.add_constraint(b[u] + b[v], max=1)
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
"""
if solver != None:
raise ValueError("Solver argument deprecated. This parameter now has to be set when calling the class' constructor")
if log != None: self._backend.set_verbosity(log)
self._backend.solve()
return self._backend.get_objective_value()
def set_min(self, v, min):
r"""
Sets the minimum value of a variable.
INPUT:
- ``v`` -- a variable (not a ``MIPVariable``, but one of its
elements).
- ``min`` -- the minimum value the variable can take.
When ``min=None``, the variable has no lower bound.
EXAMPLE::
sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable()
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])
"""
self._backend.variable_lower_bound(self._variables[v], min)
def set_max(self, v, max):
r"""
Sets the maximum value of a variable.
INPUT
- ``v`` -- a variable (not a ``MIPVariable``, but one of its
elements).
- ``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()
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
"""
self._backend.variable_upper_bound(self._variables[v], max)
def get_min(self, v):
r"""
Returns the minimum value of a variable.
INPUT:
- ``v`` -- a variable (not a ``MIPVariable``, but one of its elements).
OUTPUT:
Minimum value of the variable, or ``None`` if
the variable has no lower bound.
EXAMPLE::
sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable()
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])
"""
return self._backend.variable_lower_bound(self._variables[v])
def get_max(self, v):
r"""
Returns the maximum value of a variable.
INPUT:
- ``v`` -- a variable (not a ``MIPVariable``, but one of its elements).
OUTPUT:
Maximum value of the variable, or ``None`` if
the variable has no upper bound.
EXAMPLE::
sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable()
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
"""
return self._backend.variable_upper_bound(self._variables[v])
def solver_parameter(self, 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 are using 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 both with GLPK and CPLEX.
- ``"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
:meth:`GLPKBackend.solver_parameter
<sage.numerical.backends.glpk_backend.GLPKBackend.solver_parameter>`.
- CPLEX's parameters are identified by a string. Their
list is available `on ILOG's website
<http://publib.boulder.ibm.com/infocenter/odmeinfo/v3r4/index.jsp?topic=/ilog.odms.ide.odme.help/Content/Optimization/Documentation/ODME/_pubskel/ODME_pubskels/startall_ODME34_Eclipse1590.html>`_.
The command ::
sage: p = MixedIntegerLinearProgram(solver = "CPLEX") # optional - CPLEX
sage: p.solver_parameter("CPX_PARAM_TILIM", 60) # optional - CPLEX
works as intended.
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
"""
if value is None:
return self._backend.solver_parameter(name)
else:
self._backend.solver_parameter(name, value)
class MIPSolverException(Exception):
r"""
Exception raised when the solver fails.
"""
def __init__(self, value):
r"""
Constructor for ``MIPSolverException``.
``MIPSolverException`` is the exception raised when the solver fails.
EXAMPLE::
sage: from sage.numerical.mip import MIPSolverException
sage: MIPSolverException("Error")
MIPSolverException()
TESTS:
No continuous solution::
sage: p=MixedIntegerLinearProgram(solver="GLPK")
sage: v=p.new_variable()
sage: p.add_constraint(v[0],max=5.5)
sage: p.add_constraint(v[0],min=7.6)
sage: p.set_objective(v[0])
Tests of GLPK's Exceptions::
sage: p.solve()
Traceback (most recent call last):
...
MIPSolverException: 'GLPK : Solution is undefined'
No integer solution::
sage: p=MixedIntegerLinearProgram(solver="GLPK")
sage: v=p.new_variable()
sage: p.add_constraint(v[0],max=5.6)
sage: p.add_constraint(v[0],min=5.2)
sage: p.set_objective(v[0])
sage: p.set_integer(v)
Tests of GLPK's Exceptions::
sage: p.solve()
Traceback (most recent call last):
...
MIPSolverException: 'GLPK : Solution is undefined'
"""
self.value = value
def __str__(self):
r"""
Returns the value of the instance of ``MIPSolverException``.
EXAMPLE::
sage: from sage.numerical.mip import MIPSolverException
sage: e = MIPSolverException("Error")
sage: print e
'Error'
"""
return repr(self.value)
cdef class MIPVariable:
r"""
``MIPVariable`` is a variable used by the class
``MixedIntegerLinearProgram``.
"""
def __cinit__(self, p, vtype, dim=1, name=""):
r"""
Constructor for ``MIPVariable``.
INPUT:
- ``p`` -- the instance of ``MixedIntegerLinearProgram`` to which the
variable is to be linked.
- ``vtype`` (integer) -- Defines the type of the variables
(default is ``REAL``).
- ``dim`` -- the integer defining the definition of the variable.
- ``name`` -- A name for the ``MIPVariable``.
For more informations, see the method
``MixedIntegerLinearProgram.new_variable``.
EXAMPLE::
sage: p=MixedIntegerLinearProgram()
sage: v=p.new_variable()
"""
self._dim = dim
self._dict = {}
self._p = p
self._vtype = vtype
self._hasname = (len(name) >0)
cdef char *name_c = name
self._name = <char*>sage_malloc(len(name)+1)
strcpy(self._name, name_c)
def __dealloc__(self):
if self._name:
sage_free(self._name)
def __getitem__(self, i):
r"""
Returns the symbolic variable corresponding to the key.
Returns the element asked, otherwise creates it.
(When depth>1, recursively creates the variables).
EXAMPLE::
sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable()
sage: p.set_objective(v[0] + v[1])
sage: v[0]
x_0
"""
cdef MIPVariable s = self
cdef int j
if self._dict.has_key(i):
return self._dict[i]
elif self._dim == 1:
j = self._p._backend.add_variable(0.0, None, False, True, False, 0.0,
(str(self._name) + "[" + str(i) + "]")
if self._hasname else None)
v = LinearFunction({j : 1})
self._p._variables[v] = j
self._p._backend.set_variable_type(j,self._vtype)
self._dict[i] = v
return v
else:
self._dict[i] = MIPVariable(
self._p,
self._vtype,
dim=self._dim-1,
name = ("" if not self._hasname
else (str(self._name) + "[" + str(i) + "]")))
return self._dict[i]
def __repr__(self):
r"""
Returns a representation of self.
EXAMPLE::
sage: p=MixedIntegerLinearProgram()
sage: v=p.new_variable(dim=3)
sage: v
MIPVariable of dimension 3.
sage: v[2][5][9]
x_0
sage: v
MIPVariable of dimension 3.
"""
return "MIPVariable of dimension " + str(self._dim) + "."
def keys(self):
r"""
Returns the keys already defined in the dictionary.
EXAMPLE::
sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable()
sage: p.set_objective(v[0] + v[1])
sage: v.keys()
[0, 1]
"""
return self._dict.keys()
def items(self):
r"""
Returns the pairs (keys,value) contained in the dictionary.
EXAMPLE::
sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable()
sage: p.set_objective(v[0] + v[1])
sage: v.items()
[(0, x_0), (1, x_1)]
"""
return self._dict.items()
def depth(self):
r"""
Returns the current variable's depth.
EXAMPLE::
sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable()
sage: p.set_objective(v[0] + v[1])
sage: v.depth()
1
"""
return self._dim
def values(self):
r"""
Returns the symbolic variables associated to the current dictionary.
EXAMPLE::
sage: p = MixedIntegerLinearProgram()
sage: v = p.new_variable()
sage: p.set_objective(v[0] + v[1])
sage: v.values()
[x_0, x_1]
"""
return self._dict.values()
class LinearFunction:
r"""
An elementary algebra to represent symbolic linear functions.
"""
def __init__(self,f):
r"""
Constructor taking a dictionary or a numerical value as its argument.
A linear function is represented as a dictionary. The
values are the coefficient of the variable represented
by the keys ( which are integers ). The key ``-1``
corresponds to the constant term.
EXAMPLES:
With a dictionary::
sage: from sage.numerical.mip import LinearFunction
sage: LinearFunction({0 : 1, 3 : -8})
x_0 -8 x_3
Using the constructor with a numerical value::
sage: from sage.numerical.mip import LinearFunction
sage: LinearFunction(25)
25
"""
if isinstance(f, dict):
self._f = f
else:
self._f = {-1:f}
def dict(self):
r"""
Returns the dictionary corresponding to the Linear Function.
A linear function is represented as a dictionary. The
value are the coefficient of the variable represented
by the keys ( which are integers ). The key ``-1``
corresponds to the constant term.
EXAMPLE::
sage: from sage.numerical.mip import LinearFunction
sage: lf = LinearFunction({0 : 1, 3 : -8})
sage: lf.dict()
{0: 1, 3: -8}
"""
return self._f
def __add__(self,b):
r"""
Defining the + operator
EXAMPLE::
sage: from sage.numerical.mip import LinearFunction
sage: LinearFunction({0 : 1, 3 : -8}) + LinearFunction({2 : 5, 3 : 2}) - 16
-16 +x_0 +5 x_2 -6 x_3
"""
if isinstance(b, LinearFunction):
e = deepcopy(self._f)
for (id,coeff) in b.dict().iteritems():
e[id] = self._f.get(id,0) + coeff
return LinearFunction(e)
else:
el = deepcopy(self)
el.dict()[-1] = el.dict().get(-1,0) + b
return el
def __neg__(self):
r"""
Defining the - operator (opposite).
EXAMPLE::
sage: from sage.numerical.mip import LinearFunction
sage: -LinearFunction({0 : 1, 3 : -8})
-1 x_0 +8 x_3
"""
return LinearFunction(dict([(id,-coeff) for (id, coeff) in self._f.iteritems()]))
def __sub__(self,b):
r"""
Defining the - operator (substraction).
EXAMPLE::
sage: from sage.numerical.mip import LinearFunction
sage: LinearFunction({2 : 5, 3 : 2}) - 3
-3 +5 x_2 +2 x_3
sage: LinearFunction({0 : 1, 3 : -8}) - LinearFunction({2 : 5, 3 : 2}) - 16
-16 +x_0 -5 x_2 -10 x_3
"""
if isinstance(b, LinearFunction):
e = deepcopy(self._f)
for (id,coeff) in b.dict().iteritems():
e[id] = self._f.get(id,0) - coeff
return LinearFunction(e)
else:
el = deepcopy(self)
el.dict()[-1] = self._f.get(-1,0) - b
return el
def __radd__(self,b):
r"""
Defining the + operator (right side).
EXAMPLE::
sage: from sage.numerical.mip import LinearFunction
sage: 3 + LinearFunction({2 : 5, 3 : 2})
3 +5 x_2 +2 x_3
"""
if isinstance(self,LinearFunction):
return self.__add__(b)
else:
return b.__add__(self)
def __rsub__(self,b):
r"""
Defining the - operator (right side).
EXAMPLE::
sage: from sage.numerical.mip import LinearFunction
sage: 3 - LinearFunction({2 : 5, 3 : 2})
3 -5 x_2 -2 x_3
"""
if isinstance(self,LinearFunction):
return (-self).__add__(b)
else:
return b.__sub__(self)
def __mul__(self,b):
r"""
Defining the * operator.
EXAMPLE::
sage: from sage.numerical.mip import LinearFunction
sage: LinearFunction({2 : 5, 3 : 2}) * 3
15 x_2 +6 x_3
"""
return LinearFunction(dict([(id,b*coeff) for (id, coeff) in self._f.iteritems()]))
def __rmul__(self,b):
r"""
Defining the * operator (right side).
EXAMPLE::
sage: from sage.numerical.mip import LinearFunction
sage: 3 * LinearFunction({2 : 5, 3 : 2})
15 x_2 +6 x_3
"""
return self.__mul__(b)
def __repr__(self):
r"""
Returns a string version of the linear function.
EXAMPLE::
sage: from sage.numerical.mip import LinearFunction
sage: LinearFunction({2 : 5, 3 : 2})
5 x_2 +2 x_3
"""
cdef dict d = deepcopy(self._f)
cdef bint first = True
t = ""
if d.has_key(-1):
coeff = d.pop(-1)
if coeff!=0:
t = str(coeff)
first = False
cdef list l = sorted(d.items())
for id,coeff in l:
if coeff!=0:
if not first:
t += " "
t += ("+" if (not first and coeff >= 0) else "") + (str(coeff) + " " if coeff != 1 else "") + "x_" + str(id)
first = False
return t
def __le__(self,other):
r"""
Defines the <= operator
EXAMPLE::
sage: from sage.numerical.mip import LinearFunction
sage: LinearFunction({2 : 5, 3 : 2}) <= LinearFunction({2 : 3, 9 : 2})
5 x_2 +2 x_3 <= 3 x_2 +2 x_9
"""
return LinearConstraint(self).__le__(other)
def __lt__(self,other):
r"""
Defines the < operator
EXAMPLE::
sage: from sage.numerical.mip import LinearFunction
sage: LinearFunction({2 : 5, 3 : 2}) < LinearFunction({2 : 3, 9 : 2})
Traceback (most recent call last):
...
ValueError: The strict operators are not defined. Use <= and >= instead.
"""
return LinearConstraint(self).__lt__(other)
def __gt__(self,other):
r"""
Defines the > operator
EXAMPLE::
sage: from sage.numerical.mip import LinearFunction
sage: LinearFunction({2 : 5, 3 : 2}) > LinearFunction({2 : 3, 9 : 2})
Traceback (most recent call last):
...
ValueError: The strict operators are not defined. Use <= and >= instead.
"""
return LinearConstraint(self).__gt__(other)
def __ge__(self,other):
r"""
Defines the >= operator
EXAMPLE::
sage: from sage.numerical.mip import LinearFunction
sage: LinearFunction({2 : 5, 3 : 2}) >= LinearFunction({2 : 3, 9 : 2})
3 x_2 +2 x_9 <= 5 x_2 +2 x_3
"""
return LinearConstraint(self).__ge__(other)
def __hash__(self):
r"""
Defines a ``__hash__`` function
EXAMPLE::
sage: from sage.numerical.mip import LinearFunction
sage: d = {}
sage: d[LinearFunction({2 : 5, 3 : 2})] = 3
"""
return id(self)
def __eq__(self,other):
r"""
Defines the == operator
EXAMPLE::
sage: from sage.numerical.mip import LinearFunction
sage: LinearFunction({2 : 5, 3 : 2}) == LinearFunction({2 : 3, 9 : 2})
5 x_2 +2 x_3 = 3 x_2 +2 x_9
"""
return LinearConstraint(self).__eq__(other)
def Sum(L):
r"""
Efficiently computes the sum of a sequence of
``LinearFunction`` elements
INPUT:
- ``L`` a list of ``LinearFunction`` instances.
.. 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()
The following command::
sage: from sage.numerical.mip import Sum
sage: s = Sum([v[i] for i in xrange(90)])
is much more efficient than::
sage: s = sum([v[i] for i in xrange(90)])
"""
d = {}
for v in L:
for (id,coeff) in v._f.iteritems():
d[id] = coeff + d.get(id,0)
return LinearFunction(d)
class LinearConstraint:
"""
A class to represent formal Linear Constraints.
A Linear Constraint being an inequality between
two linear functions, this class lets the user
write ``LinearFunction1 <= LinearFunction2``
to define the corresponding constraint, which
can potentially involve several layers of such
inequalities (``(A <= B <= C``), or even equalities
like ``A == B``.
This class has no reason to be instanciated by the
user, and is meant to be used by instances of
MixedIntegerLinearProgram.
INPUT:
- ``c`` -- A ``LinearFunction``
EXAMPLE::
sage: p = MixedIntegerLinearProgram()
sage: b = p.new_variable()
sage: b[2]+2*b[3] <= b[8]-5
x_0 +2 x_1 <= -5 +x_2
"""
def __init__(self, c):
r"""
Constructor for ``LinearConstraint``
INPUT:
- ``c`` -- A linear function (see ``LinearFunction``).
EXAMPLE::
sage: p = MixedIntegerLinearProgram()
sage: b = p.new_variable()
sage: b[2]+2*b[3] <= b[8]-5
x_0 +2 x_1 <= -5 +x_2
"""
self.equality = False
self.constraints = []
if isinstance(c, LinearFunction):
self.constraints.append(c)
else:
self.constraints.append(LinearFunction(c))
def __repr__(self):
r"""
Returns a string representation of the constraint.
EXAMPLE::
sage: p = MixedIntegerLinearProgram()
sage: b = p.new_variable()
sage: print b[3] <= b[8] + 9
x_0 <= 9 +x_1
"""
if self.equality:
return str(self.constraints[0]) + " = " + str(self.constraints[1])
else:
first = True
s = ""
for c in self.constraints:
s += (" <= " if not first else "") + c.__repr__()
first = False
return s
def __eq__(self,other):
r"""
Defines the == operator
EXAMPLE::
sage: p = MixedIntegerLinearProgram()
sage: b = p.new_variable()
sage: print b[3] == b[8] + 9
x_0 = 9 +x_1
"""
if not isinstance(other, LinearConstraint):
other = LinearConstraint(other)
if len(self.constraints) == 1 and len(other.constraints) == 1:
self.constraints.extend(other.constraints)
self.equality = True
return self
else:
raise ValueError("Impossible to mix equality and inequality in the same equation")
def __le__(self,other):
r"""
Defines the <= operator
EXAMPLE::
sage: p = MixedIntegerLinearProgram()
sage: b = p.new_variable()
sage: print b[3] <= b[8] + 9
x_0 <= 9 +x_1
"""
if not isinstance(other, LinearConstraint):
other = LinearConstraint(other)
if self.equality or other.equality:
raise ValueError("Impossible to mix equality and inequality in the same equation")
self.constraints.extend(other.constraints)
return self
def __lt__(self, other):
r"""
Prevents the use of the stricts operators ``<`` and ``>``
EXAMPLE::
sage: p = MixedIntegerLinearProgram()
sage: b = p.new_variable()
sage: print b[3] < b[8] + 9
Traceback (most recent call last):
...
ValueError: The strict operators are not defined. Use <= and >= instead.
sage: print b[3] > b[8] + 9
Traceback (most recent call last):
...
ValueError: The strict operators are not defined. Use <= and >= instead.
"""
raise ValueError("The strict operators are not defined. Use <= and >= instead.")
__gt__ = __lt__
def __ge__(self,other):
r"""
Defines the >= operator
EXAMPLE::
sage: p = MixedIntegerLinearProgram()
sage: b = p.new_variable()
sage: print b[3] >= b[8] + 9
9 +x_1 <= x_0
"""
if not isinstance(other, LinearConstraint):
other = LinearConstraint(other)
if self.equality or other.equality:
raise ValueError("Impossible to mix equality and inequality in the same equation")
self.constraints = other.constraints + self.constraints
return self
