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Linear Programming (Mixed Integer)
==================================
This document explains the use of linear programming (LP) -- and of
mixed integer linear programming (MILP) -- in Sage by illustrating it
with several problems it can solve. Most of the examples given are
motivated by graph-theoretic concerns, and should be understandable
without any specific knowledge of this field. As a tool in
Combinatorics, using linear programming amounts to understanding how
to reformulate an optimization (or existence) problem through linear
constraints.
This is a translation of a chapter from the book
`Calcul mathematique avec Sage <http://sagebook.gforge.inria.fr>`_.
Definition
----------
Here we present the usual definition of what a linear program is: it
is defined by a matrix `A: \mathbb{R}^m \mapsto \mathbb{R}^n`, along
with two vectors `b,c \in \mathbb{R}^n`. Solving a linear program is
searching for a vector `x` maximizing an *objective* function and
satisfying a set of constraints, i.e.
.. MATH::
c^t x = \max_{x' \text{ such that } Ax' \leq b} c^t x'
where the ordering `u \leq u'` between two vectors means that the
entries of `u'` are pairwise greater than the entries of `u`. We also
write:
.. MATH::
\text{Max: } & c^t x\\
\text{Such that: } & Ax \leq b
Equivalently, we can also say that solving a linear program amounts to
maximizing a linear function defined over a polytope (preimage or
`A^{-1} (\leq b)`). These definitions, however, do not tell us how to
use linear programming in combinatorics. In the following, we will
show how to solve optimization problems like the Knapsack problem, the
Maximum Matching problem, and a Flow problem.
Mixed integer linear programming
--------------------------------
There is a bad news coming along with this definition of linear
programming: an LP can be solved in polynomial time. This is indeed a
bad news, because this would mean that unless we define LP of
exponential size, we can not expect LP to solve NP-complete problems,
which would be a disappointment. On a brighter side, it becomes
NP-complete to solve a linear program if we are allowed to specify
constraints of a different kind: requiring that some variables be
integers instead of real values. Such an LP is actually called a "mixed
integer linear program" (some variables can be integers, some other
reals). Hence, we can expect to find in the MILP framework a *wide*
range of expressivity.
Practical
---------
The ``MILP`` class
^^^^^^^^^^^^^^^^^^
The ``MILP`` class in Sage represents a MILP! It is also used to
solve regular LP. It has a very small number of methods, meant to
define our set of constraints and variables, then to read the solution
found by the solvers once computed. It is also possible to export a
MILP defined with Sage to a ``.lp`` or ``.mps`` file, understood by most
solvers.
Let us ask Sage to solve the following LP:
.. MATH::
\text{Max: } & x + y - 3z\\
\text{Such that: } & x + 2y \leq 4\\
\text{} & 5z - y \leq 8\\
To achieve it, we need to define a corresponding ``MILP`` object,
along with the 3 variables we need::
sage: p = MixedIntegerLinearProgram()
sage: x, y, z = p['x'], p['y'], p['z']
Next, we set the objective function
.. link
::
sage: p.set_objective(x + y + 3*z)
And finally we set the constraints
.. link
::
sage: p.add_constraint(x + 2*y <= 4)
sage: p.add_constraint(5*z - y <= 8)
The ``solve`` method returns by default the optimal value reached by
the objective function
.. link
::
sage: round(p.solve(), 2)
8.8
We can read the optimal assignation found by the solver for `x, y` and
`z` through the ``get_values`` method
.. link
::
sage: round(p.get_values(x), 2)
4.0
sage: round(p.get_values(y), 2)
0.0
sage: round(p.get_values(z), 2)
1.6
Variables
^^^^^^^^^
The variables associated with an instance of ``MILP`` belong to the
``MIPVariable`` class, though we should not be concerned with this. In
the previous example, we obtained these variables through the
"shortcut" ``p['x']``, which is easy enough when our LP is defined
over a small number of variables. This being said, the LP/MILP we will
present afterwards very often require us to associate one -- or many
-- variables to each member of a list of objects, which can be
integers, or the vertices or edges of a graph, among plenty of other
alternatives. This means we will very soon need to talk about vectors
of variables or even dictionaries of variables.
If an LP requires us to define variables named `x_1, \dots, x_{15}`, we
will this time make use of the ``new_variable`` method
.. link
::
sage: x = p.new_variable()
It is now very easy to define constraints using our `15` variables
.. link
::
sage: p.add_constraint(x[1] + x[12] - x[14] >= 8)
Notice that we did not need to define the length of our
vector. Actually, ``x`` would accept any immutable object as a key, as
a dictionary would. We can now write
.. link
::
sage: p.add_constraint(x["I am a valid key"]
... + x[("a",pi)] <= 3)
Other LPs may require variables indexed several times. Of course, it is
already possible to emulate it by using tuples like `x[(2,3)]`, though
to keep the code understandable the method ``new_variable`` accepts as
a parameter the integer ``dim``, which lets us define the dimension of
the variable. We can now write
.. link
::
sage: y = p.new_variable(dim=2)
sage: p.add_constraint(y[3][2] + x[5] == 6)
Typed variables
"""""""""""""""
By default, all the LP variables 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 the methods ``set_min`` and
``set_max``.
It is also possible to change the type of a variable after it has been
created with the methods ``set_binary`` and ``set_integer``.
Basic linear programs
---------------------
Knapsack
^^^^^^^^
The *Knapsack* problem is the following: given a collection of items
having both a weight and a *usefulness*, we would like to fill a bag
whose capacity is constrained while maximizing the usefulness of the
items contained in the bag (we will consider the sum of the items'
usefulness). For the purpose of this tutorial, we set the restriction
that the bag can only carry a certain total weight.
To achieve this, we have to associate to each object `o` of our
collection `C` a binary variable ``taken[o]``, set to 1 when the
object is in the bag, and to 0 otherwise. We are trying to solve the
following MILP
.. MATH::
\text{Max: } & \sum_{o \in L} \text{usefulness}_o \times \text{taken}_o\\
\text{Such that: } & \sum_{o \in L} \text{weight}_o \times \text{taken}_o \leq C\\
Using Sage, we will give to our items a random weight::
sage: C = 1
.. link
::
sage: L = ["pan", "book", "knife", "gourd", "flashlight"]
.. link
::
sage: L.extend(["random_stuff_" + str(i) for i in range(20)])
.. link
::
sage: weight = {}
sage: usefulness = {}
.. link
::
sage: set_random_seed(685474)
sage: for o in L:
... weight[o] = random()
... usefulness[o] = random()
We can now define the MILP itself
.. link
::
sage: p = MixedIntegerLinearProgram()
sage: taken = p.new_variable(binary=True)
.. link
::
sage: p.add_constraint(sum(weight[o] * taken[o] for o in L) <= C)
.. link
::
sage: p.set_objective(sum(usefulness[o] * taken[o] for o in L))
.. link
::
sage: p.solve()
3.1502766806530307
sage: taken = p.get_values(taken)
The solution found is (of course) admissible
.. link
::
sage: sum(weight[o] * taken[o] for o in L)
0.6964959796619171
Should we take a flashlight?
.. link
::
sage: taken["flashlight"]
1.0
Wise advice. Based on purely random considerations.
Matching
--------
Given a graph `G`, a matching is a set of pairwise disjoint edges. The
empty set is a trivial matching. So we focus our attention on maximum
matchings: we want to find in a graph a matching whose cardinality is
maximal. Computing the maximum matching in a graph is a polynomial
problem, which is a famous result of Edmonds. Edmonds' algorithm is
based on local improvements and the proof that a given matching is
maximum if it cannot be improved. This algorithm is not the hardest to
implement among those graph theory can offer, though this problem can
be modeled with a very simple MILP.
To do it, we need -- as previously -- to associate a binary variable
to each one of our objects: the edges of our graph (a value of 1
meaning that the corresponding edge is included in the maximum
matching). Our constraint on the edges taken being that they are
disjoint, it is enough to require that, `x` and `y` being two edges
and `m_x, m_y` their associated variables, the inequality `m_x + m_y
\leq 1` is satisfied, as we are sure that the two of them cannot both
belong to the matching. Hence, we are able to write the MILP we
want. However, the number of inequalities can be easily decreased by
noticing that two edges cannot be taken simultaneously inside a
matching if and only if they have a common endpoint `v`. We can then
require instead that at most one edge incident to `v` be taken inside
the matching, which is a linear constraint. We will be solving:
.. MATH::
\text{Max: } & \sum_{e \in E(G)} m_e\\
\text{Such that: } & \forall v, \sum_{e \in E(G) \atop v \sim e} m_e \leq 1
Let us write the Sage code of this MILP::
sage: g = graphs.PetersenGraph()
sage: p = MixedIntegerLinearProgram()
sage: matching = p.new_variable(binary=True)
.. link
::
sage: p.set_objective(sum(matching[e] for e in g.edges(labels=False)))
.. link
::
sage: for v in g:
... p.add_constraint(sum(matching[e]
... for e in g.edges_incident(v, labels=False)) <= 1)
.. link
::
sage: p.solve()
5.0
.. link
::
sage: matching = p.get_values(matching)
sage: [e for e,b in matching.iteritems() if b == 1] # not tested
[(0, 1), (6, 9), (2, 7), (3, 4), (5, 8)]
Flows
-----
Yet another fundamental algorithm in graph theory: maximum flow! It
consists, given a directed graph and two vertices `s, t`, in sending a
maximum *flow* from `s` to `t` using the edges of `G`, each of them
having a maximal capacity.
.. image:: media/lp_flot1.png
:align: center
The definition of this problem is almost its LP formulation. We are
looking for real values associated to each edge, which would
represent the intensity of flow going through them, under two types of
constraints:
* The amount of flow arriving on a vertex (different from `s` or `t`)
is equal to the amount of flow leaving it.
* The amount of flow going through an edge is bounded by the capacity
of this edge.
This being said, we have to maximize the amount of flow leaving
`s`: all of it will end up in `t`, as the other vertices are sending
just as much as they receive. We can model the flow problem with the
following LP
.. MATH::
\text{Max: } & \sum_{sv \in G} f_{sv}\\
\text{Such that: } & \forall v \in G, {v \neq s \atop v \neq t}, \sum_{vu \in G} f_{vu} - \sum_{uv \in G} f_{uv} = 0\\
& \forall uv \in G, f_{uv} \leq 1\\
We will solve the flow problem on an orientation of Chvatal's
graph, in which all the edges have a capacity of 1::
sage: g = graphs.ChvatalGraph()
sage: g = g.minimum_outdegree_orientation()
.. link
::
sage: p = MixedIntegerLinearProgram()
sage: f = p.new_variable()
sage: s, t = 0, 2
.. link
::
sage: for v in g:
... if v != s and v != t:
... p.add_constraint(
... sum(f[(v,u)] for u in g.neighbors_out(v))
... - sum(f[(u,v)] for u in g.neighbors_in(v)) == 0)
.. link
::
sage: for e in g.edges(labels=False):
... p.add_constraint(f[e] <= 1)
.. link
::
sage: p.set_objective(sum(f[(s,u)] for u in g.neighbors_out(s)))
.. link
::
sage: p.solve()
2.0
.. image:: media/lp_flot2.png
:align: center
Solvers
-------
Sage solves linear programs by calling specific libraries. The
following libraries are currently supported:
* `CBC <http://www.coin-or.org/projects/Cbc.xml>`_: A solver from
`COIN-OR <http://www.coin-or.org/>`_
Provided under the open source license CPL, but incompatible with
GPL. CBC can be installed through the command ``install_package("cbc")``.
* `CPLEX
<http://www-01.ibm.com/software/integration/optimization/cplex/>`_:
A solver from `ILOG <http://www.ilog.com/>`_
Proprietary, but free for researchers and students.
* `GLPK <http://www.gnu.org/software/glpk/>`_: A solver from `GNU
<http://www.gnu.org/>`_
Licensed under the GPLv3. This solver is installed by default with Sage.
* `GUROBI <http://www.gurobi.com/>`_
Proprietary, but free for researchers and students.
Using CPLEX or GUROBI through Sage
----------------------------------
ILOG's CPLEX and GUROBI being proprietary softwares, you must be in possession
of several files to use it through Sage. In each case, the **expected** (it may
change !) filename is joined.
* A valid license file
* CPLEX : a ``.ilm`` file
* GUROBI : a ``.lic`` file
* A compiled version of the library
* CPLEX : ``libcplex.a``
* GUROBI : ``libgurobi45.so``
* The library file
* CPLEX : ``cplex.h``
* GUROBI : ``gurobi_c.h``
The environment variable defining the licence's path must also be set when
running Sage. You can append to your ``.bashrc`` file one of the following :
* For CPLEX ::
export ILOG_LICENSE_FILE=/path/to/the/license/ilog/ilm/access_1.ilm
* For GUROBI ::
export GRB_LICENSE_FILE=/path/to/the/license/gurobi.lic
As Sage also needs the files library and header files the easiest way is to
create symbolic links to these files in the appropriate directories:
* For CPLEX:
* ``libcplex.a`` -- in ``SAGE_ROOT/local/lib/``, type::
ln -s /path/to/lib/libcplex.a .
* ``cplex.h`` -- in ``SAGE_ROOT/local/include/``, type::
ln -s /path/to/include/cplex.h .
* ``cpxconst.h`` (if it exists) -- in ``SAGE_ROOT/local/include/``, type::
ln -s /path/to/include/cpxconst.h .
* For GUROBI
* ``libgurobi45.so`` -- in ``SAGE_ROOT/local/lib/``, type::
ln -s /path/to/lib/libgurobi45.so libgurobi.so
* ``gurobi_c.h`` -- in ``SAGE_ROOT/local/include/``, type::
ln -s /path/to/include/gurobi_c.h .
**It is very important that the names of the symbolic links in Sage's folders**
** be precisely as indicated. If the names differ, Sage will not notice that**
**the files are present**
Once this is done, Sage is to be asked to notice the changes by calling::
sage -b