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    #9836: reviewer patch: Linear Programming Thematic Tutorial
    
    diff --git a/doc/en/thematic_tutorials/linear_programming.rst b/doc/en/thematic_tutorials/linear_programming.rst
    a b  
    1 (Mixed Integer) Linear Programming
     1Linear Programming (Mixed Integer)
    22==================================
    33
    4 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 understandabe 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.
     4This document explains the use of linear programming (LP) -- and of
     5mixed integer linear programming (MILP) -- in Sage by illustrating it
     6with several problems it can solve. Most of the examples given are
     7motivated by graph-theoretic concerns, and should be understandable
     8without any specific knowledge of this field. As a tool in
     9Combinatorics, using linear programming amounts to understanding how
     10to reformulate an optimization (or existence) problem through linear
     11constraints.
    512
    6 .. centered::
    7     *(this is a translation of a chapter from the book "Calcul mathematique avec Sage")*
     13This is a translation of a chapter from the book
     14`Calcul mathematique avec Sage <http://sagebook.gforge.inria.fr>`_.
     15
    816
    917Definition
    1018----------
    1119
    12 Set to bear once more upon our shoulders the heavy weight of mathematical formalism, let us give here 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 a quest for a vector `x` maximizing an *objective* function and satisfying a set of constraints, i.e.
     20Here we present the usual definition of what a linear program is: it
     21is defined by a matrix `A: \mathbb{R}^m \mapsto \mathbb{R}^n`, along
     22with two vectors `b,c \in \mathbb{R}^n`. Solving a linear program is
     23searching for a vector `x` maximizing an *objective* function and
     24satisfying a set of constraints, i.e.
    1325
    1426.. MATH::
    15       c^t x = \max_{x' \text{ such that}Ax'\leq b}c^t x'`
     27    c^t x = \max_{x' \text{ such that } Ax' \leq b} c^t x'
    1628
    17 (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 :
     29where the ordering `u \leq u'` between two vectors means that the
     30entries of `u'` are pairwise greater than the entries of `u`. We also
     31write:
    1832
    1933.. MATH::
    20       \text{Max : }&c^t x\\
    21       \text{ Such that : }&Ax\leq b
     34    \text{Max: } & c^t x\\
     35    \text{Such that: } & Ax \leq b
    2236
    23 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 try to fix this by showing how to solve optimization problems like the Knapsack problem, the Maximum Matching problem, and a Flow problem.
     37Equivalently, we can also say that solving a linear program amounts to
     38maximizing a linear function defined over a polytope (preimage or
     39`A^{-1} (\leq b)`). These definitions, however, do not tell us how to
     40use linear programming in combinatorics. In the following, we will
     41show how to solve optimization problems like the Knapsack problem, the
     42Maximum Matching problem, and a Flow problem.
    2443
    2544
    26 Mixed Integer Linear Programming
     45Mixed integer linear programming
    2746--------------------------------
    2847
    29 There is a bad news coming along with this definition of linear programming : a LP can be solved in polynomial time. This is indeed a bad news, because this would mean than 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 a 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.
     48There is a bad news coming along with this definition of linear
     49programming: an LP can be solved in polynomial time. This is indeed a
     50bad news, because this would mean that unless we define LP of
     51exponential size, we can not expect LP to solve NP-complete problems,
     52which would be a disappointment. On a brighter side, it becomes
     53NP-complete to solve a linear program if we are allowed to specify
     54constraints of a different kind: requiring that some variables be
     55integers instead of real values. Such an LP is actually called a "mixed
     56integer linear program" (some variables can be integers, some other
     57reals). Hence, we can expect to find in the MILP framework a *wide*
     58range of expressivity.
    3059
    31 Practically
    32 -----------
     60
     61Practical
     62---------
    3363
    3464The ``MILP`` class
    3565^^^^^^^^^^^^^^^^^^
    3666
    37 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.
     67The ``MILP`` class in Sage represents a MILP! It is also used to
     68solve regular LP. It has a very small number of methods, meant to
     69define our set of constraints and variables, then to read the solution
     70found by the solvers once computed. It is also possible to export a
     71MILP defined with Sage to a ``.lp`` or ``.mps`` file, understood by most
     72solvers.
    3873
    39 Let us ask Sage to solve the following LP :
     74Let us ask Sage to solve the following LP:
    4075
    4176.. MATH::
    42   \text{Max : }&x+y-3z\\
    43   \text{Such that : }&x+2y \leq 4\\
    44   \text{}&5z - y \leq 8\\
     77    \text{Max: } & x + y - 3z\\
     78    \text{Such that: } & x + 2y \leq 4\\
     79    \text{} & 5z - y \leq 8\\
    4580
    46 To achieve it, we need to define a corresponding ``MILP`` object, along with the 3 variables we need ::
     81To achieve it, we need to define a corresponding ``MILP`` object,
     82along with the 3 variables we need::
    4783
    48   sage: p = MixedIntegerLinearProgram()
    49   sage: x, y, z = p['x'], p['y'], p['z']
     84    sage: p = MixedIntegerLinearProgram()
     85    sage: x, y, z = p['x'], p['y'], p['z']
    5086
    51 The objective function
     87Next, we set the objective function
    5288
    5389.. link
    5490
    5591::
    5692
    57   sage: p.set_objective( x + y + 3*z )
     93    sage: p.set_objective(x + y + 3*z)
    5894
    59 And finally the constraints
     95And finally we set the constraints
    6096
    6197.. link
    6298
    6399::
    64100
    65   sage: p.add_constraint( x + 2*y <= 4 )
    66   sage: p.add_constraint( 5*z - y  <= 8 )
    67  
    68 The ``solve`` method returns by default the optimal value reached by the objective function
     101    sage: p.add_constraint(x + 2*y <= 4)
     102    sage: p.add_constraint(5*z - y <= 8)
     103
     104The ``solve`` method returns by default the optimal value reached by
     105the objective function
    69106
    70107.. link
    71108
    72109::
    73110
    74   sage: round(p.solve(),2)
    75   8.8
     111    sage: round(p.solve(), 2)
     112    8.8
    76113
    77 We can read the optimal assignation found by the solver for `x, y` and `z` through the ``get_values`` method
     114We can read the optimal assignation found by the solver for `x, y` and
     115`z` through the ``get_values`` method
    78116
    79117.. link
    80118
    81119::
    82120
    83   sage: round(p.get_values(x),2)
    84   4.0
    85   sage: round(p.get_values(y), 2)
    86   0.0
    87   sage: round(p.get_values(z), 2)
    88   1.6
     121    sage: round(p.get_values(x), 2)
     122    4.0
     123    sage: round(p.get_values(y), 2)
     124    0.0
     125    sage: round(p.get_values(z), 2)
     126    1.6
     127
    89128
    90129Variables
    91130^^^^^^^^^
    92131
    93 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 one 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, if not of dictionaries of variables.
     132The variables associated with an instance of ``MILP`` belong to the
     133``MIPVariable`` class, though we should not be concerned with this. In
     134the previous example, we obtained these variables through the
     135"shortcut" ``p['x']``, which is easy enough when our LP is defined
     136over a small number of variables. This being said, the LP/MILP we will
     137present afterwards very often require us to associate one -- or many
     138-- variables to each member of a list of objects, which can be
     139integers, or the vertices or edges of a graph, among plenty of other
     140alternatives. This means we will very soon need to talk about vectors
     141of variables or even dictionaries of variables.
    94142
    95 If a LP requires us to define variables named `x_1, \dots, x_{15}`, we will this time make use of the ``new_variable`` method
     143If an LP requires us to define variables named `x_1, \dots, x_{15}`, we
     144will this time make use of the ``new_variable`` method
    96145
    97146.. link
    98147
    99148::
    100149
    101   sage: x = p.new_variable()
     150    sage: x = p.new_variable()
    102151
    103152It is now very easy to define constraints using our `15` variables
    104153
     
    106155
    107156::
    108157
    109   sage: p.add_constraint( x[1] + x[12] - x[14] >= 8 )
     158    sage: p.add_constraint(x[1] + x[12] - x[14] >= 8)
    110159
    111 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
     160Notice that we did not need to define the length of our
     161vector. Actually, ``x`` would accept any immutable object as a key, as
     162a dictionary would. We can now write
    112163
    113164.. link
    114165
    115166::
    116167
    117   sage: p.add_constraint( x["I am a valid key"]
    118   ...                   + x[("a",pi)] <= 3 )
     168    sage: p.add_constraint(x["I am a valid key"]
     169    ...                    + x[("a",pi)] <= 3)
    119170
    120 
    121 Other LP 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
     171Other LPs may require variables indexed several times. Of course, it is
     172already possible to emulate it by using tuples like `x[(2,3)]`, though
     173to keep the code understandable the method ``new_variable`` accepts as
     174a parameter the integer ``dim``, which lets us define the dimension of
     175the variable. We can now write
    122176
    123177.. link
    124178
    125179::
    126180
    127   sage: y = p.new_variable(dim = 2)
    128   sage: p.add_constraint( y[3][2] + x[5] == 6)
     181    sage: y = p.new_variable(dim=2)
     182    sage: p.add_constraint(y[3][2] + x[5] == 6)
     183
    129184
    130185Typed variables
    131186"""""""""""""""
    132187
    133 By default, all the LP variables are assumed to be non-negativereals. 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``.
     188By default, all the LP variables are assumed to be non-negative
     189reals. They can be defined as binary through the parameter
     190``binary=True`` (or integer with ``integer=True``). Lower and
     191upper bounds can be defined or re-defined (for instance when you want
     192some variables to be negative) using the methods ``set_min`` and
     193``set_max``.
    134194
    135 It is also possible to change the type of a variable after it has been created with the methods ``set_binary`` and ``set_integer``.
     195It is also possible to change the type of a variable after it has been
     196created with the methods ``set_binary`` and ``set_integer``.
    136197
    137 Basic Linear Programs
     198
     199Basic linear programs
    138200---------------------
    139201
    140202Knapsack
    141203^^^^^^^^
    142204
    143 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 through maximizing the usefulness of the items it contains (we will here consider their sum).
     205The *Knapsack* problem is the following: given a collection of items
     206having both a weight and a *usefulness*, we would like to fill a bag
     207whose capacity is constrained through maximizing the usefulness of the
     208items it contains (we will here consider their sum). For the purpose
     209of this tutorial, we set the restriction that the bag can only carry a
     210certain total weight.
    144211
    145 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
     212To achieve this, we have to associate to each object `o` of our
     213collection `C` a binary variable ``taken[o]``, set to 1 when the
     214object is in the bag, and to 0 otherwise. We are trying to solve the
     215following MILP
    146216
    147217.. MATH::
    148   \text{Max : }&\sum_{o\in L}usefulness_o\times taken_o\\
    149   \text{Tel que : }&\sum_{o\in L} poids_o\times taken_o \leq C\\
     218    \text{Max: } & \sum_{o \in L} \text{usefulness}_o \times \text{taken}_o\\
     219    \text{Such that: } & \sum_{o \in L} \text{usefulness}_o \times \text{taken}_o \leq C\\
    150220
    151 Using \Sage, we will give to our items a random weight ::
     221Using Sage, we will give to our items a random weight::
    152222
    153   sage: C = 1
    154 
    155 
     223    sage: C = 1
    156224
    157225.. link
    158226
    159227::
    160228
    161   sage: L = ["Casserole", "Livre", "Couteau",
    162   ...        "Gourde", "Lampe de poche"]
    163 
    164 
     229    sage: L = ["pan", "book", "knife", "gourd", "flashlight"]
    165230
    166231.. link
    167232
    168233::
    169234
    170   sage: L.extend( ["divers_"+str(i) for i in range(20)] )
    171 
    172 
     235    sage: L.extend(["divers_" + str(i) for i in range(20)])
    173236
    174237.. link
    175238
    176239::
    177240
    178   sage: poids = {}
    179   sage: usefulness = {}
    180 
    181 
     241    sage: poids = {}
     242    sage: usefulness = {}
    182243
    183244.. link
    184245
    185246::
    186247
    187   sage: set_random_seed(685474)
    188   sage: for o in L:
    189   ...      poids[o]  = random()
    190   ...      usefulness[o] = random()
     248    sage: set_random_seed(685474)
     249    sage: for o in L:
     250    ...       poids[o] = random()
     251    ...       usefulness[o] = random()
    191252
    192253We can now define the MILP itself
    193254
     
    195256
    196257::
    197258
    198 
    199   sage: p = MixedIntegerLinearProgram()
    200   sage: taken = p.new_variable( binary = True )
    201 
     259    sage: p = MixedIntegerLinearProgram()
     260    sage: taken = p.new_variable(binary=True)
    202261
    203262.. link
    204263
    205264::
    206265
    207   sage: p.add_constraint(
    208   ...     sum( poids[o] * taken[o] for o in L ) <= C )
    209 
     266    sage: p.add_constraint(sum(poids[o] * taken[o] for o in L) <= C)
    210267
    211268.. link
    212269
    213270::
    214271
    215   sage: p.set_objective(
    216   ...     sum( usefulness[o] * taken[o] for o in L ) )
    217 
     272    sage: p.set_objective(sum(usefulness[o] * taken[o] for o in L))
    218273
    219274.. link
    220275
    221276::
    222277
    223   sage: p.solve()
    224   3.1502766806530307
    225   sage: taken = p.get_values(taken)
     278    sage: p.solve()
     279    3.1502766806530307
     280    sage: taken = p.get_values(taken)
    226281
    227282The solution found is (of course) admissible
    228283
     
    230285
    231286::
    232287
    233   sage: sum( poids[o] * taken[o] for o in L )
    234   0.69649597966191712
     288    sage: sum(poids[o] * taken[o] for o in L)
     289    0.69649597966191712
    235290
    236 Should we take a flashlight ?
     291Should we take a flashlight?
    237292
    238293.. link
    239294
    240295::
    241296
    242   sage: taken["Lampe de poche"]
    243   1.0
     297    sage: taken["flashlight"]
     298    1.0
    244299
    245300Wise advice. Based on purely random considerations.
    246301
     302
    247303Matching
    248304--------
    249305
    250 Given a graph `G`, a matching is a set of pairwise disjoint edges. The empty set being a matching, we naturally 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 : his algorithm is based on local improvements, and the proof that a given matching is maximum if it can not 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.
     306Given a graph `G`, a matching is a set of pairwise disjoint edges. The
     307empty set is a trivial matching. So we focus our attention on maximum
     308matchings: we want to find in a graph a matching whose cardinality is
     309maximal. Computing the maximum matching in a graph is a polynomial
     310problem, which is a famous result of Edmonds. Edmonds' algorithm is
     311based on local improvements and the proof that a given matching is
     312maximum if it cannot be improved. This algorithm is not the hardest to
     313implement among those graph theory can offer, though this problem can
     314be modeled with a very simple MILP.
    251315
    252 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 can not both belong to the matching. Hence, we are able to write the MILP we want. However, the number of inequalities can be easily decreaded by noticing that two edges can not 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 :
     316To do it, we need -- as previously -- to associate a binary variable
     317to each one of our objects: the edges of our graph (a value of 1
     318meaning that the corresponding edge is included in the maximum
     319matching). Our constraint on the edges taken being that they are
     320disjoint, it is enough to require that, `x` and `y` being two edges
     321and `m_x, m_y` their associated variables, the inequality `m_x + m_y
     322\leq 1` is satisfied, as we are sure that the two of them cannot both
     323belong to the matching. Hence, we are able to write the MILP we
     324want. However, the number of inequalities can be easily decreased by
     325noticing that two edges cannot be taken simultaneously inside a
     326matching if and only if they have a common endpoint `v`. We can then
     327require instead that at most one edge incident to `v` be taken inside
     328the matching, which is a linear constraint. We will be solving:
    253329
    254330.. MATH::
    255   \text{Max : }&\sum_{e\in E(G)}m_e\\
    256   \text{Tel que : }&\forall v, \sum_{e\in E(G)\atop v\sim e} m_e \leq 1\\
     331    \text{Max: } & \sum_{e \in E(G)} m_e\\
     332    \text{Such that: } & \forall v, \sum_{e \in E(G) \atop v \sim e} m_e \leq 1
    257333
    258334Let us write the Sage code of this MILP::
    259335
    260   sage: g = graphs.PetersenGraph()
    261   sage: p = MixedIntegerLinearProgram()
    262   sage: matching = p.new_variable(binary = True)
    263 
    264 
     336    sage: g = graphs.PetersenGraph()
     337    sage: p = MixedIntegerLinearProgram()
     338    sage: matching = p.new_variable(binary=True)
    265339
    266340.. link
    267341
    268342::
    269343
    270   sage: p.set_objective(sum( matching[e]
    271   ...                   for e in g.edges(labels = False)))
    272 
    273 
     344    sage: p.set_objective(sum(matching[e] for e in g.edges(labels=False)))
    274345
    275346.. link
    276347
    277348::
    278349
    279   sage: for v in g:
    280   ...      p.add_constraint(sum( matching[e]
    281   ...           for e in g.edges_incident(v, labels = False)) <= 1)
    282 
    283 
     350    sage: for v in g:
     351    ...       p.add_constraint(sum(matching[e]
     352    ...           for e in g.edges_incident(v, labels=False)) <= 1)
    284353
    285354.. link
    286355
    287356::
    288357
    289   sage: p.solve()
    290   5.0
    291 
    292 
     358    sage: p.solve()
     359    5.0
    293360
    294361.. link
    295362
    296363::
    297364
    298   sage: matching = p.get_values(matching)
    299   sage: [e for e,b in matching.iteritems() if b == 1]
    300   [(0, 1), (6, 9), (2, 7), (3, 4), (5, 8)]
     365    sage: matching = p.get_values(matching)
     366    sage: [e for e,b in matching.iteritems() if b == 1]
     367    [(0, 1), (6, 9), (2, 7), (3, 4), (5, 8)]
     368
    301369
    302370Flows
    303371-----
    304372
    305 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 f them having a maximal capacity.
     373Yet another fundamental algorithm in graph theory: maximum flow! It
     374consists, given a directed graph and two vertices `s, t`, in sending a
     375maximum *flow* from `s` to `t` using the edges of `G`, each of them
     376having a maximal capacity.
    306377
     378.. image:: static/lp_flot1.png
     379    :align: center
    307380
    308 .. image:: lp_flot1.png
    309    :align: center
     381The definition of this problem is almost its LP formulation. We are
     382looking for real values associted to each edge, which would
     383represent the intensity of flow going through them, under two types of
     384constraints:
    310385
    311 The definition of this problem is almost its LP formulation. We are looking for real values associted to each edge, which would representthe intensity of flow going through them, under two types of constraints:
     386* The amount of flow arriving on a vertex (different from `s` or `t`)
     387  is equal to the amount of flow leaving it.
     388* The amount of flow going through an edge is bounded by the capacity
     389  of this edge.
    312390
    313     * The amount of flow arriving on a vertex (different from `s` or `t`) is equal to the amount of flow leaving it
    314     * The amount of flow going through an edge is bounded by the capacity of this edge
    315 
    316 This being said, we but 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
     391This being said, we have to maximize the amount of flow leaving
     392`s`: all of it will end up in `t`, as the other vertices are sending
     393just as much as they receive. We can model the flow problem with the
     394following LP
    317395
    318396.. MATH::
    319   \text{Max : }&\sum_{sv\in G}f_{sv}\\
    320   \text{Tel que : }&\forall v\in G, {v\neq s\atop v\neq t}, \sum_{vu\in G}f_{vu} - \sum_{uv \in G}f_{uv} = 0\\
    321   &\forall uv\in G, f_{uv} \leq 1\\
     397    \text{Max: } & \sum_{sv \in G} f_{sv}\\
     398    \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\\
     399    & \forall uv \in G, f_{uv} \leq 1\\
    322400
    323 We will he solve the flow problem on an orienration of Chvatal's Graph, in which all the edges have a capacity of 1::
     401We will solve the flow problem on an orientation of Chvatal's
     402graph, in which all the edges have a capacity of 1::
    324403
    325   sage: g = graphs.ChvatalGraph()
    326   sage: g = g.minimum_outdegree_orientation()
    327 
    328 
     404    sage: g = graphs.ChvatalGraph()
     405    sage: g = g.minimum_outdegree_orientation()
    329406
    330407.. link
    331408
    332409::
    333410
    334   sage: p = MixedIntegerLinearProgram()
    335   sage: f = p.new_variable()
    336   sage: s, t = 0, 2
    337 
    338 
     411    sage: p = MixedIntegerLinearProgram()
     412    sage: f = p.new_variable()
     413    sage: s, t = 0, 2
    339414
    340415.. link
    341416
    342417::
    343418
    344   sage: for v in g:
    345   ...      if v != s and v != t:
    346   ...          p.add_constraint(
    347   ...              sum( f[(v,u)] for u in g.neighbors_out(v))
    348   ...             -sum( f[(u,v)] for u in g.neighbors_in(v)) == 0)
    349 
    350 
     419    sage: for v in g:
     420    ...       if v != s and v != t:
     421    ...           p.add_constraint(
     422    ...               sum(f[(v,u)] for u in g.neighbors_out(v))
     423    ...               - sum(f[(u,v)] for u in g.neighbors_in(v)) == 0)
    351424
    352425.. link
    353426
    354427::
    355428
    356   sage: for e in g.edges(labels = False):
    357   ...      p.add_constraint( f[e] <= 1 )
    358 
    359 
     429    sage: for e in g.edges(labels=False):
     430    ...       p.add_constraint(f[e] <= 1)
    360431
    361432.. link
    362433
    363434::
    364435
    365   sage: p.set_objective(sum( f[(s,u)] for u in g.neighbors_out(s)))
    366 
    367 
     436    sage: p.set_objective(sum(f[(s,u)] for u in g.neighbors_out(s)))
    368437
    369438.. link
    370439
    371440::
    372441
    373   sage: p.solve()
    374   2.0
     442    sage: p.solve()
     443    2.0
    375444
    376 .. image:: lp_flot2.png
    377    :align: center
     445.. image:: static/lp_flot2.png
     446    :align: center
     447
    378448
    379449Solvers
    380450-------
    381451
    382452Sage solves linear programs by calling specific libraries. The
    383 following libraries are currently supported :
     453following libraries are currently supported:
    384454
    385 * `CBC <http://www.coin-or.org/projects/Cbc.xml>`_: A solver from from
    386   `COIN-OR <http://www.coin-or.org/>`_ (CPL -- Free)
     455* `CBC <http://www.coin-or.org/projects/Cbc.xml>`_: A solver from
     456  `COIN-OR <http://www.coin-or.org/>`_
    387457
    388   CBC can be installed through the command ``install_package("cbc")``
     458  Provided under the open source license CPL, but incompatible with
     459  GPL. CBC can be installed through the command ``install_package("cbc")``.
    389460
    390461* `CPLEX
    391462  <http://www-01.ibm.com/software/integration/optimization/cplex/>`_:
    392   A solver from `ILOG <http://www.ilog.com/>`_ (Proprietary, but free
    393   for researchers and students)
     463  A solver from `ILOG <http://www.ilog.com/>`_
     464
     465  Proprietary and closed source, but free for researchers and students.
    394466
    395467* `GLPK <http://www.gnu.org/software/glpk/>`_: A solver from `GNU
    396   <http://www.gnu.org/>`_ (GPL3, Free)
     468  <http://www.gnu.org/>`_
    397469
    398   This solver is installed by default with Sage
     470  Licensed under the GPLv3. This solver is installed by default with Sage.
     471
    399472
    400473Installing CPLEX
    401474----------------
    402475
    403 ILOG CPLEX being Proprietary -- you must be in possession of several
     476ILOG's CPLEX being proprietary, you must be in possession of several
    404477files to use it through Sage:
    405478
    406 * A valid license file
    407 * A compiled version of the CPLEX library (usually named libcplex.a)
    408 * The header file cplex.h
     479* A valid license file.
     480* A compiled version of the CPLEX library (usually named ``libcplex.a``).
     481* The header file ``cplex.h``.
    409482
    410483The license file path must be set the value of the environment
    411 variable ILOG_LICENSE_FILE. For example, you can write ::
     484variable ``ILOG_LICENSE_FILE``. For example, you can write::
    412485
    413486    export ILOG_LICENSE_FILE=/path/to/the/license/ilog/ilm/access_1.ilm
    414487
    415 at the end of your .bashrc file.
     488at the end of your ``.bashrc`` file.
    416489
    417 As Sage also needs the files libcplex.a and cplex.h, the easiest way
    418 is to create symbolic links toward these files in the appropriate
    419 directories :
     490As Sage also needs the files ``libcplex.a`` and ``cplex.h``, the
     491easiest way is to create symbolic links to these files in the
     492appropriate directories:
    420493
    421 * libcplex.a -- in SAGE_ROOT/local/lib/, type ::
     494* ``libcplex.a`` -- in ``SAGE_ROOT/local/lib/``, type::
    422495
    423496    ln -s /path/to/lib/libcplex.a .
    424497
    425 * cplex.h -- in SAGE_ROOT/local/include/, type ::
     498* ``cplex.h`` -- in ``SAGE_ROOT/local/include/``, type::
    426499
    427500    ln -s /path/to/include/cplex.h .
    428501
    429502Once this is done, and as CPLEX is used in Sage through the Osi
    430 library, which is part of the Cbc package, you can type::
     503library, which is part of the CBC package, you can type::
    431504
    432505    sage: install_package("cbc")  # not tested
    433506
    434 or, if you had already installed Cbc ::
     507or, if you had already installed CBC::
    435508
    436     sage: install_package("cbc", force = True)  # not tested
     509    sage: install_package("cbc", force=True)  # not tested
    437510
    438511to reinstall it.
    439 
  • new file doc/en/thematic_tutorials/static/lp_flot1.png

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