Ticket #6869: MIP-now-symbolic.patch

module_list.py

@@ -873 +873 @@
     #           sources = ['sage/modules/vector_rational_sparse.pyx'],
     #           libraries = ['gmp']),
 
+    ################################
+    ## 
+    ## sage.numerical
+    ##
+    ################################
+
+
+    Extension("sage.numerical.mip",
+              ["sage/numerical/mip.pyx"],
+            include_dirs=["local/include/"],
+            libraries=["csage","stdc++"]),
+
 
     ################################
     ## 

sage/numerical/all.py

@@ -1 +1 @@
 from optimize import (find_root, find_maximum_on_interval,
                       find_minimum_on_interval,minimize,minimize_constrained,
                       linear_program, find_fit) 
+from sage.numerical.mip import *

new file sage/numerical/mip.pxd

@@ +1 @@
+cdef extern from *:
+    ctypedef double* const_double_ptr "const double*"
+ No newline at end of file

new file sage/numerical/mip.pyx

@@ +1 @@
+include '../ext/stdsage.pxi'
+
+class MIP:
+    r"""
+    The MIP class is the link between SAGE and LP ( Linear Program ) and 
+    MIP ( Mixed Integer Program ) Solvers. Cf : http://en.wikipedia.org/wiki/Linear_programming
+
+    It 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 ``MIP`` also requires the information
+    on the direction of the optimization :
+    
+    A ``MIP`` ( or ``LP`` ) is defined as a maximization 
+    if ``sense=1``, and is a minimization if ``sense=-1``
+
+    INPUT:
+        
+        - ``sense'' : 
+                 * When set to `1` (default), the ``MIP`` is defined as a Maximization 
+                 * When set to `-1`, the ``MIP`` is defined as a Minimization 
+    
+    EXAMPLES::
+
+         sage: ### Computation of a maximum stable set in Petersen's graph ###
+         sage: g=graphs.PetersenGraph()
+         sage: p=MIP(sense=1)
+         sage: b=p.newvar()
+         sage: p.setobj(sum([b[v] for v in g]))
+         sage: for (u,v) in g.edges(labels=None):
+         ...       p.addconstraint(b[u]+b[v],max=1) 
+         sage: p.setbinary(b)
+         sage: p.solve(objective_only=True)     # optional - requires Glpk or COIN-OR/CBC
+         4.0
+    """        
+
+    def __init__(self,sense=1):
+        r"""
+        Constructor for the MIP class
+
+        INPUT:
+        
+        - ``sense'' : 
+                 When set to 1, the MIP is defined as a Maximization 
+                 When set to -1, the MIP is defined as a Minimization 
+
+        EXAMPLE:
+
+            sage: p=MIP(sense=1)
+        """
+
+        try:
+             from sage.numerical.mipCoin import solveCoin
+             self.default_solver="Coin"
+        except:
+             try:
+                  from sage.numerical.mipGlpk import solveGlpk
+                  self.default_solver="GLPK"
+             except:
+                  self.default_solver=None
+        
+
+        from sage.rings.polynomial.infinite_polynomial_ring import InfinitePolynomialRing
+        from sage.rings.real_double import RealDoubleField as RR
+        P = InfinitePolynomialRing(RR(), names=('x',)); 
+        (self.x,) = P._first_ngens(1)
+
+        self.count=[0]
+        self.sense=sense
+        self.objective=None
+        self.variables={}
+        self.constraints=[]
+        self.min={}
+        self.max={}
+        self.types={}
+        self.values={}
+        self.__BINARY=1
+        self.__REAL=-1
+        self.__INTEGER=0
+
+    def __repr__(self):
+         r"""
+         Returns a short description of the MIP
+         
+         EXAMPLE:
+         
+         sage: p=MIP()
+         sage: v=p.newvar()
+         sage: p.addconstraint(v[1]+v[2],max=2)
+         sage: print p
+         Mixed Integer Program ( maximization, 2 variables, 1 constraints )
+         """
+         return "Mixed Integer Program ( "+("maximization" if self.sense==1 else "minimization")+", "+str(len(self.variables))+" variables, "+str(len(self.constraints))+" constraints )"
+
+    def newvar(self,dim=1):
+        r"""
+        Returns an instance of ``MIPVariable`` associated
+        to the current instance of ``MIP``.
+        
+        A new ``MIP`` variable ``x`` defined by :::
+
+            sage: p=MIP()
+            sage: x=p.newvar()
+
+        It behaves exactly as an usual dictionary would. It can use any key
+        argument you may like, as ``x[5]`` or ``x["b"]``, and has methods 
+        ``items()`` and ``keys()``
+        
+        Any of its fields exists, and is uniquely defined.
+
+        INPUT:
+
+        - ``dim`` ( integer ) : Defines the dimension of the dictionary
+                      If ``x`` has dimension `2`, its fields will
+                      be of the form ``x[key1][key2]``
+
+        EXAMPLE::
+
+            sage: p=MIP()
+            sage: x=p.newvar()
+            sage: y=p.newvar(dim=2)
+            sage: p.addconstraint(x[2]+y[3][5],max=2)
+        """
+        return MIPVariable(self.x,self._addElementToRing,dim=dim)
+         
+
+    def export(self,format="text"):
+         r"""
+         Exports the MIP to a string in different formats.
+         
+         INPUT:
+         
+         - ``format'' :
+                   "text" : human-readable format
+
+         sage: p=MIP()
+         sage: x=p.newvar()
+         sage: p.setobj(x[1]+x[2])
+         sage: p.addconstraint(-3*x[1]+2*x[2],max=2)
+         sage: print p.export(format="text")
+         Maximization :
+           x2 + x1
+         Constraints :
+           2.0*x2 - 3.0*x1
+         Variables :
+           x2 is a real variable (min=0.0,max=+oo)
+           x1 is a real variable (min=0.0,max=+oo)
+         """
+         if format=="text":
+              value=("Maximization :\n" if self.sense==1 else "Minimization :\n")
+              value=value+"  "+(str(self.objective) if self.objective!=None else "Undefined")
+              value=value+"\nConstraints :"
+              for c in self.constraints:
+                   value=value+"\n  "+str(c["function"])
+              value=value+"\nVariables :"
+              for v in self.variables.keys():
+                   value=value+"\n  "+str(v)+" is"
+                   if self.isinteger(v):
+                       value=value+" an integer variable"
+                   elif self.isbinary(v):
+                       value=value+" an boolean variable"                       
+                   else:
+                       value=value+" a real variable"
+                   value+=" (min="+(str(self.getmin(v)) if self.getmin(v)!= None else "-oo")+",max="+(str(self.getmax(v)) if self.getmax(v)!= None else "+oo")+")"
+              return value
+
+    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.
+
+        EXAMPLE::
+
+            sage: p=MIP()
+            sage: x=p.newvar()
+            sage: y=p.newvar(dim=2)
+            sage: p.setobj(x[3]+y[2][9]+x[5])
+            sage: p.addconstraint(x[3]+y[2][9]+2*x[5],max=2)
+            sage: p.solve() # optional - requires Glpk or COIN-OR/CBC
+            2.0
+            sage: #
+            sage: # Returns the optimal value of x[3]
+            sage: p.get_values(x[3]) # optional - requires Glpk or COIN-OR/CBC
+            0.0
+            sage: #
+            sage: # Returns a dictionary identical to x
+            sage: # containing values for the corresponding
+            sage: # variables
+            sage: x_sol=p.get_values(x)
+            sage: x_sol.keys()
+            [3, 5]
+            sage: #
+            sage: # Obviously, it also works with
+            sage: # variables of higher dimension
+            sage: y_sol=p.get_values(y)
+            sage: # 
+            sage: # We could also have tried :
+            sage: [x_sol,y_sol]=p.get_values(x,y)
+            sage: # 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.values[v] if self.values.has_key(v) else None
+                    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.values[l])
+        if len(lists)==1:
+            return val[0]
+        else:
+            return val
+                
+            
+
+    def show(self):
+        r"""
+        Prints the MIP in a human-readable way
+
+        EXAMPLE:
+
+        sage: p=MIP()
+        sage: x=p.newvar()
+        sage: p.setobj(x[1]+x[2])
+        sage: p.addconstraint(-3*x[1]+2*x[2],max=2)
+        sage: p.show()
+        Maximization :
+          x2 + x1
+        Constraints :
+          2.0*x2 - 3.0*x1
+        Variables :
+          x2 is a real variable (min=0.0,max=+oo)
+          x1 is a real variable (min=0.0,max=+oo)
+        """
+        print self.export(format="text")
+        
+    #Ok
+    def setobj(self,obj):
+        r"""
+        Sets the objective of the ``MIP``.
+
+        INPUT:
+        
+        - ``obj`` : A linear function to be optimized
+
+        EXAMPLE::
+        
+           This code solves 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 )
+
+           sage: p=MIP(sense=1)
+           sage: x=p.newvar()
+           sage: p.setobj(x[1]+5*x[2])
+           sage: p.addconstraint(x[1]+0.2*x[2],max=4)
+           sage: p.addconstraint(1.5*x[1]+3*x[2],max=4)
+           sage: p.solve()     # optional - requires Glpk or COIN-OR/CBC
+           6.6666666666666661
+           
+        """
+        self.objective=obj
+
+    def addconstraint(self,linear_function,max=None,min=None):
+        r"""
+        Adds a constraint to the MIP
+        
+        INPUT :
+
+        - ``consraint`` : : A linear function
+        - ``max``  : An upper bound on the constraint ( set to ``None`` by default )
+        - ``min``  : A lower bound on the constraint
+
+        EXAMPLE::
+        
+           This code solves 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 )
+
+           sage: p=MIP(sense=1)
+           sage: x=p.newvar()
+           sage: p.setobj(x[1]+5*x[2])
+           sage: p.addconstraint(x[1]+0.2*x[2],max=4)
+           sage: p.addconstraint(1.5*x[1]+3*x[2],max=4)
+           sage: p.solve()     # optional - requires Glpk or COIN-OR/CBC
+           6.6666666666666661
+        """
+
+        max=float(max) if max!=None else None
+        min=float(min) if min!=None else None
+        self.constraints.append({"function":linear_function,"min":min, "max":max,"card":len(linear_function.variables())})
+
+    def setbinary(self,e):
+        r"""
+        Sets a variable or a ``MIPVariable`` as binary
+
+        INPUT:
+
+        - ``e`` : An instance of ``MIPVariable`` or one of
+                  its elements
+
+        NOTE:
+
+        We recommend you to define the types of your variables after
+        your problem has been completely defined ( see example )
+
+        EXAMPLE:
+
+          sage: p=MIP()
+          sage: x=p.newvar()
+          sage: #
+          sage: # The following instruction does absolutely nothing
+          sage: # as none of the variables of x have been used yet
+          sage: p.setbinary(x)
+          sage: p.setobj(x[0]+x[1])
+          sage: p.addconstraint(-3*x[0]+2*x[1],max=2)
+          sage: #
+          sage: # This instructions sets x[0] and x[1]
+          sage: # as binary variables
+          sage: p.setbinary(x)
+          sage: p.addconstraint(x[3]+x[2],max=2)
+          sage: #
+          sage: # x[3] is not set as binary
+          sage: # as no setbinary(x) has been called
+          sage: # after its first definition
+          sage: #
+          sage: # Now it is done 
+          sage: p.setbinary(x[3])
+        """
+        if isinstance(e,MIPVariable):
+            if e.depth()==1:
+                for v in e.values():
+                    self.types[v]=self.__BINARY                
+            else:
+                for v in e.keys():
+                    self.setbinary(e[v])
+        elif self.variables.has_key(e):        
+            self.types[e]=self.__BINARY
+        else:
+            raise Exception("Wrong kind of variable..")
+
+    def isbinary(self,e):
+        r"""
+        Tests whether the variable is binary.
+
+        ( Variables are real by default )
+
+        INPUT:
+
+        - ``e`` : a variable ( not a ``MIPVariable``, but one of its elements ! )
+
+        OUTPUT:
+
+        ``True`` if the variable is binary, ``False`` otherwise
+
+        EXAMPLE:
+
+            sage: p=MIP()
+            sage: v=p.newvar()
+            sage: p.setobj(v[1])
+            sage: p.isbinary(v[1])
+            False
+            sage: p.setbinary(v[1])            
+            sage: p.isbinary(v[1])
+            True
+        """
+        # Returns an exception if the variable does not exist.. 
+        # For exemple if the users tries to find out the type of
+        # a MIPVariable or anything else
+        self.variables[e]
+
+        if self.types.has_key(e) and self.types[e]==self.__BINARY:
+            return True
+        return False
+
+    def setinteger(self,e):
+        r"""
+        Sets a variable or a ``MIPVariable`` as integer
+
+        INPUT:
+
+        - ``e`` : An instance of ``MIPVariable`` or one of
+                  its elements
+
+        NOTE:
+
+        We recommend you to define the types of your variables after
+        your problem has been completely defined ( see example )
+
+        EXAMPLE:
+
+          sage: p=MIP()
+          sage: x=p.newvar()
+          sage: #
+          sage: # The following instruction does absolutely nothing
+          sage: # as none of the variables of x have been used yet
+          sage: p.setinteger(x)
+          sage: p.setobj(x[0]+x[1])
+          sage: p.addconstraint(-3*x[0]+2*x[1],max=2)
+          sage: #
+          sage: # This instructions sets x[0] and x[1]
+          sage: # as integer variables
+          sage: p.setinteger(x)
+          sage: p.addconstraint(x[3]+x[2],max=2)
+          sage: #
+          sage: # x[3] is not set as integer
+          sage: # as no setinteger(x) has been called
+          sage: # after its first definition
+          sage: #
+          sage: # Now it is done 
+          sage: p.setinteger(x[3])
+        """
+        if isinstance(e,MIPVariable):
+            if e.depth()==1:
+                for v in e.values():
+                    self.types[v]=self.__INTEGER                
+            else:
+                for v in e.keys():
+                    self.setbinary(e[v])
+        elif self.variables.has_key(e):        
+            self.types[e]=self.__INTEGER
+        else:
+            raise Exception("Wrong kind of variable..")
+
+    def isinteger(self,e):
+        r"""
+        Tests whether the variable is integer.
+
+        ( Variables are real by default )
+
+        INPUT:
+
+        - ``e`` : a variable ( not a ``MIPVariable``, but one of its elements ! )
+
+        OUTPUT:
+
+        ``True`` if the variable is integer, ``False`` otherwise
+
+        EXAMPLE:
+
+            sage: p=MIP()
+            sage: v=p.newvar()
+            sage: p.setobj(v[1])
+            sage: p.isinteger(v[1])
+            False
+            sage: p.setinteger(v[1])            
+            sage: p.isinteger(v[1])
+            True
+        """
+        # Returns an exception if the variable does not exist.. 
+        # For exemple if the users tries to find out the type of
+        # a MIPVariable or anything else
+        self.variables[e]
+
+        if self.types.has_key(e) and self.types[e]==self.__INTEGER:
+            return True
+        return False
+
+    def setreal(self,e):
+        r"""
+        Sets a variable or a ``MIPVariable`` as real
+
+        INPUT:
+
+        - ``e`` : An instance of ``MIPVariable`` or one of
+                  its elements
+
+        NOTE:
+
+        We recommend you to define the types of your variables after
+        your problem has been completely defined ( see example )
+
+        EXAMPLE:
+
+          sage: p=MIP()
+          sage: x=p.newvar()
+          sage: #
+          sage: # The following instruction does absolutely nothing
+          sage: # as none of the variables of x have been used yet
+          sage: p.setreal(x)
+          sage: p.setobj(x[0]+x[1])
+          sage: p.addconstraint(-3*x[0]+2*x[1],max=2)
+          sage: #
+          sage: # This instructions sets x[0] and x[1]
+          sage: # as real variables
+          sage: p.setreal(x)
+          sage: p.addconstraint(x[3]+x[2],max=2)
+          sage: #
+          sage: # x[3] is not set as real
+          sage: # as no setreal(x) has been called
+          sage: # after its first definition
+          sage: # ( even if actually, it is as variables
+          sage: # are real by default ... )
+          sage: #
+          sage: # Now it is done 
+          sage: p.setreal(x[3])
+        """
+        if isinstance(e,MIPVariable):
+            if e.depth()==1:
+                for v in e.values():
+                    self.types[v]=self.__REAL                
+            else:
+                for v in e.keys():
+                    self.setbinary(e[v])
+        elif self.variables.has_key(e):        
+            self.types[e]=self.__REAL
+        else:
+            raise Exception("Wrong kind of variable..")
+
+
+    def isreal(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=MIP()
+            sage: v=p.newvar()
+            sage: p.setobj(v[1])
+            sage: p.isreal(v[1])
+            True
+            sage: p.setbinary(v[1])
+            sage: p.isreal(v[1])
+            False
+            sage: p.setreal(v[1])            
+            sage: p.isreal(v[1])
+            True
+        """
+        
+        # Returns an exception if the variable does not exist.. 
+        # For exemple if the users tries to find out the type of
+        # a MIPVariable or anything else
+        self.variables[e]
+
+        if (not self.types.has_key(e)) or self.types[e]==self.__REAL:
+            return True
+        return False
+
+
+    def solve(self,solver=None,log=False,objective_only=False):
+        r"""
+        Solves the MIP.
+
+        INPUT :
+        - ``solver'' :
+                 3 solvers should be available through this class :
+                     - GLPK ( ``solver="GLPK"`` )
+                     http://www.gnu.org/software/glpk/
+                     
+                     - COIN Branch and Cut  ( ``solver="Coin"`` )
+                     COIN-OR http://www.coin-or.org/
+                     If the spkg is installed
+                     
+                     - CPLEX  ( ``solver="CPLEX"`` )
+                     http://www.ilog.com/products/cplex/
+                     Not Implemented Yet
+                     
+                     ``solver`` should then be equal to one of ``"GLPK"``, 
+                     ``"Coin"``, ``"CPLEX"``, or ``None``.
+                     If ``solver=None`` ( default ), the default solver is used 
+                     ( Coin if available, GLPK otherwise )
+                     
+        - ``log`` : This boolean variable indicates whether progress should be printed 
+                    during the computations.
+        
+        - ``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
+        
+        EXAMPLE :
+        
+        This code solves 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 )
+
+
+           sage: p=MIP(sense=1)
+           sage: x=p.newvar()
+           sage: p.setobj(x[1]+5*x[2])
+           sage: p.addconstraint(x[1]+0.2*x[2],max=4)
+           sage: p.addconstraint(1.5*x[1]+3*x[2],max=4)
+           sage: p.solve()           # optional - requires Glpk or COIN-OR/CBC
+           6.6666666666666661
+           sage: p.get_values(x)     # optional - requires Glpk or COIN-OR/CBC
+           {1: 0.0, 2: 1.3333333333333333}
+
+           sage: ### Computation of a maximum stable set in Petersen's graph ###
+           sage: g=graphs.PetersenGraph()
+           sage: p=MIP(sense=1)
+           sage: b=p.newvar()
+           sage: p.setobj(sum([b[v] for v in g]))
+           sage: for (u,v) in g.edges(labels=None):
+           ...       p.addconstraint(b[u]+b[v],max=1) 
+           sage: p.setbinary(b)
+           sage: p.solve(objective_only=True)     # optional - requires Glpk or COIN-OR/CBC
+           4.0
+
+
+        """        
+
+        if self.objective==None:
+            raise Exception("No objective function has been defined !")
+
+        if solver==None:
+            solver=self.default_solver
+
+        if solver==None:
+             raise Exception("There does not seem to be any solver installed...\n Please visit http://www.sagemath.org/doc/tutorial/tour_LP.html for more informations")
+        elif solver=="Coin":
+             try:
+                  from sage.numerical.mipCoin import solveCoin
+             except:
+                raise NotImplementedError("Coin/CBC is not installed and cannot be used to solve this MIP\n To install it, you can type in Sage : sage: install_package('cbc')")                
+             return solveCoin(self,log=log,objective_only=objective_only)
+
+        elif solver=="GLPK":
+             try:
+                  from sage.numerical.mipGlpk import solveGlpk
+             except:
+                raise NotImplementedError("GLPK is not installed and cannot be used to solve this MIP\n To install it, you can type in Sage : sage: install_package('glpk')")                
+             return solveGlpk(self,log=log,objective_only=objective_only)
+        elif solver=="CPLEX":
+             raise NotImplementedError("The support for CPLEX is not written yet... We're seriously thinking about it, though ;-)")
+        else:
+            raise NotImplementedError("solver should be set to 'GLPK', 'Coin', 'CPLEX' or None (in which case the default one is used).")
+
+
+    def _NormalForm(self,exp):
+        r"""
+        Returns a dictionary built from the linear function
+
+        INPUT:
+
+        - ``exp`` : The expression representing a linear function
+
+        OUTPUT:
+
+        A dictionary whose keys are the id of the variables, and whose
+        values are their coefficients.
+        The value corresponding to key `-1` is the constant coefficient
+
+        EXAMPLE:
+
+            sage: p=MIP()
+            sage: v=p.newvar()
+            sage: p._NormalForm(v[0]+v[1])
+            {1: 1.0, 2: 1.0, -1: 0.0}
+        """
+        d=dict(zip([self.variables[v] for v in exp.variables()],exp.coefficients()))
+        d[-1]=exp.constant_coefficient()
+        return d
+
+    def _addElementToRing(self):
+        r"""
+        Creates a new variable from the main ``InfinitePolynomialRing``
+
+        OUTPUT:
+
+        - The newly created variable
+
+        EXAMPLE:
+
+            sage: p=MIP()
+            sage: v=p.newvar()
+            sage: p.count[0]
+            0
+            sage: p._addElementToRing()
+            x1
+            sage: p.count[0]
+            1
+        """
+        self.count[0]+=1
+        v=self.x[self.count[0]]
+        self.variables[v]=self.count[0]
+        self.types[v]=self.__REAL
+        self.min[v]=0.0
+        return v
+
+    def setmin(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=MIP()
+            sage: v=p.newvar()
+            sage: p.setobj(v[1])
+            sage: p.getmin(v[1])
+            0.0
+            sage: p.setmin(v[1],6)
+            sage: p.getmin(v[1])
+            6.0
+        """
+        self.min[v]=min
+
+    def setmax(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=MIP()
+            sage: v=p.newvar()
+            sage: p.setobj(v[1])
+            sage: p.getmax(v[1])
+            sage: p.setmax(v[1],6)
+            sage: p.getmax(v[1])
+            6.0
+        """
+        self.max[v]=max
+
+
+    def getmin(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`` is
+        the variable has no lower bound
+
+        EXAMPLE::
+
+            sage: p=MIP()
+            sage: v=p.newvar()
+            sage: p.setobj(v[1])
+            sage: p.getmin(v[1])
+            0.0
+            sage: p.setmin(v[1],6)
+            sage: p.getmin(v[1])
+            6.0
+        """
+        return float(self.min[v]) if self.min.has_key(v) else 0.0
+    def getmax(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`` is
+        the variable has no upper bound
+
+        EXAMPLE::
+
+            sage: p=MIP()
+            sage: v=p.newvar()
+            sage: p.setobj(v[1])
+            sage: p.getmax(v[1])
+            sage: p.setmax(v[1],6)
+            sage: p.getmax(v[1])
+            6.0
+        """
+        return float(self.max[v]) if self.max.has_key(v) else None
+
+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: MIPSolverException("Error")
+            MIPSolverException()
+                    
+        """
+        self.value = value
+    def __str__(self):
+        r"""
+        Returns the value of the instance of ``MIPSolverException` `
+
+        EXAMPLE:
+
+            sage: e=MIPSolverException("Error")
+            sage: print e
+            'Error'
+        """
+        return repr(self.value)
+
+class MIPVariable:
+     r"""
+     ``MIPVariable`` is a variable used by the class ``MIP``
+     """
+     def __init__(self,x,f,dim=1):
+         r"""
+         Constructor for ``MIPVariable``
+
+         INPUT:
+
+         - ``x`` is the generator element of an ``InfinitePolynomialRing``
+         - ``f`` is a function returning a new variable from the parent class
+         - ``dim`` is the integer defining the definition of the variable
+
+         For more informations, see method ``MIP.newvar``
+
+         EXAMPLE:
+
+            sage: p=MIP()
+            sage: v=p.newvar()
+            
+            
+         """
+         self.dim=dim
+         self.dict={}
+         self.x=x
+         self.f=f
+
+     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=MIP()
+             sage: v=p.newvar()
+             sage: p.setobj(v[0]+v[1])
+             sage: v[0]
+             x1
+          """
+          if self.dict.has_key(i):
+               return self.dict[i] 
+          elif self.dim==1:
+              self.dict[i]=self.f()
+              return self.dict[i]
+          else:
+               self.dict[i]=MIPVariable(dim=self.dim-1,x=self.x, f=self.f)
+               return self.dict[i]
+     def keys(self):
+         r"""
+         Returns the keys already defined in the dictionary
+
+         EXAMPLE:
+
+             sage: p=MIP()
+             sage: v=p.newvar()
+             sage: p.setobj(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=MIP()
+             sage: v=p.newvar()
+             sage: p.setobj(v[0]+v[1])
+             sage: v.items()
+             [(0, x1), (1, x2)]
+         """
+         return self.dict.items()
+     def depth(self):
+         r"""
+         Returns the current variable's depth
+
+         EXAMPLE:
+
+             sage: p=MIP()
+             sage: v=p.newvar()
+             sage: p.setobj(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=MIP()
+             sage: v=p.newvar()
+             sage: p.setobj(v[0]+v[1])
+             sage: v.values()
+             [x1, x2]
+         """
+         return self.dict.values()
+
+