====== BigObject LinearProgram<Scalar> ======
//from application [[..:polytope|polytope]]//\\
\\
 A linear program specified by a linear or abstract objective function

  ? Type Parameters:
  :: ''Scalar'': numeric type of variables and objective function
===== Properties =====

==== no category ====
{{anchor:abstract_objective:}}
  ?  **''ABSTRACT_OBJECTIVE''**
  :: Abstract objective function.  Defines a direction for each edge such that each non-empty face has a unique source and a unique sink.  The i-th element is the value of the objective function at vertex number i. Only defined for bounded polytopes.
    ? Type:
    :''[[..:common#Vector |Vector]]<Scalar>''
    ? Example:
    :: The following creates a new LinearProgram object and assigns an abstract objective to it:
    :: <code perl> > $l = cube(2)->LP(ABSTRACT_OBJECTIVE=>[1,2,3,4]);
 > print $l->ABSTRACT_OBJECTIVE;
 1 2 3 4
</code>


----
{{anchor:directed_bounded_graph:}}
  ?  **''DIRECTED_BOUNDED_GRAPH''**
  :: Subgraph of ''[[..:polytope:Polytope#BOUNDED_GRAPH |BOUNDED_GRAPH]]''. Consists only of directed arcs along which the value of the objective function increases.
    ? Type:
    :''[[..:graph:Graph |Graph]]<[[..:common#Directed |Directed]]>''


----
{{anchor:directed_graph:}}
  ?  **''DIRECTED_GRAPH''**
  :: Subgraph of ''[[..:polytope:Polytope#GRAPH |GRAPH]]''. Consists only of directed arcs along which the value of the objective function increases.
    ? Type:
    :''[[..:graph:Graph |Graph]]<[[..:common#Directed |Directed]]>''
    ? Example:
    :: The following defines a LinearProgram together with a linear objective for the centered square with side length 2. The directed graph according to the linear objective is stored in a new variable and the corresponding edges are printend.
    :: <code perl> > $c = new Vector([0, 1, 0]);
 > $p = cube(2);
 > $p->LP(LINEAR_OBJECTIVE=>$c);
 > $g = $p->LP->DIRECTED_GRAPH;
 > print $g->EDGES;
 {0 1}
 {2 3}
</code>


----
{{anchor:linear_objective:}}
  ?  **''LINEAR_OBJECTIVE''**
  :: Linear objective function. In d-space a linear objective function is given by a (d+1)-vector.  The first coordinate specifies a constant that is added to the resulting value.
    ? Type:
    :''[[..:common#Vector |Vector]]<Scalar>''
    ? Example:
    :: The following creates a new LinearProgram object and assigns a linear objective to it:
    :: <code perl> > $l = cube(2)->LP(LINEAR_OBJECTIVE=>[0,1,1]);
 > print $l->LINEAR_OBJECTIVE;
 0 1 1
</code>


----
{{anchor:maximal_face:}}
  ?  **''MAXIMAL_FACE''**
  :: Indices of vertices at which the maximum of the objective function is attained.
    ? Type:
    :''[[..:common#Set |Set]]<[[..:common#Int |Int]]>''
    ? Example:
    :: The following defines a LinearProgram together with a linear objective for the centered square with side length 2 and asks for the maximal face:
    :: <code perl> > $c = new Vector([0, 1, 0]);
 > $p = cube(2);
 > $p->LP(LINEAR_OBJECTIVE=>$c);
 > print $p->LP->MAXIMAL_FACE;
 {1 3}
</code>


----
{{anchor:maximal_value:}}
  ?  **''MAXIMAL_VALUE''**
  :: Maximum value of the objective function. Negated if linear problem is unbounded.
    ? Type:
    :''Scalar''
    ? Example:
    :: The following defines a LinearProgram together with a linear objective for the centered square with side length 2 and asks for the maximal value:
    :: <code perl> > $c = new Vector([0, 1, 0]);
 > $p = cube(2);
 > $p->LP(LINEAR_OBJECTIVE=>$c);
 > print $p->LP->MAXIMAL_VALUE;
 1
</code>
    ? Example:
    :: The following defines a LinearProgram together with a linear objective with bias 3 for the centered square with side length 4 and asks for the maximal value:
    :: <code perl> > $c = new Vector([3, 1, 0]);
 > $p = cube(2,2);
 > $p->LP(LINEAR_OBJECTIVE=>$c);
 > print $p->LP->MAXIMAL_VALUE;
 5
</code>
    ? Example:
    :: The following defines a LinearProgram together with a linear objective for the positive quadrant (unbounded) and asks for the maximal value:
    :: <code perl> > $c = new Vector([0, 1, 1]);
 > $p = facet_to_infinity(simplex(2),0);
 > $p->LP(LINEAR_OBJECTIVE=>$c);
 > print $p->LP->MAXIMAL_VALUE;
 inf
</code>


----
{{anchor:maximal_vertex:}}
  ?  **''MAXIMAL_VERTEX''**
  :: Coordinates of a (possibly not unique) affine vertex at which the maximum of the objective function is attained.
    ? Type:
    :''[[..:common#Vector |Vector]]<Scalar>''
    ? Example:
    :: The following defines a LinearProgram together with a linear objective for the centered square with side length 2 and asks for a maximal vertex:
    :: <code perl> > $c = new Vector([0, 1, -1/2]);
 > $p = cube(2);
 > $p->LP(LINEAR_OBJECTIVE=>$c);
 > print $p->LP->MAXIMAL_VERTEX;
 1 1 -1
</code>


----
{{anchor:minimal_face:}}
  ?  **''MINIMAL_FACE''**
  :: Similar to ''[[..:polytope:LinearProgram#MAXIMAL_FACE |MAXIMAL_FACE]]''.
    ? Type:
    :''[[..:common#Set |Set]]<[[..:common#Int |Int]]>''
    ? Example:
    :: The following defines a LinearProgram together with a linear objective for the centered square with side length 2 and asks for the minimal face:
    :: <code perl> > $c = new Vector([0, 1, 0]);
 > $p = cube(2);
 > $p->LP(LINEAR_OBJECTIVE=>$c);
 > print $p->LP->MINIMAL_FACE;
 {0 2}
</code>


----
{{anchor:minimal_value:}}
  ?  **''MINIMAL_VALUE''**
  :: Similar to ''[[..:polytope:LinearProgram#MAXIMAL_VALUE |MAXIMAL_VALUE]]''.
    ? Type:
    :''Scalar''
    ? Example:
    :: The following defines a LinearProgram together with a linear objective for the centered square with side length 2 and asks for the minimal value:
    :: <code perl> > $c = new Vector([0, 1, 0]);
 > $p = cube(2);
 > $p->LP(LINEAR_OBJECTIVE=>$c);
 > print $p->LP->MINIMAL_VALUE;
 -1
</code>
    ? Example:
    :: The following defines a LinearProgram together with a linear objective with bias 3 for the centered square with side length 4 and asks for the minimal value:
    :: <code perl> > $c = new Vector([3, 1, 0]);
 > $p = cube(2,2);
 > $p->LP(LINEAR_OBJECTIVE=>$c);
 > print $p->LP->MINIMAL_VALUE;
 1
</code>


----
{{anchor:minimal_vertex:}}
  ?  **''MINIMAL_VERTEX''**
  :: Similar to ''[[..:polytope:LinearProgram#MAXIMAL_VERTEX |MAXIMAL_VERTEX]]''.
    ? Type:
    :''[[..:common#Vector |Vector]]<Scalar>''
    ? Example:
    :: The following defines a LinearProgram together with a linear objective for the centered square with side length 2 and asks for a minimal vertex:
    :: <code perl> > $c = new Vector([0, 1, 0]);
 > $p = cube(2);
 > $p->LP(LINEAR_OBJECTIVE=>$c);
 > print $p->LP->MINIMAL_VERTEX;
 1 -1 -1
</code>


----
{{anchor:random_edge_epl:}}
  ?  **''RANDOM_EDGE_EPL''**
  :: Expected average path length for a simplex algorithm employing "random edge" pivoting strategy.
    ? Type:
    :''[[..:common#Vector |Vector]]<[[..:common#Rational |Rational]]>''


----
===== Methods =====

==== no category ====
{{anchor:vertex_in_degrees:}}
  ?  **''VERTEX_IN_DEGREES()''**
  :: Array of in-degrees for all nodes of ''[[..:polytope:LinearProgram#DIRECTED_GRAPH |DIRECTED_GRAPH]]'' or numbers of objective decreasing edges at each vertex
    ? Returns:
    :''[[..:common#Array |Array]]<[[..:common#Int |Int]]>''


----
{{anchor:vertex_out_degrees:}}
  ?  **''VERTEX_OUT_DEGREES()''**
  :: Array of out-degrees for all nodes of ''[[..:polytope:LinearProgram#DIRECTED_GRAPH |DIRECTED_GRAPH]]'' or numbers of objective increasing edges at each vertex
    ? Returns:
    :''[[..:common#Array |Array]]<[[..:common#Int |Int]]>''


----