<ManSection>
<Oper Arg="L" Name="PolyhedronByInequalities" Label="for IsList"/>
<Returns>a <C>Polyhedron</C> Object
</Returns>
<Description>
The function takes a list of lists <C>L</C><Math>:=[L_1, L_2, ...]</Math> where each <Math>L_j</Math> represents
an inequality and returns the polyhedron defined by them.
For example the <Math>j</Math>'th entry corresponds to the inequality
<Math>c_j+\sum_{i=1}^n a_{ji}x_i \geq 0</Math>.
</Description>
</ManSection>
<ManSection>
<Oper Arg="P, C" Name="Polyhedron" Label="for IsPolytope, IsCone"/>
<Returns>a <C>Polyhedron</C> Object
</Returns>
<Description>
The input is a polytope <C>P</C> and a cone <C>C</C>. The output is the polyhedron defined by the Minkowski sum <C>P+C</C>.
</Description>
</ManSection>
<ManSection>
<Oper Arg="L, C" Name="Polyhedron" Label="for IsList, IsCone"/>
<Returns>a <C>Polyhedron</C> Object
</Returns>
<Description>
The input is a list <C>L</C> and a cone <C>C</C>. The output is the polyhedron defined by the Minkowski sum
<C>P+C</C> where <C>P</C> is the polytope, i.e., the convex hull, defined by the points <C>L</C>.
</Description>
</ManSection>
<ManSection>
<Oper Arg="P, L" Name="Polyhedron" Label="for IsPolytope, IsList"/>
<Returns>a <C>Polyhedron</C> Object
</Returns>
<Description>
The input is a polytope <C>P</C> and a list <C>L</C>. The output is the polyhedron defined by the Minkowski sum
<C>P+C</C> where <C>C</C> is the cone defined by the rays <C>L</C>.
</Description>
</ManSection>
<ManSection>
<Oper Arg="P, C" Name="Polyhedron" Label="for IsList, IsList"/>
<Returns>a <C>Polyhedron</C> Object
</Returns>
<Description>
The input is a list <C>P</C> and a list <C>C</C>. The output is the polyhedron defined by the Minkowski sum
of the polytope defined by <C>P</C> and the cone defined by <C>C</C>.
</Description>
</ManSection>
<ManSection>
<Attr Arg="P" Name="ExternalCddPolyhedron" Label="for IsPolyhedron"/>
<Returns>cdd Object
</Returns>
<Description>
Converts the polyhedron to a cdd object. The operations of CddInterface can then be applied
on this convex object.
</Description>
</ManSection>
<ManSection>
<Attr Arg="P" Name="ExternalNmzPolyhedron" Label="for IsPolyhedron"/>
<Returns>normaliz Object
</Returns>
<Description>
Converts the polyhedron to an normaliz object. The operations of NormalizInterface can then be applied
on this convex object.
</Description>
</ManSection>
<ManSection>
<Attr Arg="P" Name="DefiningInequalities" Label="for IsPolyhedron"/>
<Returns>a list
</Returns>
<Description>
Returns the Defining inequalities of the given polyhedron.
</Description>
</ManSection>
<ManSection>
<Attr Arg="P" Name="MainRatPolytope" Label="for IsPolyhedron"/>
<Returns>a <C>Polytope</C> Object
</Returns>
<Description>
Returns the main rational polytope of the polyhedron.
</Description>
</ManSection>
<ManSection>
<Attr Arg="P" Name="MainPolytope" Label="for IsPolyhedron"/>
<Returns>a <C>Polytope</C> Object
</Returns>
<Description>
Returns the main integral polytope of the given polyhedron.
</Description>
</ManSection>
<ManSection>
<Attr Arg="P" Name="VerticesOfMainRatPolytope" Label="for IsPolyhedron"/>
<Returns>a list
</Returns>
<Description>
Returns the vertices of the main rational polytope of the polyhedron.
</Description>
</ManSection>
<ManSection>
<Attr Arg="P" Name="VerticesOfMainPolytope" Label="for IsPolyhedron"/>
<Returns>a list
</Returns>
<Description>
Returns the vertices of the main integral polytope of the given polyhedron.
</Description>
</ManSection>
<ManSection>
<Attr Arg="P" Name="TailCone" Label="for IsPolyhedron"/>
<Returns>a <C>Cone</C> Object
</Returns>
<Description>
Returns the tail cone of the polyhedron.
</Description>
</ManSection>
<ManSection>
<Attr Arg="P" Name="RayGeneratorsOfTailCone" Label="for IsPolyhedron"/>
<Returns>a list
</Returns>
<Description>
Returns the Ray Generators of the tail cone.
</Description>
</ManSection>
<ManSection>
<Attr Arg="P" Name="LatticePointsGenerators" Label="for IsPolyhedron"/>
<Returns>a list
</Returns>
<Description>
Returns the integral lattice generators of the polyhedron. The output is a list <Math>L</Math> of length <Math>3</Math>. Any integral point in
polyhedron can be written as <Math>a+mb+nc</Math>, where <Math>a\in L[1],b\in L[2],c\in L[3], m\geq 0</Math>.
</Description>
</ManSection>
<ManSection>
<Attr Arg="P" Name="BasisOfLinealitySpace" Label="for IsPolyhedron"/>
<Returns>a list
</Returns>
<Description>
Returns a basis to the lineality space of the polyhedron. I.e., a basis to the vector space that is contained in <C>P</C>.
</Description>
</ManSection>
<ManSection>
<Attr Arg="P" Name="FVector" Label="for IsPolyhedron"/>
<Returns>a list
</Returns>
<Description>
Returns a list whose <Math>i</Math>'th entry is the number of faces of dimension .
</Description>
</ManSection>
<ManSection>
<Prop Arg="P" Name="IsBounded" Label="for IsPolyhedron"/>
<Returns>true or false
</Returns>
<Description>
The input is a polyhedron P and the output is whether it is bounded or not.
</Description>
</ManSection>
<ManSection>
<Prop Arg="P" Name="IsPointed" Label="for IsPolyhedron"/>
<Returns>true or false
</Returns>
<Description>
The input is a polyhedron P and the output is whether its tail cone is pointed or not.
</Description>
</ManSection>
<#Include Label="example3">
</Section>
<Section Label="Chapter_Polyhedrons_Section_Solving_Linear_programs">
<Heading>Solving Linear programs</Heading>
The problem of solving linear programs can be solved in the gap package
<C>CddInterface</C>, which is required by <C>NConvex</C>.
<ManSection>
<Oper Arg="P, max_or_min, target_func" Name="SolveLinearProgram" Label="for IsPolyhedron, IsString, IsList"/>
<Returns>a list or fail
</Returns>
<Description>
The input is a polyhedron <C>P</C>, a string <C>max_or_min</C> <Math>\in</Math> {"max", "min"} and an objective function
<C>target_func</C>, which we want to maximize or minimize. If the linear program has an optimal solution,
the operation returns a list of two entries, the solution vector and the optimal value of the objective function,
otherwise it returns fail.
</Description>
</ManSection>
<ManSection>
<Oper Arg="P, max_or_min, target_func" Name="SolveLinearProgram" Label="for IsPolytope, IsString, IsList"/>
<Returns>a list or fail
</Returns>
<Description>
The input is a polytope <C>P</C>, a string <C>max_or_min</C> <Math>\in</Math> {"max","min"} and an objective function
<C>target_func</C>, which we want to maximize or minimize. If the linear program has an optimal solution,
the operation returns a list of two entries, the solution vector and the optimal value of the objective function,
otherwise it returns fail.
</Description>
</ManSection>
<ManSection>
<Func Arg="eqs,eqs_rhs,ineqs,ineqs_rhs[,signs]" Name="SolveEqualitiesAndInequalitiesOverIntergers" />
<Returns>a list of three matrices
</Returns>
<Description>
This function produces a basis of the system <A>eqs</A> = <A>eqs_rhs</A>
and <A>ineqs</A> >= <A>ineqs_rhs</A>. It outputs a list containing three matrices.
The first one is a list of points in a polytope, the second is the hilbert basis
of a cone. The set of solutions is then the minkowski sum of the polytope
generated by the points in the first list and the cone generated by the hilbert
basis in the second matrix. The third one is the free part of the solution polyhedron.
The optional argument <A>signs</A> must be a list of zeros and ones which length is
the number of variables. If the ith entry is one, the ith variable must be >= 0.
If the entry is 0, the number is arbitraty. Default is all zero.
</Description>
</ManSection>
<#Include Label="zsolve">
</Section>
</Chapter>
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