Linear programming

Basic Concepts and principles

A Linear programming problem in standard formulation is a problem in the form
Min Ctx
Subject to
Ax = b
x ≥ 0
Where A∈ Mmxn, and rank(A)=m, b∈Rm, C∈Rn
From now on we will call P to this problem.
C will be the costs vector, A the constraints matrix and b the constraints vector.
The feasible set
    S = {x∈Rn : Ax = b, x≥ 0}
is a polyhedral set (semispaces intersection), in particular is a convex set. The convex theory join linear programming theorems says that if P has finite optimal solution it will be into the extreme points set of S.

Extended Theory

The feasible Set
    S = {x∈Rn : Ax = b, x≥ 0}
Can be characterized in terms of its extreme points and its extreme directions
Theorem 1 (extreme points existence)
The feasible set, se has at least one extreme point.
If B is submatrix of A with dimension mxm then we can write
    A = [B, N] and the linear programming equations system can be write as follows
     BXB + NXN = b
and we can write too
     x = [XB, XN]
There exists a characterization of the extreme points of feasible Region S, There is a characterization of extreme points of feasible set S, the intuitive idea is that any extreme point is a solution of a linear system of equations constructed from a submatrix of A of range m.
Teorema 2 (Extreme points characterization)
Lets S = {x∈Rn : Ax = b, x≥ 0} ⊂ Rn.
    A∈ Mmxn, con range(A)=m, b∈Rm, C∈Rn
then     x is an extreme point if and only if ∃ B submatrix of A with r(B) = m such as
    x = [B-1b, 0]t

The following corollary gives us an upper bound for the number of extreme points.
The maximum number of extreme points of S is (n m) = n!/m!(n-m)!
(We apologize for the notation, we have no way to write combinatorial numbers)

As we shall see, not every linear programming problem has finite optimal solution, but if one then there is a characterization for it in the next theorema:

Theorem 3 (Finite optimal solutions characterization for problem P)
    x1, x2, ... xk los Extreme points of S and lets
    d1, d2, ... dr las Extreme directions of S.
P has finite optimal solution if and only if Ctdj ≥ 0, j =1,2, ..., r
And one of the extreme points is the solution for the problem P.

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