13
Fundamental theorems
Now that we have made the simplex method a reliable finite algorithm, we
can use it to deduce some striking properties of general linear programming
problems. We start by asking when a linear programming problem in stan
dard equality form,
(
*
)
maximize
c
T
x
subject to
Ax
=
b
x
≥
0
,
has a basic feasible solution.
Theorem 13.1 (Existence of basic feasible solutions)
The linear pro
gramming problem
(
*
)
has a basic feasible solution if and only if it is feasible
and the columns of the matrix
A
span.
Proof
If the linear programming problem (
*
) has a basic feasible solution,
then it is feasible, and furthermore, since the corresponding basis matrix is
invertible, the columns of
A
must certainly span.
Conversely, suppose (
*
)
is feasible and the columns of
A
span.
Applying Phase 1 of the simplex
method using the smallest subscript rule, we must terminate with optimal
value zero, because (
*
) is feasible.
Furthermore, in the body of the final
tableau, no row of coefficients of the original variables in (
*
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 Fall '08
 TODD
 Linear Programming, Optimization, linear programming problem

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