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 standard
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 program
ming 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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 '08
 TODD
 Optimization, linear programming problem

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