Numerical Methods and Optimization

250 Numerical Methods and Optimization: An Introduction

Since these identities hold for any t,wehave

¯

b

i

=

˜

b

i

, ¯a

ik

=˜a

ik

, ¯z =˜z, ¯c

k

=˜c

k

for all i ∈B,k ∈N.

Therefore, the following property holds.

Any two dictionaries of the same LP with the same basis are identical.

The initial dictionary (11.23) is, essentially, a linear system that represents the

original LP written in the standard form. The only transformations we apply

to this linear system at each subsequent iteration of the simplex method are

the elementary row operations used to express the new set of basic variables

through the remaining variables. Since applying an elementary row operation