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Introduction to Linear Optimization and Extensions with MATLAB®
book

Introduction to Linear Optimization and Extensions with MATLAB®

by Roy H. Kwon
September 2013
Intermediate to advanced content levelIntermediate to advanced
362 pages
8h 44m
English
CRC Press
Content preview from Introduction to Linear Optimization and Extensions with MATLAB®
Duality Theory 143
Since x
is optimal, the corresponding reduced costs are non-negative, i.e.,
r
N
= c
N
c
T
B
B
1
N 0.
Now, let π
= (c
T
B
B
1
)
T
, then
c A
T
π
=
c
B
c
N
B
T
N
T
π
=
c
B
c
N
B
T
N
T
(c
T
B
B
1
)
T
=
c
B
c
N
c
B
(B
1
N)
T
c
B
=
0
r
N
0.
Thus, π
is a feasible solution for the dual problem. Furthermore,
b
T
π
= (π
)
T
b = c
T
B
B
1
b = c
T
B
x
B
= c
T
x
,
so by Corollary 4.6, π
is an optimal solution for the dual.
Example 4.10
Consider the linear program
maximize 2x
1
+ x
2
subject to x
1
+ x
2
4
2x
1
+ x
2
6
x
1
0, x
2
0.
To use the simplex method on the primal, we convert to standard form
by adding slack variables x
3
and x
4
and convert the objective to a minimiza-
tion problem by multiplying the objective function by 1 (we omit the outer
negation of the minimization) to get
minimize 2x
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Publisher Resources

ISBN: 9781439862636