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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®
80 Introduction to Linear Optimization and Extensions with MATLAB
R
Now the idea is to first solve for x
B
using the first constraint and then
substitute into the objective function equation z. After the substitution, some
terms will be re-arranged from which a necessary and sufficient condition for
optimality will be obtained.
From the first constraint we have Bx
B
= b Nx
N
and so x
B
= B
1
(b
Nx
N
) and then substituting into
z = c
T
B
x
B
+ c
T
N
x
N
we get
z = c
T
B
B
1
(b Nx
N
) + c
T
N
x
N
= c
T
B
B
1
b c
T
B
B
1
Nx
N
+ c
T
N
x
N
= c
T
B
B
1
b + (c
T
N
c
T
B
B
1
N)x
N
= c
T
B
x
B
+ (c
T
N
c
T
B
B
1
N)x
N
since x
B
= B
1
b. Now z is the objective function value associated with x P
and so z = c
T
x. Also, c
T
x
= c
T
B
x
B
+ c
T
N
x
N
= c
T
B
x
B
since x
N
= 0, so
c
T
x = c
T
B
x
B
+ (c
T
N
c
T
B
B
1
N)x
N
or
c
T
x c
T
x
= (c
T
N
c
T
B
B
1
N)x
N
.
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Publisher Resources

ISBN: 9781439862636