
148 Introduction to Linear Optimization and Extensions with MATLAB
R
Table 4.4 Primal complementary
slackness for Example 4.11
Optimal primal Dual slack Product
x
∗
1
= 0 π
4
= 25 0
x
∗
2
= 5.5 π
5
= 0 0
x
∗
3
= 7 π
6
= 0 0
Table 4.5 Dual complementary
slackness for Example 4.11
Optimal dual Primal slack Product
π
∗
1
= 0 x
4
= 21.5 0
π
∗
2
= 7.5 x
5
= 0 0
π
∗
3
= 30 x
6
= 0 0
4.5.1 Complementary Slackness for Standard Form
Now consider the primal problem P in standard form:
minimize c
T
x
subject to Ax = b
x ≥ 0
with its dual D
maximize b
T
π
subject to A
T
π ≤ c
π unrestricted.
For this case, note that any feasible solution x of the primal problem P
will satisfy condition (2) of complementary slackness, ...