
138 Introduction to Linear Optimization and Extensions with MATLAB
R
maximize 5x
1
− 3x
2
+ 2x
3
subject to x
1
+ 4x
2
− 2x
3
≥ 1
2x
1
− x
2
+ x
3
= −5
x
1
+ x
2
− x
3
≤ 2
x
1
≤ 0, x
2
≥ 0, x
3
unrestricted.
Solution:
Now the primal data is such that
c =
5
−3
2
, b =
1
−5
2
, and
A =
1 4 −2
2 −1 1
1 1 −1
.
Thus, the dual cost vector is b, the right-hand side vector is c, and the
constraint coefficient matrix is
A
T
=
1 2 1
4 −1 1
−2 1 −1
.
There are three dual variables since there are three primal constraints
(excluding the restrictions, if any, on the primal variables). The first dual
variable π
1
corresponds to the first primal constraint, which is of the ≥ type,
so π
1
≤ 0, the second dual v ...