Stochastic Programming - Index of

BASIC CONCEPTS 73
which, observing that Aˆx = b by (7.14), implies
ˆx T c − b T û ≥ 0.
Since ˆx ∈Band û ∈Dwere arbitrarily chosen, we have
c T x ≥ b T u
∀x ∈B,u∈D,
and hence
inf x∈B c T x ≥ sup u∈D b T u.
✷
In view of this proposition, the question arises as to whether or when it
might happen that
inf
b T u.
x∈B cT x>sup
u∈D
Example 1.8 Consider the following primal linear program:
and its dual program
min{3x 1 +3x 2 − 16x 3 }
s.t. 5x 1 +3x 2 − 8x 3 =2,
−5x 1 +3x 2 − 8x 3 =4,
x i ≥ 0, i =1, 2, 3,
max{2u 1 +4u 2 }
s.t. 5u 1 − 5u 2 ≤ 3,
3u 1 +3u 2 ≤ 3,
−8u 1 − 8u 2 ≤−16.
Adding the equations of the primal program, we get
6x 2 − 16x 3 =6,
and hence
x 2 =1+ 8x 3 3,
which, on insertion into the first equation, yields
x 1 = 1 5 (2 − 3 − 8x 3 +8x 3 )
= − 1 5 ,
showing that the primal program is not feasible.
Looking at the dual constraints, we get from the second and third
inequalities that
u 1 + u 2 ≤ 1,
u 1 + u 2 ≥ 2,