Stochastic Programming - Index of

BASIC CONCEPTS 71
✷
Example 1.6 Considering the primal program
min c T x
s.t. Ax ≤ b,
x ≥ 0
in its standard form
min c T x
s.t. Ax + Iy = b,
x ≥ 0,
y ≥ 0
would yield the dual program
max b T u
s.t. A T u ≤ c,
u ≤ 0,
or equivalently, with v := −u,
max (−b T v)
s.t. A T v ≥−c,
v ≥ 0.
Therefore we now have the following pair of a primal and the corresponding
dual program:
min c T x max (−b T v)
s.t. Ax ≤ b, s.t. A T v ≥−c,
x ≥ 0; v ≥ 0.
✷
Example 1.7 Finally consider the primal program
max g T x
s.t. Dx ≤ f.
This program is of the same form as the dual of our standard linear
program (7.14) and—using the fact that for any function ϕ defined on some set
M we have sup x∈M ϕ(x) =− inf x∈M {−ϕ(x)}—its standard form is written
as
− min (−g T x + + g T x − )
s.t. Dx + − Dx − + Iy = f,
x + ≥ 0,
x − ≥ 0,
y ≥ 0,