Note 3: LP Duality
If the primal problem (P) in the canonical form is
min
Z
=
∑
n
j
=1
c
j
x
j
s.t.
∑
n
j
=1
a
ij
x
j
≥
b
i
∀
i
= 1
,
2
,...,m
x
j
≥
0
∀
j
= 1
,
2
,...,n,
(1)
then the dual problem (D) in the canonical form is
max
W
=
∑
m
i
=1
b
i
y
i
s.t.
∑
m
i
=1
a
ij
y
i
≤
c
j
∀
j
= 1
,
2
,...,n
y
i
≥
0
∀
i
= 1
,
2
,...,m.
(2)
We may think of
j
as a food type,
i
as a nutrition type,
c
j
as the per-unit price
of food type
j
,
b
i
as the required quantity of nutrition type
i
,
a
ij
as the quantity of
nutrition type
i
contained in each unit of food type
j
,
x
j
as the quantity of food type
j
to purchase, and
y
i
as the per-unit price to be charged for nutrition type
i
. The
primal problem may be considered as ﬁnding the least costing quantities of foods to
buy to satisfy nutritional needs, and the dual problem can be considered as ﬁnding
the most proﬁtable pricing scheme for pure nutritions that is competitive in the face
of existing prices of food types.
In essence,
(P)’s variable corresponds to (D)’s constraint, and (D)’s variable corresponds to
(P)’s constraint;
If (P) is a minimization (maximization) problem, then (D) is a maximization
(minimization) problem;
(P)’s free variable corresponds to (D)’s = constraint and its nonnegative variable
corresponds to (D)’s max
/
≤
constraint (meaning that (D) is a maximization problem
and the constraint is of the “
≤
” type) or min
/
≥
constraint; on the other hand, (D)’s
free variable corresponds to (P)’s = constraint, its nonnegative variable corresponds
to (P)’s max
/
≤
or min
/
≥
constraint.
The dual of dual is the primal itself.
In lieu of the above, if the primal is our standard form
min
Z
=
∑
n
j
=1
c
j
x
j
s.t.
∑
n
j
=1
a
ij
x
j
=
b
i
∀
i
= 1
,
2
,...,m
x
j
≥
0
∀
j
= 1
,
2
,...,n,
1