17
Duality for general linear programs
In the previous section, we introduced the powerful idea of a “certiﬁcate of
optimality” for a feasible solution
x
*
of a linear program in standard equality
form. A certiﬁcate of optimality is simply a feasible solution
y
*
for the dual
problem whose (dual) objective value equals the (primal) objective value of
x
*
. This certiﬁcate constitutes a simple way to verify that
x
*
is optimal:
we simply need to check that
x
*
is primal-feasible, that
y
*
is dual-feasible,
and that the two corresponding objective values are equal, and we then have
a proof (by the Weak duality theorem) of optimality. The Strong duality
theorem guarantees that any optimal solution of a linear program in standard
equality form does indeed have a certiﬁcate of optimality.
Many linear programs in practice are not in standard equality form. In
this section we extend the idea of a certiﬁcate of optimality to general linear
programs. We know how to transform such problems into standard equality
form, so we could always resort to this technique before looking for a cer-
tiﬁcate. However, a more appealing approach is to seek a certiﬁcate that
reﬂects the structure of the original problem, so instead we retrace the steps
of the last section, but in a more general context.
Without much loss of generality, we can write any linear program in the
form
(
P
)
maximize
c
T
x
+
d
T
w
subject to
Ax
+
Ew
=
b
Fx
+
Gw
≤
e
x
≥
0
,
w
free
,
for some given vectors
c, d, b, e
and matrices
A, E, F, G
of appropriate sizes.
The variables are the vectors
x
and
w
. As in a standard equality-form prob-
lem, each component of
x
is non-negative, whereas each component of
w
is
“free”, or in other words, unrestricted in sign.
We follow the same approach to construct the dual problem as before: we
seek upper bounds on the optimal value by taking linear combinations of the
constraints and comparing the result to the objective function. Beginning
with the equality constraints, if we take the
i
th constraint, multiply it by
any number
y
i
, and add up the results, we arrive at the equation
y
T
(
Ax
+
Ew
) =
y
T
b.
This equation must hold for any feasible pair of vectors (
x, w
). Turning to the
inequality constraints, we can multiply inequality
j
by any number
u
j
≥
0 to
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