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OR 320/520 Optimization I
10/30/07
Prof. Bland
Complementary Slackness
see BHM section 4.5
Consider a standard form linear programming problem (P) and its dual (D):
max
cx
min
yb
(
P
)
s.t.
Ax
≤
b
(
D
)
s.t.
yA
≥
c
x
≥
0
y
≥
0
Assume that
A
is
m
×
n
A pair of vectors
x
= (
x
1
, ..., x
n
)
T
, y
= (
y
1
, ..., y
m
) is
complementary
if
(a)
y
i
= 0
or
n
∑
j
=1
a
ij
x
j
=
b
i
for
i
= 1
, ..., m
and
(b)
x
j
= 0
or
m
∑
i
=1
a
ij
y
i
=
c
j
for
j
= 1
, ..., n.
The
m
+
n
conditions (a) and (b) are called
complementarity
conditions, or
com
plementary slackness
conditions.
Note that
(a)
=
⇒
yAx
=
yb
(b)
=
⇒
yAx
=
cx
.
Hence, if the pair
x, y
is complementary, then
cx
=
yb
.
Note if we introduce slack variables
x
n
+1
...x
n
+
m
in (P) and introduce surplus variables
y
m
+1
, ..., y
m
+
n
in (D), the complementarity conditions (a) and (b) become:
(a)
y
i
=
0
or
x
n
+
i
=
0
i
= 1
, ..., m
(b)
x
j
=
0
or
y
m
+
j
=
0
j
= 1
, ..., n
Complementary Slackness Theorem
1. If (
b
x,
b
y
) is complementary, then
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 Fall '07
 BLAND

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