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6.
Use Solver to maximize f(x, y, z)
=
x y z ,
subject to
λ :
4 x y
+
3 x z
+
2 y z
≤
72 ,
x
≥
0 ,
y
≥
0 ,
z
≥
0 .
Then write the KKT conditions for the same optimization problem, and solve them
analytically.
Do you get the same solution?
Since the decision variables are nonnegative in this maximization problem, the crossover
table shows that the complementary constraints are these “≥” inequalities:
x :
λ (4 y
+
3 z)
≥
y z
,
y :
λ (4 x
+
2 z)
≥
x z
,
z :
λ (3 x
+
2 y)
≥
x z
.
The optimal value is positive, we know that
x ,
y
and
z
are positive, so complementary
slackness shows that the above three inequalities hold the equations:
x :
z
3
y
4
z
y
+
=
λ
,
y :
z
2
x
4
z
x
+
=
λ
,
z :
y
2
x
3
y
x
+
=
λ
.
Since
x,
y
and
z
are positive, these equations guarantee that
λ
is positive, so
complementary slackness shows that
λ :
4 x y
+
3 x z
+
2 y z
=
72 ,
That’s four equations in four unknowns. The righthand sides of the first and second
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 Fall '10
 ERICDENARDO

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