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E160 Operations Research I
Spring 2009
Homework #5 Solution
Chapter 12
Section 12.6
1. We wish to minimize f(x, y) =
Σ
{(x
i
 x)
2
+ (y
i

y)
2
}
i = 1
i = n
∂
f/
∂
x = 2
Σ
(x
i
 x) = 0 for x* = (x
1
+ x
2
+... +x
n
)/n
i = 1
i = n
∂
f/
∂
y= 2
Σ
(y
i
 x) = 0 for y* = (y
1
+ y
2
+... +y
n
)/n
i = 1
f(x,y) is the sum of convex functions and is therefore
convex. To show that (x*, y*) is a local minimum we compute
the Hessian.
=
n
n
H
2
0
0
2
Thus H
1
= 2n >0 and H
2
= 4n
2
>0, and
(x*, y*) is a local minimum. The fact that f(x,y) is convex
now shows that (x*, y*) minimizes f(x,y) over all possible
choices of (x, y).
2. We wish to maximize f(q
1
, q
2
) = 2(q
1
1/3
+ q
2
2/3
)  q
1
 1.5q
2
∂
f/
∂
q
1
= 2q
1
2/3
/3  1 = 0 for q
1
= (2/3)
3/2
∂
f/
∂
q
2
= 4q
2
1/3
/3  1.5 = 0 for q
2
= (4/4.5)
3
The objective function is the sum of concave functions, so
we know that the above values for q
1
and q
2
will maximize
profit.
3. Let f(q
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 Spring '07
 HOCHBAUM
 Operations Research

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