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ARE211
Problem Set #11
First NPP Problem Set
Due date: Nov 27
(1) Consider the following maximization problem (solve it graphically):
max
x
1
,x
2
f
(
x
1
, x
2
) with
f
(
x
1
, x
2
) =

x
1
subject to
g
1
:

x
3
1
+
x
2
≤
0 and
g
2
:

x
3
1

x
2
≤
0.
a) What is the solution to the maximization problem?
b) Is the Mantra satisFed for the solution to part a). If yes, write the gradient of the
objective function as a postive linear combination of the gradients of the contstraints
that are satisFed with equality. If not, explain why?
c) Now, slightly change the problem and let the second constraint be
g
2
:

x
3
1

ex
1

x
2
≤
0 for
e >
0 Again, what is the solution to your problem?
d) ±or the revised problem in part c), is the Mantra satisFed.
If yes, write the gradi
ent of the objective function as a postive linear combination of the gradients of the
contstraints that are satisFed with equality. If not, explain why?
(2) Consider the following minimization problem
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 Fall '07
 Simon

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