Practice Final Exam.
15.053
May, 2002
Note
.
This exam does not cover every topic mentioned in class on the last day, each of
which may appear on the final exam.
1.
Consider the following quadratic programming problem.
Min
x
1
2
+ 2x
2
2
+ 2x
1
– 4x
2
s.t.
3x
1
+ 2x
2
≤
6
x
1
+
x
2
≥
1
x
1,
x
2
≥
0
a.
Using the following breakpoints, rewrite the original problem using the
λ
 method:
x
1
:
{0,1,2}
x
2
:
{0,1,2,3}
What are the adjacency conditions for the
λ
 method? Are they guaranteed to be satisfied
if one solves the linear program obtained by ignoring the adjacency conditions?
Briefly
justify your answer.
2.
Indicate whether the following functions are (i) convex, (ii) concave, (iii) both convex
and concave, or (iv) neither convex nor concave.
a.
f(x,y) = 3x – 7y
b.
f(x) = x
.5
for 1
≤
x
≤
5
c.
f(x) = x
2
– x
3
for –1
≤
x
≤
0
3.
A student needs to take 7 incredibly difficult subjects over the next 6 semesters.
She
has committed to completing at least two of these subjects by the end of Spring 2003, and
completing at least four of these subjects by Spring 2004.
In addition, she will not take
more than two incredibly difficult subjects in any semester.
Year
Cum.
Demand
Utility of taking 0, 1, or 2
incredibly difficult subjects in that term
0
1
2
Fall 2002
0
10
7
4
Spring 2003
2
10
8
6
Fall 2003
2
10
8
6
Spring 2004
4
10
8
6
Fall 2004
4
10
7
4
Spring 2005
7
10
5
1
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Let f(j, term) be the maximum utility solution starting with j difficult courses taken at the
beginning of term j.
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 Spring '05
 Prof.JamesOrlin
 Optimization, developer, square feet

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