MS&E 211
Fall 2007
Linear and Nonlinear Optimization
Oct 18, 2007
Prof. Yinyu Ye
Homework Assignment 3: Due 3:15pm Friday, Oct 26
There is a homework collecting box outside of professor Ye’s office (Terman 316) for you to
submit your homework. No late homework accepted!
Reading.
Read sections 6.1 to 6.7 in the text book.
Problem 1 : Sensitivity Analysis
1
Consider the following linear programming problem
Maximize
9
x
1
+ 8
x
2
+ 5
x
3
Subject to
2
x
1
+ 3
x
2
+
x
3
≤
4
5
x
1
+ 4
x
2
+ 3
x
3
≤
11
x
1
, x
2
, x
3
≥
0
Let
x
4
and
x
5
denote the slack variables for the respective functional constraints. First, we
convert the problem into a minimization problem, then we apply the simplex method. The
final simplex tableau is
0
2
0
2
1
19
x
1
1
5
0
3
1
1
x
3
0
7
1
5
2
2
a) Write the dual problem.
b) Will the optimal basis change if we change the RHS vector b from (4;11) to (6;11).
c) Will the optimal basis change if the coefficient of
x
2
in the objective function is changed
from 8 to 11.
d) Suppose that the following constraint is added to the problem:
x
1
+
x
2
+
x
3
≤
6
.
Using
sensitivity, find the new optimal solution.
1
This is a revised version of problem 6.718 in the textbook
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Problem 2 : Asset Allocation Duality
Your rich uncle Warren has been impressed with your performance at Stanford and has
decided that he’d like you in invest $10M for him.
Warren has a lot of friends currently
starting new companies that need money.
He’d like you to invest his $10M among these
companies such that you maximize the total return. There are a fixed number of companies
in which you can choose to invest your assets. Each company requires some fixed amount
of capital and has a promised return. For each company, you must decide how much of its
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 Fall '07
 YINYUYE
 Linear Programming, Optimization, business units, optimal basis change

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