ORF307: Assignment 2 Due Thursday March 9
1. Solve exercise 2.7 in the textbook and plot the sequence of points produced by the simplex method. 2. Solve exercise 2.9 in the textbook. 3. Solve exercise 2.10 in the textbook. Can you solve the followin

ORF307 (Spring 2015) Optimization
Assignment: [SOLUTIONS]: Homework 1
[SOLUTIONS]: Homework 1
Lecturer: Robert Vanderbei
1
Email: rvdb@princeton.edu
3D Version of Snells Law (20 points)
The setup is very similar to the 2D case but with a third dimension a

Solution to Homework 6
Tianqi Zhao
April 15, 2015
1
Exercise 12.1
T
The
regression
model is b = x1 + ax2 . Using the graph, we know that b = (0, 3, 1, 2) ,
1 0
1 1
A=
. Hence by the formula we know that
1 2
1 4
(x1 , x2 )T = (AT A)1 AT b = (1, 0.285

ORF 307. Optimization
Robert Vanderbei
Spring 2015.
Problem set 5 Solution
Problem 6.1
(a) x1 and x3 are the basic variables; x2 , x4 and x5 are the nonbasic ones.
0
(b) xB =
.
2
2
(c) zN
= 3 .
0
2 3
10 1 0
1 2 3
(d) B =
,N=
, B 1 N =
.
1
2
7 0 1
4 1

Solutions to ORF 307 HW2
Haotian Pang
Spring 2015
Exercise 3.1
We set up the dictionary with perturbation:
We choose x1 as the entering variable and w2 as the leaving variable:
We choose x3 as the entering variable and w3 as the leaving variable:
1
Now we

Solutions to ORF 307 HW7
Haotian Pang
Spring 2014
Exercise 11.1
Let the first column/row represent showing a and the second column/row represent showing
b by both player
! A and B. A the row player and B is the column player. We have
a a
A=
. According to

Solutions to ORF 307 HW4
Haotian Pang
Spring 2015
Exercise 5.1
There are 4 variables and 3 constraints in the primal, so there will be 3 variables and 4
constraints in the dual. Table 5.1 tells that an equality constraint corresponds to a free
variables a

ORF 307. Optimization
Robert Vanderbei
Spring 2015.
Problem set 8 Solution
Haotian Pang and Tianqi Zhao
Problem 15.9 We use the following notation: node a as USD, node b as Yen, node c as Mark and
node d as Franc
(a) At node b: rab xab + rcb xcb + rdb x

ORF 307
Homework 6 Solutions Exercise 1. We solve this problem by optimizing on the expected value of the asset to be priced. The constraints are that the discounted expected value of the known payoffs is equal to the prices of the assets for which w

ORF 307
Homework 4 Solutions Exercise 1. The Lagrangian of this problem is: L(x, 1 , 2 , ) = cT x + T Ax - T x + T (Dx - f ) 1 2 with 1 , 2 0 The Lagrange dual function is: g(1 , 2 , ) = infx = infx
T D (c D L(x, 1 , 2 , ) T Dc x
= infx
+ T Ax -

ORF 307
Homework 3 Solutions Exercise 1. 3.1 The initial dictionary for the perturbed problem is: 0 w1 = w2 = w3 =
1 2
1+
3
+10x1 -.5x1 -.5x1 -x1
-57x2 +5.5x2 +1.5x2
-9x3 +2.5x3 +.5x3
-24x4 -9x4 -x4
We pivot x1 into the basis and w2 out of the

ORF 307
Homework 1 Solutions Exercise 1. 2.1 Initialization: Write down the initial dictionary 0 5 3 +6x1 -2x1 -x1 +8x2 -x2 -3x2 +5x3 -x3 -x3 +9x4 -3x4 -2x4
w1 = w2 =
5 The largest positive coefficient in the objective is 9 for x4 . We then calcula

Solution to Homework 3
Tianqi Zhao
March 2, 2015
1
Exercise 4.1
The initial dictionary is:
Using the largest coefficient rule, the optimal solution is reached in two pivots.
First pivot: x2 enters, w3 leaves. Second pivot: w1 enters, x1 leaves.
Using the