CO 250 Assignment 1 Winter 2015
Solutions
Problem 1: Linear Algebra Review
Answer each of the following questions. For questions (a)(e), indicate if the given statements are TRUE
or FALSE. Justify your answer by either providing a proof (if the statement

C O250 I NTRODUCTION
TO
O PTIMIZATION - HW 1
Due Date: Friday January 14th, by 2PM, in the drop box outside the Tutorial Center.
Recall, late assignments will not be graded.
I MPORTANT INFORMATION :
While it is acceptable for students to discuss the cours

MATH 239
Assignment 1
DUE: NOON Friday 14 January 2011 in the drop boxes opposite the Math
Tutorial Centre MC 4067.
1. Let k and be non-negative integers such that k . Give two proofs, one
combinatorial and one using the binomial theorem, of
k
i=0
m
i
n
k

CO 250 Assignment 1 Spring 2013
Due: Friday May 17th by 11 am
Solutions should be deposited in the appropriate drop box slots (see table below, according to section
and last name), outside MC 4066.
Section 1 (R. Fukasawa, TTh 1-2:20pm):
Section 2 (J. Cher

CS371/AMATH242/CM271 - Winter 2012
Assignment 4
Date Due: Wednesday, March 28th , 2012, @9:00am, in the course assignment box (located at the pink boxes area
on the 3rd oor of MC, outside of DC-MC bridge).
Topics Covered
Numerical integration and root ndi

CO 250 Assignment 1 Fall 2013
Solutions
Problem 1: Linear Algebra Review
(12 marks: (a),(b),(c) 2 each, (d),(e) 3 each)
Parts (a),(b),(c) each have a statement. Either prove that the statement is true, or provide a counterexample that shows that the state

CO 250 Assignment 1 Spring 2013
Solutions
Problem 1: Linear Algebra Review
For each of the following statements, either prove that the statement is true, or provide a counterexample
that shows that the statement is false. (2 pts each)
1. Let a and b be ve

C O250 I NTRODUCTION TO O PTIMIZATION HW 1
Due Date: Friday, January 13th, by 2PM, in the drop box outside the Tutorial Center.
L ATE ASSIGNMENTS WILL NOT BE GRADED .
Important information: While it is acceptable to discuss the course material and the ass

CO 250 Assignment 1 Spring 2012
Solutions
Problem 1: Linear Algebra Review
For each of the following statements, either prove that the statement is true, or provide a counterexample
that shows that the statement is false.
(a) For two 2 2 matrices A and B

CO 250 Assignment 1 Spring 2014
Solutions
Problem 1: Linear Algebra Review
For each of the following statements, indicate if it is TRUE or FALSE. Justify your answer by either
providing a proof (if the statement is true) or a counterexample (if the statem

C O250/CM340 I NTRODUCTION
TO
O PTIMIZATION HW 1
Due Date: Friday May 13th, by 2PM, in the drop box outside the Tutorial Center.
Late assignments will not be graded.
Important information: While it is acceptable to discuss the course material and the assi

C O250 I NTRODUCTION
TO
O PTIMIZATION - HW 1
Due Date: Friday January 14th, by 2PM, in the drop box outside the Tutorial Center.
Recall, late assignments will not be graded.
I MPORTANT INFORMATION :
While it is acceptable for students to discuss the cours

CO 250 Assignment 1 Fall 2011
Solutions
Problem 1: Linear Algebra Review
For each of the following statements, either prove that the statement is true, or provide a counterexample
that shows that the statement is false.
1. Let a and b be vectors such that

C O250/CM340 I NTRODUCTION TO O PTIMIZATION - HW 6 SOLUTIONS
Exercise 1 (25 marks).
(a) If there is a topological ordering y and a directed cycle i1 i2 , i2 i3 , . . . , ik1 ik , ik i1 then yi1 > yi2 + 1 >
yi3 + 2 > > yik + k 1 > yi1 + k , a contradiction

CO 350 Assignment 3 Winter 2010
Due: Friday January 29 at 10:25 a.m.
Solutions are due in drop box #9 (Section 1 and drop box #10 (Section 2), outside MC
4066 by the due time. Use the appropriate slot for your section and surname.
Please acknowledge all o

C O250/CM340 I NTRODUCTION TO O PTIMIZATION - HW 2
Due Date: Friday January 21st, by 2PM, in the drop box outside the Tutorial Center.
Late assignments will not be graded.
Important information: While it is acceptable to discuss the course material and th

UNIVERSITY OF WATERLOO
FINAL EXAMINATION
FALL TERM 2010
Surname:
First Name:
Signature:
Id.#:
Section#:
Course Number
Course Title
Instructors
CO250/CM340
Introduction to Optimization
B. Guenin, L. Tunel
c
Date of Exam
December 15th, 2010
Time Period
7:30

Multiple choice
1.A system formed by the interactions of a community of organizations and their environment is referred to as
a(n)
a. interorganizational relationship.
b. organizational ecosystem.
c. collaboration network.
d. institutional environment.
AN

CO 250 Assignment 10 Spring 2014
Solutions
Problem 1: Shortest paths (Ch. 3.1)
Two examples of the shortest st-paths problem are given in the gure below. Choose one of these two examples.
Apply the algorithm from Chapter 3.1 to the example chosen by you t

CO 250 Assignment 2 Fall 2013
Solutions
Problem 1: IP Model MovingDay
(20 marks)
Graduation is coming up next week and you have accepted a position in sunny Vancouver. Now you need
to make plans for moving day. You have n items of size sj , j = 1, . . . ,

1
Problem 1: Simplex [14 Points]
(a) [3 Points] Suppose at some point of the Simplex algorithm you obtain the following problem in
canonical form. Indicate at this stage what are the set of all possible choices for the variables entering
the basis.
max
s.

CO 250 Assignment 6 Spring 2014
Solutions
Problem 1: Two Phase Method I
Consider the LP
(P )
max cT x subject to Ax = b, x 0,
where
3
1
c = 1 .
4
7
A=
2 1 4 2 1
1 0 3 1 1
2
1
b=
(a) Formulate the auxiliary problem (AuxP), and solve it using the simple

CO 250 Final Exam, Fall 2011
Page 2
#1 [16 marks]
[12] (a) Consider the following linear programming problem (P ) in standard equality form.
(P )
maximize z = 5x1 + 5x2 3x3
subject to
x 1 + 3 x2 x3 + x4
=2
x1
+ x5 = 2
x1 , x2 , x3 , x4 , x5 0
Solve (P ) u

Page 2
#1.
[15 marks]
(Simplex method)
Consider the following linear programming problem (P ) in SEF. Note that (P) is in canonical
form for the feasible basis B = cfw_3, 4.
(P )
maximize z =
subject to
3x1 + 4x2
x1 +
x1 +
x1 ,
[10]
x2 + x3
= 40
x2
+ x4 =

CO 250 - Assignment 2 solutions
Fall 2016
Page 1
Assignment #2: Solutions
Note: The solutions are written so that you understand in detail how each formulation is derived. Particularly, some inequalities only appear to explain how we derive other inequali

CO 250 - Assignment 10
Fall 2016
Page 1
Assignment #10: (Due on Friday, November 25, 11:59 pm)
Recommended reading: Sections 6.1 and 6.2 of the textbook.
Question 1
(20 points)
2
2
Consider the polyhedron P = cfw_x R2 : Ax b where A =
2
2
2
2
and b = 3

CO 250 - Assignment 4
Fall 2016
Page 1
Assignment #4: (Due on Friday, October 7, 11:59 pm)
Recommended reading: Section 2.2. 2.3, 2.5.1 (note: the textbooks treatment is significantly different
from the in-class presentation). Alternatively, we will post

CO 250 - Assignment 3
Fall 2016
Page 1
Assignment #3: (Due on Friday, September 30, 11:59 pm)
Question 1
Question
Points
1
12
2
23
3
20
Total:
55
(12 points)
(a) (4 points) Let A denote the following matrix:
2
5
6
T
T
and let b1 := 1 20 21 and let b2 := 2

CO 250 - Assignment 8
Fall 2016
Page 1
Assignment #8: (Due on Friday, November 11, 11:59 pm)
Recommended reading: Sections 4.1, 4.2, 4.3 of the textbook.
Question 1 (20 points)
Note:
Suppose (Q) and (Q0 ) are two LPs with the same sense (i.e. either they