CO350 Homework 2 Winter 2007
Due: Friday, Jan. 19, 2007. Hand in at the start of class. Problem 1: Formulation - Labour Management (Bertsimas & Freund) A manufacturing company is anticipating an increase in demand for its products. However, manageme

CO350
Assignment 1
Due: May 14 at 2pm.
Submit your assignment in drop box 2 (opposite MC4066) by 2pm. Your full name, student ID number, the course number, and the name of your instructor should all be clearly
visible on the front of your assignment. You

CO 350 Linear Optimization
TAs comments for Assignment 2
General: It seems many students are confused about basic operations in linear algebra. Below we list
some examples of mistakes of these types. Watch out for them.
(i) If A is a matrix and Ax = 0 for

CO 350 Linear Optimization
TAs comments for Assignment 4
General: Many students are still making arithmetic mistakes. Please double check all your calculations
before you hand in your assignment.
Q1 [(a) 4 marks; (b) 2 marks; (c) 4 marks]:
1. In part (a),

CO 350 Linear Optimization
TAs comments for Assignment 5
Q1 [10 marks]: Many students made claims without justication. For instance, if you want to say
that a positive multiple of the row of the leaving variable is added to the z -row, then you must justi

CO 350 Linear Optimization
TAs comments for Assignment 6
General: Many students are still confused by the multiple uses of the word degenerate. Recall that
degenerate basis and degenerate iterations are two completely dierent things.
A basis B is degenera

CO 350 Linear Optimization
TAs comments for Assignment 7
General:
Q1 [(a) 5 marks; (b) 5 marks]: This question was generally well done. However, some students
solved the system AT y = cB to nd the dual basic solution; the correct system to solve is AT y =

CO 350 Linear Optimization
TAs comments for Assignment 8
Q1 [10 marks]:
1. Some students assumed that the basic solution corresponding to B in the new LP is given by
xB + d, but did not prove this. See the solution for the details of how to prove it.
2. S

CO 350 Linear Optimization
TAs comments for Assignment 9
Q1 [10 marks]: Some students used the same slack variables for all the new constraints. This is
wrong. Each new constraint gets its own slack variable.
Q2 [10 marks]: Generally okay.
Q3 [10 marks]:

C O350 L INEAR P ROGRAMMING - A SSIGNMENT 8
Due Date: Friday July 16 at 2 PM in assignment drop box 2, in front of MC4066.
Please make sure that your name and student number, the course number, and the name of your instructor
are clearly written on the fr

CO350 Linear Optimization
This course is an introduction to linear programming.
Instructors.
Jim Geelen (jfgeelen@math), MC5020.
Mathieu Guay-Paquet (mguaypaquet@math), DC3144.
Irene Pivotto (ipivotto@math), DC3145 .
Course notes. Available from Campus

CO350 Assignment 1
Alex Fok 20300650
May 14, 2010
Instructor: Mathieu Guay-Paquet
1(a) Given
Ax=b has a solution, s
Columns of A are not linearly independent
Proof
Let K be the solution set of the system Ax=b
Let KH be the solution set of the correspondin

CO350
Assignment 1
Due: May 14 at 2pm.
Submit your assignment in drop box 2 (opposite MC4066) by 2pm. Your full name, student ID number, the course number, and the name of your instructor should all be clearly
visible on the front of your assignment. You

CO350 Assignment 2
Wing Man Fok 20300650
May 21, 2010
Instructor: Mathieu Guay-Paquet
1(a) [contrapositive] Suppose the set a1,ak+1 is linearly dependent,
Then, there exists scalars c1,ck+1 not all zero such that c1a1+
+ck+1ak+1=0(*)
Note that ck+10. Othe

C O350 L INEAR P ROGRAMMING - A SSIGNMENT 2
Due Date: Friday May 21 at 2 PM.
Please make sure that your name and student number, the course number, and the name of your instructor
are clearly written on the front of your assignment, and that all pages are

CO350 Assignment 3
Wing Man Fok 20300650
May 28, 2010
Instructor: Mathieu Guay-Paquet
1.
The dual problem is
min &12&y1+&22&y2+&8&y3s.t.
&12&y1+&7&y2+&3&y32&y1+&y3-2&2&y1+&3&y2+&y31&y1+&4&y2+&y32
The C.S. conditions are
&x1*=0 or &12&y1*+&7&y2*+&3&y3*&=2&

Following a "Balanced" Trajectory from an
Infeasible Point to an Optimal Linear
Programming Solution with a Polynomial-time
Algorithm
Robert M. Freund
WP# 3453-92-MSA
July, 1992
Following a "Balanced" Trajectory from an
Infeasible Point to an Optimal Line

Lecture notes ?: The simplex algorithm
Vincent Conitzer
1
Introduction
We will now discuss the best-known algorithm (really, a family of algorithms) for solving a linear program, the simplex algorithm. We will demonstrate it on an example. Consider again

C O350 L INEAR P ROGRAMMING - A SSIGNMENT 2
Due Date: Friday May 21 at 2 PM.
Please make sure that your name and student number, the course number, and the name of your instructor
are clearly written on the front of your assignment, and that all pages are

CO350Linear Optimization
Assignment 3
This assignment is due Friday, May 28, 2010, at 2 pm, in assignment drop box #2, in front of
MC4066. Please make sure that your full name, your student ID number, the course number, and
the name of your instructor are

CO350
Assignment 4
Due: June 11 at 2pm.
Your full name, student ID number, the course number, and the name of your instructor
should all be clearly visible on the front of your assignment.
Exercise 1: Consider the linear program:
max
2x1
subject to
x1
+

C O350 L INEAR P ROGRAMMING - A SSIGNMENT 5
Due Date: Friday June 11 at 2 PM in assignment drop box 2, in front of MC4066.
Please make sure that your name and student number, the course number, and the name of your instructor
are clearly written on the fr

CO350Linear Optimization
Assignment 6
This assignment is due Friday, June 25, 2010, at 2 pm, in assignment drop box #2, in front of
MC4066. Please make sure that your full name, your student ID number, the course number, and
the name of your instructor ar

CO350
Assignment 7
Due: July 9 at 2pm.
Your full name, student ID number, the course number, and the name of your instructor should all be
clearly visible on the front of your assignment.
Exercise 1: For the following pairs (A, b) solve the feasibility pr

C O350 L INEAR P ROGRAMMING - A SSIGNMENT 8
Due Date: Friday July 16 at 2 PM in assignment drop box 2, in front of MC4066.
Please make sure that your name and student number, the course number, and the name of your instructor
are clearly written on the fr

CO350Linear Optimization
Assignment 9
This assignment is due Friday, July 23, 2010, at 2 pm, in assignment drop box #2, in front of
MC4066. Please make sure that your full name, your student ID number, the course number, and
the name of your instructor ar

CO350
Assignment 1 Solutions
Exercise 1: (linear algebra review.) Let A Rmn and b Rm .
(a) If the system Ax = b has a solution and the columns of A are not linearly independent, then Ax = b has innitely many solutions.
Proof. Suppose that x0 Rn satises Ax

C O350 L INEAR P ROGRAMMING - S OLUTIONS TO ASSIGNMENT 2
Exercise 1. Let cfw_a1 , . . . , ak Rm be a set of linearly independent vectors.
(a) Prove that if ak+1 Rm is not in the subspace spanned by a1 , . . . , ak then the set cfw_a1 , . . . , ak+1 is
l