Assignment 5
Due: Wednesday November 27 at the BEGINNING of class
1. Given the following linear programs, write down any 3 Chvatal-Gomory cuts. Explain
how you derive each one.
(a)
maximize
subject to
2x1 + x2 x3 + 3x4 + x5
9
11
4
x1 + x2 + x4 + x5
3
7
5

Assignment 5
Due: Friday December 4 at the BEGINNING of class
1. Given the following linear programs, write down any 3 Chvatal-Gomory cuts. Explain
how you derive each one.
(a)
maximize
subject to
2x1 + x2 x3 + 3x4 + x5
9
11
4
x1 + x2 + x4 + x5
3
7
5
8
6

Assignment 3
Due: Wednesday October 23 at the BEGINNING of class
1. Given the following linear programs with feasible basis B, do the following:
(i) solve the linear program without tableau, give an appropriate certicate
(ii) solve the linear program with

UNIVERSITY OF WATERLOO
FINAL EXAMINATION
Sample
Student Name (Print Legibly)
(family name)
(given name)
Signature
Student ID Number
COURSE NUMBER
CO 227
COURSE TITLE
Introduction to Optimization
COURSE SECTION
001
DATE OF EXAM
Someday
TIME PERIOD
Sometime

CO 227 - Homework assignment 1
Fall `10
Page 1
CO 227 - Fall `10 Homework assignment #1: (Due at the beginning of the class, 10:30AM, on Monday, Oct 4th) Instructions: Please show all your work and justify your answers. Answers without proper justificatio

Assignment 1
Due: Wednesday September 28 at the BEGINNING of class
1. Pats Porsches is a local company that makes cars that look like Porsches.
There are 2 suppliers (S1,S2) that provide the steel to make the cars. Each supplier
can supply the following a

CO 250 Assignment 10 Spring 2012
Grading Scheme and Common Errors
Problem 1:
(a) Write the dual (D) of the given linear program (P), and the complementary slackness conditions for
the pair (P) and (D).
max (2, 4, 2)x
s.t.
12
7
0 1 1 =
x
9 0
0
2 2 2
=
3
9

CO 250 Assignment 10 Spring 2012
Due: Friday July 20th by 10am
Solutions are due in the drop boxes outside MC 4066 by the due time.
Section 1 (L. Sanit`, TTh 1-2:20 p.m.)
a
Section 2 (J. Knemann, MWF 10:30-11:20 a.m.)
o
Box 9 Slots 6 (AD), 7 (EI), 8 (JM),

CO 227 - Assignment 4 - Winter 2017
Due: Wednesday March 15 at the BEGINNING of class
1. Given the following linear programs, write down the corresponding dual linear program:
(a)
(b)
(c)
maximize
subject to
x1 2x2 + 3x3 4x4 + 5x5
7x1 + 9x3 + 2x4 x5
2x1 3

CO 227 - Assignment 3 - Winter 2017
Due: Wednesday February 15 at the BEGINNING of class
1. Given the following linear programs with feasible basis B, do the following:
(i) solve the linear program without tableau, give an appropriate certificate
(you sho

CO 227 - Assignment 2 - Winter 2017
Due: Wednesday February 1 at the BEGINNING of class
1. Convert the following linear programs into standard equality form:
(a)
minimize
subject to
3x1 2x2 + 7x3
x1 x2 2x3 5
3x1 + x3 2
x3 0
Solution: maximize
subject to
(

CO 250 Assignment 3 Spring 2012
Grading Scheme and Common Errors
Problem 1: LP Solutions
Let A be an m n matrix. Prove that for each b Rm , either the system of equations Ax = b has no
solution or there exists a solution x with at most m nonzero component

CO 227 - Assignment 3 - Winter 2017
Due: Wednesday February 15 at the BEGINNING of class
1. Given the following linear programs with feasible basis B, do the following:
(i) solve the linear program without tableau, give an appropriate certificate
(you sho

CO 227 - Assignment 4 - Winter 2017
Due: Wednesday March 15 at the BEGINNING of class
1. Given the following linear programs, write down the corresponding dual linear program:
(a)
maximize
subject to
Solution:
(b)
maximize
subject to
Solution:
(c)
minimiz

Assignment 2
Due: Wednesday October 9 at the BEGINNING of class
1. Convert the following linear programs into standard equality form:
(a)
minimize
subject to
2x1 + 3x2 + 4x3
8x1 + x3 2
3x2 8x3 1
x1 0
Solution: maximize
subject to
(b)
maximize
subject to
2

CO 250 Assignment 9 Spring 2012
Solutions
Problem 1:
For each of (a) and (b) do the following:
Write the dual (D) of the given linear program (P).
Use Weak Duality to prove that x is optimal for (P) and y optimal for (D).
a)
max
(2, 1, 0)x
s.t.
1 32
x
1

CO 250 Assignment 9 Spring 2012
Grading Scheme and Common Errors
Problem 1:
For each of (a) and (b) do the following:
Write the dual (D) of the given linear program (P).
Use Weak Duality to prove that x is optimal for (P) and y optimal for (D).
a)
max
(

CO 250 Assignment 9 Spring 2012
Due: Friday July 13th by 10am
Solutions are due in the drop boxes outside MC 4066 by the due time.
a
Section 1 (L. Sanit`, TTh 1-2:20 p.m.)
Section 2 (J. Knemann, MWF 10:30-11:20 a.m.)
o
Box 9 Slots 6 (AD), 7 (EI), 8 (JM),

CO 250 Assignment 3 Spring 2012
Solutions
Problem 1: LP Solutions
Let A be an m n matrix. Prove that for each b Rm , either the system of equations Ax = b has no
solution or there exists a solution x with at most m nonzero components.
Solution: If the sys

CO 250 Assignment 4 Spring 2012
Due: Friday June 1st by 10am
Solutions are due in the drop boxes outside MC 4066 by the due time.
Section 1 (L. Sanit`, TTh 1-2:20 p.m.)
a
Section 2 (J. Knemann, MWF 10:30-11:20 a.m.)
o
Box 9 Slots 6 (AD), 7 (EI), 8 (JM),
9

CO 250 Assignment 4 Spring 2012
Grading Scheme and Common Errors
Problem 1: SEF
Let A, B, D be matrices and b, c, d, f be vectors (all of suitable dimensions). Convert the following LP
with variables x and y (where x, y are vectors) into SEF,
min cT x + d

CO 250 Assignment 4 Spring 2012
Solutions
Problem 1: SEF
Let A, B, D be matrices and b, c, d, f be vectors (all of suitable dimensions). Convert the following LP
with variables x and y (where x, y are vectors) into SEF,
min cT x + dT y
s.t.
Ax By b
Dy
f
y

CO 250 Assignment 5 Spring 2012
Due: Friday June 8th by 10am Solutions are due in the drop boxes outside MC 4066 by the due time. Section 1 (L. Sanit`, TTh 1-2:20 p.m.) a Box 9 Slots 6 (AD), 7 (EI), 8 (JM), 9 (NQ), 10 (RU), 11 (V Z) Section 2 (J. Knemann,

CO 250 Assignment 5 Spring 2012
Grading Scheme and Common Errors
Problem 1: Simplex I
Consider the following linear programming problem. Start with the basis B := cfw_1, 2, put the problem
in canonical form and solve the problem using the Simplex Method.

CO 250 Assignment 5 Spring 2012
Solutions
Problem 1: Simplex I Consider the following linear programming problem. Start with the basis B := cfw_1, 2, put the problem in canonical form and solve the problem using the Simplex Method. max z = 5x1 + x2 - x3 -

CO 250 Assignment 6 Spring 2012
Due: Friday June 22nd by 10am
Solutions are due in the drop boxes outside MC 4066 by the due time.
Section 1 (L. Sanit`, TTh 1-2:20 p.m.)
a
Section 2 (J. Knemann, MWF 10:30-11:20 a.m.)
o
Box 9 Slots 6 (AD), 7 (EI), 8 (JM),

CO 250 Assignment 6 Spring 2012
Solutions
Problem 1: Simplex Geometry
Consider the LP maxcfw_cT x : Ax b, x 0, where
A :=
21
24
b :=
4
8
c :=
2
.
3
a) Convert this LP into SEF in the standard way, and then solve it via Simplex. In each iteration, for
the

CO 250 Assignment 7 Spring 2012
Due: Friday June 29th by 10am
Solutions are due in the drop boxes outside MC 4066 by the due time.
Section 1 (L. Sanit`, TTh 1-2:20 p.m.)
a
Section 2 (J. Knemann, MWF 10:30-11:20 a.m.)
o
Box 9 Slots 6 (AD), 7 (EI), 8 (JM),

CO 250 Assignment 7 Spring 2012
Solutions
Problem 1: Geometry of the simplex
Consider the following linear program
maximize c1 x1 + c2 x2
subject to
x1 +
x2
x1
x2
x1
x2
4
3
3
0
0
(a) Give a diagram showing the set of feasible solutions of the LP
(b) For e

CO 250 Assignment 8 Spring 2012
Due: Friday July 6th by 10am
Solutions are due in the drop boxes outside MC 4066 by the due time.
Section 1 (L. Sanit`, TTh 1-2:20 p.m.)
a
Section 2 (J. Knemann, MWF 10:30-11:20 a.m.)
o
Box 9 Slots 6 (AD), 7 (EI), 8 (JM),
9