CO 227 Assignment 2 Solutions
Exercise 1.9 We rst solve the cargo loading problem with the data given in Table 1.16
(below) with weight restriction W = 17 then we use this solution to solve the cargo loading
problem with W = 100.
Table 1.16
Item i wi vi v

CO 227 Assignment 4
Due: Friday November 20 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 3x2 + x4 + 3x5
x1

Assignment 2
Due: Friday October 16 at the BEGINNING of class
1. Convert the following linear programs into standard equality form:
(a)
minimize
subject to
3x1 + 2x2 7x3
x2 2x3 4
3x1 + x3 2
x1 0
Solution: maximize
subject to
(b)
maximize
subject to
3x1 2x

Assignment 5
Due: Wednesday November 28 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
3
8
7
x1 + x2 + x4 + x5
4
2
3
8

Assignment 5
Due: Wednesday April 1 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 1
Due: Friday October 2 at the BEGINNING of class
Note: Do NOT attempt to solve any of the mathematical programs you come up with for
this assignment.
1. Prof. Roh is selling copies of the latest Justin Bieber CD.
There are 3 suppliers (S1,S2,S

Assignment 3
Due: Friday October 30 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 ta

UNIVERSITY OF WATERLOO
MIDTERM EXAMINATION
WINTER TERM 2011
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
Tuesday, March 1

Assignment 2
Due: Wednesday October 10 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

Assignment 3
Due: Wednesday October 24 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 wit

Assignment 1
Due: Wednesday September 26 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

Assignment 2
Due: Wednesday October 10 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
(b)
maximize
subject to
4x1 7x2 + x3
5x1 3x2 + x3
x1 +

UNIVERSITY OF WATERLOO
MIDTERM EXAMINATION
FALL TERM 2011
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
Wednesday, Novembe

Assignment 3
Due: Friday October 30 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 ta

Assignment 2
Due: Friday October 16 at the BEGINNING of class
1. Convert the following linear programs into standard equality form:
(a)
minimize
subject to
3x1 + 2x2 7x3
x2 2x3 4
3x1 + x3 2
x1 0
(b)
maximize
subject to
6x1 4x2 + 3x3
5x1 x2 + x3
2x1 3x3
x2

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University of Waterloo
Department of Electrical and Computer Engineering
ECE 602 Introduction to Optimization
Winter, 2016
Classes: Thursdays, 8:30 - 11:20 AM, RCH 308
Instructor: Prof. Oleg Michailovich
Oce: EIT 4127
Oce hours: TBD
Contact: [email protected] kno

The Matrix Cookbook
[ http:/matrixcookbook.com ]
Kaare Brandt Petersen
Michael Syskind Pedersen
Version: November 15, 2012
1
Introduction
What is this? These pages are a collection of facts (identities, approximations, inequalities, relations, .) about ma

Optim Eng (2007) 8: 67127
DOI 10.1007/s11081-007-9001-7
E D U C AT I O N A L S E C T I O N
A tutorial on geometric programming
Stephen Boyd Seung-Jean Kim
Lieven Vandenberghe Arash Hassibi
Received: 17 March 2005 / Revised: 15 September 2005 /
Published

Optimization problem in standard form
Standard optimization problem has the following form
minimize f0 (x)
subject to fi (x) 0,
i = 1, . . . , m
hi (x) = 0,
i = 1, . . . , p
x R is the optimization variable.
f0 : Rn R is the cost or objective function.
fi

Convex optimization
Convex optimization is a mathematically rigorous and well-studied
eld, having countless practical applications.
With only a bit of exaggeration, we can say that, if you formulate a
practical problem as a convex optimization problem, th

Inner product, Euclidean norm, and angle
The standard inner product on Rn , the set of real n-vectors, is given by
hx, yi = xT y =
n
X
x i yi ,
i=1
for x, y 2 Rn .
The Euclidean norm, or `2 -norm, of x 2 Rn is dened as
kxk2 = xT x
1/2
= x2 + x2 + . . . +

Home Assignment 1
Due on January 28, 2016
Principal part (5% of the nal mark)
Exercise 1
Which of the following sets are convex? Explain your answer.
a) A slab, i.e., a set of the form cfw_x Rn | aT x .
b) A rectangle, i.e., a set of the form cfw_x Rn | i

UNIVERSITY OF WATERLOO
MIDTERM EXAMINATION
FALL TERM 2015
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
Tuesday, November

CO 227 Assignment 4
Due: Friday November 20 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)
minimize
subject to
x1

Assignment 1
Due: Friday October 2 at the BEGINNING of class
Note: Do NOT attempt to solve any of the mathematical programs you come up with for
this assignment.
1. Prof. Roh is selling copies of the latest Justin Bieber CD.
There are 3 suppliers (S1,S2,S

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

Assignment 2
Due: Friday October 14 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
(b)
maximize
subject to
6x1 4x2 + 3x3
5x1 x2 + x3
2x1 3x3

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
(b)
maximize
subject to
6x1 4x2

CO 227
Introduction To Optimization
Instructor: Patrick Roh
E-mail: [email protected]
Fall 2016
Office: MC 6494
Extension: x36200
Lecture: MWF 12:30-1:20pm, MC 4041
Course Website: The course website will be hosted on LEARN. Students are responsible
for c