CO250 I NTRODUCTION TO O PTIMIZATION , W INTER 2017
A SSIGNMENT 3 SOLUTIONS
un
e
l
Due: Friday, January 27, 2017, at 4:00 p.m.
Exercise 1 (25 marks). [Integer Programming]
Milton Industries has a number of employees ei and machines mj . On a given day, ea
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
CO 250 - Assignment 6
Fall 2016
Page 1
Assignment #6: (Due on Friday, October 28, 11:59 pm)
Recommended reading: Sections 2.6, 2.8 of the textbook.
For the Simplex Algorithm questions, you may use whichever form of the Simplex Algorithm you are
most com
CO 250 - Assignment 9
Fall 2016
Page 1
Assignment #9: (Due on Friday, November 18, 11:59 pm)
Recommended reading:
Sections 3.2.1 to 3.2.5 of the textbook.
Document posted on Learn: Minimum Cost Perfect Matching in Bipartite Graphs https:
/learn.uwaterl
CO 250 - Assignment 11
Fall 2016
Page 1
Assignment #11: (Due on Sunday, December 4th, 11:59 pm)
All answers must be justified, unless otherwise stated.
Recommended reading: Sections 7.3, 7.4 and 7.5 of the textbook.
Question 1 (18 points)
Suppose that f
CO 250 - Assignment 1
Fall 2016
Page 1
Assignment #1: (Due on Friday, September 16, 11:59 pm)
Question
Points
1
14
2
16
3
15
4
15
Total:
60
Question 1 (14 points)
This is a linear algebra review question. If you have trouble completing any part of this qu
CO 250 - Assignment 7
Fall 2016
Page 1
Assignment #7: (Due on Friday, November 4, 11:59 pm)
Recommended reading:
Sections 4.1, 4.2 of the textbook.
Document posted on Learn: Writing the Dual of a Linear Program https:/learn.uwaterloo.
ca/d2l/le/content
Page 1
#1.
[18 marks]
(Integer Programming models)
Note: Your solution should include a brief description of the main decision variables
(state the purpose of each one), and a brief description of the main constraints.
Your goal is to present a model, and
1
Problem 1: Extreme points of polyhedra
(a) Explain why x = (0, 0, 2, 0, 0)> is an extreme point of the following polyhedron.
1 0 2 0 3
4
5
xR :
x=
, x0 .
1 2 1 3 0
2
(14 marks)
(4 marks)
(b) Consider the polyhedron Q defined by the following constrain
1
Problem 1: Simplex Algorithm and the Two-Phase Method [23 Points]
Consider the following LP:
(P)
max
1
s.t.
4
2
3
(1, 3, 1,
2)x
3
4
20
x =
2
1
10
x
0.
(a) [4 Points] Suppose we would like to solve (P) using the Two-Phase Method. Write down the
auxilia
CO 250 - Assignment 5
Fall 2016
Page 1
Assignment #5: (Due on Saturday, October 15, 11:59 pm)
Recommended reading: Sections 2.3, 2.4, 2.5 of the textbook. The document posted on Learn: The
Simplex Algorithm using Dictionaries https:/learn.uwaterloo.ca/d2
CO 250 - Assignment 2
Fall 2016
Page 1
Assignment #2: (Due on Friday, September 23, 11:59 pm)
Recommended reading: Section 1.3, Section 1.4.2, Section 1.5.1.
Question 1 (20 points)
Disclaimer: This is supposed to be an exercise in modeling and thus sever
CO 250 - Assignment 3
Fall 2016
Page 1
Assignment #3: (Due on Friday, September 30, 11:59 pm)
Recommended Reading: Section 2.1 from Chapter 2, and Sections 1.4.2, 1.5.1 from Chapter 1.
Question 1
(12 points)
(a) (4 points) Let A denote the following matr
CO 250 - Assignment 1
Fall 2016
Page 1
Assignment #1: (Due on Friday, September 16, 11:59 pm)
Question 1 (14 points)
This is a linear algebra review question. If you have trouble completing any part of this question, it is
an indication that you need to r
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
CO 250
Introduction to
Optimization
September 12
There are three refineries (1, 2, 3), two distribution terminals
(Kitchener and Waterloo), and three gas stations (A, B, C).
For i cfw_1, 2, 3, refinery i produces no more than pi tons of gasoline.
For j cf
CO 485/685: THE MATHEMATICS OF PUBLIC-KEY CRYPTOGRAPHY
Fall 2016
Instructor:
Alfred Menezes (MC 5026, ajmeneze@uwaterloo.ca)
Office hours: 4:00-5:00 pm, Monday and Tuesday
Teaching assistant: Luis Ruiz-Lopez (MC 5492, luis.ruiz-lopez@uwaterloo.ca)
Office
CO 685: Course Project
A list of suggested project topics is available on the course web site. This list will grow as the semester
progresses, and will be more-or-less complete by November 15. You are also welcome to select your
own topic, however I have
CO 466/666: Continuous Optimization
Fall 2016
Problem Set 1
S. Vavasis
Handed out: 2016-Sep-16.
Due: 2016-Sep-23 in lecture.
1. Let A Rnn be a symmetric positive semidefinite matrix.
(a) Show that if x Rn is a vector such that xT Ax = 0, then Ax = 0. Ther
CO 342
Assignment 2
Due: Wednesday, September 25
1. (5 points) List all the 2-connected simple graphs on five vertices which cannot be expressed
as non-trivial 2-sums.
2. (5 points) Show that a connected regular bipartite graph with valency at least two i
C&O 342
Assignment 1: corrected version
Due: 11:30am, Wednesday September 18
1. If is an automorphism of G and u V (G), show that u and (u) have
the same valency. [If S V (G), it may be useful to use (S) to denote
the set cfw_(x) : x S.]
2. Let G be a cub
CO 342
Assignment 7
Due: Wednesday, November 6
1. (5 points) Determine which cycles in K 3,3 and K 5 are peripheral and show that, in both
cases, each edge lies in at least three peripheral cycles.
2. (5 points) Let e be an edge in a graph G and suppose t
CO 342
Assignment 7
Due: Wednesday, November 6
1. (5 points) Let G be a connected graph with vertices u and v, and suppose that there is a
uv-separating set of size k. Prove that there are vertices a and b at distance two in G which
can be separated by a
CO 342
Assignment 4
Due: Wednesday, October 9
1. (5 points) Let G be a 2-connected graph and let C be a cycle in G with greatest possible
length. Show that if e is an edge in C , then G/e is 2-connected.
2. (3 points) Construct a 2-connected graph G such