MS&E111 Introduction to Optimization Prof. Amin Saberi
HW-2 Solutions
Homework 2 Solutions Problem 1. a) The feasible region for the given LP is:
6
4
2 Feasible Region 0
-2
-4
-6 -5
-4
-3
-2
-1
0
1
2
3
4
5
b) The dual of the LP is minimize subject to -4y1

ENGR62/MS&E111 Introduction to Optimization Prof. Ben Van Roy
Spring 2008 April 11, 2008
Homework Assignment 1: Solutions
Network Routing
See the Excel solution worksheet on the course web page.
Linear Algebra Review (Chapter 2)
Problem 1 Expanding out th

ENGR62/MS&E111 Introduction to Optimization Prof. Ben Van Roy
Spring 2008 May 2, 2008
Homework 4 Solutions
Part A True or false: a) TRUE. Let x1 be a solution to Ax = b. If x2 is a second distinct solution, then it must be that x2 - x1 N (A). But here dim

MS&E111 Introduction to Optimization Prof. Amin Saberi
Spring 2006 May 1, 2006
Solutions to Assignment 1
1. a) Let xi be the number of agents working ith time shift, i.e. x2 means the number of working agents during 8am-4pm and so on. Then we can formulat

DRAFT Formulation and Analysis of Linear Programs
Benjamin Van Roy and Kahn Mason c Benjamin Van Roy and Kahn Mason September 26, 2005
1
2
Contents
1 Introduction 1.1 Linear Algebra . . . . . . . . . . 1.2 Linear Programs . . . . . . . . . 1.3 Duality . .

e62 Introduction to Optimization Prof. Ben Van Roy
Spring 2008 April 24, 2008
Homework Assignment 4: Due May 2 PART A
Consider a polyhedron P = cfw_x|Ax = b, x 0. Suppose that the matrix A has dimensions m n and that its rows are linearly independent. For

Chapter 5 Network Flows
A wide variety of engineering and management problems involve optimization of network flows that is, how objects move through a network. Examples include coordination of trucks in a transportation system, routing of packets in a co

ENGR62/MS&E111 Introduction to Optimization Prof. Ben Van Roy
Spring 2005 June 2, 2005
Final Exam: Due Friday June 3rd, 2005, at 5:00PM Instructions
Sign the honor code statement and return this hand-out when you submit your solutions. Turn in your exam t

46
Chapter 3 Linear Programs
A linear program involves optimization (i.e., maximization or minimization) of a linear function subject to linear constraints. A linear inequality constraint on a vector x N takes the form aT x b or a1 x1 + a2 x2 + . . . + aN

Chapter 4 Duality
Given any linear program, there is another related linear program called the dual. In this chapter, we will develop an understanding of the dual linear program. This understanding translates to important insights about many optimization

e62 Introduction to Optimization Prof. Ben Van Roy
Spring 2008 April 2, 2008
Homework Assignment 1: Due April 11
Network Routing
The purpose of this problem is to develop experience with Excel Solver. We will be working with the spreadsheet presented on t

ENGR62/MS&E111 Introduction to Optimization Prof. Ben Van Roy
Spring 2005 May 5, 2005
Midterm Exam: Due Friday May 6th, 2005, at 5:00PM Instructions
Sign the honor code statement and return this hand-out when you submit your solutions. Turn in your exam t

MS&E111 Introduction to Optimization Prof. Amin Saberi
Lecture 10 May 15, 2006 From Prof. Van Roy's Notes
1
Network Flows
A wide variety of engineering and management problems involve optimization of network flows that is, how objects move through a netwo

46
Chapter 3 Linear Programs
A linear program involves optimization (i.e., maximization or minimization) of a linear function subject to linear constraints. A linear inequality constraint on a vector x N takes the form aT x b or a1 x1 + a2 x2 + . . . + aN

MS&E111 Introduction to Optimization Prof. Amin Saberi
Lecture 8 May 1-3, 2006
1
Two player Zero-Sum games
In this section, we consider games in which each of two opponents selects a strategy and receives a payoff contingent on both his own and his oppone

ENGR62/MS&E111 Introduction to Optimization Prof. Ben Van Roy
Fall 2008 May, 2008
Midterm Exam
Instructions
This is a take-home, open book/notes exam. There are 4 questions. Each is worth a total of 25 points. Partial credit is possible for each question

e62 Introduction to Optimization Prof. Ben Van Roy
Spring 2008 April 17, 2008
Homework Assignment 3: Due April 25 Part A
Consider the following linear program: maximize cT x
2
1
3
0 subject to 2
3 0 5 x2
3
x1
3
2
8
a) Provide a vector c such that t

e62 Introduction to Optimization Prof. Ben Van Roy
Spring 2008 April 10, 2008
Homework Assignment 2: Due April 18
Do problems 1, 2, 3, 8, 10 from Chapter 3.

e62 Introduction to Optimization Prof. Ben Van Roy
Spring 2008 May 2, 2008
Homework Assignment 5: Due May 16 PART A
Do problems 4.6.2, 4.6.3, 4.6.4, 4.6.8, and 4.6.10 from the notes.
PART B
Problem 1. Provide numerical examples (object function and constr

e62 Introduction to Optimization Prof. Ben Van Roy
Fall 2006 May 16, 2008
Homework Assignment 6: Due May 23 PART A
Do problems 4.6.5, 4.6.6, 4.6.11, 4.6.12, 4.6.13.
PART B
Consider the following two-player, zero-sum game. Player 1 chooses action A, B or C

Max-Flow / Min-Cut Theorem
MS&E 111-Introduction to Optimization May 22, 2006
Refresher on the Duality Theorems
The Weak Duality Theorem tells us that any solution to the dual gives us an upper (lower) bound The Strong Duality Theorem tells us that at the

ENGR62/MS&E111 Introduction to Optimization Prof. Ben Van Roy
Fall 2006 November, 2006
Midterm Exam
Instructions
This is a take-home, open book/notes exam. There are 4 questions. Each is worth a total of 20 points. Partial credit is possible for each que

ENGR62/MS&E111 Introduction to Optimization Prof. Ben Van Roy
Fall 2006 November, 2006
Midterm Exam Solutions
1: The Geometry of Linear Programming
(a) Consider a polyhedron in 2-dimensional Euclidean space dened by 5 inequality constraints. i) What is th

ENGR62/MS&E111 Introduction to Optimization Prof. Ben Van Roy
Spring 2005 May 5, 2005
Midterm Exam: Due Friday May 6th, 2005, at 5:00PM Instructions
Sign the honor code statement and return this hand-out when you submit your solutions. Turn in your exam t

ENGR62/MS&E111 Introduction to Optimization Prof. Ben Van Roy
Fall 2005 December 8, 2005
Final Exam
Instructions
This is a take-home, open book/notes exam. There are 3 questions. Each has equal weight. Partial credit is possible for each question provide

Chapter 5 Network Flows
A wide variety of engineering and management problems involve optimization of network flows that is, how objects move through a network. Examples include coordination of trucks in a transportation system, routing of packets in a co

Chapter 2 Linear Algebra
Linear algebra is about linear systems of equations and their solutions. As a simple example, consider the following system of two linear equations with two unknowns: 2x1 - x2 = 1 x1 + x2 = 5. There is a unique solution, given by

ENGR62/MS&E111 Introduction to Optimization Prof. Ben Van Roy
Spring 2008 April 25, 2008
Homework Assignment 3: Solutions
Part A
(a) We have the following feasible region (shaded in yellow):
10
8
6
4
2
0 -4 -2 0 -2 2 4 6 8 10
-4
where the constraints, cou

ENGR62/MS&E111 Introduction to Optimization Prof. Ben Van Roy
Spring 2008 April 18, 2008
Homework Assignment 2 : Solutions
Solve Questions 1, 2, 3, 8 and 10 from Chapter 3. 3.1. Adding x + y 0 reduces the feasible region to cfw_[0, 0]. 3.2. The polyhedron