ORIE 321/521
Optimization II
Summer 2006
Homework # 3
Due: Wednesday, August 2, in class. Please print your name clearly on the first page of your homework. 1. #6, BHM Chapter 8. 2. #13, BHM Chapter 8 formulate only. 3. #26, BHM Chapter 8. 4. Gi

7.5
Cutting Planes
Integer programming problems are typically hard to solve. Nonetheless, the state of the art today enables IP models to be solved sufficently well, so that it is a key element of the optimization toolkit. Consider the following in

13
Column generation methods
We return to the question of solving large-scale integer programs and their linear programming relaxations. We have already studied a technique for solving large-scale (but still compactly represented) linear programs b

7.6
A useful IP formulation for a scheduling problem
Consider the following scheduling problem: there are m identical machines, and n jobs on which they must be scheduled; each job must be scheduled on exactly one machine, and must be processed to

ORIE 321/521
RECITATION 11
Spring 2007
In this recitation, we return to an application mentioned at the start of the fall semester. The registrar of a prominent university approaches you to help with the problem of scheduling final exams. At this

ORIE 321/521
Optimization II
Summer 2006
Homework # 1
Due: Tuesday, July 18, 4 p.m. in the OR homework drop box for OR321. Please print your name clearly on the first page of your homework. 1. (a) Give brief (but precise) mathematical definitions

ORIE 321/521
Optimization II
Summer 2006
Homework # 2
Due: Tuesday, July 26, 4 p.m. in the OR homework drop box for OR321. Please print your name clearly on the first page of your homework. These first few problems are "practice" problems, not to

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i , j
4) a. Let M be a very large constant (on the order of tij ). Yikj is an

ORIE 321/521
Optimization II
Summer 2006
Homework # 4
Due: Thursday, August 10, in class. Please print your name clearly on the first page of your homework. 1. #1, BHM Chapter 9. 2. #4, BHM Chapter 9. 3. #10, BHM Chapter 9. 4. #18, BHM Chapter 9.

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ORIE 361/523 : Introduction to Stochastic Processes Summer2007 Problem Set 4
Due in one week, on July 27 in class. 1. (10 points, page 266, problem 20) A transition matrix P is said to be doubly stochastic if the sum over each column equals one; that

ORIE 361/523 Homework 4
Instructor: Bikramjit Das
due July 27, 2007
Answers. 1. If i Pi,j = 1 then, rj =
1 M +1
j,
satisfies rj = m ri Pi,j i=0
1 = m ri i=0 Hence by uniqueness of stationary distribution(since the chain is irreducible and aperi

7
Integer Programming
Integer programming methods are among the most powerful optimization tools available, and are used in a wide range of applications. Whereas 30 years ago, most real-world problems were unsolvable by this approach, today integer

6.7
Stochastic dynamic programming
Dynamic programming techniques can also be applied to settings in which there is a probabilistic element to the problem specification. As a first "toy" example, consider the following betting game. Suppose that yo

ORIE 321/521/522
RECITATION 3
Spring 2007
Consider the following AMPL model file maxflow.mod for the maximum flow problem, which can be found in the MODELS folder within the amplcml folder. set nodes; param orig symbolic in nodes; param dest symbo

ORIE 321/521
RECITATION 4
Spring 2007
Consider the following AMPL model file maxflow.mod for the maximum flow problem, which can be found in the MODELS folder within the amplcml folder on your desktop. set nodes; param orig symbolic in nodes; para

8
Nonlinear optimization
Linear programs are relatively easy to solve. By contrast, as we have seen, integer programs may be much harder. To solve an integer program, we try to capitalize on our ability to solve linear programs quickly, studying li

ORIE 321/521
RECITATION 8
Spring 2007
The purpose of this recitation exercise is to solve, by a combination of hand computation, and with AMPL, the LP relaxation of the IP formulation of the traveling salesman problem (TSP) (as was discussed in th

18-7. A socialaccounting matrix is a tablethat showsthe flows from eachsector an economy in to eachothersector.Hereis sirnplefive-sector example, with blankentries indicating flows known to be zero:
I,AB I,AB H1 H2 P1 P2 total
15 ? ,)
15 25
3
130 4

ORIE 321/521
RECITATION 6
Spring 2007
In the previous recitation exercise, you computed the optimal betting strategy for a completed NCAA tournament. In this one, you will apply dynamic programming techniques to select a bet in the same style pool

ORIE 321/521/522
In this recitation we review
RECITATION 1
Spring 2007
how to download the AMPL modeling language package to your computer (though you should not need to do this now); how to use AMPL to solve simple linear and integer programs.

ORIE 321/521
RECITATION 2
Spring 2007
Write out an AMPL model for the maximum flow problem. In this model, you should let VERTICES be the name of the set used for the nodes in the input, you should let EDGES be the name for the set used to specify

1
Introduction
This course will cover methods to solve integer and linear programming problems, and present applications in which these methods are useful; whereas the emphasis of Optimization I was on linear programming, the emphasis of Optimizati

3
The maximum flow problem
Although it is a special case of the minimum-cost network flow problem, the maximum flow problem is of great importance in its own right. This problem can be motivated by the following setting. Imagine that you have a net

4
The project selection problem
One of the surprising aspects of the maximum-flow minimum-cut theorem is that while we started thinking about solving one optimization problem, we ended up solving two problems for the price of one. We now also have

6
Dynamic Programming
The next main topic is a general algorithmic technique for solving optimization problems; this is in contrast to previous topics in this course and in the fall, where we have focused on a class of optimization models, such as

6.4
Distribution of effort problems
One type of problem that can be solved via dynamic programming are socalled "distribution of effort problems". This type of problem occurs when you have some resource that is to be shared among a number of compet

ORIE 321/521
RECITATION 5
Spring 2007
The aim of this recitation is to apply dynamic programming techniques to compute (after the fact) an optimal betting strategy for a pool to wager on the outcome of the 2001 NCAA men's basketball tournament. Fo