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Mathematical Optimization, Applications & Analysis
"In recent times, modern science has developed to give mankind, for the first time in the history of the human race, a way of securing a more abundant life which does not simply consist in taking away from someone else."
-- Karl Taylor Compton
About the cover: The quote is an interesting perspective on a peaceful goal of science. History has shown us that those with advanced technology typically use it to attain what they desire from others. Optimization, which is a common theme in most mathematical modeling applications, provides us a means to improve the world we live in. Remember these words that were spoken in the early 1900s… The figure shows the various branches of the tree of optimization. The original graphic is from the NEOS web site, an operations research potpourri, at: http://www-fp.mcs.anl.gov/otc/Guide/index.html.
Welcome! This course is the capstone of your mathematical tenure at Carroll. On our journey this semester, we will explore the application of mathematics to a variety of real-world problems. We will not only use much of the mathematics you have learned in your other courses; we will also tie together and extend it as well. You might have a few questions about the course; hopefully, many will be answered below.
“What is this course about?”
We will focus on the application of applied mathematics to solving optimization problems, focusing extensively on the modeling process, effectively solving models, and sensitivity analysis of models. Our textbook will provide a basis from which we can begin our explorations. We focus most of our attention on projects selected from my experiences as both an engineer and operations research analyst, from COMAP Mathematical Competition in Modeling problems, and from UMAP Modules.
“What textbook are we using?”
The book we will be using is Mathematical Modeling, 2nd Edition, by Mark Meerschaert (Academic Press). I chose it to provide you a single source reference for the mathematical tools we will be using. Much of the material in the book you will have seen in other courses; rather than having to refer to multiple books, the text provides a unifying presentation from a new perspective.
“So… what is the layout of the course?”
We will focus our efforts on the following topics:
1) Unconstrained and Constrained Optimization,
2) Linear Programming,
3) and Integer Programming.
Throughout the semester, we will focus on understanding what type of problem we are trying to solve and how mathematics is used to solve the problem.
For each topic, we will spend a week or two introducing the material and then several weeks working on projects based upon that material.
When we introduce a new topic, I will provide reading outlines to help you pick out the major concepts we will be using. A reading outline is a set of questions for you to answer as you read the material in the book. You won’t be handing the answers in; rather your ability (or inability) to answer the questions and feedback to me will help stimulate classroom discussion.
I will assign most of the projects, but your group will select at least one project – I will provide you several good references to find one. In this class, as in the real world, the word “project” will take on its full range of meanings. Your projects will be a mix of the following:
ª Group effort with group hand-ins (AKA group projects)
ª Group effort with individual hand-ins (AKA homework sets)
ª Individual effort with individual hand-ins (AKA exams)
Additionally, each of you will be responsible for an in-class presentation of a project. These will take place during the final exam time period – 1:00 – 2:45 TUESDAY MAY, 8TH.
“What is this class discussion board thing?”
I have set up a class discussion board on the Carroll web page for this course – I will use it to electronically make materials (reading outlines, hints, supplements, URLs, etc) available to you, and you can use it to ask questions and share materials. We’ll talk more about how to access the board in class. You don’t need a password to read materials, but you need to register and use the password to post messages.
“Will we be meeting in the Computer Lab?”
I have the small computer lab (room 147 Science) on reserve for Wednesdays. We won’t be meeting there every week, so stay tuned for directions. We will use Mathematica, Excel, and GAMS (and possibly other software) for many of the projects. All software will be available on the computers in both Science Building labs (146 and 147).
“How will my grade be determined?”
Your grade will be based entirely on the projects. As projects are assigned, I will provide you a grading rubric for the assignment. If you are unhappy with your grade on a group effort project (either group or individual hand-in), you may address my comments and re-submit it once, with the following caveats:
1. The project must have been turned in on time originally.
2. You must resubmit the project within two weeks of it being handed back.
3. You are eligible to make up all but 10 of the points you missed (so if you got an 80 out of 100 on a project, and resubmit it, you are eligible for at most 90 points).
Meaning: I went to grad school with a guy that turned in all of his assignments the last week of classes, and I don’t think that is fair to those that do work throughout the semester.
“What’s your policy on late work?”
I collect projects at the beginning of class on the due date.
Late work (less than 1 week from due date) will be accepted for 1/2 credit.
Work that is more than 1 week past due date will not be accepted for credit.
“What about getting some help?”
Stop by if you need help! My office is always open; however, you may find that I’m not always the person in my office. I will be on campus MWF afternoons (1:00 – 5:00) and TR mornings (8:00 – 12:00) only. If you stop by at other times, you will find my wife inhabiting our shared office (I will be at home playing Daddy with our three-year-old daughter). If my office hours don’t work for you, let me know and we can schedule another time for me to be on campus.
“Anything else?”
I welcome your constructive comments to help me make this the best course possible. My goal is to help prepare you for the real world – where typically no one is looking over your shoulder to make sure that you are where you are supposed to be and doing what you are supposed to do. At the same time, this is still an environment where you have a safety net (i.e. I won’t fire you… but I may confront you if you are slacking). The key to your success in this course depends mainly upon your attitude, your study habits, and your desire to learn. Let’s have fun!
MA 421: Optimization SYLLABUS SPRING 2001
Lesson and Topic
Jan 17 Introduction
Jan 19 Mathematica Review Lab
Jan 24 Unconstrained Optimization
Jan 29 Multivariable Unconstrained Optimization
Jan 31 Hessian, Convexity, and Positive/Negative Definite
Feb 7 La Grange Multiplier Lab
Feb 9 COMAP Competition
Feb 12 COMAP Competition
Feb 14 Multivariable, inequality Constraints
Feb 19 President’s Day
Feb 21 Gradient Search
Feb 23 Project Presentations
Feb 26 Project Presentations
Feb 28 More Convexity
Mar 5 LP Introduction
Mar 9 COMAP Judging
Mar 19 More fun with Tableaux
Mar 26 LP Sensitivity Analysis
Mar 28 LP Sensitivity Analysis II
Apr 6 Modeling with Binary Variables II
Apr 9 Math Programming and Linear Algebra
Apr 11 Math Programming and Linear Algebra II
Apr 13 Good Friday
Apr 16 Easter Monday
Apr 18 Whole class: Extended Project Discussion
Apr 20 Cassi, Jaime,JD Status on Project
Apr 23 Jennifer and Michelle status on Project
Apr 25 Toshie and Sang status on Project
Apr 27 Duality Introduction
Apr 30 Duality Part II
May 2 Duality Part III
May 8 In Class Project 1:00 – 2:45