# Math 340: Linear Programming

## Table of Contents

## News

- Last term’s final is available, now with solutions.
- Solutions for assignment 5 are available.
- Final Exam office hours have been scheduled, see below.
- Solutions for Assignment 4 are available.
- Assignment 5 is available, due the last day of class, Friday Decemeber 2nd.
- Solutions for the midterm (v1, v2) are available.

## About

Linear Programming means maximizing or minimizing linear functions of variables subject to linear equations or inequalities (so maybe “Linear Optimization” would be a more descriptive name).

- Classes: Monday, Wednesday and Fridays from 2pm to 3pm in Mathematics Annex 1100.
Office Hours for final exam (all in LSK 300):

11am-12pm 4pm-5pm 11am-12pm 2pm-3pm

## Textbook

We will use *Linear Programming* by Vašek Chvátal. This text explains
the simplex method using the dictionary format that we will use in
class.^{1}

## Notes

You can get the course notes in a single PDF or each topic in a separate web page:

- What is Linear Programming? (Ch. 1)
^{2} - Standard form for linear programs , (Ch. 1)
- Thinking of linear programs geometrically
- Introduction to the simplex method (Ch. 2; examples by Richard Anstee)
- Multiple optimal solutions (Ch. 3; example of an unbounded LP by Richard Anstee)
- Unbounded linear programs (Ch. 3)
- The 2-phase Simplex Method and infeasible linear programs (Ch. 3)
- Degenerate pivots and cycling (Ch.3; slides of Chvátal’s example of cycling)
- Matrix formulas for dictionaries (Ch. 7, first section)
^{3} - Bland’s Rule guarantees termination (Ch. 3, Theorems 3.1 & 3.3)
- Recap of the Simplex Method
- Duality theorems (Ch. 5) , ,
- Complementary Slackness (Ch. 5)
- Coffee stain problem (by Richard Anstee)
- Theorem of the Alternative
- Marginal values (Ch. 5, p. 65)
- The Revised Simplex Method (Ch. 7) ,
- Sensitivity Analysis (Ch. 10) , (see also these examples by Richard Anstee)
- The Dual Simplex Method (Ch. 10) (for examples of dual pivoting see Anstee’s sensitivity analysis examples, or the solutions to practice quiz 5).
- Branch and Bound
- The New Forrest problem (Ch. 11)
- Some combinatorial optimization problems
- The tollbooth staffing problem
- Matrix Games (Ch. 15)
- All problems are basically the same
- The cutting stock problem

## Quizzes

## Midterm

There will be an in-class midterm on practice midterm. It is important to really try the problems before looking at the solutions! You can get solutions to the practice midterm if you’re sure you’re ready to look at them.

. Here is a## Course Outline

Each of the following topics will take approximately 2 weeks (the chapters listed are from Chvátal’s book):

- Simplex Method (chapters 1–4 & 8)
- Duality Theory (chapters 5,9)
- Revised Simplex Method (chapters 6,7)
- Sensitivity Analysis (chapter 10)

After that we will cover some of the following topics:

- Applications and some modeling techniques (chapters 11-14)
- Branch and Bound
- Game theory (chapter 15)
- Non-linear Programming and the Karush-Kuhn-Tucker conditions
- General Upper Bounding (chapter 25)

## Computer Software

On some problems you will be asked to use computer software to solve
linear programming problems. I recommend the Classic LINDO application
for Microsoft Windows. You can obtain a free evaluation copy from that
link to install on your own computer, or use the computer lab in
LSK 310. You either should choose a time with no labs or, you can work
quietly at the back of the lab even if a lab is scheduled (assuming
there are some empty computers). Your ID is the first 8 characters in
lower case of your name as recorded first name, middle name (if you
have one), final name. The password is set to capital `S`

followed by
the first 7 numbers of your student ID. You can change your password.
Access the Windows system and click on LINDO.

You don’t need to use LINDO, specially if you have prior programming experience. Other options include:

- Glpk.js and online-optimizer. These are websites, you don’t need to install anything on your own computer! The second also does sensitivity analysis (for the constraints).
- The
`MixedLinearIntegerProgram`

class in SageMath. Sage can be used without installing any software at SageMathCell. If you chose the GLPK solver, it can also do sensitivity analysis. - The
`scipy.optimize.linprog`

function from SciPy.

## Grading

The grade will be computed as 55% final; 15% midterm; 15% quizzes and 15% assignments.

- Quizzes
- These will be mostly computational problems. There will be 5 quizzes of 25 minutes in length. Practice questions will be given in advance.
- Assignments
- There will be 5 assignments. They will have an emphasis on theory. Some assignments will give computational questions and you will be able to utilize the computer Lab and the LINDO and LINGO software for Linear programming (available in the computer lab in LSK 310; you will be given accounts) or software of your choosing. Students may work together on assignments but must write up their solutions independently.
- Midterm and Final
- There will be one 50 minute midterm, it will be during class on Monday, October 31st. The final will be 3 hours long.

^{1}

Feel free to look at any other comparable text, but beware that
the simplex method might *look* fairly different, even while it
performs the same computations.

^{2}

This refers to chapters from the textbook.

^{3}

Not a typo! This is covered in Chapter 7.