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\title{An Introduction to Linear Programming}

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%\author{Midge Cozzens}
%\address{DIMACS, Rutgers University, New Brunswick, NJ 08901}
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%\email{midge6930@comcast.net}
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\author{Steven J. Miller}
\address{Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267}
\curraddr{}
\email{sjm1@williams.edu, Steven.Miller.MC.96@aya.yale.edu}
%\thanks{Partially supported by NSF Grant DMS0970067}

%\author{Wesley Pegden}
%\address{Department of Mathematics, Courant Institute, NYU, New York, NY 10012}
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%\email{pegden@math.nyu.edu}

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\chapter{Course / Book Outline}

\noindent \texttt{The following is a rough outline of ideas for a course on advanced linear algebra at Mount Holyoke for the Spring 2012 semester, which will serve as a nucleus for a textbook. The style of the book and the course will be conversational and friendly.}

\section{Motivating Questions}

\noindent Here are four questions I'm thinking about.\\

\begin{enumerate}

\item \textbf{Should the course be just linear programming, or random matrix theory too? What about the book?}

\sd There is more flexibility if we do both, more chance of showing students where things can go, but for a book it might be a bit too much (though it would of course also give instructors more flexibility). The advantage of just doing linear programming and not random matrix theory is that we can make this also a brief introduction to linear algebra, and build on that. Thus, for now, the plan for the book is \emph{not} to include much random matrix theory, at most a few pages on it at the end of the book in a `what's next' section. It would fit well there, as the rest of the book is devoted to studying fixed matrices. In fact, we could even use this as a nice bridge / transition. In the real world, we of course don't know exact values. What happens if we start drawing the entries at random? There's some nice work on this subject, leads to a quick introduction to probability, and then can end with random matrices. This is stochastic linear programming. Lots of references. A quick search yielded the following paper: http://www.jstor.org/pss/1907736; a more extensive review of the literature is needed.

\sd Goal for the course: see where the math goes, use the material as a springboard to introduce topics throughout mathematics. The goals for the book will be very similar, with the big addition of getting comfortable with mathematical modeling (a very important and marketable skill), being able to set up linear programming problems, having some idea of what resources are out there to solve them, et cetera.

\ \\

\item \textbf{Market for the book / course: Is this a general course with minimal pre-reqs, or is this a sequel to calculus?}

\sd My original thought was that, as much as possible, let's minimize calculus. This will increase marketability. We can have calculus asides. After talking with ED, it might be good to include a calculus pre-req. This will make sure students have some mathematical maturity and familiarity. Also, of course, the biggest weakness students have in calculus classes is lack of algebra skills, and that would be very hard in a course such as this.

\sd We don't need too much calculus for linear programming; at most, maybe some comments on optimizing problems and boundary issues. If we start talking about constraint matrices with random entries, however, then a bit more calculus is needed as we now must discuss probabilities. We wouldn't need too much from probability; many of the concepts students should have already seen. A big advantage of this is that many people do not know that one of the biggest applications of calculus is that areas are probabilities. We can thus motivate, after the fact, the calculus they learned.

\sd For random matrix theory, we would need more probability. As there are more than enough interesting items in linear programming, delving this deeply (in either the course or the book) seems to be a mistake, and instead we should content ourselves with references to the literature.

\ \\

\item \textbf{What linear algebra is needed?}

\sd We'll review what is needed below. This is a quick pass -- as the semester progresses and we start dealing with the material, we'll have a better sense. We can decide later for a book if this should be an introductory chapter, or an appendix.

\sd First is $A\vec{x} = \vec{b}$. The left hand side is a linear combination of columns. This is a very important idea for linear programming. We'll talk a lot about linear combinations. This is not just important in linear algebra, but also in differential equations / difference equations (linear combinations of solutions, Binet's formula for Fibonacci numbers). Another example is portfolio theory, where we want to minimize variance subject to a given expectation.

\sd Have to deal with degenerate cases: $A\vec{x} = \vec{b}$. This is not always solvable, talk about determinants.

\sd Tweaking answers: probability of an event: `almost' all matrices with real entries are invertible. Need, however, to worry about numerical stability, and this is a fascinating topic and can make for some nice asides. Maybe talk a bit about computational complexity, such as (my favorites) fast exponentiation (cryptography), Horner's algorithm (fractal geometry), Strassen's algorithm (fast matrix multiplication -- huge applications). Just `show' or briefly describe these possibilities (or maybe these are advanced topics / supplemental topics at the end of the book).

\sd Talk about the difference between real versus integer solutions. Some riddles on the number of animals satisfying given constraints, and there are infinitely many `real' solutions, but if you require the solutions to be integers (which makes sense, as no one has $e/\pi$ of a cow) then there is essentially `one' solution (or maybe up to some multiples, but we often need to have non-negative numbers). Another example is the optimal base for storing information in a computer; answer turns out to be base $e$ (a very nice calculus problem, uses almost everything you learn in Calculus I), and thus base 3 is surprisingly better than base 2 in this respect (as 3 turns out to be closer to $e$ than 2). This needs naturally into (1) the backpack problem, (2) having to take specific roads (and either go to the city or don't), (3) .... This is a very real problem / complication in the real world; you either do something or you don't, there is no temporizing. No one is half-pregnant; you don't have a flight where some people go to Tucson and some to Denver. You can, however, temporize a bit by breaking flights into legs.... Note that this leads to our current hub system.

\sd Need some combinatorics, nothing too advanced, mostly on the level of $\ncr{N}{M}$, as that arises in determining number of options, which is useful for determining how long something will take.

\ \\

\item \textbf{What is needed for Random Matrix Theory?}

\sd Though the book is almost surely going to just be linear programming and what spins off, if we do random matrix theory what is needed? Need some basic probability, the fact that integration gives areas gives probabilities, eigenvalues, and of course combinatorics. To `prove' the theorems in full detail requires a lot of advanced material, but there is no need to give a full proof; one can argue about the reasonableness of the method of moments via a comparison with Taylor series and call it a day.

\end{enumerate}

\newpage
\section{Content for Advanced Linear Algebra}

\noindent Here are three thoughts.

\begin{enumerate}

\item \textbf{Why Linear Algebra? Where is it used?}

\sd Let's start with some standard linear algebra problems to see where we began. Everyone is familiar with the boring problem of two trains leaving at different times and at different speeds and wondering when they meet. This is the classic example of a linear algebra problem, but doesn't showcase \emph{why} the subject is so important.

\sd Perhaps talk about perspective lines in art?

\sd Advanced application: cryptography: error correction / detection, such as the Hamming (7,4) code. We don't have to go into the entire theory (or maybe we do this as a supplemental chapter for a longer, more advanced course; similar comments for the next few items).

\sd Advanced application: fast Fourier transform, lots of signal processing applications.

\sd Advanced application: Method of Least Squares, fundamental in fitting.... I do have extensive notes on this, could make that an appendix if it is worthwhile including (to give a professor some flexibility in making a course).

\sd Another possibility is to talk about calculus, and the need to linearize non-linear phenomena. This is probably a good idea as it then builds on the calculus we say we require.

\sd Another item is difference equations, where we can write solutions as iterates of a matrix (this ties in with efficiency and many other items).

\sd Describe the types of solutions to linear algebra problems: zero, one (unique) or infinitely  many.

\sd Most things sadly non-linear. It is hard to solve problems exactly. Often we get approximate answers to approximate equations for the real world problem. This is something many people haven't experienced; we're so used in the first few math classes of having clean equations with clean solutions. This isn't the case in the real world. We have two conflicting items: we want our model to accurately describe the real world, and we want it to be mathematically tractable. These are competing goals.


\ \\

\item \textbf{Non-linear Equations}

\sd There are many ways to generalize from Linear Algebra. One is to go from linear equations to non-linear equations, which we discuss now.

\sd Math can be `simple' but have lots of applications. A great example is GPS systems (motivated by my son Cam asking how the computer knows where we are). Imagine you have a few satellites that tell you how far they are from you. Each one localizes you to a surface of a sphere, and then you need to find the points of intersection to uniquely determine where you are. This can tie into the Method of Least Squares (get an overdetermined system).

\sd Advanced application: Hamiltonians from Physics. Other possibilities are planetary trajectories, ballistics, .... Lots of possibilities here!

\ \\


\item \textbf{Linear Programming (and beyond)}

\sd Instead of generalizing to non-linear equations, we can generalize another way: we now want to optimize a function subject to various constraints. The most general case gives non-linear constraints, which sometimes is doable but is quite hard. We discuss now linear programming.

\sd Start with optimization with linear constraints. Much of life is optimization. We can talk about Physics (Hamiltonians, Feynman's action, ...), soap bubbles, economics, ....

\sd Main problem: given $A\vec{x}=\vec{b}$, optimize $\vec{c}^{\rm T}\vec{x}$. We'll see later that there is no loss of generality in using equality instead of inequalities in the matrix constraints. Also constraints on the variables. Typically take $\vec{x} \ge 0$, and again will see there is no loss. Here we can talk about `canonical' formulations. This is a big concept in mathematics, namely that it is sufficient to reduce certain general problems to special cases. Examples abound (find them!). One is solving the cubic -- enough to consider the depressed cubic. Another nice example is tic-tac-toe; there aren't 9 opening moves, but up to isomorphism just 3.
\sd In solving linear programming problems, we'll learn about the dual problem. Similar to the argument above (getting a canonical formulation), this is another big concept. It seems strange, but a related problem is often much easier. A great (but high level!) version of this is Poisson Summation; try and find other examples.

\sd Lots of applications to discuss. Big one is the oil industry (have a bunch of oil sources, a bunch of refineries, a bunch of markets -- where is oil shipped from). Another is the airlines (this led to the hub system we have now, figuring out what flights planes should take, but there are lots of problems because we don't know the `true' demands for people for flying from city A to city B on day D). There are a plethora of scheduling problems (can do movie theaters: what movies to buy, what screens to use, what times to start, but again, we often have to estimate demand and interaction effects, which are fascinating problems in their own right). We can also look at baseball. One possibility is describing a schedule (lots of issues, such as hard constraints, but also wanting to have big games between popular teams at popular times, save division games for the end of the season, ...). Another possibility is the fascinating problem of elimination numbers (some great papers on this, and how the press \emph{miscalculates} this!). One final problem: pure mathematics: Hales proof of the Kepler conjecture (on sphere packing) involved solving hundreds of thousands of linear programming problems.

\sd A key question is how much theory to get into. While the simplex method is not the best algorithm out there, it has a very approachable proof, which is worth doing in full detail. The proof also has a nice outline. It's broken into two parts, Phase I and Phase II (using the notation from Joel Franklin's excellent book -- need to check if this is the original notation). First, assuming we can do Phase II we prove we can do Phase I. Then, using Phase I, we prove we can do Phase II. This sounds circular; it's not -- to prove Phase I, we only need to know how to do Phase II for a related problem, and in that case it is \emph{trivial} to show Phase II is doable. This is a beautiful idea, and worth grokking fully. Other methods to solve linear programming should be mentioned and referenced, but not really discussed.

\sd Note in the various linear programming problems mentioned, many are Integer Linear Programming problems, or Binary Integer Linear Programming problems. I find formulating these a lot of fun. Specifically, so many relations that do not look linear can be made linear (such as max/min, absolute values, cut-off functions) at the cost of introducing more variables and constraints. This provides a wonderful platform to discuss trade-offs. Would you rather do lots and lots of simple problems, or a few hard problems? Most of the time, it's better to do the simpler problem, especially when we have cranks to churn out the answer. Maybe motivate solving a general polynomial to solving a chain of quadratic equations. There is of course a run-time cost for adding variables and constraints, but linear programming exists, while quadratic and higher is either harder or is not available. The motivating quote should be: \emph{What is amazing is not that the horse sings well, but that the horse sings at all.}

\sd The problems above (except maybe Hales' proof of the Kepler conjecture) involve linear programming problems, where the parameters in the constraint matrix are fixed. In the real world, we won't know these values exactly, and this leads to stochastic linear programming, where these parameters are now drawn from distributions (hopefully known). A little probability is needed here. The goal would probably be to do a few examples to give the flavor, but not proofs in general, just a literature review and an attempt to get things on people's radar.

\end{enumerate}






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\chapter{Linear Algebra Review}








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\bibitem[Da]{Da}
G. B. Dantzig, \emph{Linear Programming and Extensions}, Princeton
University Press, 1963.

\bibitem[Fr]{Fr}
J. Franklin, \emph{Mathematical Methods of Economics: Linear and
Nonlinear Programming, Fixed-Point Theorem}, Springer-Verlag, New
York, $1980$.

\bibitem[St]{St}
G. Stigler, \emph{The cost of subsistence}, The Journal of Farm
Economics, 1945.

\end{thebibliography}


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