3: The Simplex Method

Last updated
Save as PDF

Page ID: 1725

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

In Chapter 2, you learned how to handle systems of linear equations. However there are many situations in which inequalities appear instead of equalities. In such cases we are often interested in an optimal solution extremizing a particular quantity of interest. Questions like this are a focus of fields such as mathematical optimization and operations research. For the case where the functions involved are linear, these problems go under the title linear programming. Originally these ideas were driven by military applications, but by now are ubiquitous in science and industry. Gigantic computers are dedicated to implementing linear programming methods such as George Dantzig’s simplex algorithm–the topic of this chapter.

3.1: Pablo's Problem
3.2: Graphical Solutions
Before giving a more general algorithm for handling this problem and problems like it, we note that when the number of variables is small (preferably 2), a graphical technique can be used. Inequalities, such as the four given in Pablo's problem, are often called constraintsconstraints , and values of the variables that satisfy these constraints comprise the so-called feasible regionfeasible region .
3.3: Dantzig's Algorithm
In simple situations a graphical method might suffice, but in many applications there may be thousands or even millions of variables and constraints. Clearly an algorithm that can be implemented on a computer is needed. The simplex algorithm (usually attributed to George Dantzig) provides exactly that.
3.4: Pablo Meets Dantzig
3.5: Review Problems

Contributor

David Cherney, Tom Denton, and Andrew Waldron (UC Davis)