Linear Optimization

The Subject A little explanation is in order for our choice of the title Linear Opti- 1 mization (and corresponding terminology) for what has traditionally been called Linear Programming.Theword programming in this context can be confusing and/or misleading to students. Linear programming problems are referred to as optimization problems but the general term linear p- gramming remains. This can cause people unfamiliar with the subject to think that it is about programming in the sense of writing computer code. It isn¿t. This workbook is about the beautiful mathematics underlying the ideas of optimizing linear functions subject to linear constraints and the algorithms to solve such problems. In particular, much of what we d- cuss is the mathematics of Simplex Algorithm for solving such problems, developed by George Dantzig in the late 1940s. The word program in linear programming is a historical artifact. When Dantzig ?rstdevelopedthe Simplex Algorithm to solvewhat arenowcalled linear programming problems, his initial model was a class of resource - location problems to be solved for the U.S. Air Force. The decisions about theallocationswerecalled¿Programs¿bytheAirForce,andhencetheterm.

Uses the "modified Moore method" approach in which examples and proof opportunities are worked into the text in order to encourage students to develop some of the content through their own examples and arguments while they are reading the text

Introduction.- The Simplex Algorithm.- Geometry.- The Duality Theorem.- Matrix Implementation.- General Form.- Unsolvable Systems.- Geometry Revisited.- Game Theory.- Network Implementation.- Combinatorics.- Economics.- Integer Optimization.- Appendix A: Linear Algebra Overview.- Appendix B: The Equivalence of the Auxiliary and Shortcut Methods.- Appendix C: Complexity.- Appendix D: LOP Catalog.