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.