1. CS 312: Algorithm Analysis Lecture #31: Linear Programming: the Simplex Algorithm, part 2 This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License.Creative Commons Attribution-Share Alike 3.0 Unported License Slides by: Eric Ringger, with contributions from Mike Jones and Eric Mercer

2. Announcements  Homework #22  Due now  Saturday:  Review screencast on tractability (P, NP, NP-Complete, etc.)  Project #6: Linear Programming  Key for Part 1 was distributed on Thursday  Whiteboard: Saturday  Early day: Monday  Due: Wednesday  Verification suggestion: use another LP solver

3. Objectives  Understand the Simplex method  Discuss and own the pseudo-code

4. Comparison  What is the relationship between the MaxFlow algorithm and the Simplex algorithm?

5. Summary: Example from Last Time Why did the algorithm terminate?

6. Interpreting the Answer Original Problem: Final Problem: …

7. Observations  At the beginning of every round of Simplex,  The space for the transformed problem is spanned by unit vectors in the directions of the non-basic variables  The value of each non-basic variable in the current solution is 0.  i.e., the current solution is at the origin of that space  The new feasible region is defined in that space  Pivot is designed to keep our attention focused on the origin of each successive space

8. Pseudocode  The next part of the lecture derives the mathematical elements of the Simplex algorithm.  For the tutorial document and a more readable version of the algorithm, go here: http://faculty.cs.byu.edu/~ringger/CS312/rea dings/linear-programming-notes.pdfhttp://faculty.cs.byu.edu/~ringger/CS312/rea dings/linear-programming-notes.pdf

9. Algebra: Check Ratios

11. Simplex Algorithm

14. Algebra: Pivot

16. Pivot Algorithm

18. Algebra: Obj. Function Update Similarly: for each of the constraints …

19. Algebra: Obj. Function Update Similarly: for each of the constraints …

20. Pivot Algorithm

22. Assignment You’re ready to finish Project #6 now! Assignment: HW #22.5  Due Wednesday Remember Saturday’s screencast