CS 312: Algorithm Analysis

1. This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License. CS 312: Algorithm Analysis Lecture #30: Linear Programming: Intro. to the Simplex Algorithm Slides by: Eric Ringger, with contributions from Mike Jones and Eric Mercer

2. Announcements • Homework #21 (Project 6, Part 1) • Due online today by 5pm • No late submissions accepted (just like any homework asst.) • Key available online after 5pm • Important: LP Notes available on the schedule • Works through the ideas up through the Simplex algorithm • Includes the pseudo-code for Simplex (and Pivot) • Project #6: Linear Programming (Part 2) • Early day: Friday • Due: next Monday • Verification suggestion: use another LP solver

3. Objectives • Review various forms for representing an LP problem • Understand the Simplex method algebraically • Introduce the concept of a Pivot • Walk through an example to completion

4. Forms of an LP Problem • English form • Algebraic form • Standard form up to step #1 • also algebraic

5. Example: Standard Form (up to step #1)

6. Forms of an LP Problem • English form • Algebraic form • Standard form up to step #1 • also algebraic • Geometric / graphical form • Only if 2-D, maybe 3-D • Matrix-vector form • Read off directly from standard form up to step #1

7. Example: Matrix-Vector Form

8. Forms of an LP Problem • English form • Algebraic form • Standard form up to step #1 • also algebraic • Geometric / graphical form • Only if 2-D, maybe 3-D • Matrix-vector form • Read off directly from standard form up to step #1 • Padded matrix-vector form • Adapted directly from matrix-vector form • Reflects the participation of slack variables

9. Example: Padded Matrix-Vector form • Purposes: • for input to the Simplex algorithm • for computation in the Simplex algorithm

10. Forms of an LP Problem • English form • Algebraic form • Standard form up to step #1 • also algebraic • Geometric / graphical form • Only if 2-D, maybe 3-D • Matrix-vector form • Read off directly from standard form up to step #1 • Padded matrix-vector form • Adapted directly from matrix-vector form • Reflects the participation of slack variables • Standard form up to step #2 (aka “slack form”) • Also algebraic

11. Standard Form (step #2)

12. Sketch of Simplex Algorithm Start with Basic solution (the origin of the feasible region) • Pick a non-basic variable • Raise its value as high as we can • Pick a basic variable • Algebraically pivot • Repeat at #1 until we cannot raise the value of any non-basic variables

13. Basic Solution

14. Basic Solution

15. Identify a Non-Basic Variable

16. Identify a Non-Basic Variable

17. Select a Basic Variable

18. Select a Basic Variable

19. Solve for the Non-Basic Variable: “Pivot”

20. Solve for the Non-Basic Variable: “Pivot”

21. Pivot: Rewrite the Other Equations

22. Pivot: Rewrite the Other Equations

23. After the Pivot: Objective Values

24. After the Pivot: Objective Values

25. Repeat

26. Repeat

27. Repeat Again (Third time)

28. Repeat Again (Third time)

29. What next?

30. Observations • At the beginning of every round of Simplex, • The value of each non-basic variable in the current solution is 0. • The space for the transformed problem is spanned by unit vectors in the directions of the non-basic variables • 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

31. Assignments • Project #6 • Now re-read the Simplex and Pivot pseudo-code to see how these algebraic steps are performed in the code • HW #22