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CS 312: Algorithm Analysis

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

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CS 312: Algorithm Analysis

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

  2. Announcements • Homework #22 • Due now • Project #6: Linear Programming • Key for Part 1 was distributed on Monday – did you get it? • Use C# • Early day: Friday • Due: Monday • 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. Algebra: Check Ratios

  9. Simplex

  10. Simplex

  11. Simplex

  12. Simplex

  13. Algebra: Pivot Assume we’ve identified the leaving and entering variables, and , andwe’ve updated our solution (moved our current point of focusin the feasible region).

  14. Pivot Algorithm

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

  16. Assignment Finish Project #6 now Assignment: HW #22.5

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