CS 312: Algorithm Analysis

Lecture #31: Linear Programming: the Simplex Algorithm, part 2

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Slides by: Eric Ringger, with contributions from Mike Jones and Eric Mercer

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

Objectives

Understand the Simplex method Discuss and own the pseudo-code

Comparison

What is the relationship between the MaxFlow algorithm and the Simplex algorithm?

Summary: Example from Last Time

Why did the algorithm terminate?

Interpreting the Answer

Original Problem:

Final Problem:

…

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

Algebra: Check Ratios

Simplex

Simplex

Simplex

Simplex

Algebra: PivotAssume we’ve identified the leaving and entering variables, and , and

we’ve updated our solution (moved our current point of focusin the feasible region).

Pivot Algorithm

Algebra: Obj. Function Update

Similarly: for each of the constraints …

Assignment

Finish Project #6 now

Assignment: HW #22.5