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Lecture20

# Lecture20 - C&O 355 Mathematical Programming Fall 2010...

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Mathematical Programming Fall 2010 Lecture 20 N. Harvey

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The “Simplex Method” “The obvious idea of moving along edges from one vertex of a convex polygon to the next” [Dantzig, 1963] Objective Function Start Here End Here Image: http://torantula.blogspot.com/
The “Simplex Method” “The obvious idea of moving along edges from one vertex of a convex polygon to the next” [Dantzig, 1963] Algorithm Let x be any vertex of P For each neighbor y of x If c T y>c T x then Set x=y and go to start Halt Remarks The name sounds fancy, but is meaningless. In practice, very fast. Used in all LP software. In theory, we don’t know whether it’s fast or not. (Because we don’t understand the diameter of polyhedra, i.e., Hirsch Conjecture) LP: Polyhedron: This is a simplex

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Pitfalls The simplex method is very simple… …if we can handle a few issues 1. What if there are no vertices? 2. How can I find a starting vertex? 3. What are the “neighboring” vertices? 4. Does the algorithm terminate? 5. Does it produce the right answer? Algorithm Let x be any vertex of P For each neighbor y of x If c T y>c T x then Set x=y and go to start Halt LP: Polyhedron:
Issue #1 What if there are no vertices? Not all polyhedrons have vertices! Recall: Any polyhedron that does not contain a line has at least one vertex. A fix: Instead of max { c T x : Ax · b } we could solve max { c T ( u-v) : A(u-v)+w=b, u,v,w ¸ 0 }. These LPs are equivalent. The feasible region of the new LP contains no line. Summary: Can assume we’re solving an LP with a vertex. x 1 x 2 x 2 · 2 x 2 ¸ 0

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Pitfalls The simplex method is very simple… …if we can handle a few issues 1. What if there are no vertices? Can modify polyhedron so that it has a vertex. 2. How can I find a starting vertex? 3. What are the “neighboring” vertices? 4. Does the algorithm terminate? 5. Does it produce the right answer?
Issue #2 How can I find a starting vertex? This is non-trivial! As shown in Lecture 3, maximizing the LP

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Lecture20 - C&O 355 Mathematical Programming Fall 2010...

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