Math 482 (Lecture 14): The primaldual algorithm and
applications to the shortest path algorithm I
This lecture discusses Chapters 5.1 and 5.2 of the textbook.
The situation so far:
* We have the simplex method and dual simplex method.
* One can still hope for improvements (speedups) in general. One of them is the "revised
simplex method" (this is discussed in Chapter 4, but we will skip it).
* For
particular
classes of problems one can also hope the structure of the data (A,b,c) of
the LP is amenable to different methods.
* (Creativity) Motivated by this we seek to find different approaches to solving LP using,
e.g., the power of duality and complementary slackness.
* This is the viewpoint of Chapter 5: primaldual simplex was shown to work well for the
shortest path problem AND THEN generalized.
The class handout with an extended example of the primal dual algorithm. (I will hand it
out a little later in the class since I want to work through it with in class exercises.)
* The textbook's overview of the general method 104105 is, in my opinion, quite good.
Basic setup:
You are given
(P) min c'x s.t. Ax=b(≥ 0) and x≥ 0
(D) max π'b s.t. π'A≤ c' and π unconstrained.
In class discussion:
Why can we assume b is nonnegative in (P)? Why is x unconstrained
in D?
Comments:
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 Spring '08
 Staff
 Math, Optimization, Shortest path problem, LP

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