# lecture 14 - Math 482(Lecture 14 The primal-dual algorithm...

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Math 482 (Lecture 14): The primal-dual 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 (speed-ups) 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: primal-dual 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 104-105 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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lecture 14 - Math 482(Lecture 14 The primal-dual algorithm...

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