# Lecture 15 - Math 482(Lecture 15 The primal-dual algorithm and applications to the shortest path algorithm II This lecture continues discussion of

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Math 482 (Lecture 15): The primal-dual algorithm and applications to the shortest path algorithm II This lecture continues discussion of Chapter 5 of the textbook (and lecture 14). At this point, we have gone through the following iterations: 1. Given π feasible for (D) we construct a feasible point for (RP) 2. If (RP) is minimized at 0, (P) is solved r 4. Looking at π new =π+θ π r we determine θ such that this is feasible for (D). With this new feasible point of (D) we go back to 1 and iterate. Question: Why does this algorithm solve (P) (in finite time)? Brief answer: This is explained in Section 5.3 of the textbook. Let me summarize the main points of the argument: * Recall the set J of columns of A where the constraints of (D) are met with equality. (These are called the admissible columns by the textbook.) * Notice (RP) is almost the auxiliary problem (in the sense of phase 2 simplex) for (P). The only difference is that we force columns NOT in J to be in the basis of the optimal

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## This note was uploaded on 02/19/2012 for the course MATH 482 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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Lecture 15 - Math 482(Lecture 15 The primal-dual algorithm and applications to the shortest path algorithm II This lecture continues discussion of

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