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# lecture 9 - Math 482(Lecture 9 Duality II This lecture...

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Math 482 (Lecture 9): Duality II This lecture completes discussion of Section 3.1 of the textbook. Recall that given a primal problem: max c'x s.t Ax≤ b; x≥ 0 there is a dual problem min y'b s.t. y'A≥ c'; y≥ 0 We introduced the notion of shadow price . Mathematically: These are the coefficients of the slack variables y i in the final tableau at optimum. Economics interpretation: This is the additional profit for have one additional unit of resource i. (This being a math class, we can concern ourselves only with the mathematical definition, but I hope this economics interpretation is useful.) As yet unproved observation/claim: The shadow prices are the optimal solution to the dual problem. The following are some first results about duality. But first: Meta-exercise: Explain all the exercises below in terms of the UIUCbucks example. (WARNING: we have not mathematically justified the shadow price story, but it might be helpful to use it anyway as a sanity check.) In class exercise 1: (Weak Duality) Prove that if x is feasible for the primal problem, and y is feasible for the dual problem, then c'x≤ y'b. Hint: c'x≤ y'Ax (why?) ≤ ? Solution: c'x ≤ y'Ax ≤ y'b. The first inequality holds because x≥0 and c'≤ y'A. The second holds because?

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