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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:
Metaexercise:
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?
ANS: y≥0 and Ax≤ b.
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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.
 Spring '08
 Staff
 Math

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