Lecture15-2003 - Lecture XV Overview of Duality in...

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Lecture XV Overview of Duality in Mathematical Programming I. Derivation of the Dual Space A. A simple linear problem ( ) () 11 22 33 1 1 1 1 1 , 2 2 21 1 22 2 23 3 max x cx b a x ba xa x λ +++ λ λ− 1. In this framework 1 x , 2 x , and 3 x are choice variables or actions such as levels of plantings. 2. 1 λ and 2 λ are shadow values or marginal values of each resource constraint. 3. 1 c , 2 c , and 3 c are objective value coefficients for each activity. 4. 1 b and 2 b are maximum resource levels (or right-hand sides). 5. The ij a are resource use coefficients. B. Distributing and recollecting ( ) 2 2 12 21 31 3 1 3 2 1 1 2 2 23 1 3 1 b x bb x c x a x a x c x a x a xc x c a a x c a a xc a a + + −λ + λ λ λ λ+ λ+ λ+ − λ− λ + 23 2 λ 1. The transformed formulation yields the same optimum as the first formulation, but it is a minimization problem. 2. The transformed problem minimizes the value of resources used
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This note was uploaded on 11/08/2011 for the course AEB 6533 taught by Professor Moss during the Fall '08 term at University of Florida.

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