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Unformatted text preview: CMSC 451: Linear Programming Slides By: Carl Kingsford Department of Computer Science University of Maryland, College Park Linear Programming Suppose you are given: • A matrix A with m rows and n columns. • A vector ~ b of length n . • A vector ~ c of length n . Find a length n vector ~ x such that A ~ x ≤ ~ b and so that ~ c · ~ x := n X j =1 c j x j is as large as possible. Linear Algebra The matrix inequality: A ~ x ≤ ~ b in pictures: · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · × · · · · · · · · ≤ · · · · Each row of A gives coefficients of a linear expression: ∑ j a ij x j . Each row of A along with an entry of b specifies a linear inequality: ∑ j a ij x j ≤ b i . A little more general maximize X j c j x j subject to A ~ x ≤ b What if you want to minimize? What if you want to include a “ ≥ ” constraint ~ a i · ~ x ≥ b i ? What if you want to include a “=” constraint? A little more general maximize X j c j x j subject to A ~ x ≤ b What if you want to minimize? Rewrite to maximize ∑ j ( c j ) x j . What if you want to include a “ ≥ ” constraint ~ a i · ~ x ≥ b i ? What if you want to include a “=” constraint? A little more general maximize X j c j x j subject to A ~ x ≤ b What if you want to minimize? Rewrite to maximize ∑ j ( c j ) x j . What if you want to include a “ ≥ ” constraint ~ a i · ~ x ≥ b i ? Include the constraint ~ a i · ~ x ≤  b i instead....
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This note was uploaded on 01/13/2012 for the course CMSC 423 taught by Professor Staff during the Fall '07 term at Maryland.
 Fall '07
 staff

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