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# linearp - CMSC 451 Linear Programming Slides By Carl...

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CMSC 451: Linear Programming Slides By: Carl Kingsford Department of Computer Science University of Maryland, College Park

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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 .

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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?

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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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