homework-3-solutions

# homework-3-solutions - MthSc810 Mathematical Programming...

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MthSc810 – Mathematical Programming Fall 2011, Solutions to Homework #3. Exercise 1 Apply the Fourier-Motzkin elimination procedure (or the procedure you think is more appropriate) to the following n -dimensional polyhedra, to eliminate x n . P 1 = { x R 3 : P 2 = { x R 2 + : x 1 + 2 x 2 - x 3 1 x 1 + x 2 2 x 1 + 5 x 2 + 3 x 3 = - 2 x 1 - x 2 4 2 x 1 - 2 x 2 - x 3 1 x 2 5 x 1 - x 2 + x 3 3 x 1 + 2 x 2 10 } ; x 1 - x 3 3 x 2 + x 3 0 x 1 + x 2 1 x 1 0 } ; P 3 = { x [ 0 , 1 ] : a x b } (with a R n ) 1 . Solutions. P 1 : We can use the equation x 1 + 5 x 2 + 3 x 3 = - 2 to obtain x 3 = x 1 5 x 2 2 3 . Then P 1 ( x 3 ) = { x R 2 : = { x R 2 : x 1 + 2 x 2 - x 1 5 x 2 2 3 1 4 x 1 + 11 x 2 1 2 x 1 - 2 x 2 - x 1 5 x 2 2 3 1 7 x 1 - x 2 1 x 1 - x 2 + x 1 5 x 2 2 3 3 2 x 1 - 8 x 2 11 x 1 - x 1 5 x 2 2 3 3 4 x 1 + 5 x 2 7 x 2 + x 1 5 x 2 2 3 0 - x 1 - 2 x 2 2 x 1 + x 2 1 x 1 + x 2 1 x 1 0 } x 1 0 } . P 2 instead requires application of the Fourier-Motzkin elimination. P

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homework-3-solutions - MthSc810 Mathematical Programming...

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