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homework-3

# homework-3 - you ±nd an extreme point x ⋆ write the...

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MthSc810 – Mathematical Programming Fall 2011, Homework #3. Due Tuesday, September 13, 2011. 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 ). Exercise 2 Find an extreme point of the following polyhedra, or a line contained in it. If
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Unformatted text preview: you ±nd an extreme point x ⋆ , write the active linearly independent constraints at x ⋆ . 1. P = { x ∈ R n : A x ≥ b } , where A ∈ R n × n invertible, b ∈ R n . 2. P = { x ∈ R n : e ⊤ x ≥ } with e ⊤ = (1 1 ··· 1). Exercise 3 Solve problems 2.4, 2.12, and 2.22 of the textbook....
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