Math482Practicefinal.Spring2011

# Math482Practicefinal.Spring2011 - UIUC MATH 482 Practice...

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UIUC MATH 482 Practice final problems (Spring 2011) The actual final exam will ask you to solve ten of twelve problems, each worth 10 points. I will focus on material in the lectures, in class exercises, homeworks and other material presented in class. The actual final exam may either include or exclude problems similar to the ones below. 1. Let G be a graph. A subset S of the vertices is called independent if no two vertices in S are adjacent. Formulate an integer linear program that finds a maximum-size independent set. 2. Solve the following linear program. min 2 x 1 3 x 2 = z subject to x 1 + x 2 0 x 1 +2 x 2 2 x 1 , x 2 0 3. Let P be the following linear program. min x 1 +4 x 2 2 x 3 subject to x 1 x 2 = 2 x 1 +3 x 2 + x 3 = 4 x 1 , x 2 , x 3 0 Form the dual linear program D of P . Solve D . What does the optimal solution of D tell you about an optimal solution of P ? 4. Prove that the following linear program is infeasible using Farkas’ lemma.

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