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homework-9

# homework-9 - 2 Relax the third constraint and ±nd the...

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MthSc810 – Mathematical Programming Fall 2011, Homework #9. Due Tuesday , November 8, 2011, 6PM EDT. From the syllabus: homework will be penalized 50% for each day they are late. After two days, they will not be accepted. No exception. Problem 1. Given the problem min - x 1 - x 2 x 1 + 5 x 2 10 + α x 1 + x 2 6 + β x 1 , x 2 0 1. Compute the set M = { ( α, β ) R 2 : the basis B = { 1 , 2 } is optimal } . 2. For each ( α, β ) M , compute the change in objective function. 3. Is M convex? Is it a polyhedron? Is it a cone? Problem 2. Find extreme points, extreme rays, and recession cone of each of the following polyhedra: P 1 = { x R 2 : x 1 0 , x 2 1 , x 2 - αx 1 0 } for any α R ; P 2 = { x R n : n i =1 | x i | ≤ 1 } ; P 3 = { x R 2 : x 1 0 , x 2 - k 2 (( k +1) 2 - k 2 )( x 1 - k ) , k ∈ { 0 , 1 , 2 . . . , K }} . Problem 3. Consider the primal problem min - 5 x 1 - 6 x 2 s.t. x 1 + 2 x 2 6 2 x 1 + x 2 6 x 1 + x 2 α x 1 , x 2
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Unformatted text preview: 2. Relax the third constraint and ±nd the optimal solution x ⋆ associated with basis B = { 1 , 2 } ; 3. Restore (i.e., put back in) the third constraint. Find the set M of values of α for which x ⋆ is feasible. 4. For α / ∈ M , ±nd a hyperplane separating x ⋆ from the polyhedron. 5. Write the complementary slackness conditions for x ⋆ , using the original prob-lem (i.e., including the third constraint) and its dual. For what values of α do the complementary slackness conditions admit one solution? Problem 4. Solve problems 4.9, 4.16, and 4.21 of the textbook....
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