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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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 0 . 1. Write the dual in standard form.
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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 for which x is feasible. 4. For / 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 do the complementary slackness conditions admit one solution? Problem 4. Solve problems 4.9, 4.16, and 4.21 of the textbook....
View Full Document

This note was uploaded on 03/14/2012 for the course MTHSC 810 taught by Professor Staff during the Fall '08 term at Clemson.

Ask a homework question - tutors are online