s8 - CO350 LINEAR PROGRAMMING - SOLUTIONS TO ASSIGNMENT 8...

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Unformatted text preview: CO350 LINEAR PROGRAMMING - SOLUTIONS TO ASSIGNMENT 8 Exercise 1. Consider the linear program ( P ) max { c T x : Ax = b,x ≥ } where A ∈ R m × n , c ∈ R n and b ∈ R m . Let B be an optimal basis for ( P ) , let x be the associated basic solution, let ∈ R and let d be the i th column of ( A B )- 1 . Show that if we replace b i by b i + and we have x B + d ≥ then B remains an optimal basis. Solution: Let b = b + δ i , where δ i is the vector in R m having a one in position i and zero everywhere else. Then the new problem is ( P ) max { c T x : Ax = b ,x ≥ } Note the i th column of any matrix E ∈ R m × m is Eδ i . Hence d = ( A B )- 1 δ i , so d satisfies A B d = δ i . Let N = { 1 ,...,n } \ B . Define x * as x * B = x B + d and x * N = 0 . Then Ax * = A B x * B + A N x * N = A B ( x B + d )+0 = A B x B + A B d = b + δ i . Therefore x * is the basic solution for B and ( P ) . Moreover x * ≥ , so B is a feasible basis. Let ( D ) be the dual of ( P ) and let y be the dual basic solution for B and ( P ) . Note y is also the dual basic solution for B and ( P ) . As B is optimal, y is feasible for ( D ) . Changing b to b does not change the feasible region of ( D ) , hence y is also feasible for the dual of ( P ) . Therefore B is feasible and dual feasible for ( P ) , hence B is optimal for ( P ) . Exercise 2. Consider the linear program max { c T x : Ax = b,x ≥ } where A = - 3 1 0- 1 0 1 2 0 0 1 1- 2 7 0 1 2- 3 , b = - 4 3 8 , and c = [- 7 , , ,- 2 , ,- 1] T . Solve this linear program using the dual simplex method, starting with the dual feasible basis B = { 2 , 5 , 3 } . If the problem has an optimal solution, provide one. If the problem is infeasible, provide a constraint that needs to be satisfied by any vector x satisfying Ax = b but cannot be satisfied by any vector x ≥ . Show all of your work....
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This note was uploaded on 11/21/2011 for the course MATH co 350 taught by Professor Cheriyan during the Spring '09 term at Waterloo.

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s8 - CO350 LINEAR PROGRAMMING - SOLUTIONS TO ASSIGNMENT 8...

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