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samplemidtermsolutions

# samplemidtermsolutions - IEOR 162 Linear Programming Spring...

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IEOR 162 Linear Programming Spring 2007 Sample Midterm Solutions 1. a. False . A linear program may also be infeasible. b. False . There may exist optimal solutions that are not extreme points. However, we can guarantee that if there is an optimal solution, then there will exist at least one extreme point optimal solution. c. True . If the feasible region is bounded, then a linear objective function is also bounded. d. True . At any solution there are m active constraints because all constraints are equality constraints. If this question were to ask if there must be exactly m linearly independent active constraints, that would be false because in addition to the equality constraints, n - m non-negativity constraints must be at equality. 2. Standard Form: max - 6 x + 1 + 6 x - 1 + x 2 - 4 x 3 - 3 x + 4 + 3 x - 4 s.t. - 2 x 2 + x + 4 - x - 4 + s 1 = - 2 5 x + 1 - 5 x - 1 - 3 x 2 + 2 x + 4 - 2 x - 4 - e 2 = 1 - 2 x 2 - x 3 + 2 x + 4 - 2 x - 4 + s 3 = - 3 - x + 1 + x - 1 - 4 x 2 - x + 4 + x - 4 = 4 x + 1 , x - 1 , x 2 , x 3 , x + 4 , x - 4 , s 1 , e 2 , s 3 0 Note that x 1 and x 4 were unrestricted in sign in the original formulation. They are replaced in standard

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