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# hw2ea - (b Is the LP unbounded Does it have a ²nite...

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AMS 540 / MBA 540 (Fall, 2009) Estie Arkin Homework Set # 2 Due in class on Tuesday, September 22, 2009. 1). (a) Show that the set of all feasible solutions to the following linear program forms a convex set: min { cx | Ax = b, x 0 } (b) Prove that the set of optimal solutions to the linear program (of part (a)) forms a convex set. 2). An LP minimize cx subject to Ax b and x 0 was solved. Denote x * the optimal solution. Now a new constraint is added: αx β . Prove that if x * is feasible to the new constraint, then it is also an optimal solution to the new LP. 3). (From 2003 midterm) Consider a maximization linear programming problem with 4 extreme points v 1 , v 2 , v 3 , v 4 and 3 extreme directions d 1 , d 2 , d 3 , and with objective function coefficients given by the vector c , such that cv 1 = 5, cv 2 = 18, cv 3 = 4, cv 4 = 18, cd 1 = 5, cd 2 = 0, cd 3 = 2. (a). Write down the reformulated LP in terms of the extreme points and extreme directions. (b). Is the LP unbounded? Does it have a finite optimal solution? If so what is the optimal
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Unformatted text preview: (b). Is the LP unbounded? Does it have a ²nite optimal solution? If so what is the optimal z ? Does the LP have multiple optimal solutions? Explain! (Don’t just say “yes” or “no”.) 4). Consider the polyhedron P = { x | x ≥ } . (a). Show that x = is an extreme point of P . (b). Show that any x n = , x ∈ P is not an extreme point. (Recall, x n = means that at least one component of the vector x is not 0. Therefore x n = , and x ≥ does not imply that x > .) (c). What are the extreme directions of P ? Recall, an extreme direction of a set is a direction d that cannot be written as a positive combination of two distinct directions of the set, that is there do not exist directions d 1 and d 2 of the set, d 1 n = d 2 , and α 1 , α 2 > 0 such that d = α 1 d 1 + α 2 d 2 ....
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