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Unformatted text preview: ORIE 3300/5300 Prelim 1 SOLUTION Fall 2008 Problem 1 (a) (i) TRUE Recall that in the standard inequality form, we aim to maximize c T x subject to constraints Ax ≤ b and x ≥ 0. We can convert any problem to this shape by doing the following: • If we want to minimize c T x we can maximize ( c ) T x . • If we have a constraint a T x ≥ b we can change it to ( a ) T x ≤ ( b ). • If we have an equality constraint a T x = b we can replace it with two constraints a T x ≤ b and ( a ) T x ≤ ( b ). • If we have a free variable x i we can replace it with two nonnegative variables y i and z i so that x i = y i z i . (ii) FALSE This would be true if the problem was bounded. However, if the prob lem is unbounded, then the basic solution with the largest objective function value is not the optimal solution. (iii) FALSE The minimumratio rule is used to ensure that the next basic solution is feasible. (b) The solution [1 , , 2 , , 0] T is basic with the basis B = [1 , 3], then A B = 1 0 0 1 which is clearly invertible. The solution [0 , 1 , , , 0] T is basic. Let b B = [2], then A b B = 1 2 has independent column(s) and so we can extend it to a basis (all [1 , 2] , [2 , 3] , [2 , 4] and [2 , 5] are possible bases). Note that [0 , 1 , , , 0] T is degenerate. (c) Recall the corollary characterizing extreme points of a feasible region of inequality systems. It says that [1 , , 2] T is an extreme point if [1 , , 2 , , 0] T is a basic feasible solution for the system in ( b ), which we have proved before....
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 Fall '08
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