Prelim1_Solutions

Prelim1_Solutions - ORIE 3300/5300 Prelim 1 SOLUTION Fall...

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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 minimum-ratio 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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Prelim1_Solutions - ORIE 3300/5300 Prelim 1 SOLUTION Fall...

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