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Tutorial5solution

# Tutorial5solution - Tutorial 5 Outline of Solutions Q1...

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Tutorial 5: Outline of Solutions Q1. Consider a minimization linear programming problem in standard form. Let x be a basic feasible solution associated with the basis B . Prove the following: (a) If the reduced cost of every nonbasic variable is positive, then x is the unique optimal solution. (b) If x is the unique optimal solution and is nondegenerate, then the reduced cost of every nonbasic variable is positive. Solution : Consider a standard form linear program min c 0 x s.t. Ax = b x 0 . (a) Suppose ¯ c N > 0 , where N is the index set of all nonbasic variables at x and B is the index set of the basic variables. For any feasible solution y , let d = y - x . Then Ad = 0 implies d B = - B - 1 Nd N where A is partitioned using the basic and nonbasic variables as ( B | N ). The change in cost is equal to c 0 d = c 0 B d B + c 0 N d N = ( c 0 N - c 0 B B - 1 N ) d N = ¯ c 0 N d N . Because ¯ c N > 0 and d N = y N - x N = y N 0, we have ¯ c 0 N d N 0 . This shows that c 0 x c 0 y for any feasible y , i.e., x is a minimum solution. To show uniqueness, assume y is also a minimum solution, then c 0 y = c 0 x , i.e., c 0 d = 0. This implies ¯ c 0 N d N = 0. Because ¯ c N > 0 , we have d N = 0. Then d B = - B - 1 Nd N = 0 . Thus, d = 0 , i.e., y = x . This shows that x is the unique optimal solution. (b) By contradiction, we assume that ¯ c j = 0 for some j N . Construct a direction d with d j = 1, d i = 0 for i N \ { j } and d B = - B - 1 A j . Then Ad = 0 . Because x is nondegenerate, x B > 0 . Thus, there exists a θ > 0 such that x B + θ d B 0 . Since d N 0 , we have x N + θ d N 0 . Thus, ˆx

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Tutorial5solution - Tutorial 5 Outline of Solutions Q1...

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