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Unformatted text preview: 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 x s.t. Ax = b x ≥ . (a) Suppose ¯ c N > , 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 = 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 d = c B d B + c N d N = ( c N c B B 1 N ) d N = ¯ c N d N . Because ¯ c N > and d N = y N x N = y N ≥ 0, we have ¯ c N d N ≥ . This shows that c x ≤ c y for any feasible y , i.e., x is a minimum solution. To show uniqueness, assume y is also a minimum solution, then c y = c x , i.e., c d = 0. This implies ¯ c N d N = 0. Because ¯ c N > , we have d N = 0. Then d B = B 1 Nd N = . Thus, d = , 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 = . Because x is nondegenerate, x B > . Thus, there exists a....
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This note was uploaded on 12/02/2011 for the course MATH 3252 taught by Professor Tanbanpin during the Spring '10 term at National University of Singapore.
 Spring '10
 TanBanPin
 Linear Programming

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