sec4_Notes - 4 Basic feasible solutions To summarize the...

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4 Basic feasible solutions To summarize the main idea from the last section, a basis for an m -by- n matrix A is a list of numbers chosen from { 1 , 2 , , . . . , n } such that the matrix A B with columns indexed by this list is invertible. The corresponding basic solution of a system Ax = b is the unique solution of this system satisfying x j = 0 for all indices j 6∈ B . Consider now the constraints of a linear program in standard equality form: Ax = b x 0 . A basic feasible solution of this system is a feasible solution that is also basic. Such solutions play a special role in linear programming. To illustrate, consider the simple example (2.1): (4.1) maximize x 1 + x 2 subject to x 1 2 x 1 + 2 x 2 4 x 1 , x 2 0 . As before, we can sketch the feasible region: If we transform the constraints of this linear program to standard equality form, by introducing slack variables as we described in Section 2, we arrive at the following system: x 1 + x 3 = 2 x 1 + 2 x 2 + x 4 = 4 x 1 , x 2 , x 3 , x 4 0 . 26
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Feasible solutions of this system correspond one-to-one with feasible solutions of the linear program (4.1), in an obvious way. If we think of this new system as Ax = b , the matrix A has five possible bases, not including permutations: [1 , 2] , [1 , 3] , [1 , 4] , [2 , 3] , [3 , 4]. The five corresponding basic solutions are x 1 x 2 x 3 x 4 = 2 1
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sec4_Notes - 4 Basic feasible solutions To summarize the...

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