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sol5 - C O350 L INEAR O PTIMIZATION S OLUTION HW5 Exercise...

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CO350 L INEAR O PTIMIZATION - S OLUTION HW5 Exercise 6.9.7 No. The optimal solution comes from the simplex method is a basic solution. However u j and v j can not both have positive values in a basic solution. This is because in the tableau, the column corresponding to u j is the negative of the column corresponding to v j . A pivot operation will not change this property. Thus these two columns are always negative to each other in all the tableaus. Therefore these two columns are linearly dependent and can not be in a basis at the same time. Thus they can not both have positive values in a basic solution. Exercise 6.9.9 (a) Suppose there is another optimal solution ¯ x 6 = x * . If ¯ x j = 0 for all j N , then ¯ x becomes the unique basic solution for the basis B , which is identical to x * . This is a contradiction. So there must exist some j N such that ¯ x j > 0 . Since for every feasible x we have (see course note page 76) z = ¯ v + X j N ¯ c j x j , where ¯ v is the right-hand side value in the z row (which is also the optimal value), we obtain from
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