This is a rather long problem...
Consider the following:
maximize with j = 1 as it goes to n
subject to where j = 1 as it goes to n, i = 1, 2,...,m
where j = 1,2,...,n
first form its dual:
minimize with i = 1 as it goes to m
subject to with i = 1 as it goes to m, j = 1,2,...,n
where i = 1,2,...,m
Then write that minimization to a maximization in standard form:
maximize with i=1 as it goes to m
subject to with i=1 as it goes to m, j = 1,2,...,n
with i=1,2,...,m
The above process can be thought of as a Transformation T on the space of data defined by:
Let denote the optimal objective function value of the standard-form linear programming problem haing data . By strong duality and the fact that a maximation dominates a minimization, it follows that:
If the process is repeated, transformation T becomes:
and hence...
But the first and last entry in this chain of inequalities is equal, therefore all these inequalities appear equal. Although possible, this isn't always true.
THE QUESTIONS:
1. What is the error in this logic?
2. Can you state a (correct) nontrivial theorem that follows from this line of reasoning?
3. Can you give an example where the four inequalities are indeed all equalities?
Yeah... very long... please help...
Nicole