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
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:
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.
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...