Linear Programming Problem

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... (Headbang)

Nicole