5.13 Consider the following process. Starting with a linear programming prob- lem in standard form, maxiimize C.I subject to jzi by i=1,2, ,m ,20 j-1,2,..,n, first form its dual: iyi subject to i-l y20 i-1,2,..., m. Then replace the minimization in the dual with a maximization to get a new linear programming problem, which we can write in standard form as follows: maximize Ji subject to -watj--g j=1,2, . . . ,n i-1 If we identify a linear programming problem with its data, (aiy, b,, cj). the above process can be thought of as a transformation T on the space of data defined by Let(aij, bi,g) denote the optimal objective function value of the standard-form linear programming problem having data (aij, bi, cj). By strong duality together with the fact that a maximization dominates a minimization, it follows that Now if we repeat this process, we get 76 5. DUALITY THEORY and hence that But the first and the last entry in this chain of inequalities are equal. There- fore, all these inequalities would seem to be equalities

. While this out come could happen sometimes, it certainly isn't always true. What is the error in this logic? Can you state a (correct) nontrivial theorem that fol- lows from this line of reasoning? Can inequalities are indeed all equalities? you give an example where the four