 Welcome everyone. So, in the previous lecture we had studied the linear programming duality theorem and the theorem let me just state it for your convenience here. We called this problem P or the primal problem which was minimize c transpose x subject to Ax equals b and x greater than equal to 0. This was the primal problem and corresponding to it we had also what is called the dual problem that maximize the decision variable was x where the decision variable is y maximize b transpose y a transpose y less than equal to c. What was the theorem? The theorem said that two parts to it if either primal or dual is unbounded then the other is infeasible. This was something we argued very easily from the duality and the second if either primal or dual has a finite optimum optimal solution value then so does the other and these values are equal. This was the statement of the linear programming duality theorem. Let me just write the name here duality theorem of linear program. Yeah, so optimal when I say it is unbounded it means that the optimal value is unbounded or let me write it as unbounded optimal value. Alright, so now this theorem is amazingly powerful actually it is not at all evident when you look at it why this should be the case but you will see that this is actually at the heart of a lot of optimization. So I will give you an example of one it also leads to a number of unusual results. So this is an example of an unusual result it is called Parkash Lemma. Parkash Lemma says the following exactly one of following is true. So it talks of two statements and it says that exactly one of them can hold. First statement is so these both of these statements pertain to a matrix A and a vector B. So vector sorry and a vector C. So let A r m cross n and let C be in r n. Exactly one of them is true. First statement is you can solve this system. So that means there exists an X such that A X is less than equal to 0 and C transpose X is strictly less than 0. The other statement is that there exists a Y such that A transpose Y equals minus C and Y is greater than equal to 0 that is the statement of Parkash Lemma. So either you can solve this equation this system of equations has a solution the first one which says that which ask for an X X is n length vector such that A X is less than equal to 0 and C transpose X is strictly less than 0 or you can have A transpose Y equals minus C and Y is greater than equal to 0. You cannot have so none of the other cases are possible which means it is not possible that both are true it is not possible that neither is true. The only case possible is that exactly one of these is true. So now this remarkably is actually a consequence of linear programming duality. So how does this come about from linear programming duality let me show you So consider to say this optimization suppose I look at the problem minimize C transpose X subject to A X less than equal to 0. Now I have not we looked at the standard form in as we looked at the primal problem having a standard form and being a minimization problem but this problem is more in this problem has the form of your dual problem actually here. So just believe me for a moment that the dual of this one so if this is my primal then the dual of this one is actually max 0 subject to A transpose Y equals minus C and Y greater than equal to 0. So if I if you took this as you can you can always write this problem as a minimization problem put it in this in the form of this primal and then take its dual you will get back this problem. This is another either way so just believe me that for that this is if this is my primal and this is this is its dual. Now this now what happens if one is true what happens if this statement one here this statement is true. If you have if there exists an X for which A X is less than equal to 0 and C transpose X is strictly negative what would that mean? Now the feasible region here A X less than equal to 0 is actually a cone right because if I can always scale X and it will continue to remain feasible. So feasible region is a cone if I am giving getting if I have one X for which C transpose X is negative then what is going to be the minimum value of this what is going to be the optimal value of this optimization minus infinity because I can always scale scale and bring it down to minus infinity right. So so it means that the primal is unbounded if the primal is unbounded the dual has to be infeasible and dual has to be infeasible means that there cannot be a Y that satisfies the dual constraint there cannot be a Y that satisfies the dual constraints which means in effect so which means in effect that two cannot be satisfied alright. So what it so let me just write it here so if one is true then P is unbounded which means D is infeasible which means two is not true. So if one is true then two cannot be true right. Now let us let us assume one is not true okay if one is not true if one is not true which means what does that mean what is the negation of one it would mean that for all X such that AX is less than equal to 0 the negation of one if one is not true means for all X such that AX is greater than equal to sorry less than equal to 0 we have C transpose X greater than equal to 0. So for all X such that AX is less than equal to 0 C transpose X is greater than equal to 0 that would be the negation of one right if one is not true this this is what it would mean right okay then what does this say if one is not true then we have for all X such that AX is less than equal to 0 C transpose X is greater than equal to 0 what does that imply look at let us look at the optimization problems again what does it say about P the optimal value of P will be 0 right because C transpose X is always greater than equal to 0 on the feasible region and at 0 at X equal to 0 its value is 0 right. So which means so this would mean optimal value of P is equal to 0 what does that say well it says that the primal has a finite optimal value which means the dual must also have an optimal value and an optimal solution right which means which means that the dual which means the dual constraint must be satisfiable right. So if the dual must have a finite optimal value which means that the dual constraint must be satisfiable so this means optimal value of P is equal to 0 which means optimal value of D is equal to 0 right which means there must exist in particular if there is an optimal solution it means it is exactly it must be at very least feasible there exists a Y that is feasible for D which means 2 is true. So if 1 is not true we got that 2 is right. So essentially this covers all the cases you can verify this if I check one possible case was with 1 is true then it meant 2 is not true if 1 is not true then it turned out 2 has to be true. So in no way can you have that both are not true nor can you have that the case where both are true. So this sort of result is what is called a theorem of the alternative of the alternative. So it essentially says that out of a given set of alternatives only 1 can work theorem of the alternatives and it became very popular after due to Farcash himself because of this very peculiar looking result this is just one form of them you as you play around with linear programming duality you can derive your own versions of Farcash Lehmann of this kind you know this which says that either this is true or that is true.