 So, last time we were talking about linear programs and I showed you that I said I defined a linear program as simply this minimizing a linear function over a set P which was a polyhedron, P is a polyhedron. This sort of problem takes the form in general takes the form minimize C transpose X subject to AX less than equal to B. So, the constraints that define the polyhedron are these. So, the set P is X such that AX is less than equal to B. Now, this is what is called the half space representation of a polyhedron, half space representation what does that mean? So, if I give you a poly the way I am defining for you a polyhedron is through the half spaces whose intersection if I take gives me the polyhedron. So, this is for these are for example, this is my poly I have drawn the polyhedron and these are the half spaces that define. So, I am defining for you the half spaces whose intersection if you take the gives you back the polyhedron. Now, the thing that I want to show you today is that actually there is another way of representing a polyhedron which is the another way of representing the polyhedron which is which came which we saw we saw that actually through what is called the Minkowski Weyl theorem right. What did the Minkowski Weyl theorems say Minkowski Weyl theorem said that P is a polyhedron this is equivalent to the claim that saying that P is a polyhedron with at least one extreme point that is equivalent to saying that P is the convex hull of its extreme points plus plus the cone generated by its extreme rays extreme points of P and the extreme rays of P. Now, if I if I if the polyhedron is bounded then this term here which we comprise of the cone generated by the extreme rays this term is just a 0 and then P is simply the convex hull of its extreme point. So, if a polyhedron if you have a bounded polyhedron that means you have a polytope and a polytope is simply can be described as the convex hull of its extreme point. So, this set here the way is actually a poly is actually a polytope and it is every point in it can be written as the convex combination of these 5 points these extreme points right. So, so this now this way of representing a polyhedron or a polytope more specifically this way of representing a polytope as not through the half spaces that define the polytope, but rather through the extreme points these extreme points, but this way of expressing a polytope is what is called the vertex representation. Now, what I will show you now is that if you had the vertex representation of a polytope the problem linear programming would be very easy it would be very easy to know what the optimal value is and come up with at least one solution for your problem. Now, if you the so what that would mean is that the challenge in linear programming is that we do not is in coming up with the vertex representation right. So, you are given usually a problem that is in the half space representation you know the polyhedron is defined using the half spaces that that that form the faces of the polyhedron that is the and for and what we do not know is where the vertices lie they have to be computationally discovered as part of the solution method is clear. So, the all the so solution method is effectively trying to find or enumerate the vertices of the poly extreme points of the poly that is. So, let me let me just show you why this is the case that if you had the work place representation or if you more generally if you had a representation of if you had a description of all the extreme points and the extreme rays of the polyhedron then linear programming would be trivial to solve. So, let us let us just look at this. So, this basically so in order to talk about this let us first let us just get an intuitive picture first. So, I will just draw a neater diagram of the same thing here. So, I have suppose a polyhedron like this is this is my polyhedron and I am optimizing a linear function over it. And now let me plot. So, this is my polyhedron it is in some space say let us for simplicity here it is in R 2 I what I am going to plot is the contours of the linear function that I am going to optimize. So, I am going to plot contours that look better of the form that so this this contour here this is some C transpose x equal to k x is such that C transpose x is equal to k for a fixed constant k. So, what this means is take any point on this contour and the value of the objective on that for that point is k. Now, you can draw this contour for these kind of contours for various values of k. So, you have this is this is k dash this is k double dash etc etc. Now, what do you what can you observe about a contour like this. So, let us keep our focus on this one let us look at this contour what can you observe about. So, it intersects the feasible region that means the polyhedron over which you are optimizing it intersects the feasible region here. These are in this this is the interest. So, these are all the points that are in the polyhedron and give you objective value k. Now, question is are any of these optimal can any of these be optimal the the one can any of these give you the minimum value of k. Say if you think about it a little bit you will realize that it cannot be optimal. I do not need to tell you what the value of k is for that whatever be the value of k these cannot be optimal. The reason is because this is a linear function it should be that the function either increases in this direction or in this direction. So, nearby here either to the north of the function either above this contour or below the contour you should be able to find a value that is that is either greater than k or less than k. So, you should be able to get values k plus minus delta for in you know small enough delta in the by perturbing this contour a little bit and by perturbing the contour a little bit you are not going to lose feasibility you are going to still be inside you are still going to have points in in p. So, what this means is a contour like this cannot comprise of optimal solutions. So, suppose for example, since I am talking of decrease minimizing the function that is let us suppose that the function is decreasing in this direction. So, this is the direction of this is the direction of decrease of the function C transpose. So, which means that if I as if I look at a contour here or a contour here etc as I go in this direction I will keep getting lower and lower values of k. So, what this means is in order to optimize the function I need to search in this direction. So, I need to look for a smaller contour with a lower value of k and a further lower value of k and then finally it should happen that I eventually get to a point like this. What sort of point is this? This is the point where the contour just passes only through my extreme point right the contour just passes through only my extreme point. Now, if I if I look for a value of k that is even lower even lower than this then I will not I want the contour will not even intersect in the the feasible region right. So, then you are that is that is the value that is not even feasible. So, below this if I look for a contour that has value less than this is not going to intersect the feasible. So, what is this observation tell you that this observation tell you that that there must be a solution of this linear program of minimizing C transpose X over this polyhedron right. If you look at that linear program where you are minimizing C transpose X over this polyhedron there must be a solution of this linear program on an extreme point of P right. There must be at least one extreme point of P that is the solution of this linear program. Now, it can happen that there can be multiple for example you know it can happen that the contour is shaped in such a way that you know when it actually passes say for example let me draw it here the contour has a is the slope is such that it when it passes through this extreme point it actually coincides perfectly with this phase with this edge here right. And then therefore it will end up passing through this extreme point as well. So, in that case both of them will be optimal solution. In fact, all of these will be optimal solution that can also happen, but there will at least be one which is an extreme point and an optimal solution right. The main lesson here is that the thing that we expect from a linear program is that let us say consider the case where you have only optimizing over polytopes that means you have over bounded polyhedron that the kind of the nature of solutions that we expect is that there is going to be a extreme point that is that the solution right. So, now come back to this way of representing this thing that I mentioned which is representing a polyhedron using its vertices or using its extreme points alone. So, if someone gave you not these constraints like this not the constraints Ax less than equal to b, but rather a vertex representation that means a list of points that are the extreme points of your polyhedron. Then optimizing the linear function over that would be trivial. You would have to just check go through the list, enumerate the values of c transpose x for each of those extreme points output the least that would have that would have optimized the right. So, the challenge in linear programming comes down to precisely this issue that we do not know where the extreme points are and they have we do not know means that we know them implicitly we know that they are extreme points of this sort of set, but the exact formula for them or the exact values of them have to be discovered by doing some calculations and some on top of this these constraints right given these numbers right. So, they have to be discovered as part of the from the problem data. Is this clear? Any questions about this? So, now let me answer let me address the more general case suppose you have not a poly tau, but you know a general polyhedron which is which could also be unbounded and okay. So, let me address that and so let us try to let us just write out a theorem here. So, let p be a polyhedron at least one extreme point consider the linear program minimize since minimize c transpose x, so the x less than equal to b and suppose the optimal value of the LP is finite, we are talking of an LP where you are minimizing the objective. So, the optimal value of this LP is finite means that effectively it is not minus infinity. So, it is definitely greater than minus infinity that is what it that is what it means right. So, you cannot keep getting lower and lower as low value as you like that sort of thing is not possible. So, that is the sort of case we are considering. So, let p be a polyhedron with at least one extreme point and we are considering an LP whose value is finite. Is this clear? Then there exists at least one extreme point right to write p is there exists at least one extreme point of p that is an optimal solution of the LP.