 method of analysis and the topic is the local and the global optima that we will extend. In the method of analysis in the previous module, we have studied the various aspect of the other method, the main optimization and under that optimization one that will be the local optimum and this local optimum, when we say it mean that is one extreme either it is maximum either it is minimum point of the objective function and objective function as we have already studied that this is related to those twice variable that we want to decide and we want to solve either it is related to a consumer either it is related to a producer. So, local maximum is always a global maximum when objective function will be a concave or cause icon cave or when the feasible set will be the convex. So, these two features they are the basic ingredient to have a global maximum or the minimum because they give us the tangency point. So, global optimum will be the maximum or the minimum value function that available in the total analysis space and the examples are here when we say the minima and the maximum there can be the various examples that this will be one local minima, this will be one local minima, this will be one local minima and this will be one local, but when we make the level related to this, this will be the last will be the lowest among all. So, when we say this will be the global optimum or the maximum lowest among all. So, basically it is minimal, but lowest may is the biggest degree. So, we will call it global optimum minimum. And if we draw another variable x against y, then here it is possible that this is commodity x against commodity 2 or its price against its purchase. So, we have this graph of different forms in which when we look at the range here, then this will be its neighborhood range. And in this neighborhood range, this is its local, this is the optimal point, this is the second local optimum. And in all the optimals, when we will compare, if I say this is the local point, this is the point B, this is the point C, this is the point D and this is the point E. So, among these five local methods, this is also itself a local optimal. And in the optimal, since we are calculating from below, this will be the maximum, but we will calculate its mathematical condition as well. So, this is also a local optimum, but since this is the highest one, so it will get the global optimum rank as well as being local. And that is why when we did this in the form of this graph in the previous module, if we look at it, then in one graph, we can have both local and global. And if we look at it, then we will have to see that at what points we can be at a time minimum and here we can have the maximum available. When we are dealing with optimization, local and global, minima or maxima, then the most basic thing is what is our objective function. So, the basic three properties of objective function are necessary to fulfill, one of which is the most important, continuity. If we look at this graph in this graph, then this graph starts here and if we look at it in this way, then it is going upward. If I want to draw this in a second form, then if we look at it in this way, then it is possible downward as well. But the thing in which once we put the pen on our paper and we are not putting a break in it, and it is possible that it is our straight line. So, whenever we draw a graph like this or a graph or a relationship, because whenever we draw a graph, one x axis will be there and the other y axis will be there. So, in both these relationships, whenever there is no break, that will be the continuous. And in the same way, there is a non-continuous, in which we are seeing that a graph moves from x axis to here. But after that, it stops. There is a break for some time and we have started it again from there. So, if we look at it here, then this value which is getting gapped, that will be the non-continuous. And in the same way, it is possible that if we look at this graph, then here we see that we have a face of x1, x3 and x0 and similarly y1 and y2. So, here, when we show more than one variable on one graph, then this one performance is showing the other one. So, for one variable, that can be continuous and for the other, it can be continuous. But overall objective function is not following our continuity. So, the second main indicator or property which we have to fulfill in the last, that is the shape. Now, the shape of the objective function should be of this type. If we look at it, one of them we have drawn, which we have in this form. And if we look at x axis and y axis, two modalities, x1 and x2, if we draw it, then we are drawing a relationship between the two of them. Similarly, when we draw between the two of them, then if we look at it, it is almost looking like this. But there are some points where these straight lines are made. And the third point, if we look at it, then we have an inverse. So, if we look at it, then if we look at it in this form, then either we will call it convex. So, if we look at it in this form, then the cos i concave we have, that is the best solution to have for our objective function. When we say cos i concave in this form, then we can solve. But between these two points, because we look at it, the substitution is called the perfect one. And between the perfect substitution, we have the marginal rate of substitution that is constant. So, it becomes difficult to solve under the rule of our tangency. And the third curve, if I give it a name, then the C curve and the A curve are the opposite of each other. Condition of concave curve, if we look at it, then it is that if we draw a line between the two points, like it is present here, then if we draw this line, then the curve and its entire data will come below it. If we look at it, then the whole thing is on this side. So, this is the best solution. And the condition of cos i concave is this, but between these points, there is a straight line. The third one, if we join the two points between them, the straight line is drawn. And after the straight line is drawn, the curve that we look at, the entire data is on those points upward or right. And this is not possible to have a solution for our objective function, properties of feasible set. As we have already studied the properties of the objective function, so to have a proper solution of the optimization, we will study what should be the properties of the feasible set. So, the four properties of the feasible set are very important to us, one of them is non-emptiness. And if we say it in other words, then we will say that the set should not be empty. Because if that will be empty, then we do not have any solution. So, at least a minimum one variable or one variable, if we say it in other words, then that is one viable solution available with us. Second is that is the closeness. And if we look at the closeness, then this is our solution, in which we will say that we take the points of the boundary, we can count this as well. But this should come to us in a closed form, in which if we look at it, it does not take us to any form of infinity. And if I want to give you an example in this, if we have some solutions, in which we say that x is greater than or equal to 1. And on the other hand, we say that x can be less than or equal to 0. If we look at it now, then x, if we say it at 0 and say it at equal, then equal to 0, then it is clear. But in less, we have an availability, but from one side, it has become a sensor. Similarly, if we look at it in the same way, then if we say equal to 1, it will stop to 1, not below it, and it can go up. So, it is censored from below and it is censored from above. If we look at it in this way, when we have it in a condition like this, then at least one side gives it a closeness. And many of us will have a variable, which if we look at it, then we have an option in which we say that x1 is greater than or equal to x2. And that is equal to some value, suppose I say 2. So, this is a finite point or finite value. So, in such an equation, in which we can solve a finite point, that will be the closeness. The bounded, if we look at it in the same way, then this will be the set, where any one side gives us an infinity extreme point. And convexity, that curve, in which if we look at it, where every point is somewhere, we can join it through a straight line. So, it is exactly the same as if we say that if we have points anywhere, and we join them, then if we say that when we join any two points, then they can be straight lines. And if we go in this way, and if we say that we have two points here, then they can also come in the form of a straight line in front of us. If we say in analysis, then now we have decided that the objective function either strictly quasi-concave and feasible set, if we have non-convex, then we can have a form local maximum. And in this way, if we look at it, take global maximum. In the same way, there can be another function, in which the objective function is not quasi-concave, and the feasible set can be convex. We will look at its particular form now. So, in that, we have the form of global maximum. Now, if we look at it, then we have two curves that we have drawn. And in that, we have x1 on x-axis, x2 on a different commodity, we have y-axis. And if we draw and make these two points, so if we look at it, then this is our feasible set and that is the surface in which we have available options or available resources. And on the other hand, if we look here, then this is our objective function. And in the objective function, if we draw these two, then if we look at it, then the first line of objective function, this falls in the feasible set. But despite that, we have some resources left. These resources provide us the facility that we can achieve a higher level of objective. So, instead of this, we have the second objective function, which is the higher rank. If we come to it, then if we look at it, then this is the tangent above this point. But on this side, we have some resources left. And if we go further than that, then we have this option available. Now, if we look at this form, then we have two points above the tangent, either this point or either this point. So, it will be difficult for us to decide which is the viable or the best solution for us. So, when we say that in this form, whenever we have to decide, either maxima or minima, then the feasible set for us, if it is in convex form, that is, it will be present in this form, then our tangent point will be above the boundary. And above the boundary, it is with it. Now, here, the feasible set is convex. But our objective function is not our concave. Rather, it is present in the form of the causal concave. And if we look at this form, then we have more than one. If it is, then this point is coming to us, and one is coming to us. So, it asserts us that to take the best solution, we have the best option for the feasible set. And if we want to take the best option of our objective function, then it is concave. So that the one point above the tangent, then that one point can give us our viable solution. And in the same way, we see another point in which, if we look at it, then we have several times more than one point, in which we have multiple points, we can make global points. This is one form in which, if we look at it, then the feasible set is not strictly convex form and it is giving us the best solution. If this is the form, then this is the concave form. This part is coming to us in the form of a straight form. And if we look at the rest, then this shape is giving us a little bit of concave. So, if we look at it, then this is the objective function. So, objective function is not strictly concave. And the feasible set is strictly convex. The third form is the objective function, which is strictly concave, which is our best desired form. And in the same way, the feasible set is strictly convex. So, if we say this form and this form, then we have multiple forms of functions available when we plot the data. So, it comes in front of us, which gives us the point of tension or optimization. Because optimization is only possible when it comes in front of us in the form of a point of tension. Because we have to go to the saddle point of contours.