 So, I am Professor Ankur Kulkarni from Systems and Control Engineering at IIT Bombay, when I will be teaching this course, so in the IIT Bombay code is SC607, it is called optimization. In PTEL it will be called optimization from fundamentals. So optimization as a topic is mainly about choosing the best alternative out of a given set of alternatives, that is basically that is the purpose of optimization. And this is a quantitative subject, so when we have to quantitatively specify what the alternatives are, we have to quantitatively specify what it means for an alternative to be better than another alternative. So, that means we have we quantify the value or the cost or the benefit of a particular alternative. We also quantify what it means for an alternative to be an alternative, so why is it an alternative. Now, all this is very easy if you could simply list down all the alternatives and run through them and select them. So that unfortunately is not the case with most problems. Usually the number of alternatives itself is very large, how changing going from one alternative to the other, how that changes your benefit from it is also is usually complicated and unclear. As a result then one has to you need a formal approach to all of this. So that is why this is a subject in its own right. So the main elements of any optimization problem are as follows. So first is as I said we want to quantify what is the best alternative, so this quantification of alternatives is done through a function, through a real valued function. So we and by convention what we want to do is get the least value of this function by convention, this is just a convention, let us call this function f, okay. Now at least value over what set of alternatives, so the set of alternatives is called feasible region. So for example, for example your set of alternatives let us call this set S, this could be a subset of the vector space Rn, your function f could be just a function that maps Rn to R and what you are looking for is the least value of it. That means you are looking for an alternative x star what we want is an x star that belongs to S, f of x star is less than equal to f of x for all x and x, this is what we want. Now this is the ultimate goal, if you can get an x star sometimes we do not really want an x star, we just we only care about f of x star without actually explicitly saying what the x star is. So both of the x star is referred to as the optimal solution and f of x star refers to as the optimal value. Now the English meaning of the word optimal is somehow unfortunately different from both minimum and maximum, the English meaning somehow optimal can not something in between minimum and maximum, but that is not the sense in which we want to use it here. By optimal it means it solves the optimization problem, the goal of the optimization problem is to find the least value of it. Now why did we restrict to least value that is by convention, we will see later that it does not matter if it is a standard way of posing a particular problem, you can talk of maximum value also that is equivalent. Now I was saying that if you could list out all the alternatives then the problem is very easy, but usually the number of alternatives itself is huge and sometimes in many interesting cases it is actually infinite. So let us consider for example I want to find the shortest path from IIT Bombay to IIT Delhi, shortest path in some according to some measured in some way, suppose this may be measured by distance or measured by travel time whatever in measured in some units, shortest path from IIT Bombay to IIT Delhi starting from the gate of IIT Bombay to the gate of IIT Delhi. So now how many paths are there, I can so for this I will need to specify what is the environment, what is a path, I will have to give you the entire map, what sort of routes form a path, put together all of that you will realize that the number of paths is huge because pretty much at every juncture you can fork this way or that way and still somehow end up reaching IIT Delhi. So the number of alternatives is immense as you can see, but if you just do this in Google Maps right now, open up your app and do it in Google Maps you will find that in a matter of few seconds you will get a route. So it is certainly doing something more intelligent than cataloging all, searching through all the alternatives, it is probably searching through some but certainly not through all because the number of alternatives is immense right. So it is certainly doing something intelligent there, it is probably once it realizes that this alternative is bad it is also realizing that some other alternatives are bad and therefore eliminating all of them etc, etc. There is something, some intelligence is built into all of this and all of that is part of optimization. Let us consider another problem, today is the first day of registration so the problem of assigning classrooms to courses okay, hundreds of courses IIT Bombay offers hundreds of courses per semester, we have hundreds of classrooms there are about 10 to 15 different maybe 15 maybe 20 slots I do not know different slots that the courses can get offered in. Now you want to come up with a matching, you want to match courses to classrooms and slots okay, but then there are also other constraints like for example classrooms should respect the capacity, the registration okay, if you have 100 students registered you cannot see them in a classroom of size 50 whose capacity is 50, so you need a classroom of suitable size for suitable courses, some courses may require say for example lab courses can be done only in laboratories, some courses require that students be able to do programming in the class, so they have some additional requirements only certain types of classrooms can be chosen. All of these are additional requirements that may complicate the problem and in addition to all of this instructors have their own preferences, they prefer a certain slot, they prefer a certain room etc, etc. Now this whole classroom assignment problem is would be you can say well the way we can think of it is, we would like to get a classroom assignment and a slot assignment for courses that is as close to the instructor preferences as possible without violating any of these constraints, so you have to but then you see the number of alternatives here, 100 courses, 100 classrooms, the number of combinations you can make out of all and several slots, number of combinations you can make is immense, out of that you have to come up with one kind of combination for at an institute level that will be closest to the stated set of preferences of the faculty. So this is an optimization problem, it comes up in day to day life, we solve it every semester, similarly Indian railways, BST outside on the street, all of them they are also solving such problems in some or the other form during their day to day operation. So as you can see basically optimization is essentially comes up everywhere as you start thinking about it even without knowing at many times you are doing optimization, when you are deciding which way to order from, where to order food from for example you are trying to pick the best offer which app should you use to recharge your mobile bills, you are trying to sort of, you are trying to see where you will get, where you can get the best cashback or whatever, all of this is optimization without knowing.