 So, let us come back to optimization problems again. So, as I said, the way we write optimization problems, you write F which is your objective function and then we can write it as such that X belongs to a set S. So, F is called the objective function and S is called the feasible way. Now, usually this sort of way of writing an optimization problem is not very conducive for many things for both analysis as well as for computation. So, when you write that X belongs to some set, how do you even specify that set becomes a question? What does it mean that it belongs to this set? Usually a set is a collection of points, but you cannot list out all the points in the set. Usually there will be infinitely many. So, you have to describe the set in a certain way. So, the way we describe it is that we say we will put a bunch of conditions that X must satisfy. You do not actually list out all the points, but rather only list out the conditions that X must satisfy. So, the way we write it is that we write as you minimize function F such that gi of X is less than equal to 0 for all i ranging from say 1 to m and j of X is equal to 0 for all j say ranging from 1 to p. So, now you have defined an optimization problem using some additional functions. You have of course your objective function, but then you also have all these functions. These functions, so here just like F, these are functions gi and hj they are both, they are also functions from rn to r and they are called, these are referred to as constraints. Now, you can see that I have classified the constraints in a written them in a certain specific way and I have classified them into these two categories. First is that I have written out separately constraints that have less than equal to on type of requirement and another type of other constraints which have a equal to type requirement. Everything that is a less than equal to can be also written as a greater than equal to by just multiplying both sides by minus 1. So, that is not, so I do not need to separately write also a greater than equal to this. Can you, so sorry, so the less than equal to constraints, the ones that have a less than, that are to be satisfied with the less than equal to sort of requirement, these sort of constraints are called inequality constraints and the ones that are to be satisfied with equality, these are called equality constraints. Now, can someone tell me, is it possible for me to reduce everything to my problem further and write it only in terms of inequality constraints? I want to write my problem only in terms of inequality constraints. So, how do I do that? So, this is, that is the correctization, so you can, the requirement that h of j of x is equal to 0, this is equivalent actually to two different requirements, one that h j of x should be less than equal to 0 and also the, and remember both, and h j of x minus h j of x is also less, should also be less than equal to 0. So, if this and this whole, then this, so if h j of x is both less than equal to 0 and minus h j of x is also less than equal to 0, then h j of x is should be equal to 0. So, it is therefore possible to collapse this problem even further and get rid of also the equality constraints and write them as two opposing inequality constraints. However, we generally do not tend to do that and there is a good reason for this. The main reason is that the geometry of the problem is very different with the geometry of an inequality constraint is very different from that of a equality constraint. And so that is the thing I want to first, I want to sensitize you to. So, let us suppose we have, we have suppose a, say a constraint like this, say gi of x is suppose equal to negative of x minus some a the whole square. Now if gi of x is something like this, is this sort of function x minus a the whole square, then what is the region and suppose we are, so suppose we are taking this from R n to R n, so let us write this as gi of x as x minus some norm of x minus a the whole square where a is some vector in R n. So, you are fixing some vector a and then looking at this as a function of x minus negative of norm of x minus a the whole square. So, now gi of x less than equal to 0, I want to look at this region. So, I am going to look at this in this is my space R n, this is suppose my point a in that space. So, then what are all the points x such that gi of x is less than equal to 0, what are all the points? All the points, so this is actually the full space. So, let us change this, so let us suppose we have, I can instead of taking a norm of this, let us do, so I have fixed some R greater than 0, I have fixed the a in R n, I am looking at the function gi of x which is norm of x minus a the whole square minus R. Now tell me what, here is my point a, tell me what sort of region satisfies gi of x less than equal to 0. So, this is a sphere of radius R centered at a. So, this is this entire region. What if I had if I had a function h j of x is norm of x minus a the whole square minus R and I look for the region h j of x equal to 0. What sort of x's are these? This is only the shell. So, now let us, can you tell me what is the difference in these two regions? The first region is does the first, both of these sets are actually closed set. The shell itself is a closed set, the ball of radius R centered at a, with its shell is also a closed set. But the key difference is that the ball of radius R centered at a, that also has its interior included in it. Now whether the interior is, when we write an inequality constraint, we are, inequality constraint we are, what is effectively happening is that we are considering both the shell as well as the region inside it. So, as a consequence, it is implicit in the problem that what you have defined is a region. Whereas when you are talking of an equality constraint, what you have defined is really a surface. The geometry of these two is very different and this will become clearer as we go further into the course. So, the main region is that an optimization problem behaves very differently when the task of optimizing is very different from whether you are in the interior or whether you are on the boundary. If you are in the interior, you have the entire world you can explore around because there is a ball around your point which is lying completely inside your feasible region. So, your algorithm or whatever methodology you have allows you to explore in every possible direction. Where once you reach the boundary, there are only certain directions in which you can go and you have to worry about whether you are violating this or violating that. So, there is a phase change that happens when you go from interior to boundary. When we are right, that is why all methods, it is always a good idea both from a computational and analytical standpoint to know whether your constraint is truly an inequality constraint or truly an equality constraint and that is the reason why we separate these two always. Yes, it does not matter. It is a way, this will be a ball of radius, I guess square root of R, my mistake. Let me get rid of that. Now, there are other types of constraints. So, let me mention this. So, let me, so there are other types of constraints, constraints which I have, which, so for example, you can write constraints that are, for example, what can be said are bound constraints. Bound constraints means they just look at a specific variable, say X1, one variable and specify bounds on that variable saying that this variable cannot exceed a certain value capital M and cannot be greater than a certain other values. Smaller. Now, when you have bound constraints like these, you can of course think of these as two different inequality constraints, but then bound constraints are often required, the method or the type of solution algorithm you are employing, it again needs to know if there is a bound constraint. Usually that is needed and that again will become evident to you because the fact that there are, this will, if you do write this as two different inequality constraints, it will look like two different opposing inequalities of the same kind just with a different right hand side. So, it will look to the algorithm as if you have written one inequality like this and another inequality like this and this can lead to some problems. So, to avoid this, sometimes algorithms ask you to specify this separately. So, these are called bound constraints. For analysis, this usually does not matter. So, we do not, we can always absorb these as inequality constraints. There are other, there is another type of constraints which are constraints that are often either all type. So, either all type of constraints. So, when I wrote out these constraints, remember these M inequality constraints and P equality constraints, all of these constraints, remember that we want all of these to be satisfied. It is not that we have any of these or one of these or whatever. The requirement is that all of these have to be, this is a full list of things that you need to satisfy. Now, it is slightly different type of constraint will arise when I tell you that satisfy either this or that. That is a different type of constraint. Now, that also can be modeled in this sort of form, but it takes a lot of effort. You will see if you can teach that later in the course. Those are either all type constraints. And even more complicated type of constraint is then you do either or and then take a decision on top of that. So, then you, so it is if then else type of constraint. If something is satisfied, then you do this else, if that is not satisfied, then you do that. So, that would be an if then else type. We will get to this, we will look at these later in the course. So, for the moment, just we will be working with only inequality and equality constraint. So, from an engineering standpoint, we should remember that usually these, the way these arise is that, you know, an equality constraint is a hard, is a hard constraint says that this function has to be 0. So, and so value, so and so must be, so and so value is, so and so bunch of variables must satisfy a certain hard requirement. This sort of hard requirement usually comes from physical, from the physical nature of the problem. It is a, it is a law of nature that for instance that something must be equal to something. Whereas the inequality constraint is usually a human requirement. You do not want to be, you are, you have a certain tolerance for, you do not want delay to exceed a certain amount, you do not want price to exceed a certain amount, you do not want area to exceed a certain amount, etc. But you are, you are, you do not mind if it is less than that. So, that is how you, so these, the inequality constraints usually arise out of human specification. Okay. So, I will, we will stop here now.