Thread: Closed Mapping - Non-Linear Programming

Hi there
I'm studyin' some mathematics fundamentals for optimizations methods of non-linear programming.
I didn't understood the "Closed Mapping" thing, which is defined in my book ( Luenberg ) like this:

A point-to-set mapping A from X to Y is said to be closed at x ∈ X if the assumptions
i) x_k -> x, x_k ∈ X
ii) y_k -> y, y_k ∈ A(x_k)
imply
iii) y ∈ A(x)
The point-to-set map A is said to be closed on X if it is closed at each point of X

I would appreciate any help.

So this is saying, given a point x in X, take any sequence of points x_k of X converging to x. Now take any sequence of points y_k, each y_k being in the set A(x_k), and suppose this sequence converges to a point y. Then y must be in A(x). In a very loose sense, this says that the sets A(x_k) "converge" to the set A(x). (Nothing here guarantees that there exists a sequence y_k that actually converges.)

Thanks for your explanation serpicojr. But I didn't get it yet. What do you mean by "sequence of points xk of X converging to x"??
There is any graphical way to represent all those things? I think that its always easiest to understand with graphical explanations.
Sorry man, I know that's all basic concepts, but I missed all those basic concepts 'cause I was sick ><"

Here is an example. Consider the sequence x_k = 1 ⁄ k, where k is a positive integer. This is the sequence 1, ¹⁄₂, ¹⁄₃, ¹⁄₄, …

Notice that the numbers get closer and closer to 0? In fact, by choosing large enough integers k, we can make the numbers 1 ⁄ k get as close to 0 as we like. We say that the sequence 1, ¹⁄₂, ¹⁄₃, ¹⁄₄, … converges to 0.

The formal definition is this: We say that a sequence of numbers x_k converges to x iff given any ε > 0, there is a positive integer K such that |x_k − x| < ε for all integers k > K. The number ε can be viewed as a sort of “error” margin. In other words, given any specified error margin, we can make all but finitely many numbers of the sequence x_k be within that specified error margin of the limit x.

Okay, so what sort of math background do you have? At my school, in order to take nonlinear programming, you would have needed to take a sequence of courses including 3 or 4 semesters of calculus, a linear algebra course, and linear programming. In calculus--specifically, a second semester calculus course--you would have learned about sequences, limits, and convergence. So, as far as my understanding goes, you should have seen this material before this class. If this is the case, and if it's been a while since you've seen sequences and limits, then now is the time to refresh your memory. If you've never seen these topics... well, let us know.

Thank you Janet ^^
I got this part, I know about convergence. The problem is that I can't see the relationship between this and the non-linear programing.
The first thing I can conclude from that definition, is that a closed mapping is always an unlimited set - which makes no sense in this case, since we're talking about non-linear problems with their constraints...

serpicojr, I'm in mastership degree. In graduation I studied 4 calculus, and in the forth I learned convergence. As I said above, the problem is not to understand the convergence, is realise what it means in this context...
I studied 3 linear algebra ( last one I got Jordan's Canonic Form, Jordan's blocks and etc... )

I'm studyin' linear programming. I've already studied it before, but in mastership we're going more deep in the problem...

So if you understand convergence, then what don't you understand about the phrase "sequence of points x_k of X converging to x"?

Hum... I think that I get this part of convergence, its likely "in a neiborhood of x"??
I think its easiest to see it in an example, don't you know some?

Think of a neighborhood of x as being an open interval containing x, and when something says "for all neighborhoods of x", think of open intervals of x getting smaller and smaller.

So let's consider an example. This won't be coming from a nonlinear programming example, as I don't know the subject, but I'll try to make it applied.

For any real number x, let's define A(x) to be the domain of the function f(z) = sqrt(x^2 - z^2). It shouldn't be too hard to see that the domain of this function is the closed interval [-|x|,|x|]. (I could have just defined A(x) = [-|x|,|x|], but I figured defining it the way I did shows how one might "naturally" come up with this function.)

I claim that A is a closed mapping. So, let's check the definition. The set X is just the set of real numbers, so if we let x_k -> x in X, this just means x_k is a sequence of real numbers converging to x in the sense that you're already familiar. Now we're suppose to let y_k be an element of A(x_k) = [-|x_k|,|x_k|], and we're assuming that the sequence y_k converges to some number y. For A to be a closed mapping, we must show that y is an element of A(x).

Well, first I'd like to note that for any real numbers a and b, a is in the set [-|b|,|b|] if and only if |a| ≤ |b|. So, really, I'm trying to prove that given a sequence x_k converging to x, and given a sequence y_k such that |y_k| ≤ |x_k| for all k and such that y_k converges to some point y, then |y| ≤ |x|. This is easy:

|y| = lim |y_k| ≤ lim |x_k| = |x|

where these limits are as k goes to infinity. So |y| ≤ |x|, and hence y is an element of A(x) = [-|x|,|x|]. Thus A is a closed mapping.

In a rough sense, this says to me: "The function f varies by some continuous parameter, and its domain varies in a 'continuous' sense."