Optimization problems involving non-compact domains

[Inertigratus]

I have some understanding of how to solve problems involving compact domains.
Set the gradient to zero and solve for x and y, and then try to parameterize if needed to find max/min over the border of the domain.
The thing is, my book doesn't go into much detail on how to do optimize functions defined over non-compact domains. It basically says to try to narrow the domain down to a compact domain, and then optimize using the "method" described above. There are some examples too but they're not really helping me.

Does anyone have any good links to sites discussing some methods/problems?
Or if you have any tips yourself?

Oh and by compact domain I mean that it is confined and closed.
It's multivariable functions by the way.

 

[Inertigratus]

I just tried doing one thing used in one of the examples in my book.

For example if the domain is 0 < x&y < infinity, then you change that into 0 < x < a, 0 < y < b.
Then you could turn it into a function of one variable by doing f(a, y) and f(x, b).
Taking the derivative to get 'y' respective 'x' as a function of 'a' respective 'b', then plugging that into the equation and then taking the derivative with respect to 'a' respective 'b' to get the values for a and b.

Does this always work? Is it a valid method?

In the example they didn't have both 'a' and 'b' but just one of them, since the other variable was already confined.

 

[HallsofIvy]

The reason your book doesn't deal with non-compact domains is that there may NOT be a point that optimizes a given function. To take a simple example, f(x)= x has no maximum or minimum value on (0, 1).

 

[Inertigratus]

Well, it does deal with non-compact domains, but personally I think it's not explained well enough.
So how can I find out if a function has a max/min when defined over a non-compact domain?
Also, the "method" I described above, is it valid?

 

[Inertigratus]

No one has anything to add? :)