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# N-dimensional differentiation watch

1. Hey guys,

Bit of an unusual request here I suspect.

Basically, my maths knowledge is - was - A Level (further) maths standard. I don't remember all of the A Level stuff now but I'm sure I could pick it up again pretty quickly.

However, my job involves a lot of applied game theory, and I've been thinking about something recently - it could be really useful for intuitively approximating solutions to subgames if I could solve to maximise the relevant variable using calculus. So basically, what I want to do is find the maximum (inflection) point of the y axis in n-dimensional space - normally, n will be between 5 and 7.

From my basis of mathematical understanding, is it going to be possible to get to the stage of being able to do this in a few hours of learning? What path should I take?

Cheers.
2. (Original post by TheDefiniteArticle)
Hey guys,

Bit of an unusual request here I suspect.

Basically, my maths knowledge is - was - A Level (further) maths standard. I don't remember all of the A Level stuff now but I'm sure I could pick it up again pretty quickly.

However, my job involves a lot of applied game theory, and I've been thinking about something recently - it could be really useful for intuitively approximating solutions to subgames if I could solve to maximise the relevant variable using calculus. So basically, what I want to do is find the maximum (inflection) point of the y axis in n-dimensional space - normally, n will be between 5 and 7.

From my basis of mathematical understanding, is it going to be possible to get to the stage of being able to do this in a few hours of learning? What path should I take?

Cheers.
If I've understood you correctly, you have a function

where n is around about 5-7, and you wish to find the point(s) that maximize the value of f.

How you go about doing this depends on how your function f is specified. If you have a formula for f, then you can use the fact that at a stationary point of f, the partial derivatives must vanish simultaneously - so you just find all the points when the partial derivatives vanish simultaneously, calculate the value of the function at those points and then see which of them is the global maximum.

However, be aware that in dimension greater than one you can get some awkward situations that make life difficult. First of all, that condition on the partial derivatives is necessary but not sufficient for a maximum or minimum - you can get points of inflection and saddle points. More than that, you can get lines and planes (and more generally submanifolds) where the derivatives vanish simultaneously - making checking them very difficult.

If your function is not very well behaved, or if your function is not specified with a formula, then the usual way of going about maximizing the function is to use numerical techniques. These typically involve starting at some random point of the domain , using a rule of some sort to go to another point of the domain where the function is greater and repeating this until no increase occurs within a specified bound of accuracy. The this is repeated over and over again starting from other random points of the domain - in order to capture more than just local maxima.

So bottom line is: it depends on how well behaved your function is. If it's typical of the stuff I have to deal with, then a day's thinking isn't going to be enough - however there are lots of tools out there that can help - which ones you use depends on the setup that you're attempting.
3. (Original post by Gregorius)
If I've understood you correctly, you have a function

where n is around about 5-7, and you wish to find the point(s) that maximize the value of f.

How you go about doing this depends on how your function f is specified. If you have a formula for f, then you can use the fact that at a stationary point of f, the partial derivatives must vanish simultaneously - so you just find all the points when the partial derivatives vanish simultaneously, calculate the value of the function at those points and then see which of them is the global maximum.

However, be aware that in dimension greater than one you can get some awkward situations that make life difficult. First of all, that condition on the partial derivatives is necessary but not sufficient for a maximum or minimum - you can get points of inflection and saddle points. More than that, you can get lines and planes (and more generally submanifolds) where the derivatives vanish simultaneously - making checking them very difficult.

If your function is not very well behaved, or if your function is not specified with a formula, then the usual way of going about maximizing the function is to use numerical techniques. These typically involve starting at some random point of the domain , using a rule of some sort to go to another point of the domain where the function is greater and repeating this until no increase occurs within a specified bound of accuracy. The this is repeated over and over again starting from other random points of the domain - in order to capture more than just local maxima.

So bottom line is: it depends on how well behaved your function is. If it's typical of the stuff I have to deal with, then a day's thinking isn't going to be enough - however there are lots of tools out there that can help - which ones you use depends on the setup that you're attempting.
So here's an example of a function I'm thinking of, where I'd be looking for the maximum point of y:

y = xz + 2(a^2)b + c(d^3)

where x, z, a, b, c and d are all variables (this could be a weird way of writing this, I don't really remember much about notation).

Hell, I'm less confident now than I was last night that this is even a calculus problem, but I think it is.
4. (Original post by TheDefiniteArticle)
So here's an example of a function I'm thinking of, where I'd be looking for the maximum point of y:

y = xz + 2(a^2)b + c(d^3)

where x, z, a, b, c and d are all variables (this could be a weird way of writing this, I don't really remember much about notation).

Hell, I'm less confident now than I was last night that this is even a calculus problem, but I think it is.
A function like that is not going to have a global maximum. For example, if you just make d larger and larger, the function gets larger and larger.

Are you really looking to maximize a function over some bounded domain? That is, where your variables x, z, a, b, c, d each lie in some restricted range like -1 to +1?

Just to give you an illustration, if we restrict attention to a much simpler version of your function in only 2 dimensions, y = x z, this is what it looks like:

As you can see, there is a point (0,0) where the derivatives vanish, but it is neither a maximum or a minimum (it is a saddle point) and the maximum of the function over the region plotted is taken on the boundary (where the derivatives are not zero).

Best, in this case, to use numerical techniques.
5. (Original post by Gregorius)
A function like that is not going to have a global maximum. For example, if you just make d larger and larger, the function gets larger and larger.

Are you really looking to maximize a function over some bounded domain? That is, where your variables x, z, a, b, c, d each lie in some restricted range like -1 to +1?

Just to give you an illustration, if we restrict attention to a much simpler version of your function in only 2 dimensions, y = x z, this is what it looks like:

As you can see, there is a point (0,0) where the derivatives vanish, but it is neither a maximum or a minimum (it is a saddle point) and the maximum of the function over the region plotted is taken on the boundary (where the derivatives are not zero).

Best, in this case, to use numerical techniques.
I've talked to a friend about it. Essentially some of the variables are bounded, but not all, and it turns out it's not exactly a suitable method of achieving my purposes anyway.

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