How to Prove that u and Gradient(f(u)) are Colinear on the Unit Sphere?

[Calabi]

Homework Statement

Let be ##f : V \rightarrow \mathbb{R}## a ##C^{1}## function define on a neighbourhood V of the unit sphere ##S = S_{n-1}##(in ##\mathbb{R}^{n}## with its euclidian structure.).
By compacity it exists u in S with ##f(u) = max_{x \in S}f(x) = m##. My goal is to show that ##u## and ##grad(f(u))## are colinear.

Homework Equations

##f(u) = max_{x \in S}f(x)##

The Attempt at a Solution

If m is the maximum in a certain neighbourhood then the gradient is nul so the results is obvious. Then I wroght J the set of all i in ##<1, n>## with ##\frac{\partial f}{\partial i}(u) \neq 0##, if i in J that mean this equality is true on e certain neighbourhood ##V_{i}## of u so I considere the fonction ##x \in V_{i} \rightarrow \frac{x_{i}}{\frac{\partial f}{\partial i}(x)}## and try to show
that they all value a same value on u. But I don't know what to do(perhaps with the implicites function theorem.). and what to do with the nul components. I'm lost.

Could you helpme please?

Thank you in advance and have a nice afternoon.[/B]

 

[Ray Vickson]

Homework Statement

Let be ##f : V \rightarrow \mathbb{R}## a ##C^{1}## function define on a neighbourhood V of the unit sphere ##S = S_{n-1}##(in ##\mathbb{R}^{n}## with its euclidian structure.).
By compacity it exists u in S with ##f(u) = max_{x \in S}f(x) = m##. My goal is to show that ##u## and ##grad(f(u))## are colinear.

Homework Equations

##f(u) = max_{x \in S}f(x)##

The Attempt at a Solution

If m is the maximum in a certain neighbourhood then the gradient is nul so the results is obvious. Then I wroght J the set of all i in ##<1, n>## with ##\frac{\partial f}{\partial i}(u) \neq 0##, if i in J that mean this equality is true on e certain neighbourhood ##V_{i}## of u so I considere the fonction ##x \in V_{i} \rightarrow \frac{x_{i}}{\frac{\partial f}{\partial i}(x)}## and try to show
that they all value a same value on u. But I don't know what to do(perhaps with the implicites function theorem.). and what to do with the nul components. I'm lost.

Could you helpme please?

Thank you in advance and have a nice afternoon.[/B]

Please stop using a bold font in your message; it looks like you are yelling at us!

Anyway, the gradient is definitely NOT null in this problem; stationarity at an extremum applies to unconstrained optimization problems, not to problems where your variables are subject to one or more equality constraints, as they are in this case. In your problem you have the constraint ##\sum_{i=1}^n x_i^2 = 1## imposed on the variables in your function ##f(x_1, x_2, \ldots, x_n)##.

 

[Calabi]

Hello I try to keep out the bold but I can't so sorry I'm not yelling on you.

 

[Calabi]

It is possible that one of the variable of u is nul. I just say that it exists u where the maximum of f on S is reach.
And the gradient could also have a nul component which not necessarely mean that the gradient is nul.

 

[Ray Vickson]

Hello I try to keep out the bold but I can't so sorry I'm not yelling on you.

Yes you can (keep out the bold, that is); just preview your entry before you post it, to see what it will look like. If it is in bold, just use your mouse to pick out the bold part and then press the "B" button on the gray ribbon menu at the top of the input panel. Alternatively, you can manually delete the "[B|" and "[/B]' delimiters at the start and end of the bold text.

Anyway, at the optimum it might be the case that ALL components of the gradient are non-zero. Maybe some of them are zero, but maybe not---you cannnot assume anything like that. It depends very much on the form of ##f(x_1, x_2, \ldots, x_n)##

 

[Calabi]

Yes I agree. But still don't see the colinearity.

I try to make a draw and we could show that ##grad(g(u))## is orthogonal to S and I already heard that is link to implicite function.

Have you got an idea please?

 

[Ray Vickson]

Yes I agree. But still don't see the colinearity.

I try to make a draw and we could show that ##grad(g(u))## is orthogonal to S and I already heard that is link to implicite function.

Have you got an idea please?

I don't think I am allowed to say any more; I would be solving the problem if I did.

All I can suggest is that you look in your textbook, or look on-line for articles and/or tutorials on "constrained optimization problems", or something similar.

 

[Calabi]

The things is I don't see what you suggest to me.
You could say me to look an intermediar function or something else.

 

[Calabi]

Without giving me the solution the problem is not a one problem line.

 

[Calabi]

Ok it's cool I find on optimization theory some usefull information.