Re: How to solve a nonlinear non-differentiable equation in n-dimension?

From: Roger L. Bagula (rlbtftn_at_netscape.net)
Date: 09/15/04 

Date: Wed, 15 Sep 2004 01:36:39 GMT

Dear Fan Yang,
I would say that your post is
"How not to post to a news group."
1) the notation of the post is very unclear
( I had to read it and the answer several times to get an idea of what
your notation meant!).
2) the problem is "nebulous" in that the domain once understood
must contain an infinity of solutions.

One problem is symmetry:
1) even numbered dimensions ( mirror solutions?)
2) odd numbered dimensions ( single solutions?)
When a region like that is specified as the "wedge" shaped domain
in a lot of cases you will get a "mirror" on the other side of the
origin ( naturally) and to get ride of it you have to
severely hamper the functional equations.
Max( f(x(i)),0)<=0, i=1,...n in Real domain R^n ( Euclidean space of n
dimensions)
It might be easier to picture this as a cone inside of an open square in
3d for instance.
The symmetry "wedge" is
Min(f(x(i),0)>=0
on the other side of the origin.
In the complex plane you might have a solution that shows up in two
quadrants, in 3d in two octants, in 4d in 2 16th's or in only one.
The diffeomorphism isn't specified ( it could be a diffusion jet ,
second differential equation, space filling/ wedge filling , d=2 or a
dx(i)/dt=Mj*x(i) type Markov ODE).
Actually confining a function to the negative wedge like you want is
pretty difficult in functional terms ( I have a lot of experience with
parametric function in 3d for surfaces). An n-dimensional cone is
probably the easy one to get as a convex functional hull
with implicit equation like:
Sum [x(i)^2,{i,1,n-1)]=x(n)^2
 From there you can do differentiation of the kind and confining
constraints as you wish.

Let me say if you do ask questions in the future try to be more specific
and have understandable nearly universal notation.
It helps both sides and if you had the notation clearer you might have
solved it on your own.
A good standard clear notation that is universal and in plain text is
one like used in a programming language like Mathematica.
Fan Yang wrote:
> Dear all,
>
> I am facing to solve a nonlinear non-differentiable equation
> in n-dimension.
>
> The problem is to solve
> max(f(x), 0) = 0, x belongs to R(n).
> where f is a continuously differentiable map from R(n) to R(n).
>
> Seems the family of Newton methods will fail. Will bisection
> method work in this case? I only know how to use bisection
> method in 1-dimension, not the higher dimension. Or there
> exists some other more efficient methods?
>
> Thanks a lot,
>
> Fan
>
>
> 

-- 
Respectfully, Roger L. Bagula
tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 
619-5610814 :
URL :  http://home.earthlink.net/~tftn
URL :  http://victorian.fortunecity.com/carmelita/435/

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