Re: Optimizing a function on an n-torus


In article <1166181985.095647.111780@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Bill J." <nolabab@xxxxxxxxx> writes:
If I have an infinitely differentiable function f from an n-torus to R
like f:S_1^n->R, is there an efficient way to find this functions
global extrema?

Assume that the function has a well-behaved infinitely differentiable
representation as a function from (R mod Z)^n to R.

Does the answer change if there is a representation as a function from
(R mod Z)^n to R for which all of the unmixed derivatives above a
certain finite order, say m, vanish?


no. this is just smooth global optimization, hard to solve deterministically
if n gets large (any n beyond 1 might be hard, depends on the function)
and if all unmixed derivatives of higher order vanish, then because of the
interchangeability of derivatives all higher derivatives vanish, and you
have a polynomial in n variables. but even this is of no help.
such polynomials can look quite terrible and global optimization is here
as hard as if these would be only C^2 functions.
sorry, no help
peter
.

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