Optimizing a function on an n-torus

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?

.

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