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I'm doing some research on computing optimal controls in quantum mechanics, and need a numerical algorithm that I can try to adapt to my problem. I'm hoping that if I describe the problem, someone out there can point me in a good direction.

Consider a function [itex] f: \mathbb R^n \to \mathbb R [/itex] and let [itex] S=f^{-1}(0) [/itex]. I want to find

[tex] \min_{\vec x \in S} \| x \|_2 [/tex]

While we do not have any theoretical results of yet proving this, we empirically believe that S is countable and hence totally disconnected. Furthermore, within a finite radius of the origin, we believe that S is actually finite (and hence this could be seen as a combinatorial optimization problem).

Now I can find a single point in S, but the goal is to minimize the norm without knowing where the other points in S lie. I've been looking into constrained discrete simulated annealing and superficially at level set optimization, but I'm not sure if there's a better way. If anyone knows of an algorithm that is designed to handle this (or something similar) it would be much appreciated.

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# Minimization on level sets

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