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Assuming I am given a Program which can calculate the value of a continuous, infinitely differntiable (we cannot calculate these derivatives), real, positive function of two real variables which has exactly one local (and global) minimum inside a given rectangle. Moreover the function is assumed to be sub-harmonic. My goal is to find the coordinates of the minimum, as accurately as possible, while using the given program as few times as possible (it takes a long time to run).

Is there some numerical method for solving such problems?

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A steepest descent with derivatives approximated by suitable difference quotients and line search should do the job. Once your steps become small, you can use the difference of two iterates as one directional derivative and only use one more function evaluation to compute the gradient.

With every iterative method, you will have to assert that the limit is in the given rectangle!

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Yes, numerical methods are available. But, unless there is something very special about your particular problem, I would advise against writing your own software to solve it. Use numerical methods written by experts, instead. You should be able to find a suitable existing algorithm via the Plato web site.

Perhaps this page would be a good place to start.

Your rectangle would be referred to as a "bound constraint" by the optimization folks (because you are simply placing bounds on your independent variables, as opposed to more complex constraints).

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