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I am asking this question about local minima, but actually I started by trying to find the global maximum/minimum over a compact set, of a smooth function (the objective). The function has a random external input value which will determine the exact expression of the function at each simulation.

Can anyone recommend a theoretically backed criterion (or a relevant scholarly source) for estimating the number of local minima of a smooth objective over said compact domain?

Since, if I manage to find a maximum number of them which essentially realize different objective values, then I can take the smallest of these objective values and claim it to be a good guess of my global minimum, right?

Any critique or advice will be much appreciated.

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