Numerical Optimization methods?

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What kind of functions are suitable for numerical optimization methods such as Newton, Gradient Descent, ... ? Any conditions?

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There are different algorithms for different classes of functions. Hundreds of them. Generally speaking, optimization gets more difficult if the objective function is non-convex or derivatives can not be computed (for example). Not impossible; just more difficult.

If you're looking for an optimization algorithm to handle a specific class of functions then this guide might be a good place to start.