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Suppose we have a problem:

$$\max_x f(x) \quad \text{s.t.} \quad g(x)\leq 0.$$

Assume $f$ and $g$ are both continuous. Assume now the following cases:

  1. that $g(x)$ is not differentiable. We can assume what we like about $f$ though I'd prefer it to be as general as possible. If it is necessary, we can assume that $g$ is convex, or that it is piecewise differentiable.

  2. That $f(x)$ is not differentiable. Again we can assume it is convex or piecewise differentiable.

What is the generic method for solving this for local/global maximum? If $f$ and $g$ were differentiable I'd have used the KKT method. But the KKT theorem doesn't apply because it assumes they are is (continuously) differentiable, at least at the optima I think.

Note that I am not looking for numerical methods. I am looking for an exact method that can produce a closed form solution at least in the cases where this is possible.

Here is a related question.

user56834
  • 12,323
  • Hi: If I'm not mistaken, nelder-mead was the original derivative free method but I'm confident there have been additions-improvements since. This looks like a good review and it's available as pdf !!!!! https://epubs.siam.org/doi/book/10.1137/1.9780898718768 – mark leeds Dec 05 '24 at 06:02
  • @markleeds, My understanding is that that book deals with numerical methods is that correct? I am interested in closed-form solutions. – user56834 Dec 05 '24 at 06:45

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