Let $f:U \subset {\mathbb{R}^n} \to \mathbb{R}$ with $U$ being an open subset. Also, $f$ is twice continuously differentiable (i.e. a C2 function) in $U$.

I'm aware that there are necessary (Jacobian matrix must have all entries equal to zero at that point) and sufficient conditions (positive/negative-definiteness of the Hessian matrix evaluated at that point) for a point ${\mathbf{x}} = {\mathbf{x}}* \in U$ for it to be a local maximum/minimum of $f$. However, how would I go about identifying optimal points of a function when it is non-differentiable or non-twice differentiable at its optimal point? Obviously, I wouldn't be able to use the Hessian/Jacobian criteria described above to determine whether it is an optimal point or not. I would also appreciate suggested reading in dealing with this kind of functions in optimization problems.


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