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I have asked this question on math.stackexchange.com but even though I gave a bounty, I was not able to receive any answers at all, so I'm posting it here again, hoping that the question is not too basic for community.

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a function that is (at least) $C^2$. In the standard calculus classes one learns: If the gradient of $f$ at $x_0$, then $f$ has a local extremum (minimum or maximum) at $x_0$. Hence the gradient test is a necessary but not sufficient condition (as, e.g., $x\mapsto x^3$ shows). Using the (semi)positive/negative definiteness of the Hessian, one can disambiguate and identify among the previously found extremum candidates some that are local minima or maxima.

But so far this doesn't give complete classification of the extrema, as it is possible one the one-dimensional case for a test involving $n$ derivatives, and it seems that this problem is rather difficult for higher dimensions, involving potentially Morse theory, as I found it mentioned online.

Can someone clarify whether a nice, complete description (i.e. a nice characterization) of which points are minima and maxima exists in all dimension or point me to a definitive reference (that ideally also outlines the state of the art)?

There are various online references, that provide partial answers:

  • https://math.stackexchange.com/questions/2869744/what-is-the-higher-order-derivative-test-in-multivariable-calculus (but this seems to give a test involving all derivatives, but following the discussion there it seems this test is rather hard to check in practice and one has to rely on numerical computations; in the comments it is actually stated "there is not an analogue test in multivariable calculus", and this comment is backed up in the comments by a user I trust, KCd, which seems to indicate that the problem I outlined is open. Morse theory is mentioned, but not particular information is given. I would be also interested in a statement along the lines of whether for every $n$ one can construct a function such that for any "feasible" test using higher order derivatives, any "natural" test using higher order derivatives that allows "easy checking" fails.)
  • http://www.u.arizona.edu/~mwalker/MathCamp2021/UnconstrainedOptimization.pdf (this gives actually necessary and sufficient conditions, but I'm not happy with them, because they are different conditions, so the don't characterize extrema completely)
  • https://www.ripublication.com/adsa20/v15n2p11.pdf (it's from 2020, but I'm not 100% convinced this is really a legit or novel article; nonetheless it's interested to read)
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  • $\begingroup$ Even in the one-dimensional case, the description linked by you is incomplete. Indeed, it is possible that all the derivatives of $f$ at $x_0$ are $0$, and then $x_0$ can be anything: a point of local max, local min, inflection of any order, or none of those. On the other hand, the multidimensional case obviously reduces to the one-dimensional one, by considering the restrictions of $f$ to the straight lines. $\endgroup$
    – Iosif Pinelis
    Commented Oct 17, 2022 at 11:38
  • $\begingroup$ Previous comment continued: It is also clear that, in the multidimensional case, the partial derivatives (that is, the derivatives in the coordinate directions) of any order, or even a complete description of the behavior of $f$ in the coordinate directions, cannot possibly tell the whole story $\endgroup$
    – Iosif Pinelis
    Commented Oct 17, 2022 at 11:38
  • $\begingroup$ @IosifPinelis Thank you for your feedback. Could you please point me to a reference of examples that show for points of inflection of any order that all higher derivatives can vanish? For local min and max the function $f(x):=\exp(-\frac{1}{x^2})$ and $f(0):=0$ and $g(x):=−f(x)$, respectively, seem to work. Though coming up with these examples seems not to be entirely trivial. $\endgroup$
    – alhal
    Commented Oct 19, 2022 at 5:26
  • $\begingroup$ Do you know whether there exist partial results classifying points (even in the one-dimensional case)? Perhaps there is an entire theory devoted to this that I don't know. I'm not really familiar with algebraic geometry, though it seems that inflection points, for example, are usually defined in textbooks on that subject. From analysis textbook a treatment of them seems to be missing entirely (oddly, the few references that I could find via Wikipedia and the Springer Encyclopedia of Math are mostly Russian analysis textbooks). $\endgroup$
    – alhal
    Commented Oct 19, 2022 at 5:31
  • $\begingroup$ I still think that It requires Morse theory. It's not hard, but rather just too much "computacional" for a live person to verify its necessary and suficient conditions. No one in optimization will bother with results with higher order than 3, since just few algorithms generates 3 order stationary points. In your situation, It's easier to assume local minimality itself instead of doing such calculations. $\endgroup$
    – R. W. Prado
    Commented Dec 3, 2022 at 13:33

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