#Higher-order tests for extrema of multivariable functions

20 messages · Page 1 of 1 (latest)

native tiger
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So, the higher-order tests for extrema of single-variable functions and the hessian test (so, second derivative) for extrema of multivariable functions are quite well-known and studied by pretty much everyone.
However, I was wondering whether there was a variant of a higher-order test for multivariable functions, too. For example, f(x, y) = x^4 + y^4 clearly has a local minimum at the origin, yet the hessian test will obviously fail.
Is there such a test? Assuming existence and continuity of all the derivatives involved, of course. Some literature on the subject would also be quite nice!

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split panther
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How about treating the Hessian as an operator $H =
\begin{bmatrix}
\frac{\partial^2}{\partial x^2} & \frac{\partial^2}{\partial x\partial y} \
\frac{\partial^2}{\partial y\partial x} & \frac{\partial^2}{\partial y^2}
\end{bmatrix}$

dense sedgeBOT
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Djake3tooth

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split panther
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And looking at the eigenvalues of $Hf, H^2f, \dots$

dense sedgeBOT
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Djake3tooth

616165347552395271.png
split panther
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There's also the higher-order derivative tensor
https://www.quora.com/As-there-is-a-Jacobian-and-Hessian-matrix-is-there-a-third-partial-derivative-matrix

Quora
As there is a Jacobian and Hessian matrix, is there a third partial...

Answer (1 of 2): Yes, there is a third partial derivative matrix called the "third-order derivative tensor" or "third-order partial derivative matrix."

The third-order derivative tensor is a three-dimensional array of partial derivatives that describes the third-order derivatives of a scalar fun...

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But I don't know if you can extract out of that tensor if there are local minima/maxima or sadle points

split panther
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$f(x,y)=x^4+x^2y^2+y^4+xy^3$

dense sedgeBOT
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Djake3tooth

616165347552395271.png
split panther
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because $H^2$ doesn't contain every possible combination of 4 partial derivatives

dense sedgeBOT
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Djake3tooth

616165347552395271.png
blissful wigeon
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the hessian conditions are sufficient for local minima/maxima or saddle points

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if the hessian is singular at a point, the test is inconclusive and you have to analyse by inspection

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as far as I know, there is no general description of local extrema for multivariable maps

native tiger
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Ah, I see... That's a shame.

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Well, in any case, thanks to you both!

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