What is the Hessian method for determining concavity/convexity?

In summary, there is a method for determining whether a function is concave or convex by using the Hessian matrix and examining its diagonal elements. However, the details of this method, including finding the eigen values, have been forgotten by the speaker.
  • #1
OhMyMarkov
83
0
Hello Everyone!

I'm trying to remember a quick method for determining whether a function is concave or convex. There was something that involved finding the Hessian of the function, and then looking at the diagonal elements, then, I completely forgot...

What's the rest of this method, I don't remember I even had to find the eigen values...

Thanks!
 
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  • #2
OhMyMarkov said:
Hello Everyone!

I'm trying to remember a quick method for determining whether a function is concave or convex. There was something that involved finding the Hessian of the function, and then looking at the diagonal elements, then, I completely forgot...

What's the rest of this method, I don't remember I even had to find the eigen values...

Thanks!

Hi OhMyMarkov, :)

The method of using the Hessian of a function to determine the concavity/convexity is described >>here<<.

Kind Regards,
Sudharaka.
 

FAQ: What is the Hessian method for determining concavity/convexity?

What is the Concavity test by Hessian?

The Concavity test by Hessian is a mathematical method used to determine the concavity of a function at a given point. It involves calculating the Hessian matrix, which is a matrix of second-order partial derivatives of the function, and using its eigenvalues to determine the concavity.

How is the Hessian matrix calculated?

The Hessian matrix is calculated by taking the second partial derivatives of the function with respect to each of its variables and arranging them in a matrix. For example, if the function is f(x,y), the Hessian matrix would be:

H = [fxx fxy][fyx fyy]

What do the eigenvalues of the Hessian matrix represent?

The eigenvalues of the Hessian matrix represent the curvature of the function at a given point. A positive eigenvalue indicates that the function is concave up, while a negative eigenvalue indicates that the function is concave down. A zero eigenvalue indicates that the function has a saddle point at that point.

How do you use the eigenvalues to determine concavity?

If the Hessian matrix has all positive eigenvalues, then the function is concave up at that point. If the Hessian matrix has all negative eigenvalues, then the function is concave down at that point. If the Hessian matrix has both positive and negative eigenvalues, then the function has a saddle point at that point.

What is the significance of the Concavity test by Hessian?

The Concavity test by Hessian is important in calculus and optimization because it allows us to determine the behavior of a function at a given point. This information can then be used to find maximum and minimum values of the function, as well as to sketch its graph accurately.

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