Tuesday, September 1, 2020
REQUEST FOR HELP
For I know not what reason, I lay awake in the middle of the night trying to remember what a bordered Hessian is. I know what a Hessian is – a square matrix the elements of which are the second partial derivatives of a function – but I could not recall what a bordered Hessian is, a term from my distant past. Anybody know?
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6 comments:
An 18th century German soldier living in your house.
"An 18th century German soldier living in your house."
That was my guess also. But if he was bordered in addition to being boarded in your house? Maybe he is supposed to stay on one side of the room? Good luck with that.
Anonymous and Jerry Brown beat me to it. However, according to Google as well as RPW, there is such a thing:
"The bordered Hessian is a second-order condition for local maxima and minima in Lagrange problems. We consider the simplest case, where the. objective function f (x) is a function in two variables and there is one constraint of the form g(x) = b."
Has everybody got that?
Apparently, the operation of "bordering" transforms one matrix into another matrix (of smaller dimensions) by subdividing it with horizontal and vertical lines, or borders. The resulting matrix will then have matrices as its elements.
At least is what I take away from looking at this paper
https://www.researchgate.net/publication/301302098_BORDERED_MATRICES_AND_SOME_THEIR_APPLICATIONS
where near the bottom of page 3 he gives a pretty clear example of a 4x4 matrix being transformed into a 3x3 matrix via bordering.
So one would think that if the original matrix were Hessian, the incision of such borders would yield a "Hessian bordered matrix".
Wikipedia knows: https://en.wikipedia.org/wiki/Hessian_matrix#Bordered_Hessian
Here's another explanation that I like:
https://nomanarshed.wordpress.com/2014/08/09/bordered-hessian-for-optimization/
--David F.
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