Difference between a hessian and a bordered hessian

[Centurion1]

Homework Statement

I was wondering what exactly the difference between a regular (proper? is that the term) hessian is and a bordered hessian. It is difficult to find material in the book or online at this point. I mean mathmatically so that were i to do a problem i would know the layout and what differs between the two. At this point I am aware that a bordered hessian is for constrained optimizations and a proper hessian for unconstrained from there i am unaware where they differ. Theoretically and forumlaically how do they differ?

At this point i think it is something like

Homework Equations

The Attempt at a Solution

Proper Hessian

lZxx Zxyl
lZyx Zyyl

Then you find the determinant

A bordered hessian would be

l 0 Fx Fy l
l Fx Zxx Zxy l
l Fy Zyx Zyy l

Is that right because it seems too simple.

 

[Ray Vickson]

Homework Statement

I was wondering what exactly the difference between a regular (proper? is that the term) hessian is and a bordered hessian. It is difficult to find material in the book or online at this point. I mean mathmatically so that were i to do a problem i would know the layout and what differs between the two. At this point I am aware that a bordered hessian is for constrained optimizations and a proper hessian for unconstrained from there i am unaware where they differ. Theoretically and forumlaically how do they differ?

At this point i think it is something like

Homework Equations

The Attempt at a Solution

Proper Hessian

lZxx Zxyl
lZyx Zyyl

Then you find the determinant

A bordered hessian would be

l 0 Fx Fy l
l Fx Zxx Zxy l
l Fy Zyx Zyy l

Is that right because it seems too simple.

Your are right: the above matrix is a bordered Hessian. It's what you do with it afterwards that counts!

Basically, in an equality-constrained optimization problem, the Hessian matrix of the Lagrangian (not just the Hessian of the max/min objective Z) needs to be tested for positive or negative definiteness or semi-definiteness, not in the whole space, but only in tangent planes of the constraints. Using bordered Hessians is one way of doing this, but a much better way is to use so-called "projected hessians"; these are, essentially, the Hessian projected down into the lower-dimensional space of the tangent plane. Nowadays, serious optimization systems use projected Hessians, and just about the only place you will see bordered Hessians anymore is in Economics textbooks.