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Difference between a hessian and a bordered hessian

  1. Dec 13, 2012 #1
    1. The problem statement, all variables and given/known data
    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



    2. Relevant equations



    3. 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.
     
  2. jcsd
  3. Dec 14, 2012 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    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.
     
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