Lecture 6

Sometimesthisis allwedidwas

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Unformatted text preview: ∗ ∗ ∗ , ∗ , , ∗ , ∗ , ∗ ∗ ∗ , Here is what we have: ∗, ∗ ■ ∗ ■ ■ ■ ■ ■ ■ ■ So… ∗ ∗ , 0; 0; ∗ ; ; ∗ ∗ ∗ ∗ ∗ = = 0 ∗ , ∗ , , ∗ , ∗ , ∗ ; ; ∗ , ∗ ∗ , ∗ , ∗ , ∗ ∗ ∗ , 0. ∗ = , ∗, ∗ ∗ ∗ , ≡ ∗, ∗ ∗ ∗, ∗ ■ , ∗ ; , ∗ , ∗ ∗ ,∗ ∗∗ , ∗∗ , , ∗ , ∗ ,∗ ∗∗ , ∗∗ , ∗ , ∗ , ∗ ∗ This is the Hessian for our problem. Now it turns out to be convenient to note that our problem does not change, substantially, if we simply relabeled things in the following way. , , ≡ , , All we did was make the third argument of the first argument of . In this case, the Hessian for this problem looks like: ∗∗∗ ∗∗∗ ∗∗∗ ,, ,, ,, ∗∗∗ ∗∗∗ ∗∗∗ ,, ,, ,, ∗∗∗ ∗∗∗ ∗∗∗ ,, ,, ,, which is ∗∗ ∗∗ , , ∗∗ ∗∗∗ ∗∗∗ , ,, ,, ∗∗ ∗∗∗ ∗∗∗ , ,, ,, (Note that the ordering here of the variables is written in terms of our original formulation, as do the references in the matrix to , , ). ∗ This (as well as written as ∗ , , ∗ above) is referred to as a Bordered Hessian. Sometimes, this is ∗ ∗ , ∗ , ∗ ∗ , ∗ , ∗ ∗ ∗ , ∗ , , ∗ ∗ , ∗ , ∗ ∗ ∗ ∗ , ∗ , , ∗ ∗ , ∗ , ∗ ∗ All we did was make the third argument of the first argument of . In this case, the Hessian for this problem looks like: ∗∗∗ ∗∗∗ ∗∗∗ ,, ,, ,, ∗∗∗ ∗∗∗ ∗∗∗ ,, ,, ,, ∗∗∗ ∗∗∗ ∗∗∗ ,, ,, ,, which is ∗∗ ∗∗ , , ∗∗ ∗∗∗ ∗∗∗ , ,, ,, ∗∗ ∗∗∗ ∗∗∗ , ,, ,, (Note that the ordering here of the variables is written in terms of our original formulation, as do the references in the matrix to , , ). ∗ This (as well as written as ∗ , , ∗ above) is referred to as a Bordered Hessian. Sometimes, this is ∗ ∗ , ∗ , ∗ ∗ , ∗ , ∗ ∗ ∗ , ∗ , , ∗ ∗ , ∗ , ∗ ∗ ∗ ∗ , ∗ , , ∗ ∗ , ∗ , ∗ ∗ ∗∗∗ A problem we confront, however, is that our matrix ,, isn’t by itself doesn’t satisfy the sufficient conditions we laid out for being negative semidefinite – above. (The 1st principal minor of this matrix is 0 and the 2nd principal minor is 0. However, it turns out, in a two variable optimization problem that we are working with, that ∗∗∗ ,, gives us a condition which is sufficient to the sign of the determinant of determine if our critical point is a local min or max. For future reference, the determinant ∗∗∗ ,, To see our general point, consider the constraint, , 0. Since...
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