I have the following problem: $\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}\def\g{g(x_1,x_2,x_3)}\def\l{\lambda}\def\q{\begin{pmatrix}}\def\p{\end{pmatrix}}$
Find the critical points of the following constrained optimization problem:
$$\f=\1^2+\2^2+2\3^2$$
subject to $$\g=\1+\2+\3=4$$
and am required to check that these are non-degenerate and need to determine local min and maxima.
$$L = \f+\l \g$$
$$=\1^2+\2^2+2\3^2+\l\1+\l\2+\l\3-4$$
$$\n L= (2\1+\l,2\2+\l,4\3+\l)=\vec{0}$$
Which yields:
$$\1+\2+\3=4 \;\; (1)$$
$$2\1+\l=0 \;\;(2)$$
$$2\2+\l=0\;\;(3)$$
$$4\3+\l=0\;\;(4)$$
From 2),3), $\1=\2$,from 4),$\3=\frac{\2}2$
From 1) $\frac52\2=4\Rightarrow \2=\frac85$
$$\1=\frac85=\2,\3=\frac8{10},\l=-5$$
$f(\frac85,\frac85,\frac8{10})=\frac{32}{5}$
To check degeneracy, I have to look at the bordered Hessian matrix.
Can someone show me how to use this bordered Hessian matrix?
My attempt:
$$ H = \q 0 & g_{\1} & g_{\2} & g_{\3} \\ g_{\1}&L_{{\1}{\1}}&L_{{\1}{\2}}&L_{{\1}{\3}} \\ g_{\2}&L_{{\2}{\1}}&L_{{\2}{\2}}&L_{{\2}{\3}} \\ g_{\3}&L_{{\3}{\1}}&L_{{\3}{\2}}&L_{{\3}{\3}} \p $$
$$ H = \q 0 & 1 & 1 & 1 \\ 1&2&0&0 \\ 1&0&2&0 \\ 1&0&0&4 \p $$
Taking the determinant of this yields $ H = -14 \ne 0$ Hence this is not a degenerate point.
Best Answer
Expand at the first line $\det H = -\det\begin{pmatrix}1 &0 & 0\\ 1 & 2 & 0 \\ 1 &0 & 4\end{pmatrix}+\det\begin{pmatrix}1 &2 & 0\\ 1 & 0 & 0 \\ 1 &0 & 4\end{pmatrix}-\det\begin{pmatrix}1 &2 & 0\\ 1 & 0 & 2 \\ 1 &0 & 0\end{pmatrix}=-8-8-4=-20$