With frame
\begin{eqnarray*}f(x,y)&=&x^2+y^2\\ \rightarrow\,\hat H(x,y)&=&\left(\begin{array}{cc} 2 & 0 \\ 0 & 2 \end{array}\right)\to\mbox{pos.\ definite, i.e.\ minimum}\end{eqnarray*}
\begin{eqnarray*}f(x,y)&=&-x^2-y^2\\ \rightarrow\,\hat H(x,y)&=&\left(\begin{array}{cc} -2 & 0 \\ 0 & -2 \end{array}\right)\to\mbox{neg.\ definite, i.e.\ maximum}\end{eqnarray*}
\begin{eqnarray*}f(x,y)&=&x^2-y^2\\ \rightarrow\,\hat H(x,y)&=&\left(\begin{array}{cc} 2 & 0 \\ 0 & -2 \end{array}\right)\to\mbox{indefinite, i.e.\ no extremum, saddle point}\end{eqnarray*}
semi-cases difficult:
| \[f(x,y)=x^2+y^4\to\vec\nabla f=\vect{2x\\4y^3}\to \hat H(x,y)=\left(\begin{array}{cc} 2 & 0 \\ 0 & 12y^2 \end{array}\right)\] |
\(\hat H\) pos. semi-definite?\(\to\)minimum
| \[f(x,y)=x^2\to\vec\nabla f=\vect{2x\\0}\to \hat H(x,y)=\left(\begin{array}{cc} 2 & 0 \\ 0 & 0 \end{array} \right)\] |
\(\hat H\) pos. semi-definite?\(\to\)mall values \(x=0\) y-arbitrary are minima
| \[f(x,y)=x^2+y^3\to\vec\nabla f=\vect{2x\\3y^2}\to \hat H(x,y)=\left(\begin{array}{cc} 2 & 0 \\ 0 & 6y \end{array} \right)\] |
\(\hat H(0,0)\) pos. semi-definite?\(\to\)but no extremum, complicated saddle point!
| \[f(x_1,x_2)=e^{-(x_1^2+x_2^2-4x_1x_2-2x_1)}\quad \vec\nabla f=e^{-(x_1^2+x_2^2-4x_1x_2-2x_1)}\vect{-2x_1+4x_2+2\\-2x_2+4x_1}\] |
| \[\vec\nabla f=-2e^{-(x_1^2+x_2^2-4x_1x_2-2x_1)}\vect{2x_2-x_1-1\\2x_1-x_2}\begin{array}{cclcclcc} \rightarrow&2x_2-x_1&=&1\\ \to&2x_1-x_2&=&0\\ &x_2&=&2x_1&\to&4x_1-x_1&=&-1\\ &&&&&x_1&=&-\frac{1}{3}\\ &&&&&x_2&=&-\frac{2}{3}\end{array}\] |
\begin{eqnarray*}\tilde H(x_1,x_2)&=&2e^{-(x_1^2+x_2^2-5x_1x_2-2x_1)}*\\ && \left(\begin{array}{cc}-1+2(2x_2-x_1+1)^2&\quad 2+2(2x_1-x_2)(2x_2-x_1+1)\\2+2(2x_1-x_2)(2x_2-x_1+1)\quad &-1+2(2x_1-x_2)^2\end{array}\right)\end{eqnarray*}
Ignoring the positive prefactor we find
| \[\tilde H(-\frac{1}{3},-\frac{2}{3})\propto\left(\begin{array}{cc} -1 & 2 \\ 2 & -1 \end{array}\right),\] |
i.e. the Hesse matrix is negative-definite implicating that the extremum at \((-\frac{1}{3},-\frac{2}{3})\) is a maximum.
Criteria for finding extreme values in N-dimensions
© J. Carstensen (Math for MS)