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\begin{document}
\vspace{1pt}Lagrange Multipliers
Chapter 19 \ Chong/Zak
Author: Robert Beezer
History:
1999/04/07\qquad First version.
\vspace{1pt}
\subsubsection{Problems}
\vspace{1pt}
\begin{enumerate}
\item min $f(x)$, subject to \ $h_{i}(x)=0$, \ $1\leq i\leq m$, \ $%
g_{j}(x)=0$, \ $1\leq j\leq p$. \ \ We can ``roll-up'' the \ $h$'s and the \
$g$'s into vector-valued functions and compare them to the zero vector. \ \
\ Note that this problem formulation encompasses a wide variety of problems,
including linear programming. \ At first we'll stick with the equality
constraints only.\bigskip
\item Example (Edwards and Penney, 4e, Sec 14.9):
min $\ \ x^{2}+y^{2},\ $subject to $\ \ xy=1$.
Solve graphically with circles as level curves, touching up to a rotated
hyperbola at points like $(1,1)$ \ and \ $(-1,-1)$.\bigskip
\item Example (Edwards and Penney, 4e, Example 14.9.4):
Three variables, two equality constraints.
The plane \ $x+y+z=12$ \ intersects \ $z=x^{2}+y^{2}$ \ to form an ellipse.
\ Where is the lowest point on this intersection?
min $z$, \ subject to \ \ $(x+y+z-12,$\ $x^{2}+y^{2}-z)=(0,0)$. \ Draw a
picture - $(2,2,8)$ \ is the low point.\bigskip
\item Example (Uhl/Peressini/Sullivan, Problem 7.4):
Three variables, two equality constraints.
min \ $\frac{1}{x^{2}+y^{2}+z^{2}}$, \ subject to $x^{2}+2y^{2}+3z^{2}=1$, \
$x+y+z=0$.
\emph{Discuss} the appropriate picture.
\end{enumerate}
\vspace{1pt}
\subsubsection{Theory}
\vspace{1pt}
\begin{enumerate}
\item Definition: \ A point \ $x^{\ast }$ is \emph{feasible} if \ $%
h(x^{\ast })=0$.\bigskip
\item Definition: \ Given \ $h(x)=(h_{1}(x),h_{2}(x),\ldots h_{m}(x))$ \
where \ $x$ \ is an \ $n$-slot vector, the Jacobian is the \ $m\times n$ \
matrix whose rows are the gradients \ $\nabla h_{i}(x)$. \ The notation is \
$Dh(x)$. \ \ Illustrate with last example above \ (U/P/S 7.4).
\[
Dh(x)=\left(
\begin{array}{lll}
2x & 4y & 6z \\
1 & 1 & 1
\end{array}
\right)
\]
\bigskip
\item Definition: \ A feasible point \ $x^{\ast }$ is \emph{regular} if the
set of gradients \ $\{$\ $\nabla h_{i}(x)\mid $\ $1\leq i\leq m\}$ \ is
linearly independent. \ \ (Which would imply that \ $m